ODTÜ-BİLKENT Algebraic Geometry Seminar
2000 Fall Talks
2001 Spring Talks
2001 Fall Talks:
2002 Spring Talks
2002 Fall Talks
2003 Spring Talks
2003 Fall Talks
2004 Spring Talks
2004 Fall Talks
2005 Spring Talks
2005 Fall Talks
2006 Spring Talks
2006 Fall Talks
2007 Spring Talks
2007 Fall Talks
2008 Spring Talks
2008 Fall Talks
2009 Spring Talks
Abstract: Last year a 6 page proof of Hodge conjecture was deposited into the arXives. Later a 7 page revision was posted, see arXiv:0808.1402 This paper uses only the material found in chapter 0 of Griffiths and Harris' Principles of Algebraic Geometry. In this talk we will review this introductory material for the graduate students and then present the arguments of the alleged proof and ask the audience to find the error! |
Abstract: Last week we mentioned a subtle gap in the alleged proof of Hodge conjecture in arXiv:0808.1402. This week we will mention an irrecoverable gap in the proof and then give an informal survey of what is know in the Hodge conjecture front. |
Abstract: We will give an old constructive method to find the presentation of the knot group which is a knot invariant and we will finish with some illustrations. |
Abstract: There is a method of finding the group presentation of a tame knot. However, it is not an easy task to distinguish groups given their presentations, even in particular examples. Therefore, one needs to find presentation invariants. We shall first consider the Alexander matrix and elementary ideals of a given finite presentation in a general setup then restrict our attention to knot groups and get knot polynomials which happen to be knot invariants of trivial distinguishability. |
Abstract: We will present a class of toric varieties with exceptional properties. These are toric varieties corresponding to rational singularities of DE type. We show that their toric ideals have a minimal generating set which is also a Groebner basis consisting of large number of binomials of degree at most 4. |
Abstract: We will demonstrate different methods of calculating the Alexander polynomial on several examples. |
Abstract: We deal with the following generalized version of the Shapiro and Shapiro total reality conjecture: given a real curve C of genus g and a regular map C --> P1 of degree d whose all critical points are distinct and real (in C), the map itself is real up to a Mőbius transformation in the target. The generalization was suggested by B. and M. Shapiro in about 2005, after the original conjecture was proved, and it was shown that the statement does hold for g>d2/3+O(d). In the talk, we improve the above inequality to g>d2/4+O(d). |
Abstract: In this talk, after I describe algebraic automorphisms group of P1xP1, I will consider the analogous problem in the category of symplectic topology. I will present some results comparing them with the results in the study of volume preserving diffeomorphisms group. In the remaining time, I will talk about the main technique used in the proof, so called the theory of J-holomorphic curves in symplectic topology and how they are employed in this work. |
Abstract: We attempt to study/classify real Jacobian elliptic surfaces of type I or, equivalently, separating real trigonal curves in geometrically ruled surfaces. (On the way, we extend the notions of type I and being separating to make them more suitable for elliptic surfaces.) We reduce the problem to a simple graph theoretical question and, as a result, obtain a characterization and complete classification (quasi-simplicity) in the case of rational base. (The results are partially interlaced with those by V. Zvonilov.) As a by-product, we obtain a criterion for a trigonal curve of type I to be isotopic to a maximally inflected one. |
Abstract: I will talk about the fundemantal concepts in the study of Higher Chow groups, historical background and main research subjects in this field in relation with classical Hodge Theory. I will demonstrate some of these concepts and methods by discussing in a "genaralization" of Hodge conjecture (so called Hodge-D conjecture) for product of two general elliptic curves. |
Abstract: The variety of a finitely generated kG-module is a closed homogeneous subvariety of the maximal ideal spectrum of the cohomology ring of a finite group G with coefficients in an algebraically closed field k of characteristic p>0. I will give some basic definitions and properties of varieties in group cohomology. Then I will present some results on filtration of modules related to varieties. |
Abstract: We will outline the construction of pure motifs, concentrating on the Chow-Kunneth decomposition. Time permiting we intend to describe the transcendental part of the motif of a surface. This is an informal introductory talk. |
Abstract: I will discuss the limit space F of the category of coverings C of the "modular interval" as a deformation retract of the universal arithmetic curve, which is by (my) definition nothing but the punctured solenoid S of Penner. The space F has the advantage of being compact, unlike S. A subcategory of C can be interpreted as ribbon graphs, supplied with an extra structure that provides the appropriate morphisms for the category C. After a brief discussion of the mapping class groupoid of F, and the action of the Absolute Galois Group on F, I will turn into a certain "hypergeometric" galois-invariant subsystem (not a subcategory) of genus-0 coverings in C. One may define, albeit via an artificial construction, the "hypergeometric solenoid" as the limit of the natural completion of this subsystem to a subcategory. Each covering in the hypergeometric system corresponds to a non-negatively curved triangulation of a punctured sphere with flat (euclidean) triangles. The hypergeometric system is related to plane crystallography. Along the way, I will also discuss some other natural solenoids, defined as limits of certain galois-invariant genus-0 subcategories of non-galois coverings in C. The talk is intended to be informal, relaxed and audience friendly. |
2009 Fall Talks
Abstract: The aim of this talk is to give the necessary background material on vector bundles to introduce the topological K-theory. We also explain the classification theorem for vector bundles. This talk is accessible to graduate students at any level. |
Abstract: Last week we discussed the basic properties of vector bundles over a compact base space X to introduce the topological K-theory. The set of isomorphism classes of vector bundles on X forms a commutative monoid. The idea of K-theory of X is the completion of this monoid to a ring. In this talk, we will discuss basic concepts in K-theory. |
Abstract: This is going to be an introductory talk to Bott's periodicity theorem. |
Abstract: This is going to be a introductory talk to algebraic K-theory. I will introduce algebraic K-theory and discuss some basic properties of it. I will give the sketch of the proof of Swan'a theorem, which gives us the relation between topological and algebraic K-theories. |
Abstract: In this introductory talk we will define K1 of rings and discuss their basic properties. |
Abstract: As one of the topological applications of algebraic K-theory, I will introduce Wall's finiteness obstruction which is defined as the obstruction for a finitely dominated space to be homotopy equivalent to a finite CW-complex. Then, I will discuss the orbit category version of Wall's finiteness obstruction. |
Abstract: Following Rosenberg, we will describe the K theory of certain categories and talk about conditions under which we can use a more `reasonable' collection of modules instead of projective modules and still get the same K theory. This will eventually be applied to discuss Grothendieck's Riemann-Roch theorem but that may be left to the next talk if time runs up. |
Abstract: We will talk about the proof of the well-known fact that an n-dimensional sphere is an H-space if and only if n=0, 1, 3, or 7. |
Abstract: The Kontsevich moduli space of stable maps is the central object in Gromov-Witten theory. In this talk, I will discuss its birational geometry and describe how to run Mori's program on small degree examples. I will focus on a few concrete examples.This is joint work with Dawei Chen and builds on joint work with Joe Harris and Jason Starr. |
2010 Spring Talks
Abstract: We will conclude last term's seminar on K-theory with an application to algebraic geometry by developing Grothendieck's Riemann-Roch theorem. The talk will be expository and will be accessible even to those who do not remember much of last semester's talks! |
Abstract: In 1984, V. Jones introduced a new (polynomial) knot invariant by using an operator algebra. Later, it became clear that this polynomial can be obtained by several different methods. We will pick a simple approach, namely defining it by means of the slightly different Kauffman bracket polynomial. We will then consider Jones polynomials of alternating links. In the remaining time, we will finish with the proofs of Tait's conjectures (due to K. Murasugi) by using Jones Polynomial. |
Bilkent, 5 March 2010 Friday, 15:40
Deniz Kutluay-[Bilkent University]-Tait's
Conjectures
Abstract: P.G. Tait conjectured, in 1898, that a reduced alternating diagram of a knot achieves the minimum possible number of crossings for that knot (1), and writhe of such diagrams of the same knot is the same (2). We will first give K. Murasugi's proof to (1) which involves usage of Jones polynomial. We will then use the idea of taking parallels of diagrams (due to R.A. Stong) to prove (2). |
ODTU, 12 March 2010 Friday, 15:40
İnan Türkmen-[Bilkent University]- Detecting
Indecomposable Higher Chow Cycles
Abstract: Spencer Bloch defined the higher Chow in mid 80's as a "natural" extension of classical Chow groups and analysed basic properties of these groups in terms of maps to Deligne Cohomology, named regulators. There is a subgroup of higher Chow groups, group of indecomposables, of special interest. In this talk I will introduce two different methods to detect indecomposables; regulator indecomposability and filtrations on arithmetic Hodge structures. |
Bilkent, 19 March 2010 Friday, 15:40
Alexander Degtyarev-[Bilkent University]- Dihedral
covers of trigonal curves
Abstract: We classify irreducible trigonal curves in Hirzebruch surfaces that admit a dihedral cover and study geometric properties of such curves. In particular, we prove an analog of Oka's conjecture stating that an irreducible trigonal curve admits an S_3 cover if and only if it is of torus type. |
Bilkent, 26 March 2010 Friday, 15:40
Bedia Akyar-[Dokuz Eylul University]- Prismatic
sets in topology and geometry
Abstract: We study prismatic sets analogously to simplicial sets except that realization involves prisms. In particular, I will mention the examples; the prismatic subdivision of a simplicial set S and the prismatic star of S. Both have the same homotopy type as S. Moreover, I will give the role of prismatic sets in lattice gauge theory, that is, for a Lie group G and a set of parallel transport functions defining the transition over faces of the simplices, we define a classifying map from the prismatic star to a prismatic version of the classifying space of G. In turn this defines a G-bundle over the prismatic star. This is a joint work with Johan L. Dupont. |
ODTU, 9 April 2010 Friday, 15:40
Yıldıray Ozan-[ODTU]- Algebraic
K-theory in the study of regular maps in real algebraic
geometry
Abstract: After introducing some preliminary material about real algebraic varieties I will try to summarize how algebraic K-theory is used to study regular maps between real algebraic varieties. Namely, I will talk about the results of Loday and Bochnak-Kucharz which mainly show that regular maps between certain products of spherees are all null-homotopic. For example, Loday showed that any regular map from S1 x S1 to S2 is homotopically trivial, where Sk is the unit sphere in Rk+1. |
ODTU, 16 April 2010 Friday, 15:40
Ali Kemal Uncu-[TOBB ETU]- Modular
symbols on congruence subgroups of SL2(Z)
Abstract: The talk will be about finding the Fourier coefficients of a modular form of the given even weight on a congruence subgroup of SL2(Z). We will work with the Riemann surface related to the congruence subgroup of SL2(Z), define modular symbols and give the relation between modular symbols and Fourier coefficients of modular forms. |
ODTU, 30 April 2010 Friday, 15:40
Ergün Yalçın-[Bilkent University]-Koszul
Resolutions and the Lie Algebra Cohomology
Abstract: Cohomology of a Lie algebra is defined both as the cohomology of its universal algebra and via a Koszul resolution. I will introduce both of the definitions and discuss their equivalence. Then, I will show how the Lie algebra cohomology appears in the integral cohomology calculation of a group extension. |
Bilkent, 7 May 2010 Friday, 15:40
Özgün Ünlü-[Bilkent University]- Homologically
trivial group actions on products of spheres
Abstract: In this talk, I will discuss some constructions of free group actions on products of spheres with trivial action on homology. |
ODTU, 14 May 2010 Friday, 15:40
Hamza Yeşilyurt-[Bilkent University]-Rogers-Ramanujan
Functions
Abstract: We present several identities for the Rogers-Ramanujan functions along with their partition theoretic interpretations and conclude with our recent work on such identities. |
Bilkent, 28 May 2010 Friday, 15:40
Mutsuo Oka-[Tokyo University of Science]-Polar
weighted homogeneous polynomials and mixed Brieskorn
singularity
Abstract: |
2010 Fall Talks
Abstract: The Alexander module of an algebraic curve is a certain purely algebraic invariant of the fundamental group of (the complement of) the curve. Introduced by Zariski and developed by Libgober, it is still a subject of intensive research. We will describe the Alexander modules and Alexander polynomials (both over Q and over finite fields Fp ) of a special class of curves, the so called generalized trigonal curves. The rational case is closed completely; in the case of characteristic p>0, a few points remain open. (Conjecturally, all polynomials that can appear are indeed listed.) Unlike most known divisibility theorems, which rely upon the degree and the types of the singularities of the curve, our bounds are universal: essentially, the Alexander module of a trigonal curve can take but a finitely many values. |
Abstract: The curious history of Fermat's Last Theorem starts with Fermat's famous marginal commentary. The quest for the solution of this problem has created theories which affect all of mathematics. In this seminar, we will talk about Ribet's theorem which states that modularity theorem (previously known as Taniyama-Shimura conjecture) implies Fermat's Last Theorem. A central role in Ribet's proof is played by elliptic curves introduced by Frey. |
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On 13-15 October, we are having an Algebra and Number
Theory Symposium in
honor of Prof Mehpare Bilhan's retirement.
There will be no Algebraic Geometry talk this week.
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Bilkent, 22 October 2010 Friday, 15:40
Christophe Eyral-[Aarhus University] -
A short introduction to
Lefschetz theory on the topology of algebraic
varieties
Abstract: |
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29 October is Republic Day, a national day for Turkey. No
talks!
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Bilkent, 5 November 2010 Friday, 16:00
Muhammed Uludağ-[Galatasaray University] -
The Groupoid of Orientation Twists
Abstract: This is an essay to define a higher modular groupoid. The usual modular groupoid of triangulation flips admits ideal triangulations of surfaces of fixed genus and punctures as objects and flips as morphisms. The higher groupoid of orientation twists admits usual modular groupoids as its objects. |
ODTU, 12 November 2010 Friday, 15:40
İnan Utku Türkmen-[Bilkent University] - An
Indecomposable Cycle on Self Product of Sufficiently
General
Product of Two Elliptic Curves
Abstract: The group of indecomposables is too complicated to compute in general and the results in literature are cenrered around proving that this group is non-trivial or in certain cases finitely generated. In this talk I will focus on the group of indeconposables of self product of sufficiently general product of two elliptic curves, namely; CH3ind(E1x E2 x E1 x E2). I will review the results in literature related with this group and sketch an alternative proof for non-triviality of this group using a constructive method. |
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16-19 November is a religious holiday in Turkey. No talks!
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ODTU, 26 November 2010 Friday, 15:40
Mehmetcik Pamuk-[ODTU] - s-cobordism
classification of 4-manifolds
Abstract: In this talk we are going to show how one can use the group of homotopy self-equivalences of a 4-manifold together with the modified surgery of Matthias Kreck to give an s-cobordism classification of topological 4-manifolds. We will work with certain fundamental groups and give s-cobordism classification in terms of standard invariants. |
Bilkent, 3 December 2010 Friday, 15:40
Ergün Yalçın-[Bilkent University] -
Productive elements in group cohomology
Abstract: I will give the definition of a productive element in group cohomology and describe a new approach to productive elements using Dold's Postnikov decomposition theory for projective chain complexes. The motivation for studying productive elements comes from multiple complexes which is an important construction for studying varieties of modules in modular representation theory. |
ODTU, 10 December 2010 Friday, 15:40
Mustafa Kalafat-[University of Wisconsin at
Madison and ODTU] -
Hyperkahler manifolds with circle actions and the
Gibbons-Hawking Ansatz
Abstract: We show that a complete simply-connected hyperkahlerian 4-manifold with an isometric triholomorphic circle action is obtained from the Gibbons-Hawking ansatz with some suitable harmonic function. |
Bilkent, 17 December 2010 Friday, 15:40
Kürşat Aker-[Feza Gürsey] - Multiplicative
Generators for the Hecke ring of the Gelfand Pair (S(2n),
H(n))
Abstract: For
a given positive integer n, Gelfand pair (S(2n),
H(n)) resembles the symmetric group S(n) in
numerous ways. Here, H(n) is a hyperoctahedral
subgroup of the symmetric group S(2n). In this
talk, we will exhibit a new similarity between the
Hecke ring of the pair (S(2n), H(n)) and the
center of the integral group ring of S(n). |
ODTU, 24 December 2010 Friday, 15:40
Ali Sinan Sertöz-[Bilkent] - Counting
the number of lines on algebraic surfaces
Abstract: This is mostly an expository talk on the problem of counting the number of lines on an algebraic surface. The problem is to respect the rigidity of the line as opposed to accepting all rational curves as lines. Surprisingly some of the work done by Segre has not yet been matched by contemporary techniques. We will summarize what is known and speculate about what can be known! |
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31 December afternoon is no time to hold seminars on
this planet! No talks!
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2011 Spring Talks
Abstract: This term we plan to go over the interesting parts of J. McCleary's book, The User's Guide to Spectral Sequences (2nd Edition, 2001). I will begin with some motivation and basic definitions. This may take a few weeks after which many people promised to talk about the wonderful spectral sequences they have met! |
Abstract: We are continuing with J. McCleary's book, The User's Guide to Spectral Sequences (2nd Edition, 2001). I will repeat the basic definitions and work on some simple examples. |
Bilkent, 4 March 2011 Friday, 15:40
Ali Sinan Sertöz-[Bilkent University] - Basics of
Spectral Sequences III
Abstract: We are continuing with J. McCleary's book, The User's Guide to Spectral Sequences (2nd Edition, 2001). I will start with the second chapter and describe two situations where spectral sequences arise. |
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ODTU, 11 March 2011 Friday, 15:40
Ali Sinan Sertöz-[Bilkent University] - Basics of Spectral Sequences IV
Abstract: We are continuing with J. McCleary's book, The User's Guide to Spectral Sequences (2nd Edition, 2001). I will summarize the third chapter and discuss convergence of spectral sequences. |
Bilkent, 18 March 2011 Friday, 15:40
Alexander Degtyarev-[Bilkent University] - Leray-Serre
spectral sequence I
Abstract: We start exploring the geometric application of the machinery of spectral sequence. As the simplest examples, we consider the spectral sequence(s) of a filtered topological space (as a straightforward generalization of the exact sequence of a pair) and the Serre spectral sequence of a simple fibration. |
ODTU, 1 April 2011 Friday, 15:40
Alexander Degtyarev-[Bilkent
University] - Leray-Serre
spectral sequence II
Abstract: We will continue exploring the immediate consequences and applications of the Serre spectral sequence. Then we will switch to the Leray spectral sequence, which will be derived as a special case of one of the hypercohomology spectral sequences; in particular, we will show that the Leray (and hence Serre) spectral sequences are natural and retain the multiplicative structure, facts that are not immediately obvious from Serre's construction via skeletons. |
Bilkent, 8 April 2011 Friday, 15:40
Ergün Yalçın- [Bilkent University]
-The Lyndon-Hochschild-Serre spectral sequence
Abstract: Let G be a group and H be a normal subgroup of G. Then there is a spectral sequence, called LHS-spectral sequence, which converges to the cohomology of G and whose E_2 term can be expressed in terms of cohomology of H and G/H. I will show how the HLS-spectral sequence can be constructed as a spectral sequence of a double complex and then I will illustrate its usage by doing some group cohomology calculations using it. |
ODTU, 15 April 2011 Friday, 15:40
Ergün Yalçın-[Bilkent University] - Calculating
with the LHS-spectral sequence
Abstract: Let G be a group and H be a normal subgroup of G. There is a spectral sequence, called LHS-spectral sequence, which converges to the cohomology of G and whose E_2 term can be expressed in terms of cohomology of H and G/H. In last week's seminar, I showed how the LHS-spectral sequence can be constructed as a spectral sequence of a double complex. This week I will show how this spectral sequence is used to do group cohomology calculations. I plan to bring enough examples to illustrate different situations that one faces while doing calculations with spectral sequences. |
Bilkent, 22 April 2011 Friday, 14:35 (Notice the new time for this talk)
Özgün Ünlü-[Bilkent University] -Atiyah-Hirzebruch spectral sequence
Abstract: Let X be a CW complex and h be a generalized cohomology theory. Atiyah-Hirzebruch spectral sequence relates the generalized cohomology groups h_*(X) with ordinary cohomology groups with coefficients in the generalized cohomology of a point. |
ODTU, 29 April 2011 Friday, 15:40
Yıldıray Ozan-[ODTU] - On Cohomology of the Hamiltonian Gorups
Abstract: Homotopy properties of the group of Hamiltonian diffeomorphisms of symplectic manifolds are far richer than those of the diffeomorphism groups. Abrue, Anjos, Kedra, McDuff ve Reznikov are some of the authors who contributed to the theory. In this talk, I will explain basics of the theory and try to present sample arguments. |
Bilkent, 6 May 2011 Friday, 15:40
Mehmet Akif Erdal-[Bilkent University] - James Spectral Sequence
Abstract: We will construct the James spectral sequence which is a variant of Atiyah-Hirzebruch spectral sequence. |
ODTU, 13 May 2011 Friday, 15:40
Mehmetcik Pamuk-[ODTU] - An Application
of Atiyah-Hirzebruch Spectral Sequence
Abstract: |
2011 Fall Talks
Abstract: (joint
with Nermin Salepci, Université de Lyon) Trivial as it seems, this simplest case has a number of geometric applications. As a first one, we prove that any maximal real elliptic Lefschetz fibration over the sphere is algebraic. Other applications include the semi-simplicity statement for real trigonal M-curves in Hirzebruch surfaces. (One may try to speculate that products of two Dehn twists are still `tame' precisely because they are related to maximal geometric objects.) The principal tool is a description of subgroups of the modular group in terms of a certain class of Grothendieck's dessins d'enfants, followed by high school geometry. |
Abstract: The purpose of this expository talk is to lay a basis for Sinan's forthcoming account of our joint project. Recall that a quartic surface in P3 is merely a K3-surface equipped with a polarization of degree 4. Thus, I will give a gentle introduction to theory of K3-surfaces: the period space, the global Torelli theorem and surjectivity of the period map, and the implications of the Riemann--Roch theorem. I will explain how the problem of counting lines on a quartic can be reduced to a purely arithmetical question and, should time permit, give a brief account of the results obtained so far, viz. a more or less explicit description of the Picard group of the champion quartic. |
Abstract: Let G be a reductive group. A GxG-variety X is called an embedding of G if X is normal, projective, and contains G as an open dense orbit. Regular compactifications and standard embeddings are the main source of examples. In the former case, they are smooth varieties, and their equivariant cohomology has been explicitely described by Brion using GKM theory. His description relies on the associated torus embedding and the structure of the GxG-orbits. In contrast, standard embeddings constitute a much larger class of embeddings than the smooth ones, and their equivariant cohomology was, just until recently, only understood in some cases. Based on results of Renner, standard embeddings were known to come equipped with a canonical cell decomposition, given in terms of underlying monoid data. The purpose of this talk is three-fold. First, I will give an overview of the theory of group embeddings, putting more emphasis on Renner's approach, and describe the structure of the so called rational cells. Secondly, I will explain how such cellular decompositions lead to a further application of GKM theory to the study of standard embeddings. Finally, I provide a complete description of the equivariant cohomology of any rationally smooth standard embedding. The major results of this talk are part of the speaker's PhD thesis. References: PS: The speaker is supported under TUBITAK ISBAP Grant 107T897 -Matematik İşbirliği Ağı: Cebir ve Uygulamaları. |
The afternoon of 28 October is a
National Holiday.
Abstract: I will wrap up my recent investigations on lines on surfaces with a view towards settling some problems jointly with Degtyarev. |
There is no talk on 11 November
2011 due to Kurban Bayramı.
Abstract: The surgery method of classifying manifolds seeks to answer the following question: Given a homotopy equivalence of m-dimensional manifolds f: M --> N, is f homotopic to a diffeomorphism ? The surgery theory developed by Browder, Novikov, Sullivan and Wall in the 1960’s provides a systematic solution to this problem. My talk will aim to be a friendly introduction to the basic concepts of the surgery theory. |
Abstract: Splines or piecewise polynomial functions are used most commonly to approximate functions, especially by numerical analysts for approximating solutions to differential equation. Most recently, splines have also played an important role in computer graphics. That’s why it is of interest to study spline spaces. In this talk, we will discuss analyzing the piecewise functions with a specified degree of smoothness on polyhedral subdivision of region on Rn and their dimension. |
Abstract: In this talk, we review the genus zero Gromov-Witten invariants by first defining them in a brief way and then applying them in examples of dimension four and six. We also prove that the use of genus zero Gromov-Witten invariants to distinguish the symplectic structures on a smooth 6-manifold is restricted in a certain sense. |
Abstract: We give a survey of Geometric Invariant Theory for Toric Varieties, and present an application to the Einstein-Weyl Geometry. We compute the image of the Minitwistor space of the Honda metrics as a categorical quotient according to the most efficient linearization. The result is the complex weighted projective space CP_(1,1,2). We also find and classify all possible quotients. |
Abstract: We
study the fibre products of a finite number of
Kummer covers of the projective line over finite
fields. We determine the number of rational points
of the fibre product under certain conditions. We
also |
Abstract: Toric codes are some evaluation codes obtained by projective toric varieties corresponding to convex lattice polytopes. We will explain how their basic parameters are related to the torus and the number of lattice points of the polytope and introduce certain generalizations. We will also review some recent results about the minimum distance. |
Abstract: In the talk I will discuss the structure of toric variety XG equal to closure of a generic orbit of a maximal torus of a simple group G in its flag variety FG, the respective restriction map H*(FG)-->H*(XG) together with some applications. |
Abstract: How should greengrocers most efficiently stack their oranges? How about pennies on a tabletop or atoms of a single element in a crystal? More than 400 years ago Kepler conjectured that the most efficient way is the face-centered cubic packing which is well known for greengrocers nowadays. Just recently a "proof" (referees are 99% are certain) for Kepler's conjecture is given. In this talk we will give a brief history of the conjecture and related problems. By considering the problem in higher dimensions we will illustrate some special cases and their applications to different areas of mathematics. In particular, the connection between lattices and theta functions will be discussed. |
2012 Spring Talks
This semester we are going
to run a learning
seminar on Patrick
Shanahan's book, The Atiyah-Singer Index Theorem, Springer Lecture Notes in Mathematics No: 638. |
Abstract: Preliminaries will be discussed; mostly characteristic classes. |
Abstract: I will talk about the motivation for the index theorem and discuss the individual terms in the statement of the theorem. |
Abstract: We examine the consequences of applying the Atiyah-Singer Index Theorem to de Rham and Dolbeault operators. |
Abstract: In this talk we will demonstrate that the application of the Atiyah-Singer Index Theorem to Hodge operator yields the Hirzebruch signature theorem. |
Abstract: In this talk we will discuss the application of the Atiyah-Singer Index Theorem to Dirac operator. |
Abstract: In this talk we give a brief description of the ring K(X) of stable vector bundles over X. |
Abstract: In this talk we continue to give a brief description of the ring K(X) of stable vector bundles over X. |
Abstract: In this talk we will elaborate on the topological index B as covered in Shanahan's boook. |
Abstract: In this talk we will discuss pseudodifferential operators and their suitable generalizations as discussed in Shanahan's book. |
Abstract: In this talk we will discuss the construction of the index homomorphism as given in Shanahan's book. |
Abstract: In this talk we will discuss the main ideas surrounding the proof of the index theorem as given in Shanahan's book. |
2012 Fall Talks
This semester we are going
to run a learning seminar on intersection theory.
