# Abstracts

Abstracts submitted as plain text for the General Seminar. (Those submitted as pdf files are available from the schedule.)

## The Coxeter transformation on cominuscule posets

*Emine Yıldırım, 2017.12.28, 16:00*

Let *P* be a cominuscule poset which can be thought of as a
parabolic analogue of the poset of positive roots of a finite root
system. Let *J*(*P*) be the poset of order ideals of *P*.
In this talk, we will investigate the periodicity of the Coxeter
transformation on the poset *J*(*P*), and show that the
Coxeter transformation has finite order for two of the three infinite
families of cominuscule posets, and the exceptional cases. Our
motivation comes from a conjecture by Chapoton which states that the
Coxeter transformation has finite order on the poset *J*(*R*)
when *R* is the poset of positive roots of a finite root system.
Our solution is formulated in representation theory of finite
dimensional algebras, and we will further discuss the results within the
same context.

## Unimodularization of a Hopf algebra

*Atabey Kaygun, 2017.12.21, 16:00*

A unimodular Lie group is one whose adjoint representation lies in invertible matrices of determinant 1. There is a corresponding notion for Hopf algebras: a Hopf algebra with a modular pair in involution (MPI). It is an open problem whether any (finite dimensional) Hopf algebra possesses an MPI. In this talk, we are going to describe a method to extend a Hopf algebra one with an MPI.

## Galileo’nun Yargılanması: Bilginin Kaynağı Üzerine Bir Tartışma

*Cem Kamözüt, 2017.12.14, 16:00*

Galileo’nun yargılanması yaygın olarak Dünya’nın hareket edip etmediğine ilişkin bir tartışma olarak bilinir. Ancak bu yaklaşımda bazı tuhaflıklar göze çarpar. Örneğin Dünya’nın hareket ettiğini öne süren Kopernik bu görüşünü Galileo doğmadan 21 yıl önce yayınlamıştır. Üstelik kitabı yasaklanmamıştır. Öte yandan Galileo’nun Dünya’nın hareket ettiğine ilişkin sunduğu kanıt yanlıştır. 1992 yılında Katolik kilisesi yargılamadaki hatasını kabul etmiştir. Kuşkusuz Kilise Dünya’nın hareket ettiği görüşünü çok daha önce kabul etmişti.

Bu durumları daha başarıyla açıklamak için yargılamanın temelde bir astronomi problemine ilişkin olmadığını asıl tartışmanın bilginin kaynağına ilişkin olduğunu öne süreceğim. Bu nedenle Galileo’nun matematiksel yöntemle bilgi edinme iddiasının Kilise’nin yetkesini sınırlamasının yarattığı sorunlara işaret edeceğim. Son olarak 1992 yılındaki özrün aynı tartışmanın bir parçası olduğunu ve bilimin bugün kazandığı konuma karşın Kilise’ye hala özerk bir alan sağlama çabası olduğunu öne süreceğim.

## Concatenated Structure of q-abelian Codes and a Resulting Minimum Distance Bound

*Elif Saçıkara, 2017.11.30, 16:00*

For a positive integer *ℓ* and a group algebra
𝔽_{q}[*H*], a *q*-abelian code of index
*ℓ* is a 𝔽_{q}[*H*]-submodule of
𝔽_{q}[*H*]^{ℓ}, where *H* is
an abelian group of order *m*. The special case *H* :=
ℤ_{m}, where ℤ_{m} is a cyclic group of
order *m*, gives a quasi-cyclic (QC) code of index *ℓ* and
length *m**ℓ*. So, *q*-abelian codes are natural
generalization of QC codes. Sole and Ling showed that QC codes can be
decomposed as a direct sum of certain linear codes of length
*ℓ* by applying the Chinese Remainder Theorem, such a method is
called the CRT decomposition. Jensen represented a concatenated
structure of QC codes and later Guneri-Ozbudak showed that these
decompositions are equivalent. In this talk, we present a concatenated
structure of *q*-abelian codes by using the CRT decomposition of
*q*-abelian codes introduced by Jitman and Ling and we show that
both decompositions are equivalent. Concatenated structure also leads to
asymtotical goodness and provides a general minimum distance bound,
extending the analogue bound for QC codes due to Jensen.

## Fifth Moment of Modular L functions

*Eren Mehmet Kıral, 2017.11.16, 16:00*

I will talk about moments of L functions, and their relation to subconvexity results (partial progresses towards the Lindelöf hypotheses) in their historical context starting from the Riemann Zeta function and Dirichlet L-functions. I will also present our recent work with Matthew Young from Texas A&M University on the fifth moment of modular L-functions.

## On the Use of Penalty MCMC for Differential Privacy

*Sinan Yıldırım, 2017.11.09, 16:00*

We view the penalty algorithm of Ceperley and Dewing (1999), a Markov chain Monte Carlo (MCMC) algorithm for Bayesian inference, in the context of data privacy. Specifically, we study differential privacy of the penalty algorithm and advocate its use for data privacy. We show that in the simple model of independent observations the algorithm has desirable convergence and privacy properties that scale with data size. Two special cases are also investigated and privacy preserving schemes are proposed for those cases: (i) Data are distributed among several data owners who are interested in the inference of a common parameter while preserving their data privacy. (ii) The data likelihood belongs to an exponential family. The results of our numerical experiments on the Beta-Bernoulli and the logistic regression models agree with the theoretical results.

## Ergodic and Stochastic properties of
holomorphic endomorphisms of ℂℙ^{k}

*Turgay Bayraktar, 2017.11.02, 15:00*

For a holomorphic endomorphism f : ℂℙ^{k} →
ℂℙ^{k} of the complex projective space there is a unique
invariant measure μ_{f} of maximal topological entropy. In
this talk, I will survey some ergodic and stochastic properties of the
dynamical system ( ℂℙ^{k}, *f*,
μ_{f}). I will also present some recent results an open
problems in this context.

## Parallel Incremental Optimization Algorithm for Solving Partially Separable Problems in Machine Learning

*İlker Birbil, 2017.10.26, 16:00*

Consider a recommendation problem, where multiple firms are willing to cooperate to improve their rating predictions. However, the firms insists on finding a machine learning approach, which guarantees that their data remain in their own servers. To solve this problem, I will introduce our recently proposed approach HAMSI (Hessian Approximated Multiple Subsets Iteration). HAMSI is a provably convergent, second order incremental algorithm for solving large-scale partially separable optimization problems. The algorithm is based on a local quadratic approximation, and hence, allows incorporating curvature information to speed-up the convergence. HAMSI is inherently parallel and it scales nicely with the number of processors. I will conclude my talk with several implementation details and our numerical results on a set of matrix factorization problems.

## Zaman Gecikmeli ve Anahtarlamalı Doğrusal Sistemlerin Gözlemci Tabanlı Sonlu Zaman Sınırlılığı Üzerine

*Gökhan Göksu, 2017.10.19, 16:00*

Diğer Yazarlar: Ulviye Başer Ilgaz. Bu çalışmada, zaman gecikmeli ve anahtarlamalı doğrusal sistemler ele alınmıştır. Öncelikle, kapalı çevrim sisteminin sonlu zaman sınırlılık şartı elde edilmiştir. Ardından, bu şart kullanılarak, ortalama yaşam süresi yardımıyla belli bir aralık ile sınırlandırılmış zaman gecikmesine sahip anahtarlamalı doğrusal sistemler için sonlu zaman sınırlılık şartı bulunmuştur. Elde edilen kısıtlar, standart doğrusal matris eşitsizlikleri (DME) biçiminde olmadığından, bu kısıtları DME formuna çevirici bir yöntem açıklanmıştır. Kısıtlardaki doğrusal olmayan terimler için koni tamamlayıcı bir algoritma uygulanmıştır. Son olarak, önerilen yöntemlerin etkinliği ve geçerliliğini gösterme adına sayısal örnekler verilmiştir.

## Combinatorics, Model Theory and Machine Learning

*Ayhan Günaydın, 2017.10.12, 16:00*

We start by explaining VC-dimension which is a combinatorial concept introduced by Vapnik and Chervonenkis in the setting of machine learning. After giving enough examples, we relate VC-dimension with some model theoretic notions. Finally, we return to machine learning and present the use of VC-dimension in solving actual engineering problems.

We are aiming to be accessible to advanced undergraduate students and graduate students.

## Weighted Dirichlet spaces and 𝑄^{𝑝}

*Nihat Gökhan Göğüş, 2017.10.05, 16:00*

We investigate Dirichlet spaces with superharmonic weights on the open unit disk. Some results on these spaces such as the Alexander-Taylor-Ullman inequality, Carleson measures characterization and explicit formula for reproducing kernels will be explained. We also characterize the boundedness and compactness of composition operators on these spaces. There is a relation with Möbius invariant spaces of analytic functions. What we know and what can be investigated will be discussed. This talk consists of results from the joint work with G. Bao and S. Pouliasis.

## Symmetric Decompositions of Free Kleinian Groups and Hyperbolic Displacements

*İlker S. Yüce, 2017.09.28, 16:00*

I will show that every point in the hyperbolic 3-space is moved at a
distance at least
½ log(12 · 3^{k−1} − 3) by
one of the isometries of length at most *k* ≥ 2 in a
2-generator Kleinian group Γ which is torsion-free, not co-compact and
contains no parabolic. Also I will propose some lower bounds for the
maximum of hyperbolic displacements given by symmetric subsets of
isometries in purely loxodromic finitely generated free Kleinian groups.

## Eigenvalues and Linear Dynamics of Weighted Backward Shifts on Spaces of Real Analytic Functions

*Can Deha Karıksız, 2017.05.25, 15:00*

In this joint work with late Paweł Domański, we give certain conditions on linear dynamical properties of the weighted backward shift operators

B_{w} : A(Ω) → A(Ω),

with weight sequences w = (w_{n})_{n∈ℕ}, acting
on the spaces of real analytic functions A(Ω) on open subsets Ω of ℝ
containing zero, that send the monomials x^{n} to
w_{n}x^{n-1} for all n∈ℕ and the unit function to
the zero function. A useful tool for these characterizations is the
description of the point spectra and the eigenspaces of weighted
backward shifts on A(Ω).

## On Slant Geometry

*Handan Yıldırım, 2017.05.11, 15:00*

It is known that a geodesic in the Poincaré disk is a circle which is
perpendicular to the ideal boundary (i.e., unit circle). If we adopt
geodesics as lines in the Poincaré disk, we have the model of Hyperbolic
geometry. We have another class of curves in the Poincaré disk which has
an analogous property with lines in the Euclidean plane. A
*horocycle* is a circle which is tangent to the ideal boundary.
We note that a line in the Euclidean plane can be considered as the
limit of the circles when the radius tends to infinity. In the same
manner, a horocycle is also a curve which can be considered as the limit
of the circles in the Poincaré disk when the radius tends to infinity.
Hence, horocycles are also an analogous notion of lines. If we adopt
horocycles as lines, what kind of geometry do we obtain? We say that two
horocycles are *parallel* if they have common tangent point at
the ideal boundary. Under this definition, the parallel axiom is
satisfied. However, for any two fixed points in the Poincaré disk, there
exist always two horocycles passing through these points, so that the
first axiom of Euclidean geometry is not satisfied. In the case of
general dimensions, this geometry is said to be *horospherical
geometry.* On the other hand, we have another kind of curves in the
Poincaré disk which has similar properties with Euclidean lines. An
*equidistant curve* is a circle whose intersection with the ideal
boundary consists of two points. Generally, the angle between an
equidistant curve and the ideal boundary is φ ∈ (0, π/2]. Here, we
emphasize that a geodesic is an equidistant curve with φ = π/2. However,
a horocycle is not an equidistant curve, but it is a circle with φ = 0.
We call the geometry where φ = π/2 *vertical geometry* and the
geometry where φ = 0 *horizontal geometry.* And also we call the
family of geometry depending on φ *slant geometry.*

In this talk, I will give some of recent results about this geometry including joint works with Mikuri Asayama, Shyuichi Izumiya and Aiko Tamaoki.

