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. 
Muhammed Uludađ[Galatasaray]
 TBA
Abstract: 
Kadri Ýlker Berktav[ODTÜ]
 TBA
Abstract: 
Halil Ýbrahim Karakaţ[Baţkent]
 Arf Numerical Semigroups
Abstract: 
Mesut Ţahin[Hacettepe]
 TBA
Abstract: 
Tolga Karayayla[ODTÜ]
 TBA
Abstract: 
Nil Ţahin[Bilkent] 
TBA
Abstract: 
Yýldýray Ozan[ODTÜ] 
TBA
Abstract: 
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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