METU Mathematics Seminars
Spring 2012

February 23

Koc University

Kazim Buyukboduk from the Department of Mathematics at Koc University is rewarded to the Ikeda Research Award 2011. The ceremony will be at 15:30.

Deformations of Kolyvagin systems

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 of 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.

February 27 (Monday)
Room M203 at 10:30

Instituto Nacional de Matemática Pura e Aplicada (IMPA)

Arnaldo Garcia is visiting METU as a guest of the METU SIