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

(The New Yorker, Dec 7, 2020 Cover)

This semester we plan to have all our seminars online

477. Zoom, 5 February 2021, Friday, 15:40

     Caner Koca-[City University of New York] - Kähler Geometry and Einstein-Maxwell Metrics
         
     

     Abstract: A classical problem in Kähler Geometry is to determine a canonical representative in each Kähler class of a complex manifold. In this talk, I will introduce this problem in several well-known settings (Calabi-Yau, Kähler-Einstein, constant-scalar-curvature-Kähler, extremal Kähler). In light of recent examples and developments, I will elucidate a possible role of Einstein-Maxwell metrics in this problem.

      
       
     
478. Zoom, 12 February 2021, Friday, 15:40

     Yıldıray Ozan-[ODTÜ] - Liftable homeomorphisms of finite abelian p-group regular branched covers over the 2-sphere and the projective plane
         
     

     Abstract: This talk mainly is based our work joint with F. Atalan and E. Medetoğulları. In 2017 Ghaswala and Winarski classified finite cyclic regular branched coverings of the 2-sphere, where every homeomorphism of the base (preserving the branch locus) lifts to a homeomorphism of the covering surface, answering a question of Birman and Hilden. In this talk, we will present generalizations of  this result in two directions. First, we will replace finite cyclic groups with finite abelian p-groups. Second, we will replace the base surface with the real projective plane. The main tool is the algebraic characterization of such coverings in terms of the automorphism groups of these finite abelian p-groups. Due to computational insufficiencies we have complete results only for groups of rank 1 and 2. In particular, we prove that for a regular branched $A$-covering $\pi:\Sigma\rightarrow S^2$, where $A={\mathbb Z}_{p^r}\times{\mathbb Z}_{p^t}, \ 1\leq r\leq t$, all homeomorphisms $f:S^2 \to S^2$ lift to those of $\Sigma$, if and only if $t=r$ or $t=r+1$ and $p=3$. If time permits we will also present some applications to automorphisms of Riemann surfaces.

         
     
     
479. Zoom, 19 February 2021, Friday, 15:40

     Meral Tosun-[Galatasaray] - A new root system and free divisors
         
     

     Abstract: In this talk, we will  construct a root system for the minimal resolution graph of some surface singularities and we will show that the new roots give linear free divisors.

       
       
     
480. Zoom, 26 February 2021, Friday, 15:40

     Tony Scholl-[Cambridge] - Plectic structures on locally symmetric varieties
         
     

     Abstract: In this talk I will discuss a class of locally symmetric complex varieties whose cohomology seems to behave as if they are products, even though they are not. This has geometric and number-theoretic consequences which I will describe. This is joint work with Jan Nekovář (Paris).

          
     
     
481. Zoom, 5 March 2021, Friday, 15:40

     Alexander Degtyarev-[Bilkent] - 800 conics in a smooth quartic surface
     
     

     Abstract: Generalizing Bauer, define $N_{2n}(d)$ as the maximal number of smooth rational curves of degree $d$ that can lie in a smooth degree-$2n$ K3-surface in $\mathbb{P}^{n+1}$. (All varieties are over $\mathbb{C}$.) The bounds $N_{2n}(1)$ have a long history and currently are well known, whereas for $d=2$ the only known value is $N_6(2)=285$ (my recent result reported in this seminar). In the most classical case $2n=4$ (spatial quartics), the best known examples have 352 or 432 conics (Barth and Bauer), whereas the best known upper bound is 5016 (Bauer with a reference to Strømme). For $d=1$, the extremal configurations (for various values of $n$) tend to exhibit similar behavior. Hence, contemplating the findings concerning sextic surfaces, one may speculate that  -- it is easier to count *all* conics, both irreducible and reducible, but  -- nevertheless, in extremal configurations all conics are irreducible. On the other hand, famous Schur's quartic (the one on which the maximum $N_4(1)$ is attained) has 720 conics (mostly reducible), suggesting that 432 should be far from the maximum $N_4(2)$. Therefore, in this talk I suggest a very simple (although also implicit) construction of a smooth quartic with 800 irreducible conics. The quartic found is Kummer in the sense of Barth and Bauer: it contains 16 disjoint conics. I conjecture that $N_4(2)=800$ and, moreover, 800 is the sharp upper bound on the total number of conics (irreducible or reducible) in a smooth spatial quartic.

       
       
     
482. Zoom, 12 March 2021, Friday, 15:40

     Anar Dosi-[ODTU-Northern Cyprus] - Algebraic spectral theory and index of a variety
         
     

     Abstract: The present talk is devoted to an algebraic treatment of the joint spectral theory within the  framework of Noetherian modules over an algebra finite extension of an algebraically closed field. We discuss the spectral mapping theorem and analyse the index of tuples in purely algebraic case. The index function over tuples from the coordinate ring of a variety is naturally extended up to a numerical Tor-polynomial which behaves as the Hilbert polynomial and provides a link between the index and dimension of a variety.

