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

**** 2017 Fall Talks ****

 Learning seminar on K3 surfaces and  lots more

1. ODTU, 6 October 2017, Friday, 15:40

Alexander Degtyarev-[Bilkent] - Real algebraic curves with large finite number of real points

 Abstract:  (joint work in progress with Erwan Brugallé, Ilia Itenberg, and Frédéric Mangolte) Consider a real algebraic curve A$A$ in a real algebraic surface X$X$ (typically, rational) and assume that the set RA$\mathbb{R}A$ of the real points of A$A$ is finite. (Certainly, this implies that the class [A]$\left[A\right]$ of A$A$ in H2(X)${H}_{2}\left(X\right)$ is even and each point of RA$\mathbb{R}A$ is a singular point of A$A$.) Recently, quite a few researchers showed considerable interest in the possible cardinality of the finite set RA$\mathbb{R}A$. We give a partial answer (upper and lower bounds) to this question in terms of either the class [A]$\left[A\right]$ alone or the class [A]$\left[A\right]$ and genus g(A)$g\left(A\right)$; in the latter case, our bounds are often sharp. In the simplest case where A⊂P2$A\subset {\mathbb{P}}^{2}$ is a plane curve of degree 2k$2k$, we have |RA|≤k2+g(A)+1$|\mathbb{R}A|\le {k}^{2}+g\left(A\right)+1$ (sharp if g(A)$g\left(A\right)$ is small compared to k$k$) and |RC|≤32k(k−1)+1$|\mathbb{R}C|\le \frac{3}{2}k\left(k-1\right)+1$ (sharp for 1≤k≤4$1\le k\le 4$ but, most likely, not sharp in general). I will discuss the proof of the upper bounds (essentially, Petrovsky's inequality) and a few simple constructions for the lower bounds.

2. Bilkent,  13 October 2017, Friday, 15:40

Emre Can Sertöz-[Max-Planck, Leipzig] - Enumerative geometry of double spin curves

 Abstract:  This talk is about the speaker's recent PhD dissertation whose full abstract follows: This thesis has two parts. In Part I we consider the moduli spaces of curves with multiple spin structures and provide a compactification using geometrically meaningful limiting objects. We later give a complete classification of the irreducible components of these spaces. The moduli spaces built in this part provide the basis for the degeneration techniques required in the second part. In the second part we consider a series of problems inspired by projective geometry. Given two hyperplanes tangential to a canonical curve at every point of intersection, we ask if there can be a common point of tangency. We show that such a common point can appear only in codimension 1 in moduli and proceed to compute the class of this divisor. We then study the general properties of curves in this divisor. Our divisor class has small enough slope to imply that the canonical class of the moduli space of curves with two odd spin structures is big when the genus is greater than 9. If the corresponding coarse moduli spaces have mild enough singularities, then they have maximal Kodaira dimension in this range.

3. ODTÜ, 20 October 2017, Friday, 15:40

Hanife Varlı-[ODTÜ] - Perfect discrete Morse functions on connected sums

 Abstract:  Computational topology is an area between topology and computer science that applies topological techniques for problems in data and shape analysis. One of the techniques used in this area is the discrete Morse theory developed by Robin Forman as a discrete analogue of Morse theory. This theory gives a way of studying the topology of discrete objects via critical cells of discrete Morse functions.  In this talk, we will fi rst briefly mention Morse theory. Then we will talk on discrete Morse theory which will be followed by my thesis problem: composing and decomposing perfect discrete Morse functions (the most suitable functions for combinatorial and computational purposes) on connected sums of triangulated manifolds.  In this thesis, we prove that one can compose perfect discrete Morse functions on connected sums of manifolds in any dimensions. On decomposing a given perfect discrete Morse function on a connected sum, our method works in dimensions 2 and 3.

4. ODTÜ, 27 October 2017, Friday, 15:40

Çisem Güneş Aktaş-[Bilkent] - Algebraic surfaces in $\mathbb{C}{\mathbb{P}}^{3}$

 Abstract:   In this talk we give a deeper insight into the theory of K3-surfaces, which essentially boils down to the global Torelli theorem, subjectivity of period map and Riemann Roch theorem (for example, we conclude that all singular points of a K3-surface have to be simple ones). After recalling principle properties of K3-surfaces, we explain the arithmetical reduction of various classification problems, concentrating on the geometric aspects of the arithmetical restrictions appearing in the statements.

5. Bilkent, 3 November 2017, Friday, 15:40

Ergün Yalçın-[Bilkent] - Moore spaces and the Dade group

 Abstract:  Let G$G$ be a finite p$p$-group and k$k$ be a field of characteristic p$p$. A topological space X$X$ is called an n$n$-Moore space if its reduced homology is nonzero only in dimension n$n$. We call a G$G$-CW$CW$-complex X$X$ a Moore G$G$-space over k$k$ if for every subgroup H$H$ of G$G$, the fixed point set XH${X}^{H}$ is a Moore space with coefficients in k$k$. A kG$kG$-module M$M$ is called an endo-permutation module if Endk(M)$En{d}_{k}\left(M\right)$ is a permutation kG$kG$-module. We show that if X$X$ is a finite Moore G$G$-space, then the reduced homology module of X$X$ is an endo-permutationkG$kG$-module generated by relative syzygies.  We consider the Grothendieck group of finite Moore G$G$-spaces with addition given by the join operation, and relate this group to the Dade group generated by relative syzygies. In the talk I will give the necessary background on Moore G$G$-spaces and Dade group, and provide many examples to motivate the statements of the theorems.

