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

**** 2019 Spring Talks ****

1. Bilkent, 22 February 2019, Friday, 15:40

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

2. ODTÜ, 1 March 2019, Friday, 15:40

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.

3. Bilkent, 8 March 2019, Friday, 15:40

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.

4. ODTÜ, 15 March 2019, Friday, 15:40

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.

********  No talk on 22 March 2019 due to Spring Break  *****

5. Bilkent, 29 March 2019, Friday, 15:40

Yýldýray 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.

6. ******** The talk on 5 April 2019 Friday is cancelled due to the heavy schedule of the speaker**********

7. ODTÜ, 12 April 2019, Friday, 15:40

Kadri Ýlker 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.

8. ODTÜ, 19 April 2019, Friday, 15:40

Halil Ýbrahim Karakaþ-[Baþkent] - 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.

9. Bilkent, 26 April 2019, Friday, 15:40

Mesut Þahin-[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.

10. ODTÜ, 3 May 2019, Friday, 15:40

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.

11. Bilkent, 10 May 2019, Friday, 15:40

Nil Þahin-[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.

12. ODTÜ, 17 May 2019, Friday, 15:30 <<<<

Yýldýray 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 Ýkeda seminar room at the Mathematics building of ODTÜ.
Bilkent talks are
at room 141 of Faculty of Science A-building at Bilkent.

 2000-2001 Talks  (1-28) 2001 Fall Talks  (29-42) 2002 Spring Talks  (43-54) 2002 Fall Talks  (55-66) 2003 Spring Talks  (67-79) 2003 Fall Talks  (80-90) 2004 Spring Talks (91-99) 2004 Fall Talks (100-111) 2005 Spring Talks (112-121) 2005 Fall Talks (122-133) 2006 Spring Talks (134-145) 2006 Fall Talks (146-157) 2007 Spring Talks (158-168) 2007 Fall Talks (169-178) 2008 Spring Talks (179-189) 2008 Fall Talks (190-204) 2009 Spring Talks (205-217) 2009 Fall Talks (218-226) 2010 Spring Talks (227-238) 2010 Fall Talks (239-248) 2011 Spring Talks (249-260) 2011 Fall Talks (261-272) 2012 Spring Talks (273-283) 2012 Fall Talks (284-296) 2013 Spring Talks (297-308) 2013 Fall Talks (309-319) 2014 Spring Talks (320-334) 2014 Fall Talks (335-348) 2015 Spring Talks (349-360) 2015 Fall Talks (361-371) 2016 Spring Talks (372-379) 2016 Fall Talks (380-389) 2017 Spring Talks (390-401) 2017 Fall Talks (402-413) 2018 Spring Talks (414-425) 2018 Fall Talks (426-434)