          
     
     
483. Zoom, 19 March 2021, Friday, 15:40

     Remziye Arzu Zabun-[Gaziantep] - Topology of Real Schläfli Six-Line Configurations on Cubic Surfaces and in $\mathbb{RP}^3$
         
     

     Abstract: A famous configuration of 27 lines on a non-singular cubic surface in $\mathbb{CP}^3$ contains remarkable subconfigurations, and in particular the ones formed by six pairwise disjoint lines. We will discuss such six-line configurations in the case of real cubic surfaces from topological viewpoint, as configurations of six disjoint lines in the real projective 3-space, and show that the condition that they lie on a cubic surface implies a very special property which distinguishes them in the Mazurovskii list of 11 deformation types of configurations formed by six disjoint lines in $\mathbb{RP}^3$. This is joint work with Sergey Finashin.

      
       
484. Zoom, 26 March 2021, Friday, 16:00

     Türkü Özlüm Çelik-[Simon Fraser University] - Integrable Systems in Symbolic, Numerical and Combinatorial Algebraic Geometry
         
     

     Abstract: The Kadomtsev-Petviashvili (KP) equation is a universal integrable system that describes nonlinear waves. It is known that algebro-geometric approaches to the KP equation provide solutions coming from a complex algebraic curve, in terms of the Riemann theta function associated with the curve. Reviewing this relation, I will introduce an algebraic object and discuss its algebraic and geometric features: the so-called Dubrovin threefold of an algebraic curve, which parametrizes the solutions. Mentioning the relation of this threefold with the classical algebraic geometry problem, namely the Schottky problem, I will report a procedure that is via the threefold and based on numerical algebraic geometric tools, which can be used to deal with the Schottky problem from the lens of computations. I will finally focus on the geometric behaviour of the threefold when the underlying curve degenerates.

        
      
485. Zoom, 2 April 2021, Friday, 15:40

     Özhan Genç-[Jagiellonian] - Instanton Bundles on $\mathbb{P}^1 \times \mathbb{F}_1$
         
     

     Abstract: A $\mu$-stable vector bundle $\mathcal{E}$ of rank 2 with $c_1 (\mathcal{E})=0$ on $\mathbb{P}_{\mathbb{C}}^{3}$ is called mathematical instanton bundle if $\mathrm{H}^1 (\mathbb{P}^{3}, \mathcal{E}(-2))=0$. In this talk, we will study the definiton of mathematical instanton bundles on Fano 3-folds and the construction of them on $\mathbb{P}^1 \times \mathbb{F}_1$ where $\mathbb{F}_1$ is the del Pezzo surface of degree 8. This talk is based on the joint work with Vincenzo Antonelli and Gianfranco Casnati.

         
     
     
486. Zoom, 9 April 2021, Friday, 15:40

     Berrin Şentürk-[TEDU] - Free Group Action on Product of 3 Spheres
         
     

     Abstract: A long-standing Rank Conjecture states that if an elementary abelian $p$-group acts freely on a product of spheres, then the rank of the group is at most the number of spheres in the product. We will discuss the algebraic version of the Rank Conjecture given by Carlsson for a differential graded module $M$ over a polynomial ring. We will state a stronger conjecture concerning varieties of square-zero upper triangular matrices corresponding to the differentials of certain modules. By the work on free flags in $M$ introduced by Avramov, Buchweitz, and Iyengar, we will obtain some restriction on the rank of submodules of these matrices. By this argument we will show that $(\mathbb{Z}/2\mathbb{Z})^4$ cannot act freely on product of $3$ spheres of any dimensions.

          
     
     
487. Big Blue Button, 16 April 2021, Friday, 15:40

     Yankı Lekili-[Imperial College London] - A panorama of Mirror Symmetry
     
     

     Abstract: Mirror symmetry is one of the most striking developments in modern mathematics whose scope extends to very different fields of pure mathematics. It predicts a broad correspondence between two subfields of geometry - symplectic geometry and algebraic geometry. Homological mirror symmetry uses the language of triangulated categories to give a mathematically precise meaning to this correspondence. Since its announcement, by Kontsevich in ICM (1994), it has attracted huge attention and over the years several important cases of it have been established. Despite significant progress, many central problems in the field remain open. After reviewing the general features, I will survey some of my recent results on mirror symmetry (with thanks to collaborators T. Perutz, A. Polishchuk, K. Ueda, D. Treumann).

          
       
488. Zoom, 30 April 2021, Friday, 15:40
     
     Çisem Güneş Aktaş-[Abdullah Gül] - Real representatives of equisingular strata of  projective models of K3-surfaces
      
     

     Abstract: It is a wide open problem what kind of singularities a projective surface or a curve of a given degree can have. In general, this problem seems hopeless. However, in the case of K3-surfaces, the equisingular deformation classification of surfaces with any given polarization becomes a mere computation. In this talk, we will discuss projective models of K3-surfaces of different polarizations  together with the deformation classification problems.  Although it is quite common that a real variety may have no real points, very few examples of equisingular deformation classes with this property are known.  We will study an algorithm detecting real representatives in equisingular strata of projective models of K3-surfaces. Then, we will apply this algorithm to spatial quartics and find two new examples of real strata without real representatives where the only previously known example of this kind is in the space of plane sextics.

     
     

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.