ODTÜ-BİLKENT Algebraic Geometry Seminar
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**** 2021 Fall Talks ****

This semester we plan to have most of our seminars online
tentatively we now list all talks as online
check for last minute changes

489. Zoom, 1 October 2021, Friday, 16:00

     İzzet Coşkun-[UIC] - The cohomology of a general stable sheaf on a K3 surface
     
     

     Abstract: In this talk, I will describe joint work with Howard Nuer and Kota Yoshioka on computing the cohomology of the general stable sheaf in a moduli space of semistable sheaves on a K3 surface of Picard rank 1. We find sharp bounds on the Mukai vector that guarantees that the cohomology can be determined from the Euler characteristic and classify all the Mukai vectors up to rank 20 where the generic sheaf does not exhibit the expected behavior.  We make critical use of Bridgeland stability conditions in our computation.

      
       
     
490. Zoom, 8 October 2021, Friday, 15:40

     Mesut Şahin-[Hacettepe] - Linear Codes on Subgroups of Weighted Projective Tori
         
     

     Abstract: Toric varieties are interesting geometric objects lying on the crossroad of algebra, geometry and combinatorics containing a dense torus which is an algebraic group acting on the toric variety. Many champion codes obtained from toric varieties appeared in the literature. The simplest examples of toric varieties include classical and weighted projective spaces. Parameters of linear codes obtained by evaluating rational functions on a projective torus are computed in 2011 by Sarmiento, Vaz Pinto and Villarreal. This idea is transferred to weighted projective tori and some parameters are computed in 2015 by Dias and Neves. The purpose of this talk is to introduce some linear codes on toric varieties. We focus on codes obtained from certain subgroups of the weighted projective torus over a finite field, and to share some formulas for their parameters in some cases. We restrict to two dimensional case to obtain more explicit formula for the minimum distance of the code on the weighted projective torus T(1,1,a) over F_q. This is a joint work with Oğuz Yayla of METU.

         
     
     
491. Zoom, 15 October 2021, Friday, 15:40

     Oğuzhan Yürük-[TU-Berlin] - Nonnegativity of the polynomials supported on circuits
         
     

     Abstract: A real multivariate polynomial is called nonnegative if its evaluation at any given point in R^n is nonnegative. Checking the nonnegativity of a real polynomial is a not only a mathematically challenging task, but also is an effective tool both for mathematics and for sciences. Often one uses nonnegativity certificates in order to tackle this problem, i.e., easily verifiable conditions that imply the nonnegativity for a large class of polynomials. The typical nonnegativity certificates usually make use of the fact that a polynomial is nonnegative if it is a sum of squares of polynomials (SOS polynomial), however not every nonnegative polynomial is of this form. In the first part this talk, we focus on a relatively new nonnegativity certificate based on the arithmetic and geometric means (AM-GM) inequality, and we elaborate on the fact that this class of polynomials neither contains nor is contained in the class of SOS polynomials. Unlike the SOS certificates, one is only interested in the exponents that show up in the support while working with AM-GM certificates. In particular, this gives us a framework to write sufficient symbolic conditions for the nonnegativity of a given sparse polynomial in terms of its coefficients. We utilize the aforementioned AM-GM framework in the second part of the talk, and present an application to a particular problem from the chemical reaction networks theory.

       
       
     
492. Zoom, 22 October 2021, Friday, 15:40

     Alp Bassa-[Boğaziçi] - Curves over finite fields and error correcting codes
         
     

     Abstract: Historically, questions about rational points on curves over finite fields occupy a prominent place in number theory. The introduction of the zeta functions for these curves by Artin led to an increased interest in this field, which culminated in the proof of the corresponding Riemann hypothesis by Hasse and Weil in the first half of the 20th century. After a long period, interest in this field was again reawakened in the 80's, when Goppa showed how this machinery from algebraic geometry can be used in the constructions of long codes allowing reliable communication over channels in the presence of errors. Using algebraic curves it became possible to beat the best constructions known to coding theorists and in the following decades many other applications in coding theory and cryptography followed. In this talk I will talk about recent results on the number of rational points on curves of large genus and their applications in the theory of error correcting codes.

          
     
     
493. Zoom, 5 November 2021, Friday, 15:40

     Sergey Finashin-[ODTÜ] - Two kinds of real lines on real del Pezzo surfaces and invariance of their signed count

     Abstract: In his classical treatise on real cubic surfaces, Segre discovered two kinds of real lines which he called elliptic and hyperbolic. His enumeration indicated that the number of hyperbolic is greater by 3 than the number of elliptic ones independently of a real structure on the cubic surface. However this property did not receive a conceptual explanation until recently: in a joint work with V.Kharlamov we interpreted a signed count of lines as a signed count of zeroes of some vector field in a Grassmannian (and so, it is Euler’s number of the corresponding vector bundle). In the current work that I will present, we develop an alternative approach to counting lines on real del Pezzo surfaces $X$ of degrees 1 and 2 (a projective plane blown up at 8 or 7 generic points, respectively).  The two types of real lines are distinguished by certain canonical Pin-structure on the real locus $X_{\mathbb{R}}$ (this looks different from the approach of Segre, but is equivalent to it in the case of cubic surfaces). A signed count of real lines is interpreted as some lattice root enumeration, which lets us prove our invariance properties for del Pezzo of degree 1 and 2, like in the case of cubic surfaces.

       
       
     
494. Zoom, 12 November 2021, Friday, 15:40

     Berkan Üze-[Boğaziçi] - : A Glimpse of Noncommutative Motives

     Abstract: The theory of motives was conceived as a universal cohomology theory for algebraic varieties. Today it is a vast subject systematically developed in many directions spanning algebraic geometry, arithmetic geometry, homotopy theory and higher category theory. Following ideas of Kontsevich, Tabuada and Robalo independently developed a theory of “noncommutative” motives for DG-categories (such as enhanced derived categories of schemes) which encompasses the classical theory of motives and helps assemble so-called additive invariants such as Algebraic K-Theory, Hochschild Homology and Topological Cyclic Homology into a motivic formalism in the very precise sense of the word. We will review the fundamental concepts at work, which will inevitably involve a foray into the formalism of enhanced and higher categories. We will then discuss Kontsevich’s notion of a noncommutative space and introduce noncommutative motives as “universal additive invariants” of noncommutative spaces. We will conclude by offering a brief sketch of Robalo’s construction of the noncommutative stable homotopy category, which is directly in the spirit of Voevodsky’s original construction.

          
     
     
495. Zoom, 19 November 2021, Friday, 15:40

     Sadık Terzi-[ODTÜ] - Some Special Torsors and Its Relation to BMY-Inequality
         
     

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496. Zoom, 26 November 2021, Friday, 15:40

     Alexander Degtyarev-[Bilkent] - Conics on polarized K3-surfaces
         
     

     Abstract: Generalizing Barth and Bauer, denote by $N_{2n}(d)$ the maximal number of smooth degree $d$ rational curves that can lie on a smooth $2n$-polarized $K3$-surface $X\subset\mathbb{P}^n$. Originally, the question was raised in conjunction with smooth spatial quartics, which are $K3$-surfaces. The numbers $N_{2n}(1)$ are well understood, whereas the only known value for $d=2$ is $N_6(2)=285$. I will discuss my recent discoveries that support the following conjecture on the conic counts in the remaining interesting degrees. Conjecture. One has $N_2(2)=8910$, $N_4(2)=800$, and $N_8(2)=176$. The approach used does not distinguish (till the very last moment) between reducible and irreducible conics. However, extensive experimental evidence suggests that all conics are irreducible whenever their number is large enough. Conjecture. There exists a bound $N_{2n}^*(2)<N_{2n}(2)$ such that, whenever a smooth $2n$-polarized $K3$-surface $X$ has more than $N_{2n}^*(2)$ conics, it has no lines and, in particular, all conics on $X$ are irreducible. We know that $249\le N^*_6(2)\le260$ is indeed well defined, and it seems feasible that $N^*_2(2)\ge8100$ and $N^*_4(2)\ge720$ are also defined; furthermore, conjecturally, the lower bounds above are the exact values.

        
      
497. Zoom, 3 December 2021, Friday, 15:40

     Emre Coşkun-[ODTÜ] - An Introduction to Hall Algebras of Quivers
         
     

     Abstract: In this talk, we shall define and study some basic properties of Hall algebras, and prove a theorem of Ringel on the structure of the Hall algebras of Dynkin quivers.

         
     
     
498. Zoom, 10 December 2021, Friday, 15:40

     Susumu Tanabé-[Galatasaray] - Asymptotic critical values of a polynomial map
         
     

     Abstract: The bifurcation locus of a polynomial map f is the smallest subset B(f) such that f realises a local trivialisation in the neighbourhood of each point of the complement to B(f). It is known that the bifurcation locus B(f) is the union of the set of critical values f(Sing f) and the set of bifurcation values at infinity which may be non-empty and disjoint from the critical value set f(Sing f). It is a difficult task to find the bifurcation locus in the cases for a polynomial depending on more than three variables. Nevertheless, one can obtain approximations by supersets of B(f) from exploiting asymptotical regularity conditions. Jelonek and Kurdyka established an algorithm for finding a superset of B(f): the set of asymptotic critical values. In this talk, we survey the history of the research of the bifurcation locus and discuss recent results on the asymptotic critical values.

          
       
     
499. Zoom, 17 December 2021, Friday, 15:40

     Ichiro Shimada-[Hiroshima] - Computation of automorphism groups of Enriques surfaces

     Abstract: By Torelli's theorem for K3 surfaces, the automorphism group of a complex Enriques surface is determined by the Hodge structure of the covering K3 surface. However, in many cases, explicit computations are very heavy and practically infeasible. We give a method by which one can compute a finitely generated set of automorphism groups of various Enriques surfaces and their nef cones. This is a joint work with Simon Brandhorst.

          
       
500. Zoom, 24 December 2021, Friday, 15:40
     
     Kadri İlker Berktav-[ODTÜ] - Higher structures in Einstein gravity
      
     

     Abstract: This is a talk on a recent investigation about higher structures in the theory of General Relativity. The talk essentially features higher categorical constructions and their consequences in various Einstein's gravity theories. In this talk, for the sake of completeness, we shall begin with a summary of key ideas from moduli theory and derived algebraic geometry. We, indeed, overview the basics of derived algebraic geometry and its essential role in encoding the formal geometric aspects of moduli problems in physics.  With this spirit, we will then investigate higher spaces and structures in various scenarios and present some of our works in this research direction.

     
     

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.