## On a conjecture of Morgan and Mullen

*Giorgos Kapetanakis, 2017.05.04, 15:00*

Let **F**_{q} be the finite field of cardinality
*q* and **F**_{qn} its extension of
degree *n*, where *q* is a prime power and *n* is a
positive integer. A generator of the multiplicative group
**F**_{qn}^{*} is called
*primitive.* Besides their theoretical interest, primitive
elements of finite fields are widely used in various applications,
including cryptographic schemes, such as the Diffie-Hellman key
exchange.

An **F**_{q}-*normal* basis of
**F**_{qn} is an
**F**_{q}-basis of
**F**_{qn} of the form {*x*,
*x ^{q}*, …,

*x*

^{qn - 1}} and the element

*x*∈

**F**

_{qn}is called

*normal*over

**F**

_{q}. These bases bear computational advantages for finite field arithmetic, so they have numerous applications, mostly found in coding theory and cryptography. An element of

**F**

_{qn}that is simultaneously normal over

**F**

_{ql}for all

*l*|

*n*is called

*completely normal*over

**F**

_{q}.

It is well-known that primitive and normal elements exist for every
*q* and *n*. The existence of elements that are simultaneously
primitive and normal is also well-known for every *q* and *n*.

**F**urther, it is also known that for all *q* and *n*
there exist completely normal elements of
**F**_{qn} over
**F**_{q}. Morgan and Mullen [*Util. Math.,*
49:21–43, 1996], took the next step and conjectured that for any
*q* and *n*, there exists a primitive completely normal
element of **F** _{qn} over
**F**_{q}.

In order to support their claim, they provided examples for such
elements for all pairs (*q*, *n*) with *q* ≤ 97 and
*q ^{n}* < 10

^{50}. This conjecture is yet to be established for arbitrary

*q*and

*n*, but instead we have partial results, covering special types of extensions. Recently, Hachenberger [

*Des. Codes Cryptogr.,*80(3):577–586, 2016] using elementary methods, proved the validity of the Morgan–Mullen conjecture for

*q*≥

*n*

^{3}and

*n*≥ 37.

In this work, we use character sum techniques and prove the validity
of the Morgan–Mullen conjecture for all *q* and *n*,
provided that *q* > *n*. In the talk, the previous results
will briefly be presented, our proof will be outlined and possible
improvements will be discussed.

This work is joint with T. Garefalakis.

## Some mathematical questions arising from molecular biology

*Andres Aravena, 2017.04.27, 15:00*

Molecular biology is a relatively new branch of science. Despite (or maybe because of) that, it has gone through several revolutionary changes. Technological advances in the last 10 or 15 years now allow scientists to produce huge amounts of data at very low costs. Therefore this has created the need of developing new mathematical tools to analyze and understand these results.

For nearly 80 years the emphasis of molecular biology was to perform repetitive and expensive experiments that did not require advanced modeling. Today the scenario is the opposite, but many researchers lack the training to answer the meaningful questions. This presents an interesting opportunity for mathematicians who want to apply their abilities to areas that can have deep effects on health, industry and the understanding of nature.

In this talk, I will speak about the work I did with my former lab in Chile and which were the challenges we encountered. In this context, I will present some of the mathematical questions that are relevant for modern molecular biology.

## Dynamic Network Analysis of Online Interactive Platform

*Nazım Ziya Perdahçı, 2017.04.14, 15:00*

The widespread use of online interactive platforms including social networking applications, community support applications draw the attention of academics and businesses. We contend that the very nature of these platforms can be best described as a network of entangled interactions. A trans-disciplinary dynamic network analysis was conducted focusing on the largest connected component, so-called a giant component (GC), to better understand the network formation. The results of the analysis were the discovery of a phenomenon that the network has evolved over time with a particular pattern towards the emergence of the GC: The interactions of platform members coalesce into a GC with an accompanying approximately constant ratio. The network measures examined should be considered as novel and complementary metrics to those ones used in conventional web and social analytics.

## Geometry of Çark Hypersurfaces

*Ayberk Zeytin, 2017.04.06, 15:00*

In this talk, our aim is to introduce a family of hyper surfaces, named as çark hyper surfaces, which are parametrised over even integers. We will discuss hyperbolicity of these surfaces. Our interest however stems from their remarkable arithmetic properties which will also be discussed.

## Some Density Ramsey type results

*Konstantinos Tyros, 2017.03.24, 15:00*

The aim of this talk is to present the density versions of the Hales–Jewett Theorem and the Carlson–Simpson Theorem. The Hales–Jewett Theorem is one of the most representing results in Ramsey theory. Its density version was first proved by H. Furstenberg and Y. Katznelson in 1991 using Ergodic Theory. However, since then, combinatorial proofs have been discovered. The Density Hales–Jewett Theorem has as a consequence Szeméredi’s Theorem on arithmetic progressions as well as its multidimensional version. The Density Carlson–Simpson Theorem is an extension of the Density Hales–Jewett Theorem and concerns the space of the left variable words.

## Effectively closed sets

*Ahmet Çevik, 2017.03.17, 15:00*

An effectively closed set, i.e. Π^{0}_{1} class, may be viewed
as the set of infinite paths through a computable tree. Complexity of
members of effectively closed sets has resulted in rich and well
developed theory. We give recent results on Turing degrees of members of
such classes. Although some familiarity with the notion of Turing
computability would help, we shall first review basic recursion
theoretic notions and then introduce effectively closed sets with its
applications in mathematics.

## Pseudoconvex domains: where holomorphic functions live

*Sönmez Şahutoğlu, 2017.03.09, 14:00*

In this talk I will introduce a fundamental notion in several complex variables, pseudoconvexity. Nothing beyond basic knowledge in real and complex analysis is assumed.

## Brauer indecomposability of Scott modules for constrained fusion systems

*İpek Tuvay, 2017.03.02, 15:00*

The Brauer indecomposability of p-permutation modules is important for obtaining categorical equivalences between p-blocks of finite groups. Letting k be an algebraically closed field of characteristic p, P a finite p-group and F a constrained fusion system defined on P, we prove that the Scott kG-module with vertex P is Brauer indecomposable where G is the finite group realizing F which is constructed by Park. This is a joint work with Shigeo Koshitani.

## Uniform Height Bounds

*Haydar Göral, 2017.02.23, 16:00*

We first define the height function and the Mahler measure on the field of algebraic numbers. The height function measures the arithmetic complexity of an algebraic number and it has some nice properties. Combining these properties with some model theoretic ideas, we explain how to obtain some uniform height bounds in polynomial rings. Time permitting, we mention Lehmer’s problem and a geometric model theoretic approach for this problem.

## Choquet–Monge–Ampère Classes

*Sibel Şahin, 2017.01.12, 15:00*

In this talk we will consider a special class of quasi-plurisubharmonic functions, namely Choquet-Monge-Ampère classes on compact Kähler manifolds. These classes become a useful intermediate tool in the analysis of Complex Monge–Ampère operator with their small enough asymptotic capacity. We will first characterize these classes through Choquet energy and then compare them with the finite energy classes. We will see that over different singularity types the comparison between Choquet–Monge–Ampère classes and the finite energy classes yields totally different characteristics.

## On Stabilization of 𝐸_{𝑘} Envelopes

*Tuba Çakmak, 2016.12.08, 15:00*

𝐸_{𝑘} envelopes were constructed in order to find definable
nilpotent subgroups which contain the nilpotent subgroup, by considering
the fact that a double centralizer of a subgroup contains this subgroup.
A natural question about 𝐸_{𝑘} definable envelopes is
stabilization problem. It is about the decreasing sequence of
𝐸_{𝑘} envelopes stop after a specific point. In this talk after
giving some group classes have affirmative answer for stabilization
problem, I will give the details of an example which shows that
𝐸_{𝑘} envelopes do not stabilize in general.

## Index and Carlitz Rank of Permutation Polynomials

*Leyla Işık, 2016.12.01, 16:00*

Index and Carlitz rank are two important measures for the complexity
of a permutation polynomial 𝑓(𝑥) over the finite field 𝔽_{𝑞}. In
particular, for cryptographic applications we need both, a high Carlitz
rank and a high index. In this article we study the relationship between
Carlitz rank Crk(𝑓) and index Ind(𝑓). More precisely, if the permutation
polynomial is neither close to a polynomial of the form 𝑎𝑥 nor a
rational function of the form 𝑎𝑥^{−1}, then we show that Crk(𝑓)
> 𝑞−max{3Ind(𝑓),(3𝑞)^{1/2}}. Moreover we show that the
permutation polynomial which represents the discrete logarithm
guarantees both a large index and a large Carlitz rank.

## On the action of groups with TI-centralizers

*İsmail Güloğlu, 2016.11.24, 16:00*

A subgroup 𝐻 of a group 𝐺 is called a TI-subgroup if 𝐻 intersects trivially with any conjugate of 𝐻 different from 𝐻 and 𝐻 is called an STI subgroup of 𝐺 if for any normal subgroup 𝑁 of 𝐺 the subgroup 𝐻𝑁/𝑁 is TI in 𝐺/𝑁. In this talk the following result will be proven:

If 𝐴 is of prime order and acts coprimely on a finite group 𝐺 in such a way that the subroup 𝐹 of fixedpoints of 𝐴 is a STI-subgroup of 𝐺 then 𝐹 is solvable if and only if 𝐺 is solvable.

## The complexity of the topological conjugacy relation on Toeplitz subshifts

*Burak Kaya, 2016.11.17, 17:00*

In this talk, we will analyze the complexity of topological conjugacy of Toeplitz subshifts from the point of view of descriptive set theory. More specifically, we shall prove that the topological conjugacy relation on Toeplitz subshifts with separated holes is hyperfinite. This result provides a partial affirmative answer to a question asked by Sabok and Tsankov.

## On Nonsolvable Groups Whose Prime Degree Graphs Have Four Vertices and One Triangle

*Roghayeh Hafezieh, 2016.11.10, 16:00*

There is a large literature which is devoted to study the ways in which one can associate a graph with a group, for the purpose of investigating the algebraic structure using properties of the associated graph. Let 𝐺 be a finite group. The prime degree graph of 𝐺, denoted by ∆(𝐺), is an undirected graph whose vertex set is ρ(𝐺) and there is an edge between two distinct primes 𝑝 and 𝑞 if and only if 𝑝𝑞 divides some irreducible character degree of 𝐺. In general, it seems that the prime graphs contain many edges and thus they should have many triangles, so one of the cases that would be interesting is to consider those finite groups whose prime degree graphs have a small number of triangles. In this talk we will consider the case where for a nonsolvable group 𝐺, ∆(𝐺) has only one triangle and four vertices.

