**ODTÜ-BİLKENT Algebraic Geometry Seminar
(See all past talks
ordered according to speaker and
date)**

**2011 Fall Talks**

__Bilkent, 7 October 2011, Friday, 15:40__

**Alexander Degtyarev**-**[***Bilkent University***]**-**Products of two Dehn twists and real Lefschetz fibrations**

**Abstract:***(joint with Nermin Salepci, Université de Lyon)*

An object repeatedly occurring in algebraic geometry is a fibration with singular fibers. If the base is a topological disk and the number of singular fibers is finite, the topology of such a fibration can adequately be described by the so called*monodromy factorization*of the*monodromy at infinity*(boundary of the disk), regarded up to a certain collection of moves, called*Hurwitz moves*and, possibly, global conjugation. We consider the simplest nontrivial case: factorizations into product of two Artin generators in the braid group B_3 (equivalently, two Dehn twists in the mapping class group of a torus). Even here, the results obtained are quite unexpected; considering the known examples, the general case (even in B_3) is very far from its complete understanding.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.

__ODTU, 14 October 2011, Friday, 15:40__

**Alexander Degtyarev**-**[***Bilkent University***]**-**Lines on quartic surfaces**

**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**P**is merely a^{3}*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.

__Bilkent, 21 October 2011, Friday, 15:40__

**Richard Gonzales**-**[***Boğazici University***]**-**KM theory of rationally smooth group embeddings.**

**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:

*Brion, M.*''The behaviour at infinity of the Bruhat decomposition''. Comment. Math. Helv. 73, pp. 137-174 (1998).

*Gonzales, R*. ''GKM theory of rationally smooth group embeddings''. PhD thesis (2011). http://ir.lib.uwo.ca/etd/216/

*Goresky, M., Kottwitz, R., MacPherson, R*. ''Equivariant Cohomology, Koszul duality, and the localization theorem''. Invent. math. 131, pp. 25-83 (1998).

*Renner, L.*'' The H-polynomial of an Irreducible Representation''. Journal of Algebra 332, pp. 159-186 (2011).**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.**

__Bilkent, 4 November 2011, Friday, 15:40__

**Ali Sinan Sertöz**-**[***Bilkent University***]**-**An overview of counting lines on algebraic surfaces**

**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.**

__ODTU, 18 November 2011, Friday, 15:40__

**Mehmetcik Pamuk**-**[***ODTU***]**-**Surgery Method of Classifying Manifolds**

**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.

__Bilkent, 25 November 2011, Friday, 15:40__

**Selma A. Bhupal**-**[***Hacettepe University***]**-**On Piecewise Polynomial Functions and their Dimension**

**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

**R**and their dimension.^{n}

__ODTU, 2 December 2011, Friday, 15:40__

**Ahmet Beyaz**-**[***ODTU***]**-**Genus Zero Gromov-Witten Invariants**

**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.

__Bilkent,____9 December 2011, Friday, 15:40__

**Mustafa Kalafat**-**[***ODTU***]**- Geometric Invariant Theory and Einstein-Weyl Geometry

**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.

__ODTU,____16 December 2011, Friday, 15:40__

**Ferruh Özbudak**-**[***ODTU***]**- Finite number of Kummer cover and curves with many points

**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

construct expicit examples of fibre products of Kummer covers with many rational rational points, which includes a record and two new entries of the current table of the ``manypoints". This is a report on a joint work with Burcu Gulmez Temur.

__Bilkent,____23 December 2011, Friday, 15:40__

**Mesut Şahin**-**[***Karatekin University***]**-**On Toric Codes**

**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.

__ODTU, 30 December 2011, Friday, 15:40__

**Alexander Klyachko**-**[***Bilkent University***]**-**Toric and Flag varieties**

**Abstract:**In the talk I will discuss the structure of toric variety X_{G}equal to closure of a generic orbit of a maximal torus of a simple group G in its flag variety F_{G}, the respective restriction map H*(F_{G})-->H*(X_{G}) together with some applications.__Bilkent, 6 January 2012, Friday, 15:40__

**Çetin Ürtiş**-**[***TOBB***]**-**Sphere Packings, Lattices and Theta Functions**

**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.

**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.**

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2001 Fall Talks
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2002 Spring Talks
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2002 Fall Talks
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2003 Spring Talks
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2003 Fall Talks
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2005 Spring Talks
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2005 Fall Talks
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2006 Spring Talks
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2006 Fall Talks (146-157) |

2007 Spring Talks
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2007 Fall Talks
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2008 Spring Talks (179-189) |
2008 Fall Talks (190-204) |

2009 Spring Talks
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2009 Fall Talks
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2010
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2010
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2011
Spring Talks (249-260) |