6. ODTÜ, 10 November 2017, Friday, 15:40

Oğuzhan Yörük-[Bilkent] - Which K3 surfaces doubly cover an Enriques Surface

 Abstract:  K3 surfaces and Enriques Surfaces are two closely related objects in Algebraic Geometry. It is known that the unramified double cover of an Enriques Surface X$X$ defined by the torsion class KX${K}_{X}$ is an algebraic K3 surface. Also, conversely, if a K3 surface X$X$admits a fixed point free involution ι$\iota$, then the quotient surface X/ι$X/\iota$ is an Enriques Surface. In this talk we will examine, following Keum's work, under which circumstances a K3 surface will admit a fixed point free involution, hence, will cover an Enriques Surface and we will give some applications for this characterization.

7. Bilkent, 17 November 2017, Friday, 15:40

Serkan Sonel-[Bilkent] - Which K3 Surfaces with Picard number $\rho \left(X\right)\le 18$ doubly cover Enriques surfaces.

 Abstract:  In this talk, we discuss the problem of which K3 Surfaces with Picard number ρ(X)≤18$\rho \left(X\right)\le 18$ doubly cover Enriques surfaces and give the insight to the solution of the problem which is deeply related to lattice theory and integral quadratic forms. Then we give the generalization of Sertoz Theorem about the characterization of primitive embeddings of the lattices. Finally, we give our result on which indefinite even unimodular Z$\mathbb{Z}$-lattices fail to be embedded into the sublattice Λ−${\mathrm{\Lambda }}^{-}$of the K3-lattice Λ$\mathrm{\Lambda }$.

8. ODTU, 24 November 2017, Friday, 15:40

Mesut Şahin-[Hacettepe] - Vanishing Ideals of Parameterized Toric Codes

 Abstract:   We start defining subvarieties of a toric variety that are parameterized by Laurent monomials and the corresponding toric codes. We recall their vanishing ideals and give algorithms for finding their binomial generators which will be used to compute main parameters of the corresponding toric codes. We show that these ideals are lattice ideals and give an algorithm to find a basis for the corresponding lattice. Finally, we give this lattice explicitly under a mild condition. This is a joint work with Esma Baran.

9. Bilkent, 1 December 2017, Friday, 15:40

Ali Ulaş Özgür Kişisel-[ODTÜ] - The Cap Set Problem

 Abstract:  Determining the size of a largest subset of Fn3${\mathbb{F}}_{3}^{n}$ which contains no lines is called the cap set problem. I will outline what is known about this problem and report some recent progress on the asymptotic version of the problem, due to Croot, Lev, Pach and Ellenberg, Gijswijt. Croot, E. , Lev, V. F. , Pach, P. P.  Progression-free sets in Zn4${\mathbb{Z}}_{4}^{n}$ are exponentially small, Annals of Math.  185, (2017), 331---337. Ellenberg, J. , Gijswijt, D. On large subsets of Fnq${\mathbb{F}}_{q}^{n}$ with no three-term arithmetic progression, Annals of Math.  185, (2017), 339---343.

10. ODTU, 8 December 2017, Friday, 15:40

Mücahit Meral-[ODTÜ] - Semifree Hamiltonian circle actions on 6 dimensional symplectic
manifolds with non-isolated fixed point set

 Abstract:   Let (M2n,w)$\left({M}^{2n},w\right)$ be a 2n$2n$-dimensional closed symplectic manifold with a symplectic circle action. Many mathematicians tried to find some conditions on M$M$ which make a symplectic circle action Hamiltonian. Cho, Hwang and Suh discovered a condition on the 6-dimensional symplectic manifolds. In this talk, we will discuss CHS's theorem: Let (M,w)$\left(M,w\right)$ be a 6$6$-dimensional closed symplectic S1${S}^{1}$-manifold with generalized moment map  μ:M→S1$\mu :M\to {S}^{1}$. Assume that the fixed point set is not empty and dimension of each component at most 2$2$. Then the action is Hamiltonian if and only if b+2(Mξ)=1${b}_{2}^{+}\left({M}_{\xi }\right)=1$ for any regular value ξ$\xi$ of μ$\mu$.

11. Bilkent, 15 December 2017, Friday, 15:40

Özgün Ünlü-[Bilkent] - The Halperin-Carlsson conjecture

 Abstract:  The Halperin-Carlsson conjecture predicts that if an elementary abelian 2-group of rank r$r$ acts freely and cellularly on a finite CW-complex X$X$, then 2r${2}^{r}$ is less than or equal to the total dimension of the cohomology of X$X$ with coefficients in a field of charateristic 2. We will discuss some known results and some new developments related to this conjecture.

12. ODTU, 22 December 2017, Friday, 15:40

Oğuz Yayla-[Hacettepe] - The existence theory of some objects in finite geometry

 Abstract:  In this talk difference sets, Hadamard matrices, perfect sequences, cyclic irreducible codes and similar objects in finite geometry will be presented. Their relationship will be given and then their existence will be studied. If time allows some methods for their construction will be given.