## Çinlar Subgrid Scale Stress Model for Large Eddy Simulation

*Rukiye Kara, 2016.11.03, 16:00*

We construct a new subgrid scale (SGS) stress model for representing the small scale effects in large eddy simulation (LES) of incompressible flows. We use the covariance tensor for representing the Reynolds stress and include Clark’s model for the cross stress. The Reynolds stress is obtained analytically from Çinlar random velocity field, which is based on vortex structures observed in the ocean at the subgrid scale. The validity of the model is tested with turbulent channel flow computed in OpenFOAM. It is compared with the most frequently used Smagorinsky and one-equation eddy SGS models through DNS data.

## Decomposability of Fiber Bundles

*Mustafa Topkara, 2016.10.27, 16:00*

A fiber bundle is said to be "indecomposable" if it cannot be expressed as a fiber product of fiber bundles of smaller fiber dimension, and is "stably indecomposable" if its fiber product with another fiber bundle can be decomposed into fiber bundles of smaller fiber dimension. The talk will be about the relationship between these two concepts.

## Algebraic structures arising from isotonian maps between posets

*Ayesha Asloob Qureshi, 2016.10.13, 16:00*

Our main goal is to study the ideal 𝐿(𝑃, 𝑄) and toric ring 𝐾[𝑃, 𝑄] whose generators are in bijection to the isotone maps from 𝑃 to 𝑄. We examine the several algebraic properties of 𝐿(𝑃, 𝑄) including Alexander duality behaviour. The class of algebras 𝐾[𝑃, 𝑄], called isotonian, are natural generalizations of the so-called Hibi rings. We determine the Krull dimension of these algebras and for particular classes of posets 𝑃 and 𝑄 we discuss their normality behaviour. Also, we determine special classes of 𝑃 and 𝑄 for which defining ideal of 𝐾[𝑃, 𝑄] admits a quadratic Gröbner basis.

## Distinct Stein structures on contractible 4-manifolds

*Çağrı Karakurt, 2016.10.06, 16:00*

Stein manifolds are special compact subsets of affine algebraic varieties which play an important role in complex/symplectic geometry and singularity theory. Due to Eliashberg's topological characterization, the existence and uniqueness of Stein structures on 𝑛-dimensional manifolds are well-understood for 𝑛 > 4, but are more challenging when 𝑛 = 4. I will give a brief survey on these problems and talk about a recent construction done in collaboration with Oba and Ukida.

## Recursive towers of function fields over finite fields

*Seher Tutdere, 2016.05.12, 16:00*

Let 𝔽_{𝑞} be a finite field (𝑞 = 𝑝^{𝑘} with 𝑝 a prime
and 𝑘 ≥ 1 an integer) and 𝐹/𝔽_{𝑞} be an algebraic function field
of one variable with the field 𝔽_{𝑞} as its full constant field.
In this talk, I will firstly give a short introduction to recursive
towers of function fields over finite fields. Then we discuss some
results on quadratic recursive towers over the field 𝔽_{2}
(which is a joint work with Henning Stichtenoth).

## Rational points on curves over finite fields

*Henning Stichtenoth, 2016.04.28, 16:00*

This talk is about the number of rational points N(C) of an algebraic curve C over a finite field. The classical theorem by Hasse–Weil gives sharp upper and lower bounds for N(C) in terms of the genus g(C) and the cardinality q of the underlying field. However, these bounds can be improved if the genus is large with respect to q. This leads to the concept of “asymptotic bounds”.

We discuss results on asymptotic bounds, and give a partial answer to the question, for which pairs of integers (N,g) there exist curves with N(C)=N and g(C)=g.

## Minimal characteristic bisets of saturated fusion systems

*Matthew Gelvin, 2016.04.21, 17:00*

The fusion system of a finite group is an organization framework for understanding the p-local structure of that group. The data are encoded in a category with objects the subgroups of a fixed p-group S, and morphisms certain group injections that mimic conjugation homomorphisms induced by an absent ambient group. It can be seen as an algebraic structure in its own right, one which serves as a bridge between the fields of local group theory, modular representation theory, and algebraic topology.

In this talk I will discuss joint work with Sune Reeh, in which we parameterize the characteristic bisets of a saturated fusion system. A characteristic biset can be thought of a collection of abstract group elements that conjugate subgroups of S in such a way to recreate the original fusion system. It turns out that there is a simple parameterization of these bisets, and as a consequence we discover the existence of a unique minimal characteristic biset for any saturated fusion system.

## Bi-Hamiltonian Structures on a 3-Manifold

*Melike İşim Efe, 2016.04.14, 16:00*

In this talk, we will give general information on bi-Hamiltonian structures and talk about some topological obstructions, such as first Chern class and Godbillon-Vey class, on the existence of bi-Hamiltonian structures on an orientable 3-manifold.

## Intersection graphs of groups

*Selçuk Kayacan, 2016.03.24, 16:00*

For a group G the intersection graph of G is a simple graph such that its vertex set consists of the non-trivial proper subgroups of G and two distinct vertices are joined by an edge if and only if their intersection is non-trivial. In this talk, I will outline some results about the graph theoretical properties of those objects and present some speculations on possible further directions.

## Monodromy groups of real Enriques surfaces of hyperbolic type

*Sultan Erdoğan, 2016.03.17, 16:00*

I will talk about some basic definitions and facts about real Enriques surfaces. Then I will introduce an equivariant version of Donaldson's trick, used in deformation classification, that relates real Enriques surfaces and real rational surfaces. I will roughly state the result about the monodromy groups of real Enriques surfaces of hyperbolic type and (if time permits) show on a few deformation classes how the monodromy groups are found.

## Gluing the twists of a block of a group algebra

*Laurence Barker, 2016.03.10, 16:00*

For a group algebra with coefficients in a field of prime characteristic p, the representation theory decomposes into the study of p-blocks. Empirically, it appears that, up a finite amount of information, much of behaviour of a block is determined by a p-subgroup called the defect group. Some of that finite information, apparently, is the fusion system, a category which expresses some (usually not all) of the group conjugation actions on the subgroups of the defect group. Incorporating a bit more information, namely, some 2-cohomological twists on some of the automorphism groups in that category, Puig and Linckelmann have reformulated Alperin's Weight Conjecture as a p-local conjecture. The reformulation depends on the existence of a solution to a problem called the Gluing Problem. Puig has posted a manuscript on the existence. A paper of Park exhibits a counter-example to uniqueness. However, we shall suggest that the non-uniqueness is just a feature of a particular formulation of the problem. We shall introduce a category which gives rise to a canonical solution.

## Leavitt Path Algebras

*Murad Özaydın, 2016.03.03, 16:00*

LPAs (Leavitt Path Algebras) were defined recently (Abrams and Aranda Pino, 2005; Ara, Moreno and Pardo, 2007) but they have roots in the works of Leavitt in the 60s focused on understanding the extent of the failure of the IBN (Invariant Basis Number) property for arbitrary rings. A ring has IBN if any two bases of a finitely generated free module have the same number of elements. Fields, division rings, commutative rings, Noetherian rings all have IBN. However, the rings L(1,n) defined by Leavitt (1962) and their analytic cousins the C*-algebras of Cuntz (1977) are not artificial and pathological structures constructed only for the sake of providing counter examples; for instance, they implicitly come up in Signal Processing (as the algebras generated by the downsampling and upsampling operators). Moreover Leavitt's work (1962, 1965) provided important impetus for major developments in non- commutative ring theory in the 1970s by Cohn, Bergman and others.

I plan to start with the basic definitions, state some fundamental results, explain the criterion for an LPA to have IBN (joint work with Muge Kanuni Er) and indicate the ideas involved in the recent classification of the finite dimensional representations (jointly with Ayten Koc). LPAs are (Cohn) localizations of Path (or Quiver) Algebras whose finite dimensional representations are usually wild, but the category of finite dimensional representations of LPAs turn out to be tame with a very reasonable classification of all the indecomposables and the simples. All finite dimensional quotients of LPAs are also easy to describe.

## From Mackey functors to biset functors

*Olcay Coşkun, 2016.02.25, 16:00*

Mackey and biset functors provide a unified approach to representation theory of finite groups. In this talk, we give a short introduction to the theory of biset functors and discuss the induction functor from Mackey functors to biset functors.

## (Co)homology theories on diffeological spaces

*Serap Gürer, 2016.02.18, 16:00*

In this talk, we discuss "diffeological spaces" that are considered as generalizations of smooth manifolds. I will introduce (co)homology theories on diffeological spaces, I will talk about the (co)homology theories of some singular spaces in diffeological framework.

## Surfaces in 4-manifolds

*Ahmet Beyaz, 2016.02.11, 16:00*

Surfaces in a four dimensional manifold carry information about the topological, smooth or complex structures on it. This talk is an overview of their use.

## Hopf-Dihedral Cohomology of Hopf *-Algebras

*Atabey Kaygun, 2016.01.14, 16:00*

Starting from a very simple setup (an invariant trace compatible with a *-structure) I am going to derive the simplicial/cyclic/dihedral structure maps forming bases for different cohomology theories noncommutative geometers use for Hopf algebras. I will sketch some calculations on the Drinfeld-Jimbo deformation of the universal enveloping algebra of the lie algebra su(2), and time permitting, talk about its connection with L-theory.

## Admissibility and Unifiability in Modal Logics

*Çiğdem Gencer, 2015.12.29, 14:00*

Both from the theoretical viewpoint and the viewpoint of applications, inference rules in nonstandard logics give rise to many interesting problems. One of them is the determination of their admissibility. Admissible inference rules do not depend on the choice of an axiomatic systems for a given logic and constitute the greatest set of rules compatible with its derivability relation. Logic being the science of reasoning, deciding the admissibility property in such or such logic is a research topic of the utmost interest for those who want to improve the efficiency of automated deduction. The admissibility problem in modal logics is strongly related to another problem of interest, the unification problem. As the admissibility problem, the unification problem in modal logics has been motivated by automated deduction tools. Its starting point was the existence of a most general unifier for any unifiable formula in Boolean Logic. Later it was proved that in general, there are no most general unifiers for unifiable formulas in various modal logics but a finite set of maximal unifiers. These results provide a connection between unification of formulas and admissibility of inference rules. I will talk on this relationship between admissibility and unifiability.

## The complexity of topological conjugacy of pointed Cantor minimal systems

*Burak Kaya, 2015.12.24, 17:00*

In this talk, we analyze the complexity of the topological conjugacy relation on Cantor minimal systems from the point of view of descriptive set theory. We shall show that topological conjugacy of pointed Cantor minimal systems is Borel bireducible with the equality of countable sets of reals. We will also show that this is a lower bound for the Borel complexity of topological conjugacy of Cantor minimal systems. If time permits, we will cover some applications of our results to properly ordered Bratteli diagrams.