We will loosely follow the notes 3264 & All That Intersection Theory in Algebraic Geometry by David Eisenbud and Joe Harris Here is a copy of these notes to save you some Googling. Research talks from other parts of geometry will not be excluded from our program |
Abstract:
(a never ending joint project
with I. Itenberg and S. Sertoz) |
Abstract: I will start with Chapter 2 of Eisenbud-Harris notes and after a brief introduction I will describe the Chow ring of $\mathbb{G}(1,3)$, with a view toward counting the number of lines which meet four general lines in $\mathbb{P}^3$. |
Abstract: I will continue to explore the geometry of Grassmannians, after which I will start discussing the Chow ring of $\mathbb{G}(1,3)$. I hope to have time to talk about the number of lines meeting four general lines in space. |
Abstract: I will start by describing the Chow ring of $\mathbb{G}(1,3)$ and then attack the "Keynote Questions" quoted at the beginning of the chapter. |
Abstract: I will complete the multiplication table of the Chow ring of $\mathbb{G}(1,3)$ and then attack the "Keynote Questions" quoted at the beginning of the chapter. Rain or shine, I will finish my talk series this week! |
Abstract: We generalize the discussion on the intersection theory on G(1,3) given in the previous seminars to the Grassmanian variety G(k,n). We are going to define the Schubert cells and cycles and discuss their properties known as Schubert Calculus. We will determine the Chow Ring A(G(k,n)) and mention some applications to intersection theory problems. |
Abstract: We generalize the discussion on the intersection theory on G(1,3) given in the previous seminars to the Grassmanian variety G(k,n). We are going to define the Schubert cells and cycles and discuss their properties known as Schubert Calculus. We will determine the Chow Ring A(G(k,n)) and mention some applications to intersection theory problems. |
Abstract: We consider an indecomposable representation of the Klein four group over a field of characteristic two and compute a generating set for the corresponding invariant ring up to a localization. We also obtain a homogeneous system of parameters consisting of twisted norms and show that the ideal generated by positive degree invariants is a complete intersection. (joint with J. Shank) |
Abstract:
First we start with defining
rational equivalence between two cycles. Then we
define the chow group as a group of rational
equivalence classes. Then we will present essential
theorems and propositions which are developed at the
fourth chapter (D. Eisenbud and J. Harris, All That
Intersection Theory in Algebraic Geometry) to solve
the keynote question b: |
Abstract: After an introductory discussion of tropical varieties, I intend to talk about tropical intersections and in particular the tropical Grassmannian. |
7 December 2012, Friday
This week's seminar is cancelled due to the traffic of
Docent juries taking place this week.
Abstract: We generalize the discussion on the intersection theory on G(1,3) given in the previous seminars to the Grassmanian variety G(k,n). We are going to define the Schubert cells and cycles and discuss their properties known as Schubert Calculus. We will determine the Chow Ring A(G(k,n)) and mention some applications to intersection theory problems. |
Abstract: Arf Closure of a local ring corresponding to a curve branch, which carries a lot of information about the branch, is an important object of study, and both Arf rings and Arf semigroups are being studied by many mathematicians, but there is not an implementable fast algorithm for constructing the Arf closure. The main aim of this work is to give an easily implementable fast algorithm for constructing the Arf closure of a given local ring. The speed of the algorithm is a result of the fact that the algorithm avoids computing the semigroup of the local ring. Moreover, in doing this, we give a bound for the conductor of the semigroup of the Arf Closure without computing the Arf Closure by using the theory of plane branches. We also give an exposition of plane algebroid curves and present the SINGULAR library written by us to compute the invariants of plane algebroid curves. |
Abstract:
We show that a compact complex
surface together with an Einstein-Hermitian metric
of positive holomorphic bisectional curvature is
biholomorphically isometric to the complex
projective plane with its Fubini-Study metric up to
rescaling. |
2013 Spring Talks
Abstract: This is going to be an informal talk on the dimension of the Fano variety of $k$-linear subspaces of projective hypersurfaces, with emphasis on the $k=1$ case. I will losely follow the contents of Chapter 7 and 8 of Eisenbud and Harris' to-be-published book Intersection Theory in Algebraic Geometry. |
Abstract: We continue our leisurely paced learning seminar on Eisenbud and Harris' notes. I will start by reminding the definition of Chern classes as degeneracy cycles and continue with the calculation of the Chern classes of some interesting bundles. As an application I will talk about how these approaches are used to come up with the number 27, the number of lines on a smooth cubic surface in $\mathbb{P}^3$. Time permitting, I will also attempt to explain solutions to some of the keynote questions posed at the beginning of chapter 8. |
Abstract: Discovery of the gauge-theoretic invariants (Donaldson's and later Seiberg-Witten's) brought a number of fundamental discoveries completely changing the landscape of Low-dimensional topology. I will review essentials of this theory tracing its later development (Ozsvath-Szabo theory) and focusing on the applications to algebraic geometry. |
Abstract: After giving a general definition of Seiberg-Whitten invariants, their meaning in the case of Kahler surfaces will be explained. Some applications and developments will be discussed. |
Abstract: After a short review of differential topological invariants of smooth manifolds, we will discuss some applications to algebraic surfaces. As an example I will discuss the complete intersection surfaces, presented by W. Ebeling (Invent. 1990), which form a pair of nondiffeomorphic but homeomorphic surfaces. |
Abstract: Mid 90's, Broadhurst and Kreimer observed that multiple zeta values persist to appear in Feynman integral computations. Following this observation, Kontsevich proposed a conceptual explanation, that is, the loci of divergence in these integrals must be mixed Tate motives. In 2000, Belkale and Brosnan disproved this conjecture. In this talk, I will describe a way to correct Kontsevich's proposal and show that the regularized Feynman integrals in position space setting as well as their ambiguities are given in terms of periods of suitable configuration spaces, which are mixed Tate. Therefore, the integrals that are of our interest are indeed $\mathbb{Q}[1/2 \pi i]$-linear combinations of multiple zeta values. This talk is based on a joint work with M. Marcolli. |
Abstract: In this two-part talk, we will define a moduli problem, and we will discuss the solutions in a number of well-known cases. We start by defining the moduli functor. Next, we show that the Grassmannian functor is represented by the Grassmann variety of linear subspaces of projective space. After discussing the Quot scheme in very general terms, we move to the construction of the moduli space of vector bundles of given rank and degree on an algebraic curve. |
Abstract: In this two-part talk, we will define a moduli problem, and we will discuss the solutions in a number of well-known cases. We start by defining the moduli functor. Next, we show that the Grassmannian functor is represented by the Grassmann variety of linear subspaces of projective space. After discussing the Quot scheme in very general terms, we move to the construction of the moduli space of vector bundles of given rank and degree on an algebraic curve. |
Abstract: Let $G=\langle g \rangle$ be a finite group generated by $g$. Given $h\in G$, the discrete logarithm problem (DLP) in $G$ with respect to the base $g$ is computing an integer $a$ such that $h=g^a$. The security of many cryptographic protocols relies on the intractability of DLP in the underlying group. Pollard's rho method is a general purpose algorithm to solve DLP in finite groups, and runs in fully-exponential expected time of $\sqrt{|G|}$. Some special purpose algorithms, such as index calculus method, can solve DLP in finite field groups in sub-exponential time. The lack of an efficient DLP solver for elliptic curve groups has been the main reason for elliptic curve based cryptography to shine compared to finite field based cryptography and the RSA cryptosystem. Recent results show that index calculus can be modified to solve ECDLP in certain settings faster than Pollard's rho algorithm. I will discuss recent developments in using index calculus method to solve ECDLP, and some restrictions of the method that motivate many open problems in the area. |
Abstract: Paraphrasing A. Marin, we are "à la recherche de la géométrie algébrique perdue": a journey to forgotten algebraic geometry. Following Ethel I. Moody and taking her notes a bit further, I will discuss explicit equations (not just a formal construction in terms of some sheaves and their sections) describing the beautiful Bertini involution and related maps and curves. Should time permit, I will also say a few words justifying my interest in the subject: the Bertini involution can be used to produce explicit equations of the so-called maximizing plane sextics. In theory, all sextics that are still not understood can be handled in this way, but alas, sometimes Maple runs out of memory trying to solve the equations involved. |
Abstract: We analyze the topological invariants of some specific Grassmannians, the Lie group $G_2$, and give some applications. This is a joint work with Selman Akbulut. |
Bilkent, 17 May 2013, Friday, 15:40
Emre Can Sertöz-[Humboldt] - Idea
of the Moduli Space of Curves
Abstract: By considering Riemann surfaces from several different angles, we will see that there are many seemingly different ways to vary the complex structure on a surface, getting different Riemann surfaces. So we can ask "What is the most natural way to vary Riemann Surfaces?". This is what the moduli space construction answers, and we will talk about it. Also we will see why we need some extra structure on the moduli space besides the classical structures that come via a manifold (or a scheme). |
2013 Fall Talks
Abstract: In this talk, we give the classical definition of a toric variety involving the torus action and provide examples to illustrate it. We introduce two important lattices that play important roles in the theory of algebraic tori and demonstrate how they arise naturally in the toric case. Finally, we introduce affine toric varieties determined by strongly convex rational cones. |
Abstract: In this talk, we introduce fans and the (abstract) toric variety determined by a fan via gluing affine toric varieties defined by the cones in the fan. We include some examples and conclude with the correspondence between orbits of the torus action and the cones in the fan. |
