ODTÜ-BİLKENT Algebraic Geometry Seminar

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**** 2024 Spring Talks ****

This semester we plan to have all of our seminars online

541.   Zoom, 23 February 2024, Friday, 15:40

     Pınar Mete-[Balıkesir] - On some invariants of the tangent cones of numerical semigroup rings
     

     Abstract: The minimal free resolution is a very useful tool for extracting information about modules. Many important numerical invariants of a module such as Hilbert function and Betti numbers can be deduced from its minimal free resolution. Stamate gave a broad survey on these topics when the modules are the semigroup ring or its tangent cone for a numerical semigroup S. He also stated the problem of describing the Betti numbers and the minimal free resolution for the tangent cone when S is 4-generated semigroup which is symmetric. In this talk, I will first give some of our results, based on a joint work with E.E. Zengin on the problem. Then, I will talk about our ongoing study which is an application of the Apery table of the numerical semigroup to determine some properties of its tangent cone. DI. STAMATE, Betti numbers for numerical semigroup rings. Multigraded Algebra and Applications, 238, 133-157, Springer Proceedings in Mathematics and Statistics, Springer, Cham 2018.

      
       
     
542. Zoom, 1 March 2024, Friday, 15:40

     Turgay Bayraktar-[Sabancı] - Equidistribution for Zeros of Random Polynomial Systems
     

     Abstract: A classical result of Erdös and Turan asserts that for a univariate complex polynomial whose middle coefficients are comparable to the extremal ones, the zeros accumulate near the unit circle. We prove  the analogues result for random polynomial mappings with Bernoulli coefficients. The talk is based on the joint work with Çiğdem Çelik.

         
     
     
543. Zoom, 15 March 2024 Friday, 15:40
       
     Kaan Bilgin-[Amsterdam] - The Langlands – Kottwitz method for GSpin Shimura varieties and eigenvarieties
     
     

     Abstract:   Given a connected reductive algebraic group G over a number field F, the global Langlands (reciprocity) conjecture roughly predicts that, there should be a correspondence between (automorphic side) the isomorphism classes of  (cuspidal, cohomological) automorphic representations of G and (Galois side) the isomorphism classes of (irreducible, locally de-Rham) Galois representations for Gal(\bar{F} / F) taking values in the Langlands dual group of G. In the first part of this talk, I will sketch the main argument of our expected theorem/proof for (automorphic to Galois) direction of this conjecture, for G = GSpin(n,2), n odd and F to be totally real, under 3 technical assumptions (for time being), namely we assume that automorphic representations are additionally “non-CM” and “non-endoscopic” and “std-regular”. In the second part, mainly following works of Loeffler and Chenevier on overconvergent p-adic automorphic forms,  I will present an idea to remove the std-regular assumption on the theorem via the theory of eigenvarieties.

       
       
     
544. Zoom, 22 March 2024, Friday, 15:40
       
     Haydar Göral-[İYTE] - Arithmetic Progressions in Finite Fields
     
     

     Abstract:  In 1927, van der Waerden proved a theorem regarding the existence of arithmetic progressions in any partition of the positive integers with finitely many classes. In 1936, a strengthening of van der Waerden's theorem was conjectured by Erdös and Turan, which states that any subset of positive integers with a positive upper density contains arbitrarily long arithmetic progressions. In 1975, Szemeredi developed his combinatorial method to resolve this conjecture, and the affirmative answer to Erdös and Turan's conjecture is now known as Szemeredi's theorem. As well as in the integers, Szemeredi-type problems have been extensively studied in subsets of finite fields. While much work has been done on the problem of whether subsets of finite fields contain arithmetic progressions, in this talk we concentrate on how many arithmetic progressions we have in certain subsets of finite fields. The technique is based on certain types of Weil estimates. We obtain an asymptotic for the number of k-term arithmetic progressions in squares with a better error term. Moreover our error term is sharp and best possible when k is small, owing to the Sato-Tate conjecture. This work is supported by the Scientific and Technological Research Council of Turkey with the project number 122F027.

