METU Mathematics Seminars
Spring 2015

February 26 - İkeda Research Award

Boğaziçi University

Solutions of equations over finite fields - what we know about them and why they are essential even for people not interested in finite fields

Boğaziçi University

Axiomatic and functorial representation theory of finite groups

March 5

Eastern Mediterranean University

A topological proof of a version of Artin's induction theorem

March 19

Middle East Technical University
Physics Departmant

Ricci and some other geometric flows in physics.

Ricci-flow, which was used by Perelman to prove Thurston's geometrization conjecture and Poincare's conjecture, arises in low energy quantum gravity. We shall discuss this parabolic extensions of Einstein's equations and some related geometric flows and their possible applications in quantum gravity.

March 26

Bilecik Şeyh Edebali University

Some problems on half-integral weight modular forms

Modular forms have attracted attention for many years. Recently, one of the breakthrough and very significant results in pure mathematics is the proof of the Sato-Tate conjecture for non-CM modular eigenforms (even for Hilbert eigenforms) by Taylor, Barnet-Lamb, Geraghty and Harris. A special case of the Sato-Tate theorem states that signs of coefficients of integral weight Hecke eigenforms are equidistributed. That such should also be the case for half-integral weight forms was conjectured by Kohnen and Bruinier. We will explain how the Shimura lift and the Sato-Tate theorem can be exploited to obtain sign equidistribution for certain subsets of the coefficients of half-integral weight eigenforms. We will obtain the results in different notions of density. Furthermore, an explicit error bound for the Sato-Tate conjecture in the CM-case will be also given. Finally, if time permits, then we will conclude with the current position of the Bruinier-Kohnen sign equidistribution conjecture and giving some open problems in theory of automorphic forms. This is a joint work with G. Wiese (Luxembourg) and Sara Arias-de-Reyna (Luxembourg).

April 2

Tulane University

Equivariant K-theory of smooth spherical varieties

In this talk we present our work on equivariant K-theory of spherical varieties. After explaining our general result, a description of the equivariant K-rings for smooth complete spherical varieties, we present its applications to the wonderful compactifications of minimal rank symmetric varieties. In particular, we will work out our theorem in the case of complete collineations. This is joint work with Soumya Banerjee (Ben Gurion Univ.).

April 9

Koç University

Differential geometric structures in cosmology

Space-time is a 4-dimensional differentiable manifold equipped with a non-degenerate Lorentzian metric and a metric compatible but not necessarily torsion-free linear connection. General Relativity as a relativistic theory of gravity is given through Einstein’s field equations, perhaps with a cosmological term. The distribution of matter in the universe at large is usually modelled by an ideal fluid with a conserved energy-momentum tensor that occurs on the right hand side of the field equations, I will talk about maximally symmetric spaces or space-times in a cosmological context. Various cosmological scenarios and their dark components will be discussed. Finally I will say a few words on why and how General Relativity needs to be modified.

April 16

Tübitak Bilgem

Complex multiplication and its applications

The theory of complex multiplication (CM-theory) states that the singular values of suitable analytic functions are algebraic numbers with very special arithmetic properties, and the arguments of the analytic functions are essentially given as invertible ideals of orders in CM fields. A central aspect of algorithmic CM-theory is the interplay between numerical and symbolic computations, since it describes discrete objects by analytic means. With the emergence of computer algebra and the need for explicit calculations for numerous applications in different areas (e.g. algorithmic number theory, construction of algebraic curves over finite fields of low genus, primality proving and cryptography), CM-theory has become deliberately the subject to algorithmic investigations with a special view towards these applications. Among the applications are of particular interest primality proving, explicit class field theory, and cryptography.

In this talk, we give a general overview of CM-theory together with its applications, some new results and open problems.

April 21 - Tuesday, Cahit Arf Auditorium (Arf Lecture)

Institute of Advanced Study

Univalent foundations of mathematics

Today's mathematics is using foundations which have been developed in the late 19th - early 20th century. Since these foundations had been completed mathematics has grown from a field advanced by a few outstanding minds to a large enterprise involving tens of thousands of mathematicians. On this new scale of existence it is becoming impossible for mathematics to rely entirely on the old peer-review model of ensuring the correctness of the results. Univalent Foundations are new foundations of mathematics invented with the intention of being usable for the development of computer proof assistants that will facilitate the crafting and verification of complex mathematical constructions.

April 30

Hitit University

Information geometry: Why and how to apply differential geometry to statistical models?

One may describe information geometry as applying the technques of differential geometry to the probability theory. In order to do this probability distributions of a statistical model are considered to be the points of a Riemannian manifold. In this talk, I will try to explain why differential geometry is useful for statistics. Then, the concepts of the Fisher information metric and dually flat connections will be introduced. Furthermore, the close relationship between affine differential geometry and information geometry will be mentioned.

Note that, this talk is a part of the TUBITAK 1001 Project 113F296 called "Harmonic maps, affine manifolds and their applications to information geometry".

May 7

Middle East Technical University

Semi-theoretical Solution of Outer Neumann Problem Coupled with Inner Dirichlet Problem in a Circular Region

A theoretical solution is obtained to an outher Neumann problem coupled through boundary conditions with an inner Dirichlet problem in a circular region. Exterior Neumann problem is defined with Laplace equation and interior Dirichlet problem is defined with coupled advection-diffusion equations. Unique solution of advection-diffusion problem is obtained inside the region by using fundamental solutions and Divergence theorem. Exterior solution is given with Poisson's integral formula with an additive constant which is obtained from the solvability condition. The two solutions are also coupled with the boundary coditions on the circle. The theoretical part of the solution results in solving a system of coupled integral equations on the boundary. The collocation method is made use of to obtain a linear system of equations for the discretized solution of the problem.

May 14

Trinity College Dublin

On non commutative Taylor invertibility

May 28

Universal groups and large groups