18 October is Kurban Bayramı.
Abstract: We will revise the material on toric varieties with emphasis on examples and introduce some new concepts as time permits. |
Abstract: We will continue to discuss the material in Brasselet's exposition "Geometry of toric varieties", sections 5 and 6, as time permits. |
Abstract: We will complete our discussion of the material in Brasselet's exposition "Geometry of toric varieties", sections 5 and 6. |
Abstract: We will complete our discussion with more examples. |
21-24
Nov 2013 Japanese Turkish Joint Geometry Meeting,
Galatasaray University, İstanbul
Abstract: In this very introductory talk I will try to discuss the interplay between such concepts as embedded toric resolutions of singularities via Newton polygons, Viro’s combinatorial patchworking, and tropical geometry. |
Abstract: We start with the definition of normal, very ample and smooth polytopes. We next define the projective toric variety $X_A$ determined by a finite set $A$ of lattice points. When $A$ is the lattice points of a polytope $P$ we demonstrate that $X_A$ reflects the properties of $P$ best if $P$ is very ample. We also define the normal fan of $P$ and discuss the relation between the corresponding "abstract" variety $X_P$ and the embedded variety $X_A$. |
Abstract: This is a continuation of my previous talk. After a brief introduction to Hilbert’s 16$^{\rm th}$ problem, I will try to outline the basic ideas underlying Viro’s method of patchworking real algebraic varieties. |
Abstract: The aim of this talk is to introduce the so called homogeneous coordinate ring of a normal toric variety. We will see how Chow group of Weil divisors turn this ring into a graded ring. Finally we show that every normal toric variety is a categorical quotient. |
ODTU, 27 December 2013, Friday, 15:40
Mesut Şahin-[Karatekin]
- Coordinate ring of a toric variety
II
Abstract: After the promised example of "bad" quotient, I will review the correspondence between subschemes of a normal toric variety and multigraded ideals of its homogeneous coordinate ring. |
2014 Spring Talks
The first half of this semester is devoted
to toric varieties. The speaker will be mostly MESUT
SAHIN. The basic source will be the book: Toric Varieties, Cox, Little and Schenck, Graduate studies in mathematics vol 124, American Mathematical Society, 2011. |
The second half of this semester will be
devoted to deformation theory. The speaker for this topic
will be exclusively EMRE
COSKUN. He will follow the book: Deformations of Algebraic Schemes, Edoardo Sernesi, Springer-Verlag, 2006. (Grundlehren der mathematischen Wissenschaften, no. 334) |
Abstract: After recalling briefly basics of sheaf of a divisor on a normal variety, we will concentrate on the toric case. In particular, we give an explicit description of global sections of the sheaf of a torus invariant divisor. |
Abstract: We will continue with divisors and sheaves on toric varieties. Reference is chapter 4 of Cox, Little and Schenck. |
Abstract: We will talk about quasicoherent sheaves on the normal toric variety which come from multigraded modules over its Cox ring. |
Abstract: We will talk about The Toric Ideal-Variety Correspondence from Cox-Little-Schenck's book Toric Varieties, see in particular page 220. |
Abstract: .We will talk about the correspondence between closed subschemes in a normal toric variety and B-saturated homogeneous ideals in its Cox ring. |
Abstract: We
will talk about how multigraded Hilbert functions
can be used to compute dimensions of toric codes and
list some basic properties of multigraded Hilbert
functions. |
Abstract: We
will give a nice formula for the dimension of toric
complete intersection codes. We also give a bound on
the multigraded regularity of a zero dimensional
complete intersection subscheme of a projective
simplicial toric variety. The latter is important to
eliminate trivial codes. |
Abstract: In this series of lectures, we will develop deformation theory of functors of Artin rings. After discussing extensions of algebras over a fixed base ring, we will develop the theory of functors of Artin rings. These occur as 'local' versions of various moduli problems, and can give information about the local structure (e.g. smoothness, dimension) of moduli spaces near a point. We apply the theory to concrete examples of moduli problems, such as invertible sheaves on a variety, Hilbert schemes and Quot schemes. |
Abstract: Last
week we defined the |
ODTU, 9 May 2014, Friday, 15:40
Emre Coskun-[ODTÜ] - Deformation Theory 3
Abstract: This week we will start formal deformation theory. This will be the content of chapter 2 in Sernesi's book. |
Bilkent, 16 May 2014, Friday, 15:40
Emre Coskun-[ODTÜ] - Deformation Theory 4
Abstract: Last time we discussed briefly Schlessinger's theorem. We will continue from there. |
Bilkent, 21 May 2014, Wednesday,
15:40
Caner Koca-[Vanderbild] - The Monge-Ampere Equations and Yau's Proof of
the Calabi Conjecture
Abstract: The resolution of Calabi's Conjecture by S.-T. Yau in 1977 is considered to be one of the crowning achievements in mathematics in 20th century. Although the statement of the conjecture is very geometric, Yau's proof involves solving a non-linear second order elliptic PDE known as the complex Monge-Ampere equation. An immediate consequence of the conjecture is the existence of Kähler-Einstein metrics on compact Kähler manifolds with vanishing first Chern class (better known as Calabi-Yau Manifolds). In this expository talk, I will start with the basic definitions and facts from geometry to understand the statement of the conjecture, then I will show how to turn it into a PDE problem, and finally I will highlight the important steps in Yau's proof. |
ODTU, 23 May 2014, Friday, 15:40
Emre Coskun-[ODTÜ] - Deformation Theory 5
Abstract: We will discuss the closing remarks of deformation theory for this semester. |
Bilkent, 27 May 2014, Tuesday,
15:40
Caner Koca-[Vanderbilt] - Einstein's Equations on Compact Complex
Surfaces
Abstract: After a brief review of Einstein's Equations in General Relativity and Riemannian Geometry, I will talk about one of my results: The only positively curved Hermitian solution to Einstein's Equations (in vacuo) is the Fubini-Study metric on the complex projective plane. |
Bilkent, 3 June 2014, Tuesday,
15:40
Caner Koca-[Vanderbilt] - Extremal Kähler Metrics and Bach-Maxwell
Equations
Abstract: Extremal Kähler metrics are introduced by Calabi in 1982 as part of the quest for finding "canonical" Riemannian metrics on compact complex manifolds. Examples of such metrics include the Kähler-Einstein metrics, or more generally, Kähler metrics with constant scalar curvature. In this talk, I will start with an expository discussion on extremal metrics. Then I will show that, in dimension 4, these metrics satisfy a conformally-invariant version of the classical Einstein-Maxwell equations, known as the Bach-Maxwell equations, and thereby are related to physics (conformal gravity) in a surprising and mysterious way. |
2014 Fall Talks
We start with two talks on the recent
developments on "Lines on Surfaces." After that we run a
learning seminar on Dessins
d'enfants. We will mostly follow the following
book: |
Abstract: This is a joint project with I. Itenberg and S. Sertöz. I will discuss the recent developments in our never ending saga on lines in nonsingular projective quartic surfaces. In 1943, B. Segre proved that such a surface cannot contain more than 64 lines. (The champion, so-called Schur's quartic, has been known since 1882.) Even though a gap was discovered in Segre's proof (Rams, Schütt), the claim is still correct; moreover, it holds over any field of characteristic other than 2 or 3. (In characteristic 3, the right bound seems to be 112.) At the same time, it was conjectured by some people that not any number between 0 and 64 can occur as the number of lines in a quartic. We tried to attack the problem using the theory of K3-surfaces and arithmetic of lattices. Alas, a relatively simple reduction has lead us to an extremely difficult arithmetical problem. Nevertheless, the approach turned out quite fruitful: for the moment, we can show that there are but three quartics with more than 56 lines, the number of lines being 64 (Schur's quartic) or 60 (two others). Furthermore, we can prove that a real quartic cannot contain more than 56 real lines, and we have an example realizing this bound. We can also construct quartics with any number of lines in {0; : : : ; 52; 54; 56; 60; 64}, thus leaving only two values open. Conjecturally, we have a list of all quartics with more than 48 lines. (The threshold 48 is important in view of another theorem by Segre, concerning planar sections.) There are about two dozens of species, all but one 1-parameter family being projectively rigid. |
Abstract: This is the second part of the previous talk. See the above abstract. |
Abstract: With this talk we start our series of talks on "Girondo and Gonzalez-Diez, Introduction to Compact Riemann Surfaces and Dessins d'Enfants, London Mathematical Society Student Texts 79, Cambridge University Press, 2012." The first chapter is on Riemann surfaces with an emphasis on computable examples. |
Abstract: We continue with the topology of Riemann surfaces. |
Abstract: We will finish the first chapter on compact Riemann surfaces. The main topic this week will be function fields on Riemann surfaces. |
Abstract: We will start by discussing the consequences of the Uniformization Theorem of compact Riemann surfaces and continue by discussing the groups which uniformize Riemann surfaces of genus greater than one. Expect lots of pictures. |
Abstract: This week we start with hyperbolic geometry. |
Abstract: We will continue with the fundamental group of compact Riemann surfaces and, time permitting, proceed with the existence of meromorphic functions on such surfaces. |
Abstract: We will start talking about Fuchsian groups. |
ODTU, 28 November 2014, Friday, 15:40
Özgür Kişisel-[ODTÜ] - Riemann surfaces and discrete groups - V
Abstract: We will talk about automorphisms of Riemann surfaces. |
Bilkent, 5 December 2014, Friday, 15:40
Özgür Kişisel-[ODTÜ] - Riemann surfaces and discrete groups - VI
Abstract: We will talk about the moduli space of compact Riemann surfaces and conclude our discussion of chapter 2. |
ODTU, 12 December 2014, Wednesday,
15:40
Sinan Sertöz-[Bilkent] - Belyi's Theorem-I
Abstract: We will describe the content of what is known as Belyi's theorem and prove the hard part which is actually easier than the easy part! |
Bilkent, 19 December 2014, Friday, 15:40
Sinan Sertöz-[Bilkent] - Belyi's Theorem-II
Abstract: Last week we discussed the content of Belyi's theorem and worked out an example. So it is only this week that we start to prove the first part of Belyi's theorem: If a compact Riemann surface is defined over the field of algebraic numbers, then it has a meromorphic function which ramifies over exactly three points. This is know as the hard part, and the converse is known as the easy part even though the converse is more involved! |
ODTU, 26 December 2014, Tuesday,
15:40
Sinan Sertöz-[Bilkent] - Belyi's Theorem-III
Abstract: This week we will prove that if a compact Riemann surface admits a meromorphic function which ramifies over at most three points, then it is defined over the field of algebraic numbers. This was first proved by Weil in 1956. We will present a modern proof following Girondo and Gonzalez-Diez. |
ODTÜ-BİLKENT Algebraic
Geometry Seminar
(See all past talks ordered according
to speaker and date)
2015 Spring Talks
We will mainly continue our
learning seminar on Dessins
d'enfants. We follow the following book: |
Abstract: Counting lines on surfaces of fixed degree in projective space is a topic in algebraic geometry with a long history. The fact that on every smooth cubic there are exactly 27 lines, combined in a highly symmetrical way, was already known by 19th century geometers. In 1943 Beniamino Segre stated correctly that the maximum number of lines on a smooth quartic surface over an algebraically closed field of characteristic zero is 64, but his proof was wrong. It has been corrected in 2013 by Slawomir Rams and Matthias Schütt using techniques unknown to Segre, such as the theory of elliptic fibrations. The talk will focus on the generalization of these techniques to quartics admitting isolated ADE singularities. |
Ferruh Özbudak-[ODTÜ] - | Perfect nonlinear and quadratic maps on finite fields and some connections to finite semifields, algebraic curves and cryptography |
Abstract: Let |
Ali Sinan Sertöz-[Bilkent] - Belyi's Theorem-IV
Abstract: In
the previous talks, the proof of Belyi's theorem was
completed modulo a finiteness criterion. In this
talk we will prove that criterion. Namely, we
will prove that a compact Riemann surface |
Davide Cesare Veniani-[Leibniz
University of Hanover] -
An introduction to elliptic fibrations -
part I: Singular Fibres
Abstract: The
theory of elliptic fibrations is an important tool
in the study of algebraic and complex surfaces. The
talk will focus on Kodaira's classification of
possible singular fibres. I will construct some
examples of rational and K3 elliptic surfaces to
illustrate the theory, coming from pencils of plane
cubics and lines on quartic surfaces. |
Davide Cesare Veniani-[Leibniz
University of Hanover] -
An introduction to elliptic fibrations - part II:
Mordell-Weil group and torsion sections
Abstract: Given
an elliptic surface, the set of sections of its
fibration forms a group called the Mordell-Weil
group. After recalling the main concepts from part
I, I will expose the main properties of this group,
with a special focus on torsion sections. I will
give two constructions on quartic surfaces which
appear naturally in the study of the enumerative
geometry of lines, where torsion sections play a
prominent role. |
Ali Sinan Sertöz-[Bilkent] - Belyi's Theorem-V
Abstract: This
is the last talk in our series of talks on Belyi's
theorem. In this talk I will outline the proof of
the fact that a compact Riemann surface |
Ali Sinan Sertöz-[Bilkent] - Exit Belyi, enter dessins d'enfants
Abstract: This week I will first clarify some of the conceptual details of the proof of Belyi's theorem that were left on faith last week. After that we will start talking about dessins d'enfants. |
Ali Sinan Sertöz-[Bilkent] - From dessins d'enfants to Belyi pairs
Abstract: We will describe the process of obtaining a Belyi pair starting from a dessin d'enfant. |
Ali Sinan Sertöz-[Bilkent] - Calculating the Belyi function associated to a dessin
Abstract: I will go over the calculation of the Belyi pair corresponding to a particular dessin given in the book, see Example 4.21. Time permitting, I will briefly talk about constructing a dessin from a Belyi pair. |
Ali Sinan Sertöz-[Bilkent] - From Belyi pairs to dessins
Abstract: We will talk about obtaining a dessin from a Belyi function. |
Alexander Klyachko-[Bilkent] - Exceptional Belyi coverings
Abstract: (This is a joint project with Cemile Kürkoğlu.) Exceptional covering is a connected Belyi coverings uniquely determined by its ramification scheme. Well known examples are cyclic, dihedral, and Chebyshev coverings. We add to this list a new infinite series of rational exceptional coverings together with the respective Belyi functions. We shortly discuss the minimal
field of definition of a rational exceptional
covering and show that it is either Existing theories give no upper bound on degree of the field of definition of an exceptional covering of genus 1. It is an open question whether the number of such coverings is finite or infinite. Maple search for an exceptional
covering of |
Alexander Degtyarev-[Bilkent] - Dessins d'enfants and topology of algebraic curves
Abstract: I will give a brief introduction into the very fruitful interplay between Grothendieck's dessins d'enfants, subgroups of the modular group, and topology and geometry of trigonal curves/elliptic surfaces/Lefschetz fibrations. |
ODTÜ-BİLKENT Algebraic Geometry
Seminar
**** 2015 Fall Talks ****
|
Alexander Degtyarev-[Bilkent] - Lines
on smooth quartics
Abstract: In
1943, B. Segre proved that a smooth quartic surface in
the complex projective space cannot contain more than
64 lines. (The champion, so-called Schur's quartic,
has been known since 1882.) Even though a gap was
discovered in Segre's proof (Rams, Schütt, 2015), the
claim is still correct; moreover, it holds over any
field of characteristic other than 2 or 3. (In
characteristic 3, the right bound seems to be 112.) At
the same time, it was conjectured that not any number
between 0 and 64 can occur as the number of lines in a
quartic. |
Ali Sinan Sertöz-[Bilkent] - The
basic theory of elliptic surfaces-I
Abstract: This term we will be running a learnin seminar on elliptic surfaces with a view toward "lines on quartic surfaces". We will be mainly following Miranda's classical notes but other sources will not be excluded. |
Ali Sinan Sertöz-[Bilkent] - The basic theory of elliptic surfaces-II
Abstract: We continue our learning seminar talk on elliptic surfaces. We will also mention how this topic shows up in the search for lines on quartic surfaces in . |
Ergün Yalçın-[Bilkent] - Group actions on spheres with rank one isotropy
Abstract: Actions of finite groups on spheres can be studied in various different geometrical settings, such as (A) smooth G-actions on a closed manifold homotopy equivalent to a sphere, (B) finite G-homotopy representations (as defined by tom Dieck), and (C) finite G-CW complexes homotopy equivalent to a sphere. These three settings generalize the basic models arising from unit spheres S(V) in orthogonal or unitary G-representations. In the talk, I will discuss the group theoretic constraints imposed by assuming that the actions have rank 1 isotropy (meaning that the isotropy subgroups of G do not contain , for any prime ). This is joint work with Ian Hambleton. |
Özgün Ünlü-[Bilkent] - Free
group actions on products of spheres
Abstract: In this talk we will discuss the problem of finding group theoretic conditions that characterizes the finite groups which can act freely on a given product of spheres. The study of this problem breaks up into two aspects: (1) Find group theoretic restrictions on finite groups that can act freely on the given product. (2) Construct explicit free actions of finite groups on the given product. I will give a quick overview of the first aspect of this topic. Then I will discuss some recently employed methods of constructing such actions. |
Recep Özkan-[ODTÜ] Concrete sheaves and continuous spaces
Abstract: This
is a talk from the speaker's recent dissertation.