## Homology and cohomology theories graded over monoidal categories

*Mehmet Akif Erdal, 2015.12.24, 16:00*

This talk is based on a joint work with Özgün Ünlü. I will first discuss the monoid actions on sets, which is via action from both sides, but different then the usual biactions in terms of equivariance of maps. Using this view point, we will be able to associate a new action to a given monoid action on a set, so called inverse action, which generalizes the notion of inverse of group actions. In the second part I will discuss categorification of these definitions. I will define action of a monoidal categories on a category and the inverse of such action, analogues to the monoid case. As an application, "homology and cohomology theories graded over a monoidal category" will be introduced, which generalizes the usual definitions of (co)homology and (co)homotopy theories existing in the literature. This allow us to view (co)homology and (co)homotopy theories as fixed points of an action of monoidal category on a category.

## On Extension Properties of Pluricomplex Green Functions

*Zeynep Kurşungöz, 2015.12.17, 16:00*

On Reinhardt domains containing the origin, the pluricomplex Green function with logarithmic pole at 0 is polyradial. This property of the function can be used to find it using an associated convex function in logarithmic coordinates. We use this relation to prove extension properties of pluricomplex Green functions on Reinhardt domains.

## A refinement of Alperin's Conjecture

*İpek Tuvay, 2015.12.10, 16:00*

In this talk, a refinement of Alperin's Conjecture involving the blocks of the endomorphism algebra of the permutation module formed by the cosets of a p-subgroup will be presented. Then the proof of the conjecture in two special cases will be given. This is a joint work with Laurence Barker.

## Simply laced Dynkin diagrams

*Tolga Etgu, 2015.12.03, 16:00*

I’ll talk about a truly exceptional class of trees which shows up in various areas of mathematics.

## Coinvariants of modular *p*-groups

*Müfit Sezer, 2015.11.19, 16:00*

We consider the ring of coinvariants for a modular representation of
a cyclic group of prime order *p*. We show that the classes of the
terminal variables in the coinvariants have nilpotency degree *p*
and that the coinvariants are a free module over the subalgebra
generated by these classes. An incidental result we have is a
description of a Gröbner basis for the Hilbert ideal and a decomposition
of the corresponding monomial basis for the coinvariants with respect to
the monomials in the terminal variables.

## Homology classes of lifts of simple curves on surfaces

*Ingrid Irmer, 2015.11.13, 13:00*

The following question has arisen from studying mapping class groups:
Suppose *S* is a surface and *S*^{∼} is a finite
index regular cover of *S*. Do the lifts of simple curves from *S*
generate H_{1}(*S*^{∼},ℤ)?

This talk will discuss some of the background to this question, and explain why the answer to this question is “No” for large complexity covers.

## Quantum Groups and Factorization

*Münevver Çelik, 2015.11.05, 16:00*

Quantum groups are noncommutative, noncocomutative Hopf algebras with
some additional structure on them. Roughly speaking, Drinfeld's quantum
double corresponds to LU-decomposition of a quantum group. We will
introduce the bialgebra M_{p,q}(*n*) and give a
partial result concerning the factorization of it into simpler pieces to
ease the computations.

## Polyhedral Omega: A new linear Diophantine system solver

*Zafeirakis Zafeirakopoulos, 2015.10.22, 16:00*

Polyhedral Omega is a new algorithm for solving linear Diophantine systems (LDS), i.e., for computing a multivariate rational function representation of the set of all non-negative integer solutions to a system of linear equations and inequalities. Polyhedral Omega combines methods from partition analysis with methods from polyhedral geometry. In particular, we combine MacMahon’s iterative approach based on the Omega operator and explicit formulas for its evaluation with geometric tools such as Brion decomposition and Barvinok’s short rational function representations. In this way, we connect two branches of research that have so far remained separate, unified by the concept of symbolic cones which we introduce. The resulting LDS solver Polyhedral Omega is significantly faster than previous solvers based on partition analysis and it is competitive with state-of-the-art LDS solvers based on geometric methods. Most importantly, this synthesis of ideas makes Polyhedral Omega by far the simplest algorithm for solving linear Diophantine systems available to date. This is joint work with Felix Breuer.

## Rational points on horocycles and incomplete Gauss sums

*Emek Demirci, 2015.05.28, 16:00*

In this seminar I will talk about the connection between the limiting distributions of rational points on horocycle flows and the value distribution of incomplete Gauss sums. A key property of the horocycle flow on a finite-area hyperbolic surface is that long closed horocycles are uniformly distributed. We embed rational points on such horocycles on the modular surface and investigate their equidistribution properties. On the other hand, it is well known that the classical Gauss sums can be evaluated in closed form depending on the residue class of the number of terms in the sum modulo 4. This is not the case for the incomplete Gauss sums, where we restrict the range of summation to a sub-interval and study their limiting behavior at random argument as the number of terms goes to infinity. If the time permits, I also establish an analogue of the weak invariance principle for incomplete Gauss sums.

## Fredholmness of Some Toeplitz Operators

*Başak Koca, 2015.05.21, 16:00*

In this talk we consider Fredholmness of Toeplitz operators with continuous symbol in several variables (in polydisc and unit ball cases) by using C*- algebraic methods. Then, as an application for our results, we will give some conditions for solvability of a class of integral equations.

## A twisted version of the classifying space functor

*Aslı Güçlükan, 2015.05.14, 16:00*

In this talk, we compare the classifying space functor from the category of small categories to the category of topological spaces to the composition of the three functors namely the inclusion of the category of small categories in the category of simplicially enriched categories, a cofibrant replacement functor, and the geometric realization of a bar construction. To make this comparison we introduce the notion of a twisted natural transformation. This is a joint work in progress with Özgün Ünlü.

## Asymptotics of the spectral gaps for the Mathieu operator

*Berkay Anahtarcı, 2015.05.08, 13:00*

The Hill operator on the real line, is self-adjoint and its spectrum has a band-gap structure, that is, the intervals of continuous spectrum alternate with spectral gaps. The endpoints of these gaps are eigenvalues of the same differential operator L but considered on the interval [0,π] with periodic or antiperiodic boundary conditions.

Considering the Hill operator in case of the specific potential v(x)=2a⋅cos(2x), so called the Mathieu operator, we provide precise asymptotics of the spectral gaps of L.

Our result extends the asymptotic formula of Harrell--Avron--Simon by providing more asymptotic terms.This is joint work with Plamen Djakov.

## Dynamics of free group automorphisms

*Çağlar Uyanık, 1015.05.07, 16:00*

The study of outer automorphism group of a free group Out(F) is closely related to study of Mapping Class Groups. The Out(F) analog of a pseudo-Anosov homeomorphism is called a fully irreducible. We will talk about types of fully irreducible elements, geometric and hyperbolic, and prove that they act on the space of geodesic currents with north-south dynamics. As an application, we will give a criterion for subgroups of Out(F) to contain a hyperbolic fully irreducible element using ping-pong type arguments.

## Logarithmic capacity and rational lemniscates

*Stamatis Pouliasis, 2015.04.30, 16:00*

First we shall present some basic facts about logarithmic capacity, analytic capacity and Green function in the complex plane. Motivated by considerations related to the semi-additivity property of analytic capacity, we will study the lemniscates of good rational functions. We will prove a reflection principle for the harmonic measure of rational lemniscates and we will give estimates for their capacity and the capacity of their components. This is joint work with Thomas Ransford.

## Lefschetz pencils on symplectic 4-manifolds

*Çağrı Karakurt, 2015.04,09, 16:00*

Motivated from Lefschetz hyperplane theorem, Donaldson showed that every symplectic 4-manifold admits the structure of a pencil whose fibers are symplectic submanifolds. Since then these pencils have been widely used to translate geometric problems in symplectic geometry into problems in mapping class groups. This talk will be a survey about Lefschetz pencils and their relation with the topology of symplectic 4-manifolds.

## On the size of support of *p*-harmonic measure in
space

*Murat Akman, 2015.04.06, 11:00*

A function *u* is said to be p-harmonic, fixed
1<*p*<∞, in an open set
Ω⊂ℝ^{n} if *u* is a weak solution
to the *p*-Laplace equation
∇⋅(|∇*u*|^{p-2}∇*u*)=0
in Ω. This pde is nonlinear elliptic equation in divergence form
and when *p*=2, it is the Laplace equation which is linear elliptic
equation.

In this talk we study the size of the support of *p*-harmonic
measure associated with a positive *p*-harmonic function in
Ω⊂ℝ^{n} with certain boundary values. We
first discuss a recent work on “natural generalization” of a
well-known result of Jones and Wolff for harmonic measure to the
*p*-harmonic setting when *p*≥*n*. We then study
singular sets for *p*-harmonic measure on “flat”
domains. Finally, we propose some questions which is known in the
harmonic setting but not known in the *p*-harmonic setting for ≠2.

## Binomial arithmetical rank of toric ideals of graphs

*Anargyros Katsampekis, 2015.03.19*

Toric ideals arise naturally in problems from diverse areas of mathematics, including algebraic statistics, integer programming, dynamical systems and graph theory. A basic problem in the theory of toric ideals is to determine the least number of polynomials needed to generate the toric ideal up to radical. This number is commonly known as the arithmetical rank of a toric ideal. A usual approach to this problem is to restrict to a certain class of polynomials and ask how many polynomials from this class can generate the toric ideal up to radical. Restricting the polynomials to the class of binomials we arrive at the notion of the binomial arithmetical rank of a toric ideal. In the talk we study the binomial arithmetical rank of the toric ideal associated with a finite graph in two cases:

- The graph is bipartite.
- The toric ideal is generated by quadratic binomials.

## Formal groups

*Piotr Kowalski, 2015.03.12*

I will discuss the notions of a formal group and a formal map in the following two contexts:

- Ax-Schanuel problems.
- Hasse-Schmidt derivations and their model theory.

## Radial limits of partial theta and similar series

*Kağan Kurşungöz, 2015.03.05*

Consider unilateral series in a single variable where the exponent is
a polynomial, and the coefficients are periodic. Quadratic polynomials
correspond to partial theta series. Such series converge inside the unit
disk. We compute limits and asymptotic expansions of those series as the
variable tends radially to a root of unity. The method is suitable for
automation and part of the computations is based on the idea of the
*q*-integral.

## Interlacing polynomials, Ramanujam graphs and the Kadison-Singer problem

*Mohan Ravichandran, 2015.02.26*

I will talk about recent work of Adam Marcus, Daniel Spielman and Nikhil Srivastava (MSS) where they use ideas from combinatorics and spectral graph theory to settle several important problems in computer science and mathematics. Two of these are constructions of Ramanjuan graphs(expander graphs with the best possible expansion constants) and a solution of the famous Kadison-Singer problem.

Ramanujan graphs were first constructed by Lubotzy, Philips and Sarnak and independently by Margulis using deep ideas from number theory and geometric group theory respectively. MSS provided a new construction in 2013, using more or less elementary ideas, a construction that further removes important numerical restrictions in all prior work.

The Kadison-Singer problem was one of the thorniest problems in operator algebras till it was settled by this trio, also in 2013. The problem is further related to dozens of problems in areas as disparate as C* algebras, signal processing and harmonic analysis. Notably, a majority of the mathematical community expected a negative solution to the problem, an expectation that was confounded by Marcus, Spielman and Srivastava.

The techniques of their proofs are both remarkably elementary, requiring little more than linear algebra and extremely powerful. I expect to even be able to sketch their proofs within the time allocated to the talk.

