ODTÜ-BİLKENT Algebraic Geometry Seminar

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**** 2022 Spring 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

501.   Zoom, 18 February 2022, Friday, 15:40

     Deniz Kutluay-[Indiana] - Winding homology of knotoids
     
     

     Abstract: Knotoids were introduced by Turaev as open-ended knot-type diagrams that generalize knots. Turaev defined a two-variable polynomial invariant of knotoids generalizing the Jones knot polynomial to knotoids. We will give a construction of a triply-graded homological invariant of knotoids categorifying the Turaev polynomial, called the winding homology. Forgetting one of the three gradings gives a generalization of the Khovanov knot homology to knotoids. We will briefly review the basics of the theory of knotoids and also explain the notion of categorification which plays an important role in contemporary knot theory -- no prior knowledge will be assumed.

      
       
     
502. Zoom, 25 February 2022, Friday, 15:40

     Turgay Bayraktar-[Sabancı] - Universality results for zeros of random holomorphic sections
         
     

     Abstract: In this talk, I will present some recent results on the asymptotic expansion of the Bergman kernel associated with sequences of singular Hermitian holomorphic line bundles $(L_p,h_p)$ over compact Kähler manifolds.  As an application,  I will also present several universality results regarding the equidistribution of zeros of random holomorphic sections in this geometric setup. The talk is based on the joint work with Dan Coman and George Marinescu.

         
     
     
503. Zoom, 4 March 2022, Friday, 15:40

     Ilia Itenberg-[imj-prg] - Real enumerative invariants and their refinement
         
     

     Abstract: The talk is devoted to several real and tropical enumerative problems. We suggest new invariants of the projective plane (and, more generally, of toric surfaces) that arise as results of an appropriate enumeration of real elliptic curves. These invariants admit a refinement (according to the quantum index) similar to the one introduced by Grigory Mikhalkin in the rational case. We discuss tropical counterparts of the elliptic invariants under consideration and establish a tropical algorithm allowing one to compute them. This is a joint work with Eugenii Shustin.

       
       
     
504. Zoom, 11 March 2022, Friday, 15:40

     Alexander Degtyarev-[Bilkent] - Towards 800 conics on a smooth quartic surfaces
         
     

     Abstract: This will be a technical talk where I will discuss a few computational aspects of my work in progress towards the following conjecture. Conjecture: A smooth quartic surface in $\mathbb{P}^3$ may contain at most $800$ conics. I will suggest and compare several arithmetical reductions of the problem. Then, for the chosen one, I will discuss a few preliminary combinatorial bounds and some techniques used to handle the few cases where those bounds are not sufficient. At the moment, I am quite confident that the conjecture holds. However, I am trying to find all smooth quartics containing 720 or more conics, in the hope to find the real quartic maximizing the number of  real lines and to settle yet another conjecture (recall that we count all conics, both irreducible and reducible). Conjecture: If a smooth quartic $X\subset\mathbb{P}^3$ contains more than 720 conics, then $X$ has no lines; in particular, all conics are irreducible. Currently, similar bounds are known only for sextic $K3$-surfaces in $\mathbb{P}^4$. As a by-product, I have found a few examples of large configurations of conics that are not Barth--Bauer, i.e., do not contain a $16$-tuple of pairwise disjoint conics or, equivalently, are not Kummer surfaces with all 16 Kummer divisors conics.

          
     
     
505. Zoom, 18 March 2022, Friday, 15:40

     Matthias Schütt-[Hannover] - Finite symplectic automorphism groups of supersingular K3 surfaces

     Abstract: Automorphism groups form a classical object of study in algebraic geometry. In recent years, a special focus has been put on automorphisms of K3 surface, the most famous example being Mukai’s classification of finite symplectic automorphism groups on complex K3 surfaces. Building on work of Dolgachev-Keum, I will discuss a joint project with Hisanori Ohashi (Tokyo) extending Mukai’s results to fields positive characteristic. Notably, we will retain the close connection to the Mathieu group $M_{23}$ while realizing many larger groups compared to the complex setting.