After he summarizes the historical background and the
recent developments in the field he will motivate his
dissertation problems. Time permitting he will talk
about the ideas behind the proof of his main theorem. |
Cem Tezer-[ODTÜ] - Anosov
diffeomorphisms : Revisiting an old idea
Abstract: Introduced by D. V. Anosov as the discrete time analogue of geodesic flows on Riemann manifolds of negative sectional curvature, Anosov diffeomorphisms constitute one of the leitmotivs of contemporary abstract dynamics. It is conjectured that these diffeomorphisms occur on very exceptional homogeneous spaces. The speaker will delineate the basic facts and briefly mention his own recent work towards settling this conjecture. |
Haydar Göral-[Université Lyon 1] - Primality via Height Bound
Abstract: Height functions are of fundamental importance in Diophantine geometry. In this talk, we obtain height bounds for polynomial ring over the field of algebraic numbers. This enables us to test the primality of an ideal. Our approach is via nonstandard methods, so the mentioned bounds will be ineffective. We also explain the tools from nonstandard analysis. |
Alperen Ergür-[Texas A&M] - Tropical Varieties for Exponential Sums
Abstract: We
define a variant of tropical varieties for exponential
sums. These polyhedral complexes can be used to
approximate, within an explicit distance bound, the
real parts of complex zeroes of exponential sums. We
also discuss the algorithmic efficiency of tropical
varieties in relation to the computational hardness of
algebraic sets. Our proof involves techniques from
basic complex analysis, inequalities and some recent
probabilistic estimates on projections that might be
of interest to analyst. |
Ali Ulaş Özgür Kişisel-[ODTÜ]- Moduli space of elliptic curves
Abstract: The aim of this talk is to view the moduli space of elliptic curves in different contexts. After briefly discussing the classical setting, we will see how it can be viewed as an orbifold and as an algebraic stack. |
Mesut Şahin-[Hacettepe] - On
Pseudo Symmetric Monomial Curves
Abstract: In this talk,
we introduce monomial curves, toric ideals and
monomial algebras associated to 4-generated pseudo
symmetric numerical semigroups. We give a
characterization of indispensable binomials of
these toric ideals, and of these monomial algebras
to have strongly indispensable minimal graded free
resolutions. We also discuss when the tangent
cones of these monomial curves at the origin are
Cohen-Macaulay in which case Sally's conjecture
will be true. |
ODTÜ-BİLKENT Algebraic Geometry
Seminar
(See all past talks ordered according
to speaker and date)
**** 2016 Spring Talks ****
The theme of this term is |
Alexander Degtyarev-[Bilkent] - Skeletons
Abstract: This is
section 1.2. In a sense it is the heart of the
book: It explains how boring algebra can be
translated into the intuitive language of
pictures. (Of course, then it turns out that
pictures are not so easy, either, but that’s
another story.) |
Alexander Degtyarev-[Bilkent] - Skeletons-II
Abstract: This
talk is a continuation of the previous week's talk. |
Ali Sinan Sertöz-[Bilkent]
- Elliptic Surfaces
Abstract: We
will give an introduction to the concepts of elliptic
surfaces. We will mainly follow the order of Section
3.2 of the book. |
Ali Sinan Sertöz-[Bilkent]
- Elliptic Surfaces and Weierstrass theory
Abstract: We will talk
about the Weierstrass theory and the j-invariant
of elliptic surfaces. |
Alexander Degtyarev-[Bilkent] - Trigonal
curves and monodromy
Abstract: We will
discuss the simple analytic (Calculus 101)
properties of the -invariant
and the way how it affects the singular fibers.
Then, we will start the discussion of trigonal
curves, fundamental groups, the braid monodromy,
and its relation to the -invariant. |
Alexander Degtyarev-[Bilkent] - Trigonal curves and monodromy - II
Abstract: We will
discuss the fundamental groups, braid monodromy,
Zariski--van Kampen theorem, and the relation
between the braid monodromy, dessins, and the -invariant,
implying that the monodromy group is one of genus
zero and imposing strong restrictions on the
fundamental group. |
Alexander Degtyarev-[Bilkent] - Trigonal
curves and monodromy - III
Abstract: We continue the description of the braid monodromy of a trigonal curve and its relation to the dessin. The principal result is the fact that the monodromy group is a subgroup of genus zero. As an immediate application, we will discuss the dihedral coverings ramified at trigonal curves (equivalently, torsion of the Mordell—Weil group of an elliptic surface) and a trigonal curve version of the so-called Oka conjecture. |
ODTÜ, 22 April 2016, Friday, 15:40
Cancelled in favour of 4th Cemal Koç
Algebra Days at METU
Alexander Degtyarev-[Bilkent] - Trigonal
curves and monodromy: further applications
Abstract: As yet another
application of the relation between the braid
monodromy and -invariant,
we will derive certain universal bounds for the
metabelian invariants of the fundamental group of
a trigonal curve. |
ODTÜ-BİLKENT
Algebraic Geometry Seminar
(See all past talks ordered according
to speaker and date)
**** 2016 Fall Talks ****
|
Alexander Degtyarev-[Bilkent] - Lines
in K3 surfaces
Abstract: The unifying
theme of this series of talks is the classical
problem of counting lines in the projective models
of -surfaces
of small degree. Starting with such classical
results as Schur's quartic and Segre's bound
(proved by Rams and Schütt) of lines
in a nonsingular quartic, I will discuss briefly
our recent contribution (with I. Itenberg and A.
S. Sertöz), i.e., the complete classification of
nonsingular quartics with many lines. Most quartics found (in an
implicit way) in our work are ``new'', attracting
the attention of experts in the field (Rams,
Schütt, Shimada, Shioda, Veniani). For example,
one of them turned out an alternative nonsingular
quartic model of the famous Fermat surface raising
the natural question if there are other such
models. An extensive search (Shimada, Shioda)
returned no results, and we show that, although
there are over a thousand singular models,
only two models
are smooth! Taking this line of research slightly
further, one can classify all smooth quartic
models of singular -surfaces
of small discriminant, arriving at a remarkable
alternative characterisation of Schur's
quartic---the champion carrying lines:
it is also the (only) smooth quartic of the
smallest possible discriminant, which is .
Going even further, we can study other projective
models of small degree; counting lines in these
models, we arrive at the following conjectures:
These conjectures are still wide
open; I only have but a few examples. |
Alexander Degtyarev-[Bilkent] - Lines in K3
surfaces-II
Abstract: This
is the continuation of last week's talk. |
Ali Sinan Sertöz-[Bilkent] - Introduction
to complex K3 surfaces
Abstract: We will start
reviewing and explaining as the case might be some
introductory concepts in K3 surface theory. The
level will be introductory so it is a good
opportunity so jump on the "wagon". |
Ali Sinan Sertöz-[Bilkent] - K3
surfaces and lattices
Abstract:We will introduce some basic concepts of lattice theory that are used to understand K3 surfaces with a view towards Torelli type theorems. |
Ali Sinan Sertöz-[Bilkent]- K3 lattice of a K3 surface
Abstract: We will
continue our series on K3 surfaces by examining
the cohomology of K3 surfaces and finding out how
this cohomology structure characterizes the
surface. |
Abstract: In this talk
we discuss the problem of classifying complex
non-special simple quartics up to equisingular
deformation by reducing the problem to an
arithmetical problem about lattices. On this
arithmetical side, after applying Nikulin's
existence theorem, our computation based on the
Miranda-Morrison's theory computing the genus
groups. We give a complete description of
equisingular strata of non-special simple
quartics. Finally we give ideas of the proof
of our principal result. |
Çisem Güneş-[Bilkent] - Classification
of simple quartics up to equisingular deformation-II
Abstract: This is the
continuation of last week's talk. |
Oğuzhan Yörük-[Bilkent] - Which
K3 surfaces of Picard rank 19 cover an Enriques surface?
Abstract: The
parities of the entries of the transcendental
lattice of a K3 surface determine,
in most cases, if covers an
Enriques surface or not. We will summarize what is
known about this problem and talk about the missing
case when . |
Alexander Degtyarev-[Bilkent] - Projective
models of -surfaces
Abstract: Now, that we
know everything about abstract -surfaces,
I will try to take a closer look at Saint-Donat's
seminal paper and
share my findings. This |
Ali Ulaş Özgür Kişisel - [ODTÜ] - Arithmetic
of K3 surfaces
Abstract: I'll try to
outline some of the results in the survey paper of
M. Schütt with the same title. |
ODTÜ-BİLKENT
Algebraic Geometry Seminar
(See all past talks ordered
according to speaker and date)
**** 2017 Spring Talks ****
|
Ali Sinan Sertöz-[Bilkent] - On the moduli of K3 surfaces
Abstract: We will
discuss the main line of ideas involved in the
proofs of the Torelli theorems for K3 surfaces as
outlined by Huybrechts in his recent book "Lectures
on K3 Surfaces." |
Ali Sinan Sertöz-[Bilkent] - On
the moduli of K3 surfaces-II
Abstract: This is going to be a continuation of last week's talk. In particular we will talk about the ideas involved around proving the Global Torelli Theorem for K3 surfaces. Most proofs will be referred to the literature but we will try to relate the concepts involved. |
Ali Ulaş Özgür Kişisel-[ODTU] - Tropical curves
Abstract: In this talk, we will discuss several approaches to defining tropical curves and the theory of linear systems on tropical curves. |
Ali Ulaş Özgür Kişisel-[ODTU] - Tropical curves-II
Abstract: In this talk, we will continue our discussion of several approaches to defining tropical curves and the theory of linear systems on tropical curves. |
Emre Coşkun-[ODTU] - The Beilinson spectral sequence
Abstract: We overview
the Beilinson spectral sequence and its
applications in the construction of sheaves and
vector bundles. |
Abstract: I will explain
the proof of my conjectures (reported earlier in
this seminar) on the maximal number of straight
lines in sextic surfaces in ,
(42 lines) and octic surfaces/triquadrics in ,
(36 lines). I will also try to make it clear that
the complexity of the problem decreases when the
polarization grows. The asymptotic bound for
K3-surfaces in large projective spaces is 24
lines, all constituting fiber components of an
elliptic pencil. |
Mesut Şahin-[Hacettepe] - Lattice ideals and toric codes
Abstract: I
will briefly recall basics of toric varieties over
finite fields and evaluation codes on them. Then, we
will see that some vanishing ideals of subvarieties
are lattice ideals. Using this, we characterize
whether they are complete intersections or not. In the
former case; dimension, length and regularity of the
code will be understood easily. |
Nil Şahin-[Bilkent] - On Pseudo Symmetric Monomial Curves
Abstract: After giving
basic definitions and concepts about symmetric and
pseudo symmetric numerical semigroups, we will
focus on 4-generated pseudo symmetric numerical
semigroups/monomial curves. Determining the
indispensable binomials of the defining ideal, we
will give characterizations under which the
tangent cone is Cohen-Macaulay. If time permits,
determining minimal graded free resolutions of the
tangent cones, we’ll show that “If the 4 generated
pseudo symmetric numerical semigroup S is
homogeneous and the corresponding tangent cone is
Cohen Macaulay, then S is also Homogeneous
type. |
Alexander Klyachko-[Bilkent] - Transformation of cyclic words into Lie elements
Abstract: Let be
a complex vector space and be
its tensor algebra. We are primarily
concerned with Lie subalgebra generated
by commutators of elements in and
graded by degrees of the tensor components. where is a primitive root of unity of degree , is -cycle in symmetric group acting on by permutation of tensor factors. The majorization index of permutation is defined as follows The operators and satisfy the following equations Clearly, action of on a monomial in produces a cyclic word. It may be less straightforward that action of on a monomial gives Lie element where summation runs over all permutations and . It should be emphasised that cyclic permutation of arguments in adds only a phase factor equal to -th root of unity. |
Özgün Ünlü-[Bilkent] - Semi-characteristic classes
Abstract: In this talk,
I will first give basic definitions and theorems
about semi-characteristic classes. Secondly, I
will discuss some applications of
semi-characteristic classes. |
Çisem Güneş Aktaş-[Bilkent] - An Introduction to Nikulin's Theory of
Discriminant Forms-I
Abstract: In this talk I
will first recall some basic definitions and
notions about lattices. Then I will introduce
fundamentals of Nikulins's theory of discriminant
forms. Finally, I will discuss some principal
applications of this theory and give an example in
the particular case of K3-lattice. |
Çisem Güneş Aktaş-[Bilkent] - An Introduction to Nikulin's Theory of
Discriminant Forms-II
Abstract: This is the
continuation of last week's talk. |
ODTÜ-BİLKENT
Algebraic Geometry Seminar
(See all past talks ordered
according to speaker and date)
**** 2017 Fall Talks ****
|
Alexander Degtyarev-[Bilkent] - Real algebraic curves with large finite number
of real points
Abstract: (joint work
in progress with Erwan Brugallé, Ilia Itenberg,
and Frédéric
Mangolte) In the simplest case where is
a plane curve of degree ,
we have (sharp
if is
small compared to )
and (sharp
for but,
most likely, not sharp in general). |
Emre Can Sertöz-[Max-Planck,
Leipzig] - Enumerative geometry of double spin curves
Abstract: This talk is
about the speaker's recent PhD dissertation whose
full abstract follows: This thesis has two parts. In Part I we consider the moduli spaces of curves with multiple spin structures and provide a compactification using geometrically meaningful limiting objects. We later give a complete classification of the irreducible components of these spaces. The moduli spaces built in this part provide the basis for the degeneration techniques required in the second part. In the second part we consider a series of problems inspired by projective geometry. Given two hyperplanes tangential to a canonical curve at every point of intersection, we ask if there can be a common point of tangency. We show that such a common point can appear only in codimension 1 in moduli and proceed to compute the class of this divisor. We then study the general properties of curves in this divisor. Our divisor class has small enough slope to imply that the canonical class of the moduli space of curves with two odd spin structures is big when the genus is greater than 9. If the corresponding coarse moduli spaces have mild enough singularities, then they have maximal Kodaira dimension in this range. |
Hanife Varlı-[ODTÜ] - Perfect discrete Morse functions on connected
sums
Abstract: Computational
topology is an area between topology and computer
science that applies topological techniques for
problems in data and shape analysis. One of the
techniques used in this area is the discrete Morse
theory developed by Robin Forman as a discrete
analogue of Morse theory. This theory gives a way
of studying the topology of discrete objects via
critical cells of discrete Morse functions. In this talk, we will first
briefly mention Morse theory. Then we will talk on
discrete Morse theory which will be followed by my
thesis problem: composing and decomposing perfect
discrete Morse functions (the most suitable
functions for combinatorial and computational
purposes) on connected sums of triangulated
manifolds. In this thesis, we prove that one
can compose perfect discrete Morse functions on
connected sums of manifolds in any dimensions. On
decomposing a given perfect discrete Morse
function on a connected sum, our method works in
dimensions 2 and 3. |
Çisem Güneş Aktaş-[Bilkent] - Algebraic surfaces in
Abstract: In this talk we give a deeper insight into the theory of K3-surfaces, which essentially boils down to the global Torelli theorem, subjectivity of period map and Riemann Roch theorem (for example, we conclude that all singular points of a K3-surface have to be simple ones). After recalling principle properties of K3-surfaces, we explain the arithmetical reduction of various classification problems, concentrating on the geometric aspects of the arithmetical restrictions appearing in the statements. |
Ergün Yalçın-[Bilkent] - Moore spaces and the Dade group
Abstract: Let be a finite -group and be a field of characteristic . A topological space is called an -Moore space if its reduced homology is nonzero only in dimension . We call a --complex a Moore -space over if for every subgroup of , the fixed point set is a Moore space with coefficients in . A -module is called an endo-permutation module if is a permutation -module. We show that if is a finite Moore -space, then the reduced homology module of is an endo-permutation-module generated by relative syzygies. We consider the Grothendieck group of finite Moore -spaces with addition given by the join operation, and relate this group to the Dade group generated by relative syzygies. In the talk I will give the necessary background on Moore -spaces and Dade group, and provide many examples to motivate the statements of the theorems. |
Abstract: K3 surfaces
and Enriques Surfaces are two closely related
objects in Algebraic Geometry. It is known that
the unramified double cover of an Enriques Surface defined
by the torsion class is
an algebraic K3 surface. Also, conversely, if a K3
surface admits
a fixed point free involution ,
then the quotient surface is
an Enriques Surface. In this talk we will examine,
following Keum's work, under which circumstances a
K3 surface will admit a fixed point free
involution, hence, will cover an Enriques Surface
and we will give some applications for this
characterization. |
Serkan Sonel-[Bilkent] - Which K3 Surfaces with Picard number doubly
cover Enriques surfaces.
