Spring/ Summer 2022

Room: SCI 103

The seminar is hybrid in-person and via Zoom. If you would like to attend the seminar via Zoom, please send an email to Asgar Jamneshan at ajamneshan@ku.edu.tr for the link.

April 19, 2022, 14:30-15:40

Tobias Fritz  (University of Innsbruck, Austria)

Categorical probability and the de Finetti theorem

Probability theory and statistics are usually developed based on Kolmogorov’s axioms of probability space as a foundation. In this talk, I will present an alternative foundation that is currently under development, and I will argue that it improves over the traditional one in various respects. This approach is formulated in terms of category theory, and it makes Markov kernels instead of probability spaces into the fundamental primitives. Its abstract nature also implies that no measure theory is needed. After introducing the approach itself, I will summarize our proof of the de Finetti theorem in terms of it.

April 26, 2022, 14:30-15:40

Alp Eren Yilmaz (Koç University)

Steiner triple systems

A Steiner triple system or order v; STS(v); is a pair (S; T) where S is a v element set and T is a set of 3 element subsets of S (called block) with the property that each pair of elements from S is contained in exactly one block. Steiner triple systems are widely used as experimental designs and are closely related to many other mathematical structures like error-correcting codes, graph decompositions, and affine and projective planes. In this talk, we will give the necessary and sufficient conditions for the existence ofSteiner triple systems. Kirkman proved in 1847 that Steiner triple system exists if and only if v ≡ 1 or 3 (mod 6). In this talk,  we will present the proofs given by Bose in 1939 for the case v ≡ 3 (mod 6) and by Skolem in 1958 for the case v ≡ 1 (mod 6).

May 10, 2022, 14:30 – 15:40

Celal Umut Yaran (Koç University)

Weil Reciprocity on Compact Riemann Surfaces

The Weil reciprocity law is a result of Andre Weil for algebraic curves over algebraically closed
fields. We can generalize this result on compact Riemann surfaces by using Tate symbols. In this
talk, for each pair of meromorphic functions f and g on a Riemann surface X, we will associate a
complex line bundle (f, g) with connection. Then we will built a correspondence between the
holonomy effect on these complex line bundles and associated moderate symbols on (f, g). After
introducing the prerequisites and the statement of the law, we will summarize a geometric proof
for Weil reciprocity law on compact Riemann surfaces which depends on this correspondence and
the principal construction ”le symbole modere” due to P. Deligne.

May 17, 2022, 14:30 – 15:40

Nihan Tanısalı (Koç University)

Szemerédi’s theorem has been strengthened in many different directions. In this presentation, we will consider a version of it on a finite vector space. We  will  use some of the important tools in Additive combinatorics, such as discrete Fourier analysis and Gowers norms. We will explain their connections, and discuss how they are used to prove the version.

May 24, 2022, 14:30 – 15:40

Tolga Temiz (Koç University)

Exotica Problem for 4-manifolds with boundary via Knot Traces

Some difficult problems for closed smooth 4-manifolds become rather tractable when posed about smooth 4-manifolds with boundary. One such problem is the problem of exotica, and in this talk, I will talk about a specific type of 4-manifolds called knot traces and a related result regarding the exotica problem together with a sketch of its proof.

June 21, 2022, 14:30 – 15:40

Gregory Seregin (Oxford University)

Axisymmetric Solutions to the Navier-Stokes Equations

I’ll discuss recent regularity results for axisymmetric solutions to the Navier-Stokes equations.Among them are slightly supercritical assumptions providing smoothness, which, for example, exclude Type I blowups of axisymmetric solutions.

June 28, 2022, 14:30 – 15:40

Sebastián Donoso (University of Chile)

Multiplicative actions and applications

In this talk, I will discuss recurrence problems for actions of the multiplicative semigroup of integers.  Answers to these problems have consequences in number theory and combinatorics, such as understanding whether Pythagorean trios are partition regular. I will present in general terms the questions, strategies from dynamics to address them and mention some recent results we obtained. This is joint work with Anh Le, Joel Moreira, and Wenbo Sun.