## The classification problem for Toeplitz subshifts

*Burak Kaya, 2014.01.08, 16:00*

Under appropriate encoding and identification, the study of the relative complexity of various classification problems can be seen as the study of the corresponding definable equivalence relations on Polish spaces. In this talk, we will introduce the framework provided by descriptive set theory to study such classification problems and analyze the complexity of the isomorphism relation of certain topological dynamical systems. We will also discuss the connections with group theory and try to understand the complexity of classification problem for infinite finitely generated simple amenable groups.

## Mann Property

*Haydar Göral, 2014.12.26, 16:00*

In this talk we study the pair (K,G) where K is an algebraically closed field and G is a multiplicative subgroup of K* with the Mann property. The main examples of this property comes from number theory. The reason of the naming like this is that H. Mann proved that the roots of unity has the Mann property. The theory of the pair is axiomatised by L. van den Dries and A. Günaydın and they prove that the pair (K,G) is stable. We first characterize the independence in the pair and this allows us to characterize the definable groups in (K,G).

## Modular Forms and Elliptic Curves over Number Fields

*Haluk Şengün, 2014.12.25, 16:00*

The celebrated connection between elliptic curves and weight 2 newforms over the rationals has a conjectural extension to general number fields. For example, over odd degree totally real fields, one knows how to associate an elliptic curve to a weight 2 newform with integer Hecke eigenvalues. Conversely, very recent work of Freitas, Hun and Siksek show that over totally real fields, most elliptic curves are modular (in fact, over real quadratic fields, "all" are modular).

Beyond totally real fields, we are at a loss at associating elliptic curves to weight 2 newforms. The best one can do is to "search" for the elliptic curve. In joint work with X.Guitart (Essen) and M.Masdeu (Warwick), we generalize Darmon's conjectural construction of algebraic points on elliptic curves to general number fields and then use this conjectural construction to analytically construct the elliptic curve starting from a weight 2 newform over a general number field, under some hypothesis. In the talk, I will start with a discussion of the first paragraph and then will sketch our method.

## Deterministic Model of Optimal Growth as an Optimization Problem

*Ayşegül Yıldız Ulus, 2014.12.18, 16:00*

Discrete time optimal growth model is one of the important models of
the theory of economics. This simple and elegant model is described by
the presence of a social planner who maximizes the infinite sum of
discounted utilities of consumption subject to a convex one sector
production set. One useful approach to solve this problem is dynamic
programming which is in fact the study of a dynamic optimization problem
through the analysis of value functional equations. In finite horizon,
these types of problems can be solved by means of Lagrange multipliers
method. In infinite horizon optimal growth models, these multipliers
will typically belong to an infinite dimensional decision space.
Therefore, the questions whether the Lagrange multipliers exist and
whether they can be represented by a summable sequence arise. These
problems have been overcome by extending the Lagrangean to infinite
dimensional spaces and sufficient conditions for Lagrangean to be
represented by a summable sequence of multipliers are then provided. In
this talk, I will present the problem in general form and give the
extension of the Kuhn-Tucker theorem where the multipliers are
represented in (ℓ^{∞})', the dual space of
ℓ^{∞}, and then give sufficient conditions for having an
ℓ^{1} representation of the multipliers.

## Graph designs and resolvable designs

*Selda Küçükçiftçi, 2014.12.11, 16:00*

Given a collection of graphs ℋ, an
ℋ-decomposition of a graph 𝐺 is a decomposition
of the edges of 𝐺 into isomorphic copies of graphs in
ℋ. The copies of 𝐻∈ ℋ in the
decomposition are called blocks. Such a decomposition is called
resolvable if it is possible to partition the blocks into
classes 𝑃_{𝑖} such that every point of
𝐺 appears exactly once in some block of each
𝑃_{𝑖}. Moreover a class is called
uniform if every block of the class is isomorphic to the same
graph from ℋ.

In this talk, we will present famous problems in the area of graph designs and explain recent results on uniformly resolvable ℋ-decomposition problems.

## Automorphism group of a tree // *Bir ağacın
otomorfizm grubu*

*Muhammed Uludağ, 2014.12.02 (Salı), 16:30*

One may think that the group of automorphisms of a tree, being a
profinite group, is something which resembles the absolute Galois group
and as such is not good for your mental health. Here what we mean by a
tree is a regular tree, that is a tree with fixed vertex degrees (=d).
Denote this tree by A_{d}. In case d is at least 3, the
automorphism group is uncountable. My aim is to describe each element of
this group as a "shuffle". Then I will give another description as a
"twist". It is very easy to see each element explicitly in these
descriptions, but the group multiplication is somewhat messy. To this
end, I will have to endow the tree with the structure of a plane tree
(nevertheless the automorphisms will be tree automorphisms, not plane
tree automorphisms) After this I will define the boundary of the tree
and look at the action of the group on this boundary. Plane tree
structure induces a cyclic order structure on the boundary of the tree.
If we glue those points which are not separated by a third point, then
we obtain a space which is homeomorphic to the circle. The automorphism
group does not respect the order structure but nevertheless acts on the
circle "in some way". My ultimate aim is to understand this action. //
*Ağaç otomorfizmi denince, mutlak Galois grubu gibi akla zarar bir
pro-sonlu grup hayale gelir. Burada ağaçtan kastımız, muntazam, yani
köşe dereceleri sabit (=d), sonsuz bir ağaçtır. Bu ağacı A _{d}
ile gösterelim. Şayet d büyükeşit üçse, ağacın otomorfizm grubu
sayılamaz adet eleman barındırır. Benim amacım bu grubun her bir elemanı
bir "kayma" şeklinde tasvir etmektir. Sonra da her bir elemanı bir
"burma" şeklinde tasvir edeceğim. Bu tasvirlerde elemanları son derece
açıkbir şekilde görmek kolay, ama grup çarpımını uygulamak pek de kolay
değil. Bunu yapmak için ağaca bir "düzlem ağacı" yapısı giydirmem
gerekecek (ama otomorfizmler düzlem ağacının değil, yine soyut ağacın
otomorfizmleri olacak). Sonra da ağacın sınırını tanımlayıp otomorfizm
grubunun bu sınır üzerindeki etkisine bakacağım. Düzlem ağacı yapısı, bu
sınır üzerinde doğal bir devirli sıralama bağıntısı verir. Sınırda,
aralarında üçüncü bir nokta olmayan noktaları yapıştırırsak, çembere
homeomorf bir uzay elde ederiz. Otomorfizm grubu bu çember üzerinde "bir
şekilde" etkir. Nihai amacımı bu etkiyi biraz anlamaya çalışmaktır.*

## Complex Multiplication and Cryptographic Applications

*Osmanbey Uzunkol, 2014.11.27, 16:00*

Complex multiplication combines the three Gaussian A's (Algebra, Analysis and Arithmetic) by means of a beautiful interplay between the geometry of abelian varieties with complex multiplication and the arithmetic of corresponding CM number fields. It has become subject to algorithmic investigations with different applications to algorithmic algebraic number theory and cryptography ranging from primality proving, elliptic curve cryptography, group- and pairing-based cryptography to the recent privacy enhancing techniques in cloud computation security.

In this talk, we give a survey of algorithmic and cryptographic aspects of complex multiplication together with some new results and some open problems

## On the motivic Galois group

*İlhan İkeda, 2014.11.19, 11:40*

We shall discuss certain problems related with the motivic Galois groups.

## Finite non-associative division algebras: semifields

*Michel Lavrauw, 2014.11.13, 16:00*

Shortly after the classification of finite fields by E. H. Moore, announced in 1893 at the International Mathematical Congres in Chicago, mathematicians turned their attention to algebraic systems (K,+,•), satisfying a slightly weaker set of axioms compared to a finite field. For instance, L. E. Dickson and J. Wedderburn studied finite domains, omitting the axiom: (K*,•) is commutative. In 1905 they proved that there is no such thing as a non-commutative finite domain. This is now known as Wedderburn's little theorem. Similarly, Dickson studied algebraic systems (K,+,•) omitting the axiom of associativity (K,+,•) as well: non-associative division algebras. In this case however there do exist proper examples, i.e. which are not fields. This study was continued by his student A. A. Albert, and picked up shortly after by many other mathematicians, including E. Artin, M. A. Zorn, L. A. Skornyakov, R. H. Bruck, D. E. Knuth, E. Kleinfield, I. Kaplansky, to name but a few. Nowadays these algebras are called semifields (a term introduced by D. E. Knuth in 1965), and semifields are studied up to isotopisms instead of isomorphisms. We will explain the BEL-configuration from [1] and report on the structural consequences on the set of isotopism classes of semifields from [2].

- [1] S. Ball, G. Ebert, M. Lavrauw. A geometric construction of finite semifields. J. Algebra (2007)
- [2] M. Lavrauw and J. Sheekey. On BEL-configurations and finite semifields. arXiv1402.2486

## An Asymptotic Birch and Swinnerton-Dyer Conjecture

*Kazım Büyükboduk, 2014.11.07, 16:30*

The conjecture of Birch and Swinnerton-Dyer (BSD) is one of the Clay
Millennium problems that links the arithmetic invariants of an elliptic
curve to its analytic invariants. Most of this talk will be devoted to
explaining the content of this conjecture and stating an
**asymptotic** variant. As time permits, I will sketch a
proof of this asymptotic BSD conjecture for CM elliptic curves (and more
generally, for CM elliptic modular forms) along the anticyclotomic
tower. The proof relies on the Iwasawa theoretic study of the
Beilinson-Kato elements and a reciprocity law that relates them to
relevant L-functions.

## A Gentle Introduction to Teichmüller Theory

*Özgür Evren, 2014.10.30, 16:00*

Teichmüller theory is primarily concerned about studying the moduli for hyperbolic structures on a given topological surface. It is subject where different branches of mathematics such as complex analysis, hyperbolic geometry, algebraic topology, low dimensional topology and several others intersect. In this talk, we will motivate and define the Teichmüller space and introduce several geometries on this space. Time permitting, we will discuss certain results about the comparison of these geometries and some open problems on the same topic.

## The *p* ranks of Prym varieties

*Ekin Özman, 2014.10.23, 16:00*

In this talk we will start with basics of moduli space of curves,
coverings of curves, *p*-ranks and mention the differences in
characteristic 0 and positive characteristics. Then we'll define Prym
variety which is a central object of study in arithmetic geometry like
Jacobian variety. The goal of the talk is to understand various
existence results about Prym varieties of given genus, *p*-rank and
characteristics of the base field. This is joint work with Rachel Pries.

## Hopf-cyclic cohomology of quantized enveloping algebras

*Serkan Sütlü, 2014.10.10, 16:00*

Given a coalgebra coextension, we introduce a filtration whose associated spectral sequence computes the Hochschild cohomology of the coextension. We then apply this cohomological machinery to compute the Hopf-cyclic cohomology of a quantized enveloping algebra of a Lie algebra. Time permitting, we will discuss as an application of this method, the Hopf-cyclic cohomology of Connes-Moscovici Hopf algebras. (Based on a joint work with A. Kaygun.)