Abstract: In
this talk, we discuss the problem of which K3 Surfaces
with Picard number doubly
cover Enriques surfaces and give the insight to the
solution of the problem which is deeply related to
lattice theory and integral quadratic forms. Then we
give the generalization of Sertoz Theorem about the
characterization of primitive embeddings of the
lattices. Finally, we give our result on which
indefinite even unimodular -lattices
fail to be embedded into the sublattice of
the K3-lattice . |
Mesut Şahin-[Hacettepe] - Vanishing Ideals of Parameterized Toric Codes
Abstract: We start
defining subvarieties of a toric variety that are
parameterized by Laurent monomials and the
corresponding toric codes. We recall their
vanishing ideals and give algorithms for finding
their binomial generators which will be used to
compute main parameters of the corresponding toric
codes. We show that these ideals are lattice
ideals and give an algorithm to find a basis for
the corresponding lattice. Finally, we give this
lattice explicitly under a mild condition. |
Ali Ulaş Özgür Kişisel-[ODTÜ] - The Cap Set Problem
Abstract: Determining
the size of a largest subset of which
contains no lines is called the cap set problem. I
will outline what is known about this problem and
report some recent progress on the asymptotic
version of the problem, due to Croot, Lev, Pach
and Ellenberg, Gijswijt.
|
Mücahit Meral-[ODTÜ] - Semifree Hamiltonian circle actions on 6
dimensional symplectic
manifolds with non-isolated fixed point set
Abstract: Let be
a -dimensional
closed symplectic manifold with a symplectic circle
action. Many mathematicians tried to find some
conditions on which
make a symplectic circle action Hamiltonian. Cho,
Hwang and Suh discovered a condition on the
6-dimensional symplectic manifolds. In this talk, we
will discuss CHS's theorem: Let be
a -dimensional
closed symplectic -manifold
with generalized moment map .
Assume that the fixed point set is not empty and
dimension of each component at most .
Then the action is Hamiltonian if and only if for
any regular value of . |
Özgün Ünlü-[Bilkent] - The Halperin-Carlsson conjecture
Abstract: The
Halperin-Carlsson conjecture predicts that if an
elementary abelian 2-group of rank acts
freely and cellularly on a finite CW-complex ,
then is
less than or equal to the total dimension of the
cohomology of with
coefficients in a field of charateristic 2. We will
discuss some known results and some new developments
related to this conjecture. |
Oğuz Yayla-[Hacettepe] - The existence theory of some objects in finite
geometry
Abstract: In this talk
difference sets, Hadamard matrices, perfect
sequences, cyclic irreducible codes and similar
objects in finite geometry will be presented.
Their relationship will be given and then their
existence will be studied. If time allows some
methods for their construction will be given. |
ODTÜ-BİLKENT
Algebraic Geometry Seminar
(See all past talks ordered
according to speaker and date)
**** 2018 Spring Talks ****
|
Ali Ulaş Özgür Kişisel-[ODTU] - Introduction to minimal model program. MMP for
surfaces
Abstract: In this talk,
the general strategy of the minimal model program
will be outlined. Some well-known results about
the classification of surfaces will be rephrased
in this setting. |
Ali Ulaş Özgür Kişisel-[ODTU] - Cone and contraction theorems for surfaces
Abstract: We
will first review notions of ample and nef divisors
and several numerical criteria. Afterwards, we will
discuss the cone and contraction theorems for the case
of surfaces. |
Ali Ulaş Özgür Kişisel-[ODTU] - The Logarithmic Category
Abstract: The
minimal model program in higher dimensions necessarily
involves singular varieties since a minimal model for
a smooth variety doesn't have to be smooth. Iitaka's
philosophy is that considering logarithmic pairs, each
containing a variety together
with a normal crossing boundary divisor ,
is essential when dealing with problems involving such
singular varieties. The goal of this talk will be to
explain this generalization. |
Tolga Karayayla-[ODTU] - Singularities
Abstract: In this talk I will give the descriptions and characterizations of the types of singularities which arise in Minimal Model Program, namely terminal singularities, canonical singularities, log terminal singularities and log canonical singularities. |
Ali Ulaş Özgür Kişisel-[ODTU] - Vanishing theorems
Abstract: We will
discuss Kodaira Vanishing Theorem and its various
generalizations. |
Abstract: We will state
the cone and contraction theorems for dimensions
greater than or equal to three and discuss their
proofs. |
Ali Ulaş Özgür Kişisel-[ODTU] -Flips
Abstract: We
will discuss the definition of flips and some
examples, together with results and conjectures about
their existence and termination. |
Ali Ulaş Özgür Kişisel-[ODTU] - Existence and termination of flips
Abstract: We will discuss results and conjectures about the existence and termination of flips. Some of these developments are relatively recent. |
Alexander Degtyarev-[Bilkent] - Can a smooth sextic have more than 72 tritangents?
Abstract: After
a brief introduction to the history of the subject, I
will motivate the conjecture that a smooth plane
sextic curve cannot have more than 72 tritangents,
i.e., lines intersecting the curve with even
multiplicity at each point. (A stronger conjecture is
that the number of tritangents is 72 or at most 68,
with all values taken.) I will also put the problem
into a larger context and discuss the known results
and a few steps towards the proof of this conjecture. |
Melih Üçer-[Bilkent] - Miyaoka-Yau inequality in higher dimensions
Abstract: Miyaoka-Yau
inequality is a classical inequality that concerns
the Chern numbers of a minimal algebraic surface
of general type, together with a rigid geometric
characterization of the case of equality. Namely,
an algebraic surface satisfies equality if and
only if it is a quotient of the unit ball.
Corresponding result for higher-dimensional smooth |
Abstract: In this talk,
we will discuss the paper "Insufficiency of
The Brauer-Manin Obstruction Applied to Etale
Covers" by Bjorn Poonen. Firstly, we will
explain Hasse principle and Brauer groups. Then,
we will construct a nice (smooth, projective and
geometrically integral) 3-fold and we will
show that Brauer-Manin obstruction doesn't
explain failure of Hasse principle in this case. |
Abstract: In the minimal
model program it is known that there exist many
examples of rationally connected varieties, such
as smooth Fano varieties. In this talk I will
present a paper by Janos Kollar, mainly concerned
with the etale fundamental groups of
separably rationally connected varieties. |
ODTÜ talks are either at Hüseyin Demir Seminar
room or at Gündüz İkeda seminar room at the Mathematics
building of ODTÜ.
Bilkent talks are at room 141 of
Faculty of Science A-building at Bilkent.
ODTÜ-BİLKENT
Algebraic Geometry Seminar
(See all past talks ordered
according to speaker and date)
**** 2018 Fall Talks ****
|
Ali Sinan Sertöz-[Bilkent] - K3 covers of Enriques surfaces
Abstract: I last talked
on this subject on 2001 when I talked about Keum's
1990 work on the problem. There has been some
activity on the subject since then which I want to
talk about. I will explain the problem and
summarize what has been done so far and prepare
the audience for the next two talks where the
speakers will explain their most recent
contributions to the subject. |
Serkan Sonel-[Bilkent ] - K3 covers of Enriques surfaces with Picard rank
18 and 19
Abstract: In this talk,
we partially determine the necessary and
sufficient conditions on the entries of the
intersection matrix of the transcendental lattice
of algebraic K3 surface with Picard number 18 ≤
ρ(X) ≤ 19 for the surface to doubly cover an
Enriques surface. |
Oğuzhan Yörük-[Bilkent] - Parity arguments on K3 covers of Enrique
surfaces with Picard rank 19
Abstract: Last two talks
of the seminar were mostly on the theoretical
parts of the subject. This time, we introduce some
computational arguments by using equivalence of
parities of the transcendental lattice of K3
surfaces, right after a brief reminding of what
was talked on previous two talks to warm up. Then,
we will apply this idea to reduce the number of
cases and time spent on showing which K3 surfaces
of Picard number 18 &19 cover an Enriques
surface. |
Alexander Degtyarev-[Bilkent] - A few further remarks on Enriques surfaces
Abstract: I will
continue the subject of the previous talks, viz. a
characterization of the K3-surfaces covering an
Enriques surface. I will: |
Emre Coşkun-[ODTÜ] - Serre's GAGA (Géometrie Algébrique et Géométrie
Analytique)
Abstract: Serre's famous
theorem known as "GAGA" (Géometrie Algébrique et
Géométrie Analytique - Algebraic Geometry and
Analytic Geometry) is a fundamental result in
algebraic geometry. It basically says that the
theory of complex analytic subvarieties of
projective space and the theory of algebraic
subvarieties of projective space coincide. In this
series of lectures, we shall start with the
fundamentals of complex analytic geometry and then
move toward the proof of GAGA. |
Emre Coşkun-[ODTÜ] - Serre's GAGA-II
Abstract: Serre's famous
theorem known as "GAGA" (Géometrie Algébrique et
Géométrie Analytique - Algebraic Geometry and
Analytic Geometry) is a fundamental result in
algebraic geometry. It basically says that the
theory of complex analytic subvarieties of
projective space and the theory of algebraic
subvarieties of projective space coincide. In this
series of lectures, we shall start with the
fundamentals of complex analytic geometry and then
move toward the proof of GAGA. |
Emre Coşkun-[ODTÜ] - Serre's GAGA-III
Abstract: Serre's famous
theorem known as "GAGA" (Géometrie Algébrique et
Géométrie Analytique - Algebraic Geometry and
Analytic Geometry) is a fundamental result in
algebraic geometry. It basically says that the
theory of complex analytic subvarieties of
projective space and the theory of algebraic
subvarieties of projective space coincide. In this
series of lectures, we shall start with the
fundamentals of complex analytic geometry and then
move toward the proof of GAGA. |
Yıldıray Ozan-[ODTÜ] - Manifolds Admitting No Real Projective Structure
Abstract: In this talk
first, we will define and give basic results about
real projective structures on smooth
manifolds. Then we will discuss such
structures on two and three manifolds. Next we
will mention the 2015 result by D. Cooper and W.
