# Uluslararası Cebirsel GEometri Neşesi

Organizers: Alp Bassa, Türkü Özlüm Çelik, Özgür Esentepe, Ali Özgür Kişisel
Meeting Times: Wednesday 8:00 (PDT) 11:00 (EST) 18:00 (TSİ)
UCGEN is an online seminar around topics near algebraic geometry. As our first few speakers suggest, near algebraic geometry is a broad term which includes representation theory, commutative and homological algebra, arithmetic geometry and number theory and really anything that touches algebraic geometry.

Hosted by IMBM.
Contact: turkuozlum[at]gmail[dot]com

### Upcoming Talks

September 23, 2020. Homological mirror symmetry for chain type invertible polynomials Umut Varolgüneş (Stanford University)
September 30, 2020. Higher Structures in Physics Kadri Ilker Berktav (Middle East Technical University)
October 7, 2020. Monomial Ideals of Graphs and Their Syzygies Selvi Kara (University of South Alabama)
October 14, 2020. Explicit Vologodsky Integration for Hyperelliptic Curves Enis Kaya (University of Groningen)
October 21, 2020. TBA Irem Portakal (Otto von Guericke University Magdeburg)
October 28, 2020. TBA Özhan Genç (Jagiellonian University)
November 4, 2020. TBA Oğuz Şavk (Boğaziçi University)
November 11, 2020. TBA Bahar Acu (Northwestern University)
November 18, 2020. TBA Ayşegül Öztürkalan (Abdullah Gül University)
November 25, 2020. TBA Özlem Ejder (Boğaziçi University)
December 2, 2020. TBA Sema Güntürkün (University of Massachusetts Amherst)
December 9, 2020. TBA İzzet Coşkun (University of Illinois at Chicago)
December 16, 2020. TBA Burçin Güneş (Sabancı University)
December 23, 2020. TBA Seçkin Adalı (Koç University)

### Details

Speaker: Enis Kaya
Title: Explicit Vologodsky Integration for Hyperelliptic Curves
Date: October 14, 2020
Abstract: Let X be a curve over a p-adic field with semi-stable reduction and let ω be a meromorphic 1-form on X. There are two notions of p-adic integration one may associate to this data: the Berkovich–Coleman integral which can be performed locally; and the Vologodsky integral with desirable number-theoretic properties. In this talk, we present a theorem comparing the two, and describe an algorithm for computing Vologodsky integrals in the case that X is a hyperelliptic curve. We also illustrate our algorithm with a numerical example computed in Sage. This talk is partly based on joint work with Eric Katz.

Speaker: Selvi Kara
Title: Monomial Ideals of Graphs and Their Syzygies
Abstract: Given a homogeneous ideal $I$ in a polynomial ring $R=k[x_1, \ldots, x_n],$ we can describe the structure of $I$ by using its minimal free resolution. All the information related to the minimal free resolution of $I$ is encoded in its Betti numbers. However, it is a difficult problem to express Betti numbers of any homogeneous ideal in a general way. Due to this difficulty, it is common to focus on coarser invariants of $I$ or particular classes of ideals.

In this talk, we consider monomial ideals associated to graphs. We will discuss the Castelnuovo-Mumford regularity, projective dimension, and extremal Betti numbers of such ideals and provide formulas for these invariants in terms of the combinatorial data of their associated graphs. Results presented in this talk are from joint works with Biermann, O’Keefe, Lin, and Casiday.

Title: Higher Structures in Physics
Date: September 30, 2020
Abstract: This is an overview of higher structures in physics. In this talk, we intend to outline the basics of derived algebraic geometry and its essential role in encoding the formal geometric aspects of moduli spaces of solutions to certain differential equations. Throughout the talk, we always study objects with higher structures in a functorial perspective, and we shall focus on algebraic local models for those structures. To be more precise, we shall be interested in derived geometric constructions and higher spaces for certain moduli problems associated with classical field theories and their defining equations, the so-called Euler-Lagrange equations.
To this end, the talk is organized into two main parts: In the first part of the talk, we shall revisit the naïve and algebro-geometric definition of a classical field theory together with some examples, and then we will establish the connection between classical field theories and moduli problems. In the second part of the talk, we first recall the basic aspects of moduli theory in a categorical perspective and explain how higher-categorical notions like stacks come into play to overcome certain technical problems naturally arising in many moduli problems. In the spirit of these discussions, we shall also give some examples from gauge theory and Einstein gravity.