## Physics on networks, the graph Laplacian and discrete analysis

*Ayşe Erzan, 15.05.2014, 16:00*

The Laplace operator is perhaps the most central object in physics - from the harmonic oscillator (the basis for "zeroth order" modelling at least half of physical phenomena) to diffusion (the other half!). Many phenomena of physical, biological or social interest take place on discrete lattices (networks) which are not embedded in metric spaces, and the graph Laplacian emerges as the appropriate operator here. I will illustrate this with a simple example from Statistical Physics: We normally expand fluctuations of some field in terms of the eigenfunctions of the Laplace operator (the harmonic functions on d-dimensional Euclidean spaces or lattices), and can systematically eliminate the "smaller wavelength fluctuations in order to probe certain properties of the physical system. The challenge is to be able to do a similar operation on arbitrary networks. However, obtaining the eigenvectors of the Laplace operator on even perfectly regular networks such as Cayley trees of arbitrarily large lattice size can be daunting. Some attempts in this direction and many open questions will be discussed.

## Equivariant multiplicities, Chow–Künneth formulas and motivic decompositions

*Richard Gonzales, 2014.04.28, 14:30*

In this introductory talk, we study algebraic varieties with torus action (T-varieties). We focus on (1) the notion of equivariant multiplicities at isolated fixed points and (2) equivariant Chow-Künneth formulas for T-linear varieties, a class objects that contains spherical varieties (e.g. toric varieties). Finally, we present a notion of algebraic rational cell, with applications to the equivariant Chow groups (and Chow motives) of certain T-linear varieties.

## Groups with a recursively enumerable irreducible word problem

*Gabriela Aslı Rino Nesin, 2014.04.22, 13:00*

I will be presenting my paper of the same title from FCT2013. The word problem for finitely generated groups has been extensively studied; the irreducible word problem is a closely related concept, although little seems to be known about finitely generated groups with a recursively enumerable irreducible word problem. After giving a brief introduction of the terminology and geometric intuition behind some of the definitions and previous results, I will present the main result of the paper, which is that for finitely generated groups, having a recursively enumerable irreducible word problem with respect to every finite generating set is equivalent to having a recursive word problem.

Time permitting, I will also show that there are groups for which recursive enumerability of the irreducible word problem does depend on the choice of finite generating set, and explore the relationship with the number of ends of a group.

## On Brauer indecomposability of Scott modules for some families of groups

*İpek Tuvay, 2014.04.17, 16:00*

Brauer indecomposablility of p-permutation modules carries importance in verifying categorical equivalence between p-blocks of finite groups which is predicted by Broué's abelian defect group conjecture. In this talk, we will present the proof of Brauer indecomposability for some particular type of p-permutation modules: Scott modules for some families of groups constructed by Sejong Park in the context of fusion systems.

## On Martingale Theory

*Mahir Hasansoy, 2014.04.10, 16:00*

We discuss some problems on Martingales related to Analysis, Stochastic Analysis, and Financial Mathematics.

## Disjointness in Hypercyclicity

*Özgür Martin, 2014.03.20, 16:00*

In this talk, I will outline what is known about the notion of disjointness in hypercyclicity and its contrast with the dynamics of a single operator.

## On groups of zero entropy

*Kıvanç Ersoy, 2014.03.06, 16:00*

In this talk we will give some background on concepts of growth and entropy for groups and discuss the following question: Does there exist a finitely generated group with trivial center such that identity has zero entropy? We will prove some partial results and give some examples. This is a joint work with Dikran Dikranjan.

## The invariant subspace problem via compact-friendly-like operators: a survey

*Mert Çağlar, 2014.02.13, 16:00*

The celebrated result of Victor Lomonosov on the existence of non-trivial closed hyperinvariant subspaces for compact Banach space operators gave rise to the introduction of a new notion in the area of positivity, that of compact-friendliness, which motivated a huge amount of research pertaining to the invariant subspace problem for positive operators on Banach lattices. The talk aims to survey these results in connection with the ones obtained during the past three years in collaboration with Tunç Mısırlıoğlu of İstanbul Kültür University.

## Zariski Topology of an Abstract Group

*Dikran Dikranjan, 2014.01.23, 16:00*

This topology was implicitly introduced by A. Markov in 1944 and later explicitly introduced by R. Bryant, under the name verbal topology. In the last ten years a wealth of new papers appeared, where this topology is given the name Zariski topology for its striking similarity with the Zariski topology studied in algebraic geometry. Actually, one of the main direction of research on this topic (pursued by Baumslag, Myasnikov and Remeslennikov) has as principal objective the development of a counterpart of Algebraic Geometry in abstract groups.

This cycle of lectures will be dedicated to another direction, namely the one undertaken by Markov himself towards the solution of one of his problems: the existence of non-discrete Hausdroff group topologies on the infinite groups. To this end one can make use of another topology, introduced again implicitly by Markov. This topology was explicitly introduced by Dikranjan and Shakhmatov under the name Markov topology. In these terms, Markov's problem can be formulated as follows: is the Markov topology of an infinite group always non-discrete. It is easy to see that the Markov topology is finer than the Zariski topology. Markov showed that they coincide for countable groups and asked if this is always the case. The first infinite group with discrete Markov topology was built by Shelah in 1980 under the assumption of the Continuum Hypothesis. Shortly afterwards Ol'shankij gave an example of a countable group with discrete Zariski topology. Finally, some attention will be dedicated to a recently relevant progress in this line obtained by Ol'shankij and his school, in building groups with preassigned properties of the Zariski topology.

## Elementary introduction to infinite dimensional Lie algebras

*Arzu Boysal, 2013.12.27, 16:00*

This is an elementary introduction to infinite dimensional Lie algebras. We will give two constructions, and relate them to certain geometric problems.

## On growth of random groups of intermediate growth

*Mustafa Gökhan Benli, 2013.12.13, 16:00*

The growth of a finitely generated group is an asymptotic invariant
and measures the behaviour of sizes of balls in its Cayley graph. While
examples of groups with polynomial and exponential growth are
ubiquitous, the first examples of groups for which the growth is
intermediate were constructed by R. Grigorchuk in 1984. We study the
growth of typical groups from this family and find that, in the sense of
category, a generic group exhibits oscillating growth with no universal
upper bound. At the same time, from a measure-theoretic point of view
(*i.e.* almost surely relative to an appropriately chosen
probability measure), the growth function is bounded by
e^{nα} for some α<1. This is
joint work with R. Grigorchuk and Y. Vorobets.

## İlk Transandant Sayı Örnekleri, Liouville Sayıları ve Transandant Sayıların Mahler Sınıflandırması

*Gülcan Kekeç, 2013.12.05, 16:00*

Bu konuşmada, ilk olarak, J. Liouville tarafından 1844 yılında verilen ve transandant sayıların varlığının ilk ispatı olan Liouville teoreminden, ve Liouville teoreminin sonucu olarak karşımıza çıkan, ilk transandant sayı örnekleri olan Liouville sayılarından bahsedilecektir. Daha sonra, 1932 yılında, K. Mahler tarafından ortaya konan ve transandant sayıları S, T, U gibi birbirine yabancı üç sınıfa ayıran Mahler sınıflandırmasından bahsedilip, Liouville sayıları ile U-sayıları arasındaki ilişki vurgulanacaktır. (U-sınıfının elemanlarına U-sayıları denir.) Son olarak, benim Mahler'in U-sayıları ile ilgili elde etmiş olduğum bazı sonuçlar verilecektir.

## Historical achievements of topological K-theory and recent applications

*Turgur Önder, 2013.11.28, 16:00*

Topological K-theory is perhaps the most prominent example of a generalized cohomology theory, i.e it satisfies all the axioms in the axiomatic definition given by Eilenberg and Steenrod for a cohomology theory except for the dimension axiom. Among its impressive achievements in the past one can name some important and deep theorems of mathematics like Atiyah-Singer index theorem, Adams' solution of the vector field problem on spheres, solution of Hopf invariant one problem, etc. In this talk for a general audience, we will give a brief overview of the basic notions of this theory and mention some of the important applications in the past. We will also present more recent applications in geometry and topology.

## Different Approaches to Tropical Geometry and A Point-Line Incidence Structure

*M. Hakan Güntürkün, 2013.11.21, 16:00*

This will be an introductory talk on Tropical Algebraic Geometry. We will introduce tropical algebra that has two operations, max and plus. The set of extended real numbers with these operations forms a semifield which is called “Tropical Semifield”. We will show how to sketch the tropical lines on the tropical semifield. Then we will introduce a central concept in tropical algebraic geometry, “Amoeba”. This was first mentioned in 1994 by Gelfand, Kapranov and Zelevinsky. This might be seemed as a bridge between classical algebraic geometry and tropical geometry. Sometimes it is helpful to use the tropical counterparts of the algebraic varieties since we can use combinatorics extensively on these simpler objects. We will sketch the tropical line by using amoebas. Then we will pass to a field theoretic notion, Puiseux Series to explain the tropical varieties. We will give the essential definitions and sketch the tropical line by using Puiseux series. Then if time permits, we will try to explain higher dimensional tropical linear varieties. We will also give the relation with the tropical varieties and the analytification of a variety in sense of Berkovich.

Then we will give a point-line incidence structure called k-nets. We will give some open problems related to these Sylvester-Gallai type structures. We will explain how to prove the nonexistence of one case by using tropical algebraic geometry. We will tropicalize the lines and the points and get a contradiction to the existence of the structure.

## Semilinear vs semialgebraic groups

*Pantelis E. Eleftheriou, 2013.11.14, 16:00*

Let *M* be an ordered vector space over an ordered division ring
*D*. A subset *X* of *M*^{n} is called
**semilinear** if it is a boolean combination of sets
defined by linear equations and inequalities with coefficients from
*D*. A **semilinear group** is a group whose domain
and the graph of its multiplication are semilinear sets. We prove that
every semilinear group is semilinearly isomorphic to a quotient by a
lattice, exemplifying a strong connection to real Lie groups. We then
investigate analogous theorems for semialgebraic groups, as well as for
other groups definable in o-minimal structures.

(*Konuşmacı için sağladıkları konaklama desteğinden dolayı İstanbul
Matematiksel Bilimler Merkezi'ne teşekkür ederiz.*)

## Geometry as made rigorous by Euclid and Descartes

*David Pierce (Mimar Sinan Güzel Sanatlar Üniversitesi),
2013.10.31, 16:00*

For Immanuel Kant (born 1724), the discovery of mathematical proof by Thales of Miletus (born around 624 BCE) is a revolution in human thought. Modern textbooks of analytic geometry often seem to represent a return to prerevolutionary times. The counterrevolution is attributed to René Descartes (born 1596). But Descartes understands ancient Greek geometry and adds to it. He makes algebra rigorous by interpreting its operations geometrically.

The definition of the real numbers by Richard Dedekind (born 1831) makes a rigorous converse possible. David Hilbert (born 1862) spells it out: geometry can be interpreted in the ordered field of real numbers, and even in certain countable ordered fields.

In modern textbooks, these ideas are often entangled, making the
notion of proof practically meaningless.
I propose to disentangle the ideas by means of Book I of Euclid's
*Elements* and Descartes's *Geometry.*

## Dengelenmiş Ağlar

*Burak Yıldıran Stodolsky (İstanbul Teknik Üniversitesi),
2013.10.25 (Cuma), 16:00*

İki insan veya topluluk arasındaki ilişkileri sadece sevgi veya nefret olarak sınıflandırılım. Bu halde sosyologlar insan grupları arasında hangi ilişkilerin dengeli (değişmesi zor) oldugunu bize söylüyorlar. Amacımız herhangi bir insan grubu arasındaki ilişkileri graf teorik bir ağ olarak görüp, bu ağı hızlı bir şekilde nasıl dengeleyeceğimizi bulmak olacak. Bu soruyla ilgili graf teorik bir yaklaşımı inceleyeceğiz.

