ODTÜ-BİLKENT Algebraic
Geometry Seminar
(See all past talks
ordered according to speaker
and date)
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**** 2021 Spring Talks ****
(The New Yorker, Dec 7, 2020 Cover)
This semester we plan to have all our seminars online
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. |
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. 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
$p=3$. |
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. |
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. |
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). |
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. |
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
$\mathbb{RP}^3$. |
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. |
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. |
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. |
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). |
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.