# Uluslararası Cebirsel GEometri Neşesi

Organizers: Alp Bassa, Türkü Özlüm Çelik, Özgür Esentepe, Ali Özgür Kişisel
Meeting Times: Wednesday 10:00 (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
Click here to subscribe to our mailing list for upcoming talks and zoom links, here for a list of past talks and here for our YouTube channel.

### Upcoming Talks

December 2, 2020. On the Eisenbud-Green-Harris conjecture. Sema Güntürkün (Amherst College)
December 9, 2020. Brill-Noether Theorems for moduli spaces of sheaves on surfaces İzzet Coşkun (University of Illinois at Chicago)
December 16, 2020. TBA Burçin Güneş (Sabancı University)
January 6, 2021. TBA Sümeyra Sakallı (Max Planck Institute)
January 13, 2021. TBA Levent Doğan (Technical University of Berlin)
January 20, 2021. TBA Kürşat Sözer (University of Lille)
January 27, 2021. TBA Emre Şen (University of Iowa)
February 3, 2021. TBA Oğuzhan Yürük (TU Braunschweig)
February 10, 2021. TBA Ilke Çanakçı (Free University of Amsterdam)
February 17, 2021. TBA Sinem Odabaşı (Universidad Austral de Chile)
February 24, 2021. TBA Büşra Sert (TU Dresden)
March 3, 2021. TBA Emrah Sercan Yılmaz (Boğaziçi University)
March 10, 2021. TBA TBA
March 17, 2021. TBA TBA
March 24, 2021. TBA TBA
March 31, 2021. TBA Nursel Erey (Gebze Technical University)
April 7, 2021. TBA Olgur Çelikbaş (West Virginia University)

### Details

Speaker: İzzet Coşkun
Institution: University of Illinois at Chicago
Title: Brill-Noether Theorems for moduli spaces of sheaves on surfaces
Abstract: In this talk, I will describe several results on the cohomology of the general sheaf in a moduli space of sheaves on a projective surface. I will discuss joint work with Jack Huizenga on rational surfaces such as Hirzebruch surfaces and joint work with Howard Nuer and Kota Yoshioka on K3 surfaces.

Speaker: Sema Güntürkün
Institution: Amherst College
Title: On the Eisenbud-Green-Harris conjecture.
Abstract: A generalization of the Macaulay’s theorem on the growth of Hilbert functions of homogeneous ideals in $K[x_1,\ldots, x_n]$ is conjectured by Eisenbud, Green and Harris in the 90s. The conjecture, also known as the EGH conjecture, states that the lex-plus-powers ideals show an extremal behavior among the homogeneous ideals containing regular sequences in terms of their Hilbert functions.  In this talk, our focus will be on a case of the EGH conjecture for the homogeneous ideals containing a regular sequence of quadratic forms. This is a joint work with Mel Hochster.

Institution: Middle East Technical University
Title: Higher Structures in Einstein Gravity
Abstract: This is a talk on a recent investigation about higher structures in the theory of General Relativity. It can be also seen as a direct sequel of the previous talk “Higher Structures in Physics.” However, for the sake of completeness, the talk will include a brief summary of key ideas from the aforementioned talk. In that respect, we shall begin with revisiting the basics of moduli theory and derived algebraic geometry. Next, we will report some relevant constructions and results from our work encoding various stacky formulations of Einstein Gravity.

Speaker: Ayşegül Öztürkalan
Institution: Abdullah Gül University
Title: Loops in moduli spaces of real plane projective curves
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 come from strata of singular curves and originates from spaces of linear equivalent real divisors on a real cubic curve.

Speaker: Bahar Acu
Institution: ETH Zurich
Title: Understanding symplectic fillings of contact manifolds via algebraic varieties
Abstract: This talk is an attempt for a (pandemic-conscious) invitation to contact topology via an algebro-geometric approach with the caveat that we admit having little to no understanding of many concepts in algebraic geometry. A very useful strategy in studying topological manifolds is to factor them into smaller pieces. Briefly, an "open book decomposition" on an $n$-dimensional manifold (the open book) is a type of fibration over a circle that helps us study our manifold in terms of its $(n-1)$-dimensional fibers (the pages) and $(n-2)$-dimensional boundary of these fibers (the binding). Open books provide a natural framework for studying the topological properties of a geometric phenomenon called "contact structures" on smooth manifolds. In this talk, we aim to provide an exposition of results, some of which are fruits of several joint works, concerning "symplectic fillings" of contact manifolds given by certain classes of algebraic varieties using their "supporting" open books.

Speaker: Oğuz Şavk
Institution: Boğaziçi University
Title: Brieskorn spheres, homology cobordism and homology balls
Abstract: A classical question in low-dimensional topology asks which homology $3$-spheres bound homology $4$-balls. This question is fairly addressed to Brieskorn spheres $\Sigma(p,q,r)$. Since they are defined to be links of singularities $x^p+y^q+z^r=0$, Brieskorn spheres are algebro-geometric originated $3$-manifolds. Over the years, Brieskorn spheres also have been the main objects for the understanding of the algebraic structure of the integral homology cobordism group. In this talk, we will present several families of Brieskorn spheres which do or do not bound integral and rational homology balls. Also, we will investigate their positions in both integral and rational homology cobordism groups.

Speaker: Özhan Genç
Institution: Jagiellonian University
Title: Ulrich Trichotomy on del Pezzo Surfaces
Abstract: A vector bundle $\mathcal{E}$ on a projective variety $X$ in $\mathbb{P}^N$ is Ulrich if $H^∗(X,E(−k))$ vanishes for $1 ≤k ≤\dim(X)$. It has been conjectured by Eisenbud and Schreyer that every projective variety carries an Ulrich bundle. Even though this conjecture has not been proved or disproved, another interesting question is worth considering: classify projective varieties as Ulrich finite, tame or wild type with respect to families of Ulrich bundles that they support. In this talk, we will show that this trichotomy is exhaustive for certain del Pezzo surfaces with any given polarization. This talk is based on a joint work with Emre Coşkun.

Speaker: İrem Portakal
Institution: Otto von Guericke University Magdeburg
Title: Rigid toric matrix Schubert varieties
Abstract: In this talk, we introduce the usual torus action on matrix Schubert varieties. In the toric case we show that these varieties arise from a bipartite graph. We study the first order deformations of toric matrix Schubert varieties and we prove that it is rigid if and only if the three-dimensional faces of its associated (edge) cone are all simplicial.

Speaker: Enis Kaya
Institution: University of Groningen
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
Institution: University of South Alabama
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.

Institution: Middle East Technical University
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.

### Past Talks

November 25, 2020.
Higher Structures in Einstein Gravity Kadri İlker Berktav (Middle East Technical University)
November 18, 2020. Loops in moduli spaces of real plane projective curves Ayşegül Öztürkalan (Abdullah Gül University)
November 11, 2020. Understanding symplectic fillings of contact manifolds via algebraic varieties Bahar Acu (ETH Zurich)
November 4, 2020. Brieskorn spheres, homology cobordism and homology balls Oğuz Şavk (Boğaziçi University)
October 28, 2020. Ulrich Trichotomy on del Pezzo Surfaces Özhan Genç (Jagiellonian University)
October 21, 2020. Rigid toric matrix Schubert varieties Irem Portakal (Otto von Guericke University Magdeburg)
October 14, 2020. Explicit Vologodsky Integration for Hyperelliptic Curves Enis Kaya (University of Groningen)
October 7, 2020. Monomial Ideals of Graphs and Their Syzygies Selvi Kara (University of South Alabama)
September 30, 2020. Higher Structures in Physics Kadri Ilker Berktav (Middle East Technical University)
September 23, 2020. Homological mirror symmetry for chain type invertible polynomials Umut Varolgüneş (Stanford University)
September 16, 2020. An algebro-geometric view on mirror symmetry Hülya Argüz (University of Versailles Saint-Quentin-En-Yvelines)
September 9, 2020. Kloosterman Sums for SL3 Long Word Element Mehmet Kıral (RIKEN AIP)
September 2, 2020. Uniformization of the moduli space of generalized $\mathcal{D}$-elliptic sheaves Özge Ülkem (Heidelberg University)
August 26, 2020. p-adic analytic actions on Fukaya categories and iterates of symplectomorphisms Yusuf Barış Kartal (Princeton University)
August 19, 2020. Separating Periods of Quartic Surfaces Emre Sertöz (Leibniz University Hannover)
August 12, 2020. Cluster Categories Emine Yıldırım (Queen's University)