ODTÜ-BİLKENT Algebraic Geometry Seminar
(See all past talks ordered according to speaker and date)

**** 2015 Fall Talks ****

361. Bilkent, 2 October 2015, Friday, 15:40
     

     Alexander Degtyarev-[Bilkent] - Lines on smooth quartics
         
     

     Abstract: In 1943, B. Segre proved that a smooth quartic surface in the complex projective space cannot contain more than 64 lines. (The champion, so-called Schur's quartic, has been known since 1882.) Even though a gap was discovered in Segre's proof (Rams, Schütt, 2015), the claim is still correct; moreover, it holds over any field of characteristic other than 2 or 3. (In characteristic 3, the right bound seems to be 112.) At the same time, it was conjectured that not any number between 0 and 64 can occur as the number of lines in a quartic. We tried to attack the problem using the theory of $K3$-surfaces and arithmetic of lattices. This relatively simple reduction has lead us to an extremely difficult arithmetical problem. Nevertheless, the approach turned out quite fruitful: for the moment, we have a complete classification of smooth quartics containing more than 52 lines. As an immediate consequence of this classification, we have the following:   -- an alternative proof of Segre's bound 64;   -- Shur's quartic is the only one with 64 lines;   -- a real quartic may contain at most 56 real lines;   -- a real quartic with 56 real lines is also unique;   -- the number of lines takes values {0,...,52,54,56,60,64}. Conjecturally, we have a complete list of all quartics with more than 48 lines; there are about two dozens of species, most projectively rigid. I will discuss methods used in the proof and a few problems that are still open, e.g., the minimal fields of definition, triangle-free configurations, lines in singular quartics, etc. This subject is a joint work in progress with Ilia Itenberg and Sinan Sertöz.

         

     
     
     
362. ODTÜ, 9 October 2015, Friday, 15:40

     Ali Sinan Sertöz-[Bilkent] - The basic theory of elliptic surfaces-I
         
     

     Abstract: This term we will be running a learnin seminar on elliptic surfaces with a view toward "lines on quartic surfaces". We will be mainly following Miranda's classical notes but other sources will not be excluded.

         

363. Bilkent, 16 October 2015, Friday, 15:40

     Ali Sinan Sertöz-[Bilkent] - The basic theory of elliptic surfaces-II

            

     Abstract:  We continue our learning seminar talk on elliptic surfaces. We will also mention how this topic shows up in the search for lines on quartic surfaces in $\mathbb{P}^3$.

         

364. ODTÜ, 23 October 2015, Friday, 15:40

     Ergün Yalçın-[Bilkent] - Group actions on spheres with rank one isotropy

            

     Abstract:   Actions of finite groups on spheres can be studied in various different geometrical settings, such as (A) smooth G-actions on a closed manifold homotopy equivalent to a sphere, (B) finite G-homotopy representations (as defined by tom Dieck), and (C) finite G-CW complexes homotopy equivalent to a sphere. These three settings generalize the basic models arising from unit spheres S(V) in orthogonal or unitary G-representations. In the talk, I will discuss the group theoretic constraints imposed by assuming that the actions have rank 1 isotropy (meaning that the isotropy subgroups of G do not contain $\mathbb{Z}/p \times \mathbb{Z}/p$, for any prime $p$). This is joint work with Ian Hambleton.

         

365. Bilkent, 6 November 2015, Friday, 15:40

     Özgün Ünlü-[Bilkent] - Free group actions on products of spheres
     

            

     Abstract:    In this talk we will discuss the problem of finding group theoretic conditions that characterizes the finite groups which can act freely on a given product of spheres. The study of this problem breaks up into two aspects: (1) Find group theoretic restrictions on finite groups that can act freely on the given product. (2) Construct explicit free actions of finite groups on the given product. I will give a quick overview of the first aspect of this topic. Then I will discuss some recently employed methods of constructing such actions.

         

366. ODTÜ, 13 November, Friday, 15:40

     Recep Özkan-[ODTÜ] Concrete sheaves and continuous spaces

            

     Abstract:   This is a talk from the speaker's recent dissertation. After he summarizes the historical background and the recent developments in the field he will motivate his dissertation problems. Time permitting he will talk about the ideas behind the proof of his main theorem.

         

367. Bilkent, 20 November 2015, Friday, 15:40

     Cem Tezer-[ODTÜ] - Anosov diffeomorphisms : Revisiting an old idea
     

            

     Abstract: Introduced  by D. V. Anosov as  the discrete time analogue   of  geodesic flows  on Riemann manifolds of negative  sectional curvature,  Anosov diffeomorphisms  constitute one of the leitmotivs  of  contemporary abstract dynamics.  It is  conjectured that these  diffeomorphisms occur on very exceptional homogeneous spaces. The  speaker will delineate the basic facts and  briefly mention his  own recent work towards settling this conjecture.

         

368. ODTÜ, 27 November 2015, Friday, 15:40 (Joint with METU Mathematics Seminars)

     Haydar Göral-[Université Lyon 1] - Primality via Height Bound

            

     Abstract:   Height functions are of fundamental importance in Diophantine geometry. In this talk, we obtain height bounds for polynomial ring over the field of algebraic numbers. This enables us to test the primality of an ideal. Our approach is via nonstandard methods, so the mentioned bounds will be ineffective. We also explain the tools from nonstandard analysis.

         

369. Bilkent, 4 December 2015, Friday, 15:40

     Alperen Ergür-[Texas A&M] -  Tropical Varieties for Exponential Sums

            

     Abstract:   We define a variant of tropical varieties for exponential sums. These polyhedral  complexes can be used to approximate, within an explicit distance bound, the real parts of complex zeroes of exponential sums. We also discuss the algorithmic efficiency of tropical varieties in relation to the computational hardness of algebraic sets. Our proof involves techniques from basic complex analysis, inequalities and some recent probabilistic estimates on projections that might be of interest to analyst. This is joint work with Maurice Rojas and Grigoris Paouris.

         

370. ODTÜ, 11  December  2015, Friday, 15:40

     Ali Ulaş Özgür Kişisel-[ODTÜ]- Moduli space of elliptic curves

            

     Abstract:   The aim of this talk is to view the moduli space of elliptic curves in different contexts. After briefly discussing the classical setting, we will see how it can be viewed as an orbifold and as an algebraic stack.

         

371. Bilkent,  18 December 2015, Friday, 15:40

     Mesut Şahin-[Hacettepe] - On Pseudo Symmetric Monomial Curves
     

            

     Abstract:   In this talk, we introduce monomial curves, toric ideals and monomial algebras associated to 4-generated pseudo symmetric numerical semigroups. We give a characterization of indispensable binomials of these toric ideals, and of these monomial algebras to have strongly indispensable minimal graded free resolutions. We also discuss when the tangent cones of these monomial curves at the origin are Cohen-Macaulay in which case Sally's conjecture will be true. Joint with Nil Şahin of Bilkent University. Supported by Tubitak No:114F094.

             

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.