Speaker: Umut Varolgüneş
Institution: Stanford University
Special Time: 9PDT, 12EST, 7TSi
Title: Homological mirror symmetry for chain type invertible polynomials
Abstract: I will start by giving a quick introduction to classical and symplectic Picard-Lefschetz theory. Then, I will explain the homological mirror symmetry (HMS) conjecture regarding invertible polynomials. Finally, I will sketch the A-side computation that goes into proving HMS in the chain type case. This is joint work with A. Polishchuk.

Speaker: Hülya Argüz
Institution: University of Versailles Saint-Quentin-En-Yvelines
Title: An algebro-geometric view on mirror symmetry
Abstract: Mirror symmetry is a phenomenon discovered by string theorists, which relates physical theories obtained using different deformation families of Calabi-Yau manifolds. An algebro--geometric approach to mirror symmetry, which uses tropical and log geometric tools to construct such families of Calabi--Yau manifolds, is provided by the Gross-Siebert program. In this talk we will review the most recent advances in this program, and particularly report on our joint work with Mark Gross.

Speaker: Mehmet Kıral
Institution: RIKEN AIP
Title: Kloosterman Sums for SL3 Long Word Element
Date: September 9, 2020
Time: 8:00 EST, 15:00 TSi Note the time change.
Abstract: Using the reduced word decomposition of the long word element of the Weyl group element of SL3, we give a nice expression for the long word Kloosterman sum. First classical Kloosterman sums, their importance, and matrix formulation will be introduced. This is joint work with Maki Nakasuji of Sophia University (Tokyo).

Speaker: Özge Ülkem
Institution: Heidelberg University
Title: Uniformization of the moduli space of generalized $\mathcal{D}$-elliptic sheaves
Date: September 2, 2020
Abstract: Drinfeld defined the notion of elliptic modules, which are now called Drinfeld modules, as an analogue of elliptic curves in the function field setting. To prove the Langlands correspondence in this context, Drinfeld studied moduli spaces of elliptic sheaves. The categories of elliptic sheaves and Drinfeld modules are equivalent under certain conditions. Since then, many generalizations of elliptic sheaves have been studied, such as $\mathcal{D}$-elliptic sheaves defined by Laumon, Rapoport and Stuhler and Frobenius-Hecke sheaves defined by Stuhler. In this talk we will give a brief introduction to the function field world and introduce a new generalization of elliptic sheaves, called generalized $\mathcal{D}$-elliptic sheaves. We will state a uniformization theorem for the moduli space of the latter and talk about the proof if time permits. This builds on work of Laumon-Rapoport-Stuhler, of Hartl and of Rapoport-Zink.

Speaker: Yusuf Barış Kartal
Institution: Princeton University
Title: p-adic analytic actions on Fukaya categories and iterates of symplectomorphisms
Date: August 26, 2020
Abstract: A theorem of Bell, Satriano and Sierra state that for a given smooth complex surface $X$ with an automorphism $\phi$ the set of natural numbers $n$ such that $\mathscr{Ext}^i(\mathscr{F},(\phi^*)^n(\mathscr{F}'))\neq 0$ is a union of finitely many arithmetic progressions and finitely many other numbers. Due to homological mirror symmetry conjecture, one can expect a symplectic version of this statement. In this talk, we will present such a theorem for a class of symplectic manifolds and symplectomorphisms isotopic to identity. The technique is analogous to its algebro-geometric counterpart: namely we construct p-adic analytic action on a version of the Fukaya category, interpolating the action of the iterates of the symplectomorphism.

Speaker: Emre Sertöz
Institution: Leibniz University Hannover
Title: Separating Periods of Quartic Surfaces
Date: August 19, 2020
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.

In this talk, we introduce an effective height function for periods of quartic surfaces defined over algebraic numbers. We also determine the minimal distance between periods of bounded height on a single surface. We use these results to prove heuristic computations of Picard groups that rely on approximations of periods. Moreover, we give explicit Liouville type numbers that can not be the ratio of two periods of a quartic surface. This is ongoing work with Pierre Lairez (Inria, France).

Speaker: Emine Yıldırım
Institution: Queen's University
Title: Cluster Categories
Date: August 12, 2020
Abstract: Cluster Categories are introduced to understand cluster dynamics from the representation theory point of view. The subject has its roots in two important results in the literature that give us a glimpse of a relationship between cluster dynamics and representation theory. The first is that there is an one-to-one correspondence between the cluster variables of a finite type cluster algebra and the almost positive roots of the corresponding root system. The second is a well-known result by Gabriel that classifies finite representation type quivers by using positive roots of the corresponding root system. In this talk, after giving an overview of cluster categories, I will talk about a recent joint work with Charles Paquette on the generalization of discrete cluster categories.
September 2, 2020. Uniformization of the moduli space of generalized $\mathcal{D}$-elliptic sheaves Özge Ülkem (Heidelberg University)