ODTÜBÝLKENT Algebraic Geometry Seminar
(See all past talks
ordered according to speaker
and date)
**** 2019 Spring Talks ****
Ali Sinan Sertöz[Bilkent]
 Arf Rings I
Abstract: The aim of
these two talks is to discuss the background and
the content of Arf's 1946 paper on the
multiplicity sequence of an algebraic curve
branch. I will start by giving the geometric and
algebraic descriptions of a singular branch for a
curve, describe its multiplicity sequence obtained
until it is resolved by blow up operations. Du Val
defines some geometrically significant steps of
the resolution process and shows that if the
multiplicity sums up to those points are known
then the whole multiplicity sequence can be
recovered by a simple algorithm. However all this
information must be encoded at the very beginning
in the local ring of the branch. The problem is
then to decipher this data. This week I will mostly describe
the background and explain what is involved in
actually finding these numbers. Arf's original article "Une
interpretation algebrique de la suite des ordres
de multiplicite d'une branche algebrique",
together with my English translation can be found
on: http://sertoz.bilkent.edu.tr/arf.htm

Ali Sinan Sertöz[Bilkent]
 Arf Rings II
Abstract: I will first
describe the structure of the local ring of a
singular branch and explain how the blow up
process affects it. Then I will describe, aprés
Arf, how the multiplicity sequence can be
recovered, not from this ring but from a slightly
larger and nicer ring which is now known as the
Arf ring. The process of finding this nicer ring
is known as the Arf closure of this ring. Finally
I will explain how Arf answered Du Val's question
of reading off the multiplicity sequence from the
local ring. 
Alexander Degtyarev[Bilkent]
 Tritangents to sextic curves via
Niemeier lattices
Abstract: I will address
the following conjecture (and some refinements
thereof): “A smooth plane curve of degree 6 has at
most 72 tritangents.” After a brief introduction
to the subject and a survey of the known results
for the other polarized K3surfaces, I will
explain why the traditional approach does not work
and suggest a new one, using the embedding of the
Néron—Severi lattice of a K3surface to an
appropriate Niemeier lattice. I will also discuss
the pros and contras of several versions of this
approach and report the partial results obtained
so far. 
Alexander Degtyarev[Bilkent]
 Positivity and sums of squares of
real polynomials
Abstract: I will
discuss the vast area of research (in which I am
not an expert) related to Hilbert's 17th problem,
namely, positivity of real polynomials vs. their
representation as sums of squares (SOS). As is
well known, "most" PSD (positive semi definite)
forms in more than two variables are not SOS of
polynomials, although they are SOS of rational
function. I will consider a few simplest classical
counterexamples, and then I will outline the
construction part of our recent paper (in
collaboration with Erwan Brugallé, Ilia Itenberg,
and Frédéric Mangolte). Thinking that we were
dealing with Hilbert's 16th problem (widely
understood, i.e., topology of real algebraic
varieties), we constructed real plane algebraic
curves with large finite numbers of real points.
These curves provide new lower bounds on the
denominators needed to represent a PSD ternary
form as a SOS of rational functions. 
Yýldýray Ozan[ODTÜ]
 Equivariant Cohomology and
Localization after Anton Alekseev
Abstract: We will try to
present the notes by Anton Alekseev on Equivariant
Localization, mainly focusing on $S^1$actions.
First, we will introduce Stationary Phase Method.
Then we will define equivariant $S^1$cohomology
and present a proof of the localization theorem
suggested by E. Witten. If time permits,
finally we will end by the DuistermaatHeckman
formula and its proof. 
