ODTÜ-BİLKENT Algebraic Geometry Seminar
(See all past talks
ordered according to speaker
and date)
**** 2019 Fall Talks ****
İlker Berktav-[ODTÜ] -
Formal Moduli Problems and Classical Field
Theories
|
Abstract: This is an
introductory talk to the concept of a formal
moduli problem in sense of Lurie and it's
essential role in encoding the formal geometric
aspects of derived moduli spaces of solutions to
the certain moduli problems. To be more
specific, we shall be interested in a sort
of formal moduli problem describing a classical
field theory on a base manifold M in the
sense that it defines a derived moduli space of
solutions to the certain differential equations on
an open subset U of M, namely the
Euler-Lagrange equations, arising from a local
action functional defined on the space of fields
on U, see Costello and Gwilliam. The
outline of this talk is as follows: References
|
Yıldıray Ozan-[ODTÜ] -
A filtration on the Borel-Moore Homology of
Wonderful Compactification
of Some Symmetric Spaces
|
Abstract: After giving
some motivation we will introduce basic objects
mentioned in title and the tools we will be
using. Then we will give some examples and
state main results. If time permits, we will
try to sketch a proof of the results. |
Alexander Degtyarev-[Bilkent]
- Linear subspaces in algebraic
varieties
|
Abstract: (partially
joint with I. Itenberg and J. Ch. Ottem) |
Ali Ulaş Özgür Kişisel-[ODTÜ]
- Random Real Algebraic Plane Curves
|
Abstract: There has
been growing interest in recent years on random
objects in algebraic geometry. The expected number
of real roots of a univariate polynomial has been
studied for different probability measures on the
space of polynomials, by many authors. After
discussing some of these results, I will switch to
multivariate polynomials and survey some of the
known results regarding the expected number of
connected components of a real algebraic plane
curve and their expected volumes. Finally, I will
present some of our recent results with Turgay
Bayraktar regarding the expected depth of a real
algebraic plane curve. |
Muhammed Uludağ-[Galatasaray]
- Jimm, a fundamental involution
|
Abstract: Dyer's outer
automorphism of PGL(2,Z) induces an involution of
the real line, which behaves very much like a kind
of modular function. It has some striking
properties: it preserves the set of quadratic
irrationals sending them to each other in a
non-trivial way and commutes with the Galois
action on this set. It restricts to an highly
non-trivial involution of the set unit of norm +1
of quadratic number fields. It conjugates the
Gauss continued fraction map to the so-called
Fibonacci map. It preserves harmonic pairs of
numbers inducing a duality of Beatty partitions of
N. It induces a subtle symmetry of Lebesgue's
measure on the unit interval. |
Alexander Degtyarev-[Bilkent]
- Linear subspaces in algebraic
varieties. II: Niemeier lattices
|
Abstract: Using the
arithmetical reduction suggested in the previous
talk, I will prove the two new theorems stated
there, viz., “the number of tritangents to a
smooth plane sextic is at most 72”, and “the
number of 2-planes in a smooth cubic 4-fold is at
most 405” (joint with I. Itenberg and J.Ch.
Ottem). To this end, we will embed the
appropriately modified lattice of algebraic cycles
to a Niemeier lattice and study certain
configurations of square 4 vectors in the latter.
I will try to explain the advantages of this
approach and outline the principal techniques used
in counting square 4 vectors. |
Turgay Akyar-[ODTÜ]
- Clifford's Theorem on Special
Divisors
|
Abstract: It is very
well known that for a non-special divisor $D$, the
dimension of a linear system $|D|$ on a
smooth projective curve over $\mathbb{C}$ depends
only on the degree of $D$. On the other hand, if
$D$ is special, we do not have such a dependence.
After giving some facts about linear systems
on curves we will see a classical theorem
mainly concerning with the extremal behavior of
the dimension $r(D)$ of a complete special linear
system $|D|$. |
|
Abstract: Zariski-van
Kampen theorem expresses the fundamental group of
the complement of an algebraic curve on
$\mathbb{C}^2$ in terms of generators and
monodromy relations. Therefore, the Alexander
module of the curve is also (almost) expressed in
terms of generators and monodromy relations. As
far as the Alexander module of an $n$-gonal curve
is concerned, the group of monodromy relations is
a subgroup of the Burau group $Bu_n$, which is a
certain subgroup of $GL(n-1, \mathbb{Z}[t,1/t])$.
For trigonal curves ($n=3$ case), Degtyarev gave a
characterization of the monodromy groups: the
monodromy group of a trigonal curve (except a
trivial exceptional case) must be a finite index
subgroup of $Bu_3$ whose image under the special
epimorphism $Bu_3 \longrightarrow \;
PSL(2,\mathbb{Z})$ is of genus $0$ and conversely,
most of such subgroups appear as monodromy groups
of trigonal curves. However, this class of
subgroups is still too large, hence it is not
feasible to look at them all and determine their
Alexander modules. In this talk, I plan to speak
about a recently discovered method by which, given
an abstract module over $\mathbb{Z}[t,1/t]$, one
can determine whether or not it appears as the
Alexander module of a trigonal curve. With this
method, it should be feasible to determine all the
Alexander modules. |
Serkan Sonel-[Bilkent]
- On K3 surfaces covering an Enriques
surface
|
Abstract: We will
continue the subject of the previous talks, viz. a
characterization of the K3-surfaces covering an
Enriques surface. Following Nikulin, we will: |
Mesut Şahin-[Hacettepe]
- Rational points of subgroups inside a
toric variety over a finite field
|
Abstract: We talk about
counting rational points of subgroups of the torus
lying inside a toric variety over a finite field,
explaining its implications for the evaluation
codes on these subgroups. |
Halil İbrahim Karakaş-[Başkent]
- A decomposition of partitions and
numerical sets
|
Abstract: The aim of
this work is to exhibit a decomposition of
partitions of natural numbers and numerical sets.
In particular, we obtain a decomposition of a
sparse numerical set into the so called hook
semigroups which turn out to be primitive. Since
each Arf semigroup is sparse, we thus obtain a
decomposition of any Arf semigroup into primitive
numerical semigroups. |
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.