       
       
     
506. Zoom, 25 March 2022, Friday, 15:40

     Emre Can Sertöz-[Hannover] - Heights, periods, and arithmetic on curves

     Abstract:  The size of an explicit representation of a given rational point on an algebraic curve is captured by its canonical height. However, the canonical height is defined through the dynamics on the Jacobian and is not particularly accessible to computation. In 1984, Faltings related the canonical height to the transcendental "self-intersection" number of the point, which was recently used by van Bommel-- Holmes--Müller (2020) to give a general algorithm to compute heights. The corresponding notion for heights in higher dimensions is inaccessible to computation. We present a new method for computing heights that promises to generalize well to higher dimensions. This is joint work with Spencer Bloch and Robin de Jong.

          
     
     
507. Zoom, 1 April 2022, Friday, 15:40

     Halil İbrahim Karakaş-[Başkent] - Arf Partitions of Integers
         
     

     Abstract: The colection of partitions of positive integers, the collection of Young diagrams and the collection of numerical sets are in one to one correspondance with each other. Therefore any concept in one of these collections has its counterpart in the other collections. For example the concept of Arf numerical semigroup in the collection of numerical sets, gives rise to the concept of Arf partition of a positive integer in the collection of partitions. Several characterizations of Arf partitions have been given in recent works. In this talk we wil characterize Arf partitions of maximal length of positive integers. This is a joint work with Nesrin Tutaş and Nihal Gümüşbaş from Akdeniz University.

      
       
508. Zoom, 8 April 2022 Friday, 15:40

     Yıldıray Ozan-[ODTÜ] - Picard Groups of the Moduli Spaces of Riemann Surfaces with Certain Finite Abelian Symmetry Groups
         
     

     Abstract: In 2021, H. Chen determined all finite abelian regular branched covers of the 2-sphere with the property that all homeomorphisms of the base preserving the branch set lift to the cover, extending the previous works of Ghaswala-Winarski and Atalan-Medettoğulları-Ozan. In this talk, we will present a consequence of this classification to the computation of Picard groups of moduli spaces of complex projective curves with certain symmetries. Indeed, we will use the work by K. Kordek already used by him for similar computations. During the talk we will try to explain the necessary concepts and tools following Kordek's work.

        
      
509. Zoom, 15 April 2022, Friday, 15:40

     Ali Ulaş Özgür Kişisel-[ODTÜ] - An upper bound on the expected areas of amoebas of plane algebraic curves
         
     

     Abstract:The amoeba of a complex plane algebraic curve has an area bounded above by $\pi^2 d^2/2$. This is a deterministic upper bound due to Passare and Rullgard. In this talk I will argue that if the plane curve is chosen randomly with respect to the Kostlan distribution, then the expected area cannot be more than $\mathcal{O}(d)$. The results in the talk will be based on our joint work in progress with Turgay Bayraktar.

         
     
     
510. Zoom, 22 April 2022, Friday, 15:40

     Muhammed Uludağ-[Galatasaray] - Heyula
         
     

     Abstract: This talk is about the construction of a space H and its boundary on which the group PGL(2,Q) acts. The ultimate aim is to recover the action of PSL(2,Z) on the hyperbolic plane as a kind of boundary action.

       
     
511. Zoom, 29 April 2022, Friday, 15:40

     Melih Üçer-[Yıldırım Beyazıt] - Burau Monodromy Groups of Trigonal Curves

     Abstract:  For a trigonal curve on a Hirzebruch surface, there are several notions of monodromy ranging from a very coarse one in S_3 to a very fine one in a certain subgroup of Aut(F_3), and one group in this range is PSL(2,Z).  Except for the special case of isotrivial curves, the monodromy group (the subgroup generated by all monodromy actions) in PSL(2,Z) is a subgroup of genus-zero and conversely any genus-zero subgroup is the monodromy group of a trigonal curve (This is a result of Degtyarev). A slightly finer notion in the same range is the monodromy in the Burau group Bu_3. The aforementioned result of Degtyarev imposes obvious restrictions on the monodromy group in this case but without a converse result. Here we show that there are additional non-obvious restrictions as well and, with these restrictions, we show the converse as well.

          
      
     

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.