Goldman that the smooth manifold does
not admit any real projective structure (the first
known example in dimension three), and we will
generalize this result to all higher
dimensions. If time permits, we will mention
different type of examples of smooth manifolds
with no real projective structure. |
Nil Şahin-[Bilkent] - One dimensional Gorenstein Local Rings with
decreasing Hilbert Function
Abstract: In this talk,
starting from Rossi's conjecture stating "Hilbert
function of a one dimensional Gorenstein Local
Ring is non-decreasing", I will give a little
history of the recent works in this subject and
talk about Oneto, Strazzanti and Tamone's work
that constructs infinitely many one-dimensional
Gorenstein Local rings that decreases at some
level. |
ODTÜ talks are either at Hüseyin Demir Seminar
room or at Gündüz İkeda seminar room at the Mathematics
building of ODTÜ.
Bilkent talks are at room 141 of
Faculty of Science A-building at Bilkent.
ODTÜ-BİLKENT Algebraic Geometry Seminar
(See all past talks
ordered according to speaker
and date)
**** 2019 Spring Talks ****
Ali Sinan Sertöz-[Bilkent]
- Arf Rings I
Abstract: The aim of
these two talks is to discuss the background and
the content of Arf's 1946 paper on the
multiplicity sequence of an algebraic curve
branch. I will start by giving the geometric and
algebraic descriptions of a singular branch for a
curve, describe its multiplicity sequence obtained
until it is resolved by blow up operations. Du Val
defines some geometrically significant steps of
the resolution process and shows that if the
multiplicity sums up to those points are known
then the whole multiplicity sequence can be
recovered by a simple algorithm. However all this
information must be encoded at the very beginning
in the local ring of the branch. The problem is
then to decipher this data. This week I will mostly describe
the background and explain what is involved in
actually finding these numbers. Arf's original article "Une
interpretation algebrique de la suite des ordres
de multiplicite d'une branche algebrique",
together with my English translation can be found
on: http://sertoz.bilkent.edu.tr/arf.htm
|
Ali Sinan Sertöz-[Bilkent]
- Arf Rings II
Abstract: I will first
describe the structure of the local ring of a
singular branch and explain how the blow up
process affects it. Then I will describe, aprés
Arf, how the multiplicity sequence can be
recovered, not from this ring but from a slightly
larger and nicer ring which is now known as the
Arf ring. The process of finding this nicer ring
is known as the Arf closure of this ring. Finally
I will explain how Arf answered Du Val's question
of reading off the multiplicity sequence from the
local ring. |
Alexander Degtyarev-[Bilkent]
- Tritangents to sextic curves via
Niemeier lattices
Abstract: I will address
the following conjecture (and some refinements
thereof): “A smooth plane curve of degree 6 has at
most 72 tritangents.” After a brief introduction
to the subject and a survey of the known results
for the other polarized K3-surfaces, I will
explain why the traditional approach does not work
and suggest a new one, using the embedding of the
Néron—Severi lattice of a K3-surface to an
appropriate Niemeier lattice. I will also discuss
the pros and contras of several versions of this
approach and report the partial results obtained
so far. |
Alexander Degtyarev-[Bilkent]
- Positivity and sums of squares of
real polynomials
Abstract: I will
discuss the vast area of research (in which I am
not an expert) related to Hilbert's 17th problem,
namely, positivity of real polynomials vs. their
representation as sums of squares (SOS). As is
well known, "most" PSD (positive semi definite)
forms in more than two variables are not SOS of
polynomials, although they are SOS of rational
function. I will consider a few simplest classical
counterexamples, and then I will outline the
construction part of our recent paper (in
collaboration with Erwan Brugallé, Ilia Itenberg,
and Frédéric Mangolte). Thinking that we were
dealing with Hilbert's 16th problem (widely
understood, i.e., topology of real algebraic
varieties), we constructed real plane algebraic
curves with large finite numbers of real points.
These curves provide new lower bounds on the
denominators needed to represent a PSD ternary
form as a SOS of rational functions. |
Yıldıray Ozan-[ODTÜ]
- Equivariant Cohomology and
Localization after Anton Alekseev
Abstract: We will try to
present the notes by Anton Alekseev on Equivariant
Localization, mainly focusing on $S^1$-actions.
First, we will introduce Stationary Phase Method.
Then we will define equivariant $S^1$-cohomology
and present a proof of the localization theorem
suggested by E. Witten. If time permits,
finally we will end by the Duistermaat-Heckman
formula and its proof. |
Kadri İlker Berktav-[ODTÜ]
- Towards the Stacky Formulation of
Einstein Gravity
Abstract: This talk,
which essentially consists of three parts, serves
as a conceptional introduction to the formulation
of Einstein gravity in the context of derived
algebraic geometry. The upshot is as follows: we
shall first outline how to describe the notion of
a (pre-)stack $\mathfrak{X}$, by using the
functor-of-points type approach, manifestly given
as a certain groupoid-valued sheaf over a site
$\mathcal{C}$, and present main ingredients of the
homotopy theory of stacks in a relatively
succinct and naive way. In that respect, one in
fact requires to adopt certain simplicial
techniques in order to recast the
notion of a stack in the language of homotopy
theory. This homotopical treatment, on the other
hand, is essentially based on so-called the
model structure on the 2-category $Grpds$ of
groupoids. In the second part of the talk, we
shall revisit main aspects of 2+1 dimensional
vacuum Einstein gravity on a pseudo-Riemannian
manifold $M$ especially in the context of Cartan
geometry, and investigate, in the case of
$M=\Sigma\times (0,\infty)$ with vanishing
cosmological constant and $\Sigma$ being a closed
Riemann surface of genus $g>1$, the equivalence
of the quantum gravity with a gauge theory
established in the sense that the moduli space
$\mathcal{E}(M)$ of such a 2+1 dimensional
Einstein gravity is isomorphic to that of
flat Cartan $ISO(2,1)$-connections, denoted by
$\mathcal{M}_{flat}$. As an analyzing a classical
field theory with an action functional
$\mathcal{S}$ boils down to the study of the
moduli space of solutions to the corresponding
field equations, the notion of a stack in
fact provides an alternative and elegant way of
recording and organizing the moduli data. In the
final part, we shall briefly discuss (i)
how to construct the appropriate stacks
associated to $\mathcal{E}(M) $ and
$\mathcal{M}_{flat}$ respectively, and (ii)
how to extend the isomorphism that essentially
captures the equivalence of the quantum gravity
with a gauge theory in the above setup to an isomorphism
of associated stacks. |
Halil İbrahim Karakaş-[Başkent]
- Arf Numerical Semigroups
Abstract: Parametrizations
have been given for Arf numerical semigroups with
small multiplicity ($m\leq 10$) and arbitrary
conductor. In this talk, I will give a
characterization of Arf numerical semigroups in
terms of the Apery sets, and use that
characterization to parametrize Arf numerical
semigroups with multiplicity 11 and 13. I will
also share some observations about Arf numerical
semigroups with prime multiplicity. |
Mesut Şahin-[Hacettepe]
- Evaluation codes defined on subsets
of a toric variety
Abstract:
In this talk, we review algebraic
methods for studying evaluation codes defined on
subsets of a toric variety. The key object is the
vanishing ideal of the subset and its Hilbert
function. We reveal how invariants of this ideal
such as multigraded regularity and multigraded
Hilbert polynomial relate to parameters of the
code. Time permitting, we share the nice
correspondence between subgroups of the maximal
torus and lattice ideals as their vanishing
ideals. |
Tolga Karayayla-[ODTÜ]
- Singular fiber products of rational
elliptic surfaces and fixed
point free group actions on their desingularizations
Abstract: Schoen has
shown that a fiber product of two relatively
minimal rational elliptic surfaces with section is
a simply connected Calabi-Yau 3-fold if the fiber
product is smooth and the same is true for the
desingularization of the fiber product by small
resolutions in the case that the singularities are
ordinary double points. I will describe the small
resolution process and talk about lifting
automorphisms on the fiber product to
automorphisms of the desingularization. I will
discuss the problem of constructing fixed point
free finite group actions on such
desingularizations. The quotient of the 3-fold by
such group actions give rise to non-simply
connected Calabi-Yau 3-folds. The problem on the
existence of such group actions on smooth fiber
products was solved by previous works of Bouchard,
Donagi and the speaker. |
Nil Şahin-[Bilkent] -
k-sparse numerical semigroups
Abstract: In this talk,
I will present k-sparse numerical semigroups as a
generalization of sparse numerical semigroups
using the recent paper "On k-sparse numerical
semigroups" by Guilherme Tizziotti and Juan
Villanueva. |
Yıldıray Ozan-[ODTÜ] -
An Obstruction for Algebraic Realization of
Smooth Closed Manifolds with Prescribed Algebraic
Submanifolds and Some Examples
Abstract: First I will
review the history of Algebraic Realization
Problem of Smooth Manifolds starting from
Seifert's 1936 result to Tognoli and to Akbulut
and King. Then I will introduce some tools typical
to the subject like algebraic homology and
strongly algebraic vector bundles. Finally,
I will present a result (joint with one of my
former masters' student Arzu Celikten) which
introduces an obstruction for a topological vector
bundle to admit a strongly algebraic structure.
Using this obstruction we will construct examples
of manifolds promised in the title. |
ODTÜ talks are either at Hüseyin Demir Seminar room or
at Gündüz İkeda seminar room at the
Mathematics building of ODTÜ.
Bilkent talks are at room 141 of Faculty of Science
A-building at Bilkent.
ODTÜ-BİLKENT Algebraic Geometry Seminar
(See all past talks ordered
according to speaker and date)
**** 2019 Fall Talks ****
İlker Berktav-[ODTÜ] - Formal Moduli Problems and Classical Field
Theories
Abstract: This is an introductory talk to
the concept of a formal
moduli problem in sense of
Lurie and it's essential role in encoding the
formal geometric aspects of derived moduli spaces
of solutions to the certain moduli problems. To be
more specific, we shall be interested in a
sort of formal moduli problem describing a
classical field theory on a base manifold M in
the sense that it defines a derived moduli space
of solutions to the certain differential equations
on an open subset U of M,
namely the Euler-Lagrange equations, arising from
a local action functional defined on the space of
fields on U, see
Costello and Gwilliam. The outline of this talk is
as follows: References
|
Yıldıray Ozan-[ODTÜ] - A filtration on the Borel-Moore Homology of
Wonderful Compactification
of Some Symmetric Spaces
Abstract: After giving
some motivation we will introduce basic objects
mentioned in title and the tools we will be
using. Then we will give some examples and
state main results. If time permits, we will
try to sketch a proof of the results. |
Alexander Degtyarev-[Bilkent] - Linear subspaces in algebraic varieties
Abstract: (partially
joint with I. Itenberg and J. Ch. Ottem) |
Ali Ulaş Özgür Kişisel-[ODTÜ] - Random Real Algebraic Plane Curves
Abstract: There has been growing interest in
recent years on random objects in algebraic
geometry. The expected number of real roots of a
univariate polynomial has been studied for
different probability measures on the space of
polynomials, by many authors. After discussing
some of these results, I will switch to
multivariate polynomials and survey some of the
known results regarding the expected number of
connected components of a real algebraic plane
curve and their expected volumes. Finally, I will
present some of our recent results with Turgay
Bayraktar regarding the expected depth of a real
algebraic plane curve. |
Muhammed Uludağ-[Galatasaray] - Jimm, a fundamental involution
Abstract: Dyer's outer
automorphism of PGL(2,Z) induces an involution of
the real line, which behaves very much like a kind
of modular function. It has some striking
properties: it preserves the set of quadratic
irrationals sending them to each other in a
non-trivial way and commutes with the Galois
action on this set. It restricts to an highly
non-trivial involution of the set unit of norm +1
of quadratic number fields. It conjugates the
Gauss continued fraction map to the so-called
Fibonacci map. It preserves harmonic pairs of
numbers inducing a duality of Beatty partitions of
N. It induces a subtle symmetry of Lebesgue's
measure on the unit interval. |
Alexander Degtyarev-[Bilkent] - Linear subspaces in algebraic varieties. II:
Niemeier lattices
Abstract: Using the
arithmetical reduction suggested in the previous
talk, I will prove the two new theorems stated
there, viz., “the number of tritangents to a
smooth plane sextic is at most 72”, and “the
number of 2-planes in a smooth cubic 4-fold is at
most 405” (joint with I. Itenberg and J.Ch.
Ottem). To this end, we will embed the
appropriately modified lattice of algebraic cycles
to a Niemeier lattice and study certain
configurations of square 4 vectors in the latter.
I will try to explain the advantages of this
approach and outline the principal techniques used
in counting square 4 vectors. |
Turgay Akyar-[ODTÜ] - Clifford's Theorem on Special Divisors
Abstract: It is very
well known that for a non-special divisor ,
the dimension of a linear system on
a smooth projective curve over depends
only on the degree of .
On the other hand, if is
special, we do not have such a dependence. After
giving some facts about linear systems
on curves we will see a classical theorem
mainly concerning with the extremal behavior of
the dimension of
a complete special linear system . |
Abstract: Zariski-van
Kampen theorem expresses the fundamental group of
the complement of an algebraic curve on in
terms of generators and monodromy relations.
Therefore, the Alexander module of the curve is
also (almost) expressed in terms of generators and
monodromy relations. As far as the Alexander
module of an -gonal
curve is concerned, the group of monodromy
relations is a subgroup of the Burau group ,
which is a certain subgroup of .
