ODTÜBİLKENT
Algebraic Geometry Seminar
(See all past talks ordered
according to speaker and date)
**** 2017 Fall Talks ****

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) In the simplest case where is
a plane curve of degree ,
we have (sharp
if is
small compared to )
and (sharp
for but,
most likely, not sharp in general). 
Emre Can Sertöz[MaxPlanck,
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. 
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 first
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. 
Çisem Güneş Aktaş[Bilkent]  Algebraic surfaces in
Abstract: In this talk we give a deeper insight into the theory of K3surfaces, 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 K3surface have to be simple ones). After recalling principle properties of K3surfaces, we explain the arithmetical reduction of various classification problems, concentrating on the geometric aspects of the arithmetical restrictions appearing in the statements. 
Ergün Yalçın[Bilkent]  Moore spaces and the Dade group
Abstract: Let be a finite group and be a field of characteristic . A topological space is called an Moore space if its reduced homology is nonzero only in dimension . We call a complex a Moore space over if for every subgroup of , the fixed point set is a Moore space with coefficients in . A module is called an endopermutation module if is a permutation module. We show that if is a finite Moore space, then the reduced homology module of is an endopermutationmodule generated by relative syzygies. We consider the Grothendieck group of finite Moore 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 spaces and Dade group, and provide many examples to motivate the statements of the theorems. 
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 defined
by the torsion class is
an algebraic K3 surface. Also, conversely, if a K3
surface admits
a fixed point free involution ,
then the quotient surface 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. 
Serkan Sonel[Bilkent]  Which K3 Surfaces with Picard number doubly
cover Enriques surfaces.
Abstract: In
this talk, we discuss the problem of which K3 Surfaces
with Picard number 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 lattices
fail to be embedded into the sublattice of
the K3lattice . 
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. 
Ali Ulaş Özgür Kişisel[ODTÜ]  The Cap Set Problem
Abstract: Determining
the size of a largest subset of 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.

Mücahit Meral[ODTÜ]  Semifree Hamiltonian circle actions on 6
dimensional symplectic
manifolds with nonisolated fixed point set
Abstract: Let be
a dimensional
closed symplectic manifold with a symplectic circle
action. Many mathematicians tried to find some
conditions on which
make a symplectic circle action Hamiltonian. Cho,
Hwang and Suh discovered a condition on the
6dimensional symplectic manifolds. In this talk, we
will discuss CHS's theorem: Let be
a dimensional
closed symplectic ^{}manifold
with generalized moment map .
Assume that the fixed point set is not empty and
dimension of each component at most .
Then the action is Hamiltonian if and only if for
any regular value of . 
Özgün Ünlü[Bilkent]  The HalperinCarlsson conjecture
Abstract: The
HalperinCarlsson conjecture predicts that if an
elementary abelian 2group of rank acts
freely and cellularly on a finite CWcomplex ,
then is
less than or equal to the total dimension of the
cohomology of with
coefficients in a field of charateristic 2. We will
discuss some known results and some new developments
related to this conjecture. 
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. 