July 26, 2022, 14:30 – 15:40

Henrik Kreidler (University of Wuppertal)

Pure Koopmanism: An operator theoretic approach to dynamical systems

Dynamical systems play a key role in modern mathematics and
come in many variations, e.g., in the form of continuous
transformations on topological spaces or measure-preserving maps on
probability spaces. In this talk we discuss a simple, but effective
operator theoretic approach to such systems known as the Koopman
linearization.

Fall 2021-22

 October 19, 2021, 14:30-15:40 Asgar Jamneshan  (Koç University) Extensions of the ergodic Roth and Szemeredi theorems and aspects of uniformity I will give an introduction to ergodic Ramsey theory and then present some recent results. October 26, 2021, 14:30-15:40 Türkü Özlüm Çelik  (Boğaziçi University) Integrable systems from computational algebro-geometric perspective The Kadomtsev-Petviashvili (KP) hierarchy is a universal one among all integrable systems. This talk aims to make an excursion into the study of KP hierarchy with a view through computational algebraic geometry. The emphasis will be on exploiting modern tools in symbolic, numerical and combinatorial algebraic geometry to investigate solutions of the differential equations. November 02, 2021, 17:30 – 18:40 Pieter Spaas (UCLA) Group actions and equivalence relations Group actions of a countable group on a measure space give rise to interesting dynamical systems, and studying the associated orbit equivalence relations has been the subject of a lot of study over many years. In this talk we will consider some natural constructions and discuss classification results for such equivalence relations. In particular, we will study how the behavior of certain central sequences implies structural results on the involved equivalence relations and their products. All necessary notions will be introduced during the talk. This is partially based on joint work with Adrian Ioana. November 09, 2021, 14:30-15:40 Haydar Göral (İzmir Institute of Technology) Classification of Quadratic Number Fields Using Diophantine Equations In this talk, we consider some Diophantine equations over number fields. Recall that a number field is a finite field extension of the field of rational numbers. We show that the finiteness of the set of all solutions of certain Diophantine equations determines quadratic number fields among all number fields. Along the way, we explain our motivation, we give many examples and relate the results to nonstandard analysis and arithmetic geometry. November 23, 2021, 14:30-15:40 Or Shalom (Einstein Institute of Mathematics) A structure theorem for Gowers-Host-Kra seminorms for non-finitely generated countable abelian groups of unbounded torsion Furstenberg’s famous proof of Szemeredi’s theorem leads to a natural question about the convergence and limit of some multiple ergodic averages. In the case of $\mathbb{Z}$-actions these averages were studied by Host-Kra and Ziegler. They show that the limiting behavior of such multiple ergodic average is determined on a certain factor that can be given the structure of an inverse limit of nilsystems (i.e. rotations on a nilmanifold). This structure result can be generalized to $\mathbb{Z}^d$ actions (where the average is taken over a Folner sequence), but the non-finitely generated case is still open. The only progress prior to our work is due to Bergelson Tao and Ziegler, who studied actions of the infinite direct sum $\mathbb{Z}/p\mathbb{Z}$. In our work we generalize this further to the case where the sum is taken over different primes (the most interesting case is when the multiset of primes is unbounded). We will explain how this case is significantly different from the work of Bergelson Tao and Ziegler by describing a new phenomenon that only happens in these settings. Moreover, we will discuss a generalized version of nilsystems that plays a role in our work and some corollaries. If time allows we will also discuss the group actions of other abelian groups. November 30, 2021, 14:30-15:40 Özge Ülkem (Galatasaray University) Generalized D-elliptic sheaves and their moduli space Elliptic curves play a fundamental role in algebraic number theory. In the 1970’s Drinfeld defined analogues of elliptic curves in the function field setting, which are now called Drinfeld modules. Later on he defined a categorically equivalent notion, called elliptic sheaves, and studied their moduli space to prove Langlands correspondence. Since then many generalizations of Drinfeld modules and elliptic