For trigonal curves ( case),
Degtyarev gave a characterization of the monodromy
groups: the monodromy group of a trigonal curve
(except a trivial exceptional case) must be a
finite index subgroup of whose
image under the special epimorphism is
of genus and
conversely, most of such subgroups appear as
monodromy groups of trigonal curves. However, this
class of subgroups is still too large, hence it is
not feasible to look at them all and determine
their Alexander modules. In this talk, I plan to
speak about a recently discovered method by which,
given an abstract module over ,
one can determine whether or not it appears as the
Alexander module of a trigonal curve. With this
method, it should be feasible to determine all the
Alexander modules. |
Serkan Sonel-[Bilkent] - On K3 surfaces covering an Enriques surface
Abstract: We will continue the subject of
the previous talks, viz. a characterization of the
K3-surfaces covering an Enriques surface. Following Nikulin, we will: |
Mesut Şahin-[Hacettepe] - Rational points of subgroups inside a toric
variety over a finite field
Abstract: We talk about
counting rational points of subgroups of the torus
lying inside a toric variety over a finite field,
explaining its implications for the evaluation
codes on these subgroups. |
Halil İbrahim Karakaş-[Başkent] - A decomposition of partitions and numerical sets
Abstract: The aim of
this work is to exhibit a decomposition of
partitions of natural numbers and numerical sets.
In particular, we obtain a decomposition of a
sparse numerical set into the so called hook
semigroups which turn out to be primitive. Since
each Arf semigroup is sparse, we thus obtain a
decomposition of any Arf semigroup into primitive
numerical semigroups. |
ODTÜ talks are either at Hüseyin Demir Seminar
room or at Gündüz İkeda seminar room at the Mathematics
building of ODTÜ.
Bilkent talks are at room 141 of
Faculty of Science A-building at Bilkent.
ODTÜ-BİLKENT
Algebraic Geometry Seminar
(See all past talks ordered
according to speaker and date)
Refresh this page to see recent changes, if any
**** 2020 Spring Talks ****
Emre Coşkun-[ODTÜ] - Quiver Representations I
Abstract: In this series of talks, we shall
introduce quivers and their representations and
discuss their basic properties. We shall also
discuss and prove (if time permits) Gabriel's
theorem, which gives a complete classification of
quivers of finite type. |
Emre Coşkun-[ODTÜ] - Quiver Representations
II
Abstract: In this series
of talks, we shall introduce quivers and their
representations and discuss their basic
properties. We shall also discuss and prove (if
time permits) Gabriel's theorem, which gives a
complete classification of quivers of finite type. |
Davide Cesare Veniani-[Stuttgart] - Free
involutions on ihs manifolds
Abstract: Irreducible
holomorphic symplectic manifolds are one of the
building blocks of kähler manifolds with vanishing
first Chern class. In dimension 2 they are called
K3 surfaces. Free involutions on K3 surfaces are
quite interesting because they connect this class
of surfaces with another class, namely Enriques
surfaces. I will talk about a formula for the
number of free involutions on a K3 surface (joint
work with I. Shimada), the classification of K3
surfaces without any free involution (joint work
with S. Brandhorst and S. Sonel) and the
generalization to higher dimensions (joint work
with S. Boissière). |
Abstract: In
this talk, we shall introduce quivers and their
representations and discuss their basic
properties. We shall also discuss and prove (if
time permits) Gabriel's theorem, which gives a
complete classification of quivers of finite type. |
Ayşegül Öztürkalan-[AGÜ] -
Loops in moduli spaces of real plane
projective sextics
Abstract: The space of
real algebraic plane projective curves of a fixed
degree has a natural stratification. The strata of
top dimension consists of non-singular curves and
are known up to curves of degree 6. Topology and,
in particular, fundamental groups of individual
strata have not been studied systematically. We
study the stratum formed by non-singular sextics
with the real part consisting of 9 ovals which lie
outside each other and divide the set of complex
points. Apparently this stratum has one of the
most complicated fundamental groups. In the talk I
will study its subgroups which originate from
spaces of linear equivalent real divisors on a
real cubic curve and tell the connections. |
Kadri İlker Berktav-[ODTÜ] - Symplectic Structures on Derived Schemes
Abstract: This is an
overview on the basic aspects of so-called on
(affine) derived -schemes
with being
a field of characteristic 0. In this talk, we
always study objects with higher structures in a
functorial perspective, and we shall focus on
local models for those structures. To this end, in
the first part of the talk, the basics of
commutative differential graded -algebras
(cdgas) and their cotangent complexes will be
introduced. Using particular cdgas as local
models, we shall introduce the notion of a
(closed) p-form of degree k on
an affine derived -scheme
with the concept of a non-degeneracy.
As a particular case, we shall eventually define on
an affine derived -scheme,
and outline the construction of a Darboux-like local
model for together
with some examples. These will be the main topics
of interest in the second part of the talk. |
Alexander Degtyarev-[Bilkent] - The global Torelli theorem for cubic 4-folds and
its applications
Abstract: Undoubtedly,
in theory of K3-surfaces the principal tool of
study making the theory tractable is the global
Torelli theorem (essentially stating that the
isomorphism class of a surface is determined by
that of its Hodge structure), together with the
surjectivity of the period map (a description of
the realizable Hodge structures). There are a few
other classes of analytic varieties (most notably
curves, from which the name originates, or Abelian
surfaces) for which similar statements hold. I
will try to discuss the version of the global
Torelli theorem/surjectivity of the period map for
cubic 4-folds in (mostly
due to Clair Voisin). Then, I will discuss a
recent application of these statements to the
classification of large configurations of 2-planes
in cubic 4-folds. |
Muhammed Uludağ-[Galatasaray] - Mapping class groupoids and Thompson's groups
Abstract: (Joint work
with Ayberk Zeytin) |
James D. Lewis-[Alberta] - The Hodge Conjecture
Abstract: We introduce the classical Hodge
conjecture and formulate a birational version. We
then show how this birational version is used to
formulate the Hodge conjecture for higher K-groups
of smooth quasiprojective varieties. |
ODTÜ talks are either at Hüseyin Demir Seminar
room or at Gündüz İkeda seminar room at the Mathematics
building of ODTÜ.
Bilkent talks are at room 141 of
Faculty of Science A-building at Bilkent.
ODTÜ-BİLKENT
Algebraic Geometry Seminar
(See all past talks ordered
according to speaker and date)
Refresh this page to see recent changes, if any
****
2020 Fall Talks ****
This semester we plan to have
all our seminars on Zoom
Alexander Degtyarev-[Bilkent] - Counting 2-planes in cubic 4-folds in
Abstract: (work in progress joint with I. Itenberg and J.Ch. Ottem) We use the global Torelli theorem
for cubic 4-folds (C. Voisin) to establish the
upper bound of 405 2-planes in a smooth cubic
4-fold. The only champion is the Fermat cubic. We
show also that the next two values taken by the
number of 2-planes are 357 (the champion for the
number of *real* 2-planes)
and 351, each realized by a single cubic. To
establish the bound(s), we embed the appropriately
modified lattice of algebraic cycles to a Niemeier
lattice and estimate the number of square 4
vectors in the image. The existence is established
my means of the surjectivity of the period map.
According to Schütt and Hulek, the second best
cubic with 357 planes can be realized as a
hyperplane section of the Fermat cubic in . |
Emre Can Sertöz-[Max
Planck-Bonn] - Separating periods of quartic surfaces
Abstract: Kontsevich--Zagier periods form a
natural number system that extends the algebraic
numbers by adding constants coming from geometry
and physics. Because there are countably many
periods, one would expect it to be possible to
compute effectively in this number system. This
would require an effective height function and the
ability to separate periods of bounded height,
neither of which are currently possible. This is ongoing work with Pierre
Lairez (Inria, France). |
Sinan Ünver-[Koç] - Infinitesimal regulators
Abstract: We will describe a construction of
infinitesimal invariants of thickened
one dimensional cycles in three dimensional space,
which are the simplest cycles that are not in
the Milnor range. The construction also
allows us to prove the infinitesimal version of
the strong reciprocity conjecture for thickenings
of all orders. Classical analogs of our
invariants are based on the dilogarithm function
and our invariant could be seen as their
infinitesimal version. Despite this analogy,
the infinitesimal version cannot be obtained
from their classical counterparts through a
limiting process. |
Kâzım İlhan İkeda-[Boğaziçi] - Yoga of the Langlands reciprocity and
functoriality principles
Abstract: I shall describe my reflections on
the Langlands reciprocity and functoriality
principles. Those principles of Langlands are one
of the fundamental driving forces of current
mathematical research. Here, the term ``yoga''
appearing in the seminar title, which is
introduced and used extensively by Grothendieck,
means ``meta-theory''. of is defined and studied in the papers by E. Serbest and the author, where is a certain topological group constructed using Fontaine-Wintenberger theory of fields of norms. Fix and define the non-commutative topological group , which depends only on , by the ``restricted free topological product'' Here, and denote the numbers of real and the pairs of complex conjugate embeddings of the global field in . Note that, . Let denote the hypothetical Langlands group of . The existence problem of is one major conjecture in Langlands Program. For , an embedding determines a continuous homomorphism unique up to conjugacy, which in return defines a continuous homomorphism unique up to conjugacy, for each . Fixing one such morphism for each , the collection defines a unique continuous homomorphism which is compatible with Arthur's proposed construction of . Let be a connected, quasisplit reductive group over . There is a bijection between the set of ``-parameters'' of over and the set whose elements are the collections consisting of local -parameters of over for each . Note that, assuming the local reciprocity principle for over for all , the set is in bijection with the set whose elements are the collections of local -packets of over for each . As global admissible -packets of over are the restricted tensor products of local -packets of over , by Flath's decomposition theorem, we end up having the following theorems Theorem 1. Let be
a connected quasisplit reductive group over the
number field .
Assume that the local Langlands reciprocity
principle for over holds.
Then, there exists a bijection satisfying the ``naturality'' properties. and Theorem 2. Let and be
connected quasisplit reductive groups over the
number field .
Let be an -homomorphism. Assume that the local Langlands reciprocity principle for over holds. Then, the -homomorphism induces a map (lifting) from the global admissible -packets of over to the global admissible -packets of over satisfying the ``naturality'' properties. |
Deniz Ali Kaptan-[Alfred
Renyi] - The Methods of Goldston-Pintz-Yıldırım and
Maynard-Tao, and results on prime gaps
Abstract: The breakthrough method of
Goldston, Pintz and Yıldırım and its subsequent
refinement by Maynard and Tao effected a giant
leap in our understanding of prime gaps. I will
give an overview of the evolution of the ideas
involved in these methods, describing various
applications along the way. |
Ayberk Zeytin-[Galatasaray] - Continued Fractions and the Selberg zeta
function of the modular curve
Abstract: Selberg zeta function of a Riemann
surface X is known to encode the discrete spectrum
of the Laplacian on X via the Selberg trace
formula. In this talk, following Lewis-Zagier, we
will explain how one obtains the Selberg zeta
function of the modular curve as the Fredholm
determinant of an appropriate operator on an
appropriate Banach space. Along the way, we will
discuss the close relationship between the
operators in question and continued fractions.
Should time permit, we will mention some ongoing
work, partly joint with M. Fraczek, B. Mesland and
M.H. Şengün. |
Mustafa Kalafat-[Nesin
Math Village] - On special submanifolds of the Page space
Abstract: Page manifold is the underlying
differentiable manifold of the complex surface,
obtained out of the process of blowing up the
complex projective plane, only once. This space is
decorated with a natural Einstein metric, first
studied by D.Page in 1978. In this talk, we study some
classes of submanifolds of codimension one and two
in the Page space. These submanifolds are totally
geodesic. |
Alexander Degtyarev-[Bilkent] - Lines in singular triquadrics
Abstract: (joint work in progress with
Sławomir Rams) |
Ali Sinan Sertöz-[Bilkent] - From Calculus to Hodge
Abstract: This is an expository talk mainly for the young Complex Geometry students. I will start with the tangent line to a real parabola, pass to the complex case and then to the projective case. After giving informal descriptions of the de Rham and Dolbeault cohomologies, which are related by the Hodge decomposition theorem, I will describe the Hodge Conjecture with integer coefficients which is known to be false in general despite the strong evidence in its favor given by the Lefschetz (1,1)-theorem. It is known that some torsion integral Hodge classes may exist which are not algebraic. The existence of non-torsion integral Hodge classes contradicting the Hodge conjecture were constructed recently (30 years ago!) by Kollar. I want to end the talk discussing this example and its possible variants. |
Ali Ulaş Özgür Kişisel-[ODTÜ] - On complex 4-nets
Abstract: Nets are certain special line
arrangements in the plane and they occur in
various contexts related to algebraic geometry,
such as resonance varieties, homology of Milnor
fibers and fundamental groups of curve
complements. We will investigate nets in the
complex projective plane .
Let and be
integers. An -net
is a pencil of degree algebraic
curves in with
a base locus of exactly points,
which degenerates into a union of lines times.
It was conjectured that the only -net
is a -net
called the Hessian arrangement. I will outline our
proof together with A. Bassa of this conjecture. |
Sefa Feza Arslan-[Mimar
Sinan] - Apery table, microinvariants and the regularity
index
Abstract: In this talk, I will first explain
the concepts of Apery table of a numerical
semigroup introduced by Cortedellas and Zarzuela (Tangent
cones of numerical semigroup rings. Contemp.
Math. 502, 45–58 (2009)) and the
microinvariants of a local ring introduced by Juan
Elias (On the deep structure of the blowing-up
of curve singularities. Math. Proc. Camb.
Philos. Soc. 131, 227–240 (2001)). We use
these concepts to give some partial results about
a conjecture on the regularity index of a local
ring and to give some open problems. |
ODTÜ
talks are either at Hüseyin Demir Seminar room or at Gündüz İkeda seminar room at the Mathematics
building of ODTÜ.
Bilkent talks are at room 141 of
Faculty of Science A-building at Bilkent.
Zoom talks are online.