## Cebirsel ve Diferansiyel Topolojinin İktisattaki Uygulamaları

*Burak Ünveren (Yıldız Teknik Üniversitesi, İktisat Bölümü),
2013.10.10, 16:00*

Birbiriyle etkileşim halinde olan çok sayıda bireyden oluşan bir toplumda her bireyin öznel hedefine ulaşmak için elinden gelenin en iyisini yaptığı varsayılsın. Ancak genel olarak (hem matematiksel kurguda hem de gerçek hayatta) bir birey için en akılcı davranışın ne olacağı, dolayısıyla nasıl davranacağı, başka bireylerin aldığı kararlara bağlıdır. Bireysel anlamda tüm bireylerin aldığı en akılcı kararların birbiriyle tutarlı ve uyumlu olmasına denge durumu denir. Dengenin varlığı, sayısı ve dışsal koşullara sürekli ilintili olması gibi çok önemli birçok konunun cevabı cebirsel ve diferansiyel topolojinin araçları ile elde edilmiştir. İktisatın sosyal olayları incelemedeki en önemli ve belki de yegane aracı denge kavramıdır ve bu alandaki en kayda değer gelişmeler sunulacaktır.

## Geometrik Topoloji Nedir?

*Barış Coşkunüzer (Koç Üniversitesi), 2013.10.03, 16:00*

Bu konuşmada, matematiğin geometrik topoloji alanındaki temel yapı ve soruları tanıtıp, ardından Perelman'ın çözdüğü bir milyon dolar ödüllü Poincaré Sanısı'ndan bahsedeceğiz.

## Exotic spheres

*Burak Özbağcı (Koç Üniversitesi), 2013.09.26, 16:00*

In this general audience talk I will outline what is known about exotic spheres.

## Harmonic functions and large deviations for random walk in a random environment

*Atilla Yılmaz (Boğaziçi Üniversitesi), 2013.01.04, 16:00*

Consider random walk in any finite dimension. It has a typical average velocity. Conditioning it to have some other velocity corresponds to tilting its transition kernel with some exponential term. This is formalized in the framework of the theory of large deviations which deals with exponential rates of decay of probabilities of rare events. The same connection holds also in the context of random walk in a random environment (RWRE), except that it is not enough to just tilt the (now random) transition kernel by a deterministic exponential term. One has to perform an additional change of measure involving a harmonic function for the tilted (random) transition kernel. This talk will not assume any prior knowledge of large deviations or RWRE.

## Disjoint Dynamics of Linear Operators

*Özgür Martin (Miami University, Ohio), 2013.01.04, 14:30*

Linear Dynamics is a young and rapidly evolving branch of functional analysis. It is mainly concerned with the behavior of iterations of linear transformations. A continuous linear transformation T on a topological vector space X is said to be hypercyclic provided there is some vector in X with a dense orbit. Having dense orbits is one of the main ingredients in the most widely known definitions of chaos.

The aim of this talk is to compare hypercyclicity with the notion of disjointness introduced by Bernal-Gonzalez and independently by Bes and Peris. It turns out that some well known results about a single hypercyclic operator fail to hold true for disjoint hypercyclic operators.

## Arithmetic Nullstellensatz and Nonstandard Methods

*Haydar Göral (Université Claude Bernard Lyon-1), 2012.12.28, 16:00*

In this talk we find height bounds for polynomial rings over integral domains. We apply nonstandard methods and hence our constants will be ineffective. Furthermore we consider unique factorization domains and possible bounds for valuation rings and arithmetical functions.

## Two problems on cyclic codes and their relation to curves over finite fields

*Cem Güneri (Sabancı Üniversitesi), 2012.12.21, 16:00*

We will start by introducing cyclic codes and describing the classical connection between the weights of cyclic codes and families of Artin-Schreier type curves over finite fields. Then we will present two problems about cyclic codes and our results related to them. One of these results was obtained with Ferruh Özbudak and the second one was very recently obtained together with Gary McGuire. Along the way, we will provide the necessary definitions and important results on curves over finite fields. So, it is hoped that this will be an accessible talk for a general audience.

## Invariants of Legendrian Knots

*Sinem Çelik Onaran (Hacettepe Üniversitesi), 2012.12.14, 16:00*

After a brief introduction on contact 3-manifolds and knots in contact 3-manifolds, we will focus on a class of knots called Legendrian knots. We will review known invariants for these knot types and we will define new invariants for Legendrian knots. We will discuss the applications and related open problems related to the new invariants.

## Heun Functions and Their Uses in Physics

*Mahmut Hortaçsu (MSGSÜ Fizik Bölümü), 2012.11.30, 16:00*

Most of the theoretical physics known today is described by using a small number of differential equations. If we study only linear systems, different forms of the hypergeometric or the confluent hypergeometric equations often suffice to describe this problem. These equations have power series solutions with simple relations between consecutive coefficients and can be generally represented in terms of simple integral transforms. If the problem is nonlinear, one often uses one form of the Painlevé equation. There are important examples, however, where one has to use more complicated equations. An example often encountered in quantum mechanics is the hydrogen atom in an external electric field, the Stark effect. One often bypasses this difficulty by studying this problem using perturbation methods. If one studies certain problems in astronomy or general relativity, encounter with the Heun equation is inevitable. This is a general equation whose special forms take names as Mathieu, Lamé and Coulomb spheroidal equations. Here the coefficients in a power series expansions do not have two way recursion relations. We have a relation at least between three or four different coefficients. A simple integral transform solution also is not obtainable. Here I will try to introduce this equation whose popularity increased recently, mostly among theoretical physicists, and give some examples where the result can be expressed in terms of solutions of this equation.

## Variations on a theme by Aldama

*Cédric Milliet (Galatasaray Üniversitesi) , 2012.11.23, 16:00*

Let G be a group, and f(x,y) a formula in the language of groups. We
say that f(x,y) has the independence property if for all natural number
n, we can find elements a_{0}, a_{1}, …,
a_{n} and (b_{I}) _{I ⊆ n+1} in G such
that f(a_{i},b_{I}) holds if and only if i ∈ I. The
group G is said to be ‘without the independence property’ if
no formula has the independence property. Such groups are gaining much
attention nowadays. We shall review recent results of Shelah and Aldama
concerning infinite abelian and nilpotent subgroups of a group G without
the independence property, and address the following open question: is a
soluble subgroup S of G contained in a soluble definable subgroup of G?

## Real Analytic Gradient Systems and a Robot Navigation Problem

*Ferit Öztürk (Boğaziçi Üniversitesi), 2012.11.16, 16:00*

Dynamical systems for real analytic gradient vector fields have many features surprisingly similar to dynamical systems corresponding to Morse functions. We will exhibit some of these features, mention Thom's conjecture in this context and its solution. We will relate this discussion to a robot navigation problem about a number of circular robots in a 2-dimensional disk, each with a goal position and a unique one is allowed to move at a time. This is a joint work with Işıl Bozma.

## Constructing Quantum Groups

*Özgür Kişisel (Orta Doğu Teknik Üniversitesi, Kuzey Kıbrıs
Kampusu), 2012.11.02, 16:00*

Quantum groups are the group objects of noncommutative geometry. They are known to have applications in diverse areas of mathematics such as knot theory, representation theory and mathematical physics. The two main techniques for constructing quantum groups use deformations and bicrossed products. We will attempt to propose another method for constructing quantum groups and give some examples.

## Oracles and Revelations

*Alexandre Borovik (University of Manchester), 2012.11.02, 14:30*

This is joint work with Şükrü Yalçınkaya. The talk will discuss some approaches to a systematic development of black box algebra. The talk will be at an elementary level.

## On a class of nonlinear Schrödinger equations

*Burak Gürel (Boğaziçi Üniversitesi), 2012.10.19, 16:00*

In this expository talk we will introduce the so called almost cubic nonlinear Schrödinger equation and present some of the results obtained so far. These are related to global existence, finite time blow up, existence of solutions of certain types and integrability. Most of the results depend upon the function space we work in. Thus, we shall first discuss relevant spaces, albeit briefly.

## Polynomials over finite fields: conjectures, results, open problems

*Alev Topuzoğlu (Sabancı Üniversitesi), 2012.10.12, 16:00*

We will first describe some of the well-known conjectures about polynomials over finite fields. We will then focus on the subclass of permutations, discuss some open problems and present recent work towards their solutions.

An alternative approach to the study of permutations, introduced by Aksoy, Çeşmelioğlu, Meidl and Topuzoğlu, has proven to be successful in tackling problems in diverse areas. We shall explain, for instance, a recent application in uniform distribution mod 1 of an infinite sequence of real numbers. This new approach brings together, naturally, new open problems, which we will also describe.

## Fréchet spaces of global analytic functions

*Aydın Aytuna (Sabancı Üniversitesi), 2012.09.28, 16:00*

This will be an expository talk. I will define and say a few things
about nuclear Fréchet spaces and look at some examples, paying special
attention to ‘power series spaces’. However the main actor
of the talk is *O(M)*; the space of analytic functions on a Stein
manifold *M* with the topology of uniform convergence on compact
subsets. I will refer to the question of: ‘How much and what sort
of information does *O(M)* carry about the complex analytic
structure of *M* ?’. This problem seems to be intractable
however I will have something to say in the case when *O(M)* is
(isomorphic to) a power series space. Finally (if time permits) I will
look at applications of certain functional analysis techniques in some
concrete problems of complex analysis.

## Arakelov theory and Hecke correspondences on modular curves

*Ricardo Menares (Pontificia Universidad Católica de Valparaíso),
2012.06.14, 16:30*

In the context of arithmetic surfaces, J.-B. Bost defined a
generalised Arithmetic Chow Group (ACG) using the Sobolev space
L^{2}_{1}. We study the behaviour of these groups under
pull-back and push-forward and we prove a projection formula. We use
these results to define an action of the Hecke operators on the ACG of
modular curves and to show that they are self-adjoint with respect to
the arithmetic intersection product. The decomposition of the ACG in
eigencomponents which follows allows us to define new numerical
invariants, which are refined versions of the self-intersection of the
dualizing sheaf. Using the Gross-Zagier formula and a calculation due
independently to Bost and Kühn we compute these invariants in terms of
special values of L series.

## Group automorphisms as dynamical systems

*Thomas Ward (University of East Anglia),
2012.05.02, 16:00*

This is an overview of some of the problems involved in classifying group automorphisms from the point of view of dynamical systems. Many of the questions reduce to problems in number theory.