sheaves have been worked out. In the first part of this talk we will form the function field and classical setting and discuss similarities between them. Then, we define Drinfeld modules, discuss the analogy between elliptic curves and Drinfeld modules. In the second part we will define a new generalization of elliptic sheaves, called generalized D-elliptic sheaves and talk on their moduli space and of the uniformization of the latter if time permits. December 07, 2021, 16:00-17:10 Özlem Ejder (Boğaziçi University) Isolated Points on Modular Curves One of the oldest areas of mathematics is the study of integer or rational solutions to polynomial equations with integer coefficients and it remains active till today. The most natural question we can ask about such an equation is whether its set of rational solutions is finite or infinite. This can be determined by the genus of the curve defined by such equations. In particular, if the genus is greater than one, there are finitely many rational points on a curve.  What happens when one allows for solutions involving  square-roots of integers or cubic roots? Perhaps in general all complex numbers that are roots of a degree d polynomial? We call such solutions of degree 2,3 or d in general. In this talk, we will discuss when a curve has infinitely many degree d points focusing particularly on points on modular curves. December 14, 2021, 14:30-15:40 Luka Milićević (Mathematical Institute of the Serbian Academy of Sciences and Arts) Bilinear Bogolyubov Argument in Abelian Groups Bilinear Bogolyubov argument states that if we start with a dense set A of a product U x V of finite vector spaces and carry out sufficiently many steps where we replace every row or every column of A by the set difference of it with itself, then inside the resulting set we obtain a bilinear variety of codimension bounded in terms of the density of A. In this talk, we focus on a generalization of this theorem to arbitrary finite abelian groups. Along the way we also discuss analogues of a few linear-algebraic facts as well as quasirandomness of analogues of bilinear varieties in the setting of finite abelian groups. December 21, 2021, 14:30-15:40 Faruk Temur (İzmir Institute of Technology) Discrete fractional integrals, lattice points on thin arcs, and diophantine approximation  The study of discrete fractional integral operators began with an article ofArkhipov and Oskolkov on boundedness of certain multipliers, and for over thirty years concentrated on cases with translation invariant or quasitranslation invariant phase polynomial, as these cases are amenable to the Fourier transform and the Hardy-Littlewood circle method. Recently in joint work with E. Sert, we introduced methods from arithmetic to study discrete fractional integral operators along quadratic bivariate polynomials in their full generality. This is achieved by combining the information regarding distribution of lattice points on conics gleaned via classical theory of binary quadratic forms and sieving with very careful partitioning of sums under question. This effort as a side benefit yields new results on various conjectures on concentration of lattice points on conics and makes connections to diophantine approximation.  In this talk we will give a brief summary of these results on fractional integrals and lattice points, together with their mathematical context and main ideas of their proofs. December 28, 2021, 14:30-15:40 Oğuz Şavk (Boğaziçi University) Homology 3-spheres bounding contractible 4-manifolds and homology 4-balls A central problem in low-dimensional topology asks which homology 3-spheres bound contractible 4-manifolds and homology 4-balls. In this talk, we address this problem for plumbed 3-manifolds and we present the classical and new results. Our approach is based on Mazur’s famous argument and its generalization, they together provide a unification of all recognized results. January 04, 2022,16:00-17:10 Önder Türk (Institute of Applied Mathematics at METU) Modal analysis of linear elasticity equations in the incompressiblelimit A modal analysis approach for approximating the linear elasticity equations with the incompressibility constraint will be presented. The spectrum of the linear elasticity operator is approximated using an optimally convergent stabilized finite element method that is based on a variational multiscale approach. The convergence of the approximate solutions to the true ones in appropriate norms is proved both theoretically and numerically.