Kadri Ýlker Berktav[ODTÜ]
 Towards the Stacky Formulation of
Einstein Gravity
Abstract: This talk,
which essentially consists of three parts, serves
as a conceptional introduction to the formulation
of Einstein gravity in the context of derived
algebraic geometry. The upshot is as follows: we
shall first outline how to describe the notion of
a (pre)stack $\mathfrak{X}$, by using the
functorofpoints type approach, manifestly given
as a certain groupoidvalued sheaf over a site
$\mathcal{C}$, and present main ingredients of the
homotopy theory of stacks in a relatively
succinct and naive way. In that respect, one in
fact requires to adopt certain simplicial
techniques in order to recast the
notion of a stack in the language of homotopy
theory. This homotopical treatment, on the other
hand, is essentially based on socalled the
model structure on the 2category $Grpds$ of
groupoids. In the second part of the talk, we
shall revisit main aspects of 2+1 dimensional
vacuum Einstein gravity on a pseudoRiemannian
manifold $M$ especially in the context of Cartan
geometry, and investigate, in the case of
$M=\Sigma\times (0,\infty)$ with vanishing
cosmological constant and $\Sigma$ being a closed
Riemann surface of genus $g>1$, the equivalence
of the quantum gravity with a gauge theory
established in the sense that the moduli space
$\mathcal{E}(M)$ of such a 2+1 dimensional
Einstein gravity is isomorphic to that of
flat Cartan $ISO(2,1)$connections, denoted by
$\mathcal{M}_{flat}$. As an analyzing a classical
field theory with an action functional
$\mathcal{S}$ boils down to the study of the
moduli space of solutions to the corresponding
field equations, the notion of a stack in
fact provides an alternative and elegant way of
recording and organizing the moduli data. In the
final part, we shall briefly discuss (i)
how to construct the appropriate stacks
associated to $\mathcal{E}(M) $ and
$\mathcal{M}_{flat}$ respectively, and (ii)
how to extend the isomorphism that essentially
captures the equivalence of the quantum gravity
with a gauge theory in the above setup to an isomorphism
of associated stacks. 
Halil Ýbrahim Karakaţ[Baţkent]
 Arf Numerical Semigroups
Abstract: Parametrizations
have been given for Arf numerical semigroups with
small multiplicity ($m\leq 10$) and arbitrary
conductor. In this talk, I will give a
characterization of Arf numerical semigroups in
terms of the Apery sets, and use that
characterization to parametrize Arf numerical
semigroups with multiplicity 11 and 13. I will
also share some observations about Arf numerical
semigroups with prime multiplicity. 
Mesut Ţahin[Hacettepe]
 Evaluation codes defined on subsets
of a toric variety
Abstract:
In this talk, we review algebraic
methods for studying evaluation codes defined on
subsets of a toric variety. The key object is the
vanishing ideal of the subset and its Hilbert
function. We reveal how invariants of this ideal
such as multigraded regularity and multigraded
Hilbert polynomial relate to parameters of the
code. Time permitting, we share the nice
correspondence between subgroups of the maximal
torus and lattice ideals as their vanishing
ideals. 
Tolga Karayayla[ODTÜ]
 Singular fiber products of rational
elliptic surfaces and fixed
point free group actions on their desingularizations
Abstract: Schoen has
shown that a fiber product of two relatively
minimal rational elliptic surfaces with section is
a simply connected CalabiYau 3fold if the fiber
product is smooth and the same is true for the
desingularization of the fiber product by small
resolutions in the case that the singularities are
ordinary double points. I will describe the small
resolution process and talk about lifting
automorphisms on the fiber product to
automorphisms of the desingularization. I will
discuss the problem of constructing fixed point
free finite group actions on such
desingularizations. The quotient of the 3fold by
such group actions give rise to nonsimply
connected CalabiYau 3folds. The problem on the
existence of such group actions on smooth fiber
products was solved by previous works of Bouchard,
Donagi and the speaker. 
Nil Ţahin[Bilkent] 
ksparse numerical semigroups
Abstract: In this talk,
I will present ksparse numerical semigroups as a
generalization of sparse numerical semigroups
using the recent paper "On ksparse numerical
semigroups" by Guilherme Tizziotti and Juan
Villanueva. 
Yýldýray Ozan[ODTÜ] 
An Obstruction for Algebraic Realization of
Smooth Closed Manifolds with Prescribed Algebraic
Submanifolds and Some Examples
Abstract: First I will
review the history of Algebraic Realization
Problem of Smooth Manifolds starting from
Seifert's 1936 result to Tognoli and to Akbulut
and King. Then I will introduce some tools typical
to the subject like algebraic homology and
strongly algebraic vector bundles. Finally,
I will present a result (joint with one of my
former masters' student Arzu Celikten) which
introduces an obstruction for a topological vector
bundle to admit a strongly algebraic structure.
Using this obstruction we will construct examples
of manifolds promised in the title. 
ODTÜ talks are either at Hüseyin Demir Seminar room or
at Gündüz Ýkeda seminar room at the
Mathematics building of ODTÜ.
Bilkent talks are at room 141 of Faculty of Science
Abuilding at Bilkent.
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