## Öklid Algoritması, Çarklar ve Sınıf Grupoidi

*Muhammed Uludag (Galatasaray Universitesi),
2012.04.27, 14:30*

Farklı uzunlukta iki çubuk alalım. Bu çubukların uzunluklarını kıyaslamanın doğal yolu, Öklid algoritmasını uygulamaktır, sonuç olarak bir sürekli kesir elde ederiz. Öklid algoritmasını cebirsel işlemlere kodlanması, modüler grubu verir. (Özel bazı şartları taşıyan kıyaslamaların kodlaması kalandaş modüler altgrupları verir). Bu cebirsel işlemler, hiperbolik düzlemin bir dönüşüm grubu olarak, veya bu uzayın içinde oturan sonsuz bir ağacın otomorfizmleri olarak yorumlanabilir. Hiperbolik düzlemi (ağacı), modüler grubun sonsuz mertebeli tek bir elemanının ürettiği Z-altgruba bölecek olursak, halkaları (taban kenarlı çarkları) elde ederiz. (Genel olarak ağacın modüler grubun bir altgrubuna bölümü bir kurdela çizge verir). Bu Z-altgruplara aynı zamanda birer belirsiz ikili kuadratik form (bikf) tekabül eder. Z-altgrupların modüler gruptaki eşlenik sınıflarınaysa bikf'lerin bildik modüler grubu etkisi altındaki denklik sınıfları tekabül eder. Bir çarka "takla (flip)" attırarak, yeni bir çark elde edebiliriz. Soru: Nesneleri çarklar(=bikf'ler) olan ve morfizmleri taklalar tarafından üretilen "sınıf grupoid"inin temel grubu nedir?

not: Konuşma ingilizce olabilir.

## A Tate Cohomology version of the Lyndon-Hochschild-Serre Spectral Sequence

*Matteo Paganin
(Sabanci University),
2012.04.20, 15:00–17:00*

In the first part of this talk, I will sketch the ideas and definitions of Cohomology for Profinite Groups and I will point out the differences with the usual Group Cohomology. In the second part, I will introduce an extension of the Hochschild-Serre Spectral Sequence for a profinite group G and an open subgroup H that also takes into account the Tate Cohomology of the group G/H. Under certain conditions, this spectral sequence converges to zero. In the case of a finite Galois extension of local fields, with Prof. David Vauclair of the University of Caen Basse-Normandie, we proved that the results obtained provide a different interpretation of the Reciprocity Map of the Local Class Field Theory.

## On Hurwitz equivalence classes of elements of PSL(2,Z)

*Nermin Salepci
(Institut Camille Jordan),
2012.04.13, 16:00*

(Joint work with A. Degtyarev) We will discuss algebraic counterpart of the problem of classification of certain class of geometric objects and discuss some solutions of the algebraic problem.

## Cohomology of Bianchi Groups

*M. Haluk Şengün (University of Warwick), 2012.04.13, 14:00*

Bianchi groups are groups of the form SL(2,R) where R is the ring of integers of an imaginary quadratic field. They arise naturally in the study of hyperbolic 3-manifolds and of certain generalizations of the classical modular forms (called Bianchi modular forms) for which they assume the role of the classical modular group SL(2,Z). In this latter sense, the study of Bianchi groups is fundamental for developing Langlands' programme for GL(2) beyond totally real fields.

The overall goal of this talk is to give the audience an overview of some of the fundamental problems in the arithmetic aspects of the theory of Bianchi groups. After giving the necessary background, I will start with a discussion of the problem of understanding the behavior of the dimensions of the cohomology of Bianchi groups and their congruence subgroups. Next, I will focus on the amount of the torsion that one encounters in the cohomology . Finally, I will discuss the arithmetic significance of these torsion classes.

## On the non-abelian global class field theory

*K. İlhan İkeda (Yeditepe University), 2012.04.06, 14:00*~~16:00~~

Let K be a global field. The aim of this talk is to discuss the possibility of constructing the non-abelian version of global class field theory of K by "glueing" the non-abelian local class field theories of K_\nu in the sense of Koch, for each henselian prime v, following Chevalley's philosophy of ideles, and further discuss the relationship of this theory with the global reciprocity principle of Langlands.

## Non-abelian local reciprocity map

*Erol Serbest (Yeditepe University), 2012.03.30, 16:00*

In a series of papers, together with K.İ. İkeda, extending the ideas of Fesenko and Koch we have constructed the non-abelian local reciprocity map of a local field K as a certain topological isomorphism between the absolute Galois group of K and a certain topological group which depends on K and whose definition involves Fontaine-Wintenberger theory of field of norms. In this talk, we shall briefly discuss the construction of this map.

## Combinatorial vs. Analytic Methods in Integer Partions

*Kağan Kurşungöz (Sabancı University), 2012.03.23, 16:00*

We will review some classical results like Euler's partition theorem, the q-binomial theorem, or the pentagonal number theorem and compare their analytic and combinatorial proofs. Then we will look at some more recent theorems which do not yet admit purely analytic or purely combinatorial proofs. We will describe some open problems as time allows. The talk will be fairly accessible.

## Using generating functions to count graphs

*Atabey Kaygun (Bahçeşehir University), 2012.03.16, 16:00*

In this talk we will explain how a very applied problem from bio-mathematics lead us to a graph counting problem, and how we (partially) solved this problem using generating functions.

## An overview of Ulrich bundles

*Emre Coşkun (Tata Institute), 2012.03.08, 17:00*

Ulrich bundles on a smooth projective variety are those vector bundles whose pushforward to a projective linear space of the same dimension as the base space is trivial. They appear in relation to representations of hypersurfaces as determinants or Pfaffians, and representations of generalized Clifford algebras.

In this talk, we will define Ulrich bundles, explore some of their basic properties, and discuss some recent developments on the existence problem of stable Ulrich bundles on some projective surfaces with prescribed Chern classes.

## Deformations of Kolyvagin Systems

*Kazım Büyükboduk (Koç University), 2012.03.02, 16:00*

Mazur's theory of Galois deformations, inspired by Hida's earlier work on families of modular forms, has led to the resolution of many important problems in Number Theory: Wiles and Taylor/Wiles proved Taniyama-Shimura conjecture (to conclude with the proof of FLT), Buzzard/Taylor and Taylor used it to prove many cases of strong Artin conjecture as part Langlands programme. In this talk, I will first give a general outline of Mazur's abstract theory and explain how it is used to attack concrete arithmetic problems. At the end, I will talk about a recent result that Kolyvagin systems (which Mazur and Rubin prove to exist for mod p Galois representations) do often deform to a big Kolyvagin system for the "Universal Galois deformation" representation. I will touch upon important applications of this result in arithmetic.

## Schanuel Property for additive power series

*Piotr Kowalski (Bilgi University), 2012.02.24, 16:00*

I will discuss a version of Schanuel's Conjecture for a field of Laurent power series in positive characteristic replacing the field of complex numbers and a non-algebraic additive power series replacing the exponential map. The paper can be obtained from http://www.springerlink.com/content/21441308762613lq/.

## On Graph Factorizations: Hamilton Cycle Decompositions and the Hamilton-Waterloo Problem

*Sibel Özkan (GYTE), 2012.02.17, 16:00*

This lecture is in the field of graph theory and will serve as an introduction to resolvable graph decompositions; graph factorizations. Methods, applications, and different types of graph decompositions, particularly cycle decompositions, will be discussed. Hamilton cycles are particularly popular in the field, and this popularity rises from optimization problems. Main focus of the talk will be on Hamilton cycle decompositions and a relatively new method based on graph homomorphisms and edge-coloring will be introduced. Other types of interesting cycle decomposition problems, namely Oberwolfach Problem and the Hamilton-Waterloo problem, will also be introduced and new results on those problems will be discussed.

## Model theory of multiplicative subgroups of fields

*Ayhan Günaydın (Universidade de Lisboa), 2012.01.18, 16:00*

We are aiming to give an overview of a model theorist's take on certain number theoretic topics of Diophantine nature. Our approach is based on the study of ‘small’ subgroups of the multiplicative group of a field. Here, ‘small’ is quite a technical term which happens to be the model theoretic counter-part of ‘finite rank’. We present some number theoretic properties of such groups, followed by their model theoretic consequences.

## Definable subgroups

*Tuna Altınel (Université Claude Bernard Lyon-1), 2012.01.05,
16:00*

The notion of definable set is fundamental in model theory. In an algebraic structure, say a group, substructures, say subgroups, that also enjoy the property of being definable tend to be more interesting and tractable from the model theorist's viewpoint. In the case of a non definable substructure, one tries to find a definable "envelope" sharing algebraic properties with the substructure in question.

In my talk, I will go over the notion of definability and give examples. This will be followed by the presentation of a recent result, joint with Paul Baginski, on definable envelopes of nilpotent subgroups of groups with the minimal condition on centralizers (Mc-groups).

Finally, I will make a few remarks about the Fitting subgroup and the solvable radical of an arbitrary group.

## Imaginaries in fields with additional structure

*Martin Hils (Paris VII), 2011.12.21,
14:30*

An imaginary in a first order structure M is a finite tuple modulo a definable equivalence relation. In some cases (e.g. real closed or algebraically closed fields), every imaginary is interdefinable with a tuple from the structure, and M is said to eliminate imaginaries. In general, one aims to classify imaginaries up to interdefinability.

In this talk, I will give an overview on what is known about imaginaries in fields with extra structure (valuation, derivation, automorphism etc.) Using the classification of imaginaries in algebraically closed valued fields (Haskell-Hrushovski-Macpherson) together with results on higher amalgamation in valued fields of residue characteristic 0, I will indicate how imaginaries of "finite rank" may be classified in valued fields with automorphism.

## Commutators and bounded cohomology of a group

*Mustafa Korkmaz (ODTÜ), 2012.12.16*

For an element x of a group [G,G], the commutator length cl(x) of x is defined as the minimal number n such that x can be written as a product of n commutators. There is also the notion of bounded cohomology of G. After introducing these two seemingly unrelated concepts, I will discuss them for the mapping class group of a surface.

## On a theorem of I. M. Isaacs

*İsmail Güloğlu (Doğuş Üniversitesi), 2011.11.17,
16:00*

It is well known that a finite group admitting an automorphism of prime order which act fixed-point freely is nilpotent. In this talk we shall study the structure of finite groups G admitting an automorphism a of prime order p with p being coprime to |G|, so that any element of prime order which is fixed by a lies in the center of G . This is a question motivated by a result of Isaacs which deals with groups having an automorphism fixing every element of it of prime order.

## Calabi-Yau submanifolds of G_{2} manifolds

*Barış Efe, 2011.11.03, 16:00*

Akbulut and Salur suggested the study of Calabi-Yau submanifolds of
G_{2} manifolds that come from a certain process. In this talk
we will describe this process and use it to obtain several self-mirror
Calabi-Yau submanifolds of Joyce manifolds.

## Path algebras—Incidence algebras

*Müge Kanuni (Boğaziçi Ü.), 2011.10.27*

We will briefly define the two algebraic structures; path algebras and incidence algebras defined on two discrete objects; directed graphs and partially ordered sets, respectively. The interplay of these structures and further applications in literature will be surveyed in this talk.

## On explicit towers over finite fields

*Arnaldo Garcia, 2011.10.13, 16:00*

The construction of towers of function fields over finite fields has attracted much attention after Ihara showed that the so-called Hasse-Weil bound (which is equivalent to the validity of the Riemann Hypothesis in this context) was weak when the genus of the function field is much bigger than the cardinality of the finite field. Very little was known on the asymptotic behaviour of the ratios of (number of rational places) over (genus), except for finite fields of square or cubic cardinalities.

The main point of this talk is to introduce a tower with excellent asymptotic behaviour over any nonprime finite field. This is work in progress with Bassa, Beelen and Stichtenoth.

The speaker's visit to Turkey is supported by TÜBİTAK.