          
     
     
545. Zoom, 29 March 2024, Friday, 15:40
       
     Büşra Karadeniz Şen-[GTU] -Boundaries of the dual Newton polyhedron may describe the singularity
     
     

     Abstract:  We are dealing with a hypersurface $X\subset \mathbb{C}^3$ having non-isolated singularities.We construct an embedded toric resolution of $X$ using some specific vectors in its dual Newton polyhedron. To do this, we first define the profile of a full dimensional cone and we establish a relation between the jet vectors and the integer points in the profile. This is a part of the joint work with C. Plénal and M. Tosun. References [1] A. Altintaş Sharland, C. O. Oğuz, M. Tosun and Z.aferiakopoulos, An algorithm to find nonisolated forms of rational singularities, In preparation. [2] C. Bouvier and G. Gonzalez-Sprinberg, Systéme générateur minimal, diviseurs essentiels et G-désingularisations de variétés torique, Tohoku Math. J., 47, 125-149, 1995. [3] B. Karadeniz Şen, C. Plénat and M. Tosun, Minimality of a toric embedded resolution of singularities after Bouvier-Gonzalez-Sprinberg, Kodai Math J., accepted, 2024.

       
       
     
546. Zoom, 5 April 2024, Friday, 15:40
       
     Enis Kaya-[KU Leuven] - p-adic Integration Theories on Curves
     
     

     Abstract:  For curves over the field of p-adic numbers, there are two notions of p-adic integration: Berkovich-Coleman integrals which can be performed locally, and Vologodsky integrals with desirable number-theoretic properties. These integrals have the advantage of being insensitive to the reduction type at p, but are known to coincide with Coleman integrals in the case of good reduction. Moreover, there are practical algorithms available to compute Coleman integrals. Berkovich-Coleman and Vologodsky integrals, however, differ in general. In this talk, we give a formula for passing between them. To do so, we use combinatorial ideas informed by tropical geometry. We also introduce algorithms for computing Berkovich-Coleman and Vologodsky integrals on hyperelliptic curves of bad reduction. By covering such a curve by certain open spaces, we reduce the computation of Berkovich-Coleman integrals to the known algorithms on hyperelliptic curves of good reduction. We then convert the Berkovich-Coleman integrals into Vologodsky integrals using our formula. We illustrate our algorithm with a numerical example. This talk is partly based on joint work with Eric Katz from the Ohio State University.

          
     
     
547. Zoom, 19 April 2024, Friday, 15:40
       
     Ichiro Shimada-[Hiroshima] - Real line arrangements and vanishing cycles
     
     

     Abstract: We investigate the topology of a double cover of a complex affine plane branching along a nodal real line arrangement.We define certain topological 2-cycles in the double plane using the real structure of the arrangement.These cycles resemble vanishing cycles of Lefschetz.We then  calculate their intersection numbers.

      
       
548. Zoom, 26 April 2024, Friday, 15:40
       
     Yaacov Kopeliovich -[Connecticut] - The fundamental group of a Riemann surface from Riemann existence theorem
     
     

     Abstract:  It is well known that fundamental curves above complex numbers  have 2g generators where g is the genus of the curve with one non trivial relation that is the commutation relation. Surprisingly I haven’t found a proof of this well known fact in the literature. In this talk I will attempt to fill in the gap and show how this can be shown in a straightforward matter.

        
      
     
549. Zoom, 3 May 2024, Friday, 15:40
       
     Meirav Amram-[Shamoon] -  Interesting methods to classify algebraic curves and surfaces
     
     

     Abstract:  We present a few algebraic, geometric and topological methods that we use in the classification of algebraic curves and surfaces. We speak about a few invariants of the classification as well. We discuss degeneration of algebraic surfaces, the calculation of fundamental groups and some computational methods that help with these calculations.

     
     
     
550. Zoom, 10 May 2024, Friday, 15:40
       
     Alexander Degtyarev-[Bilkent] - Singular real plane sextic curves with smooth real part
     
     

     Abstract: For a change, I will give a detailed proof of one of our joint results announced in an earlier talk, viz. the fact that the equisingular equivariant deformation type of a real plane sextic curve with smooth real part is determined by its real homological type (in the most naïve meaning of the term); this theorem has been used to obtain a complete classification of such curves. The principal goal is introducing the newer generation into the fascinating theory of K3-surfaces, real aspects thereof, and algebra/number theory involved. This is a joint work in progress with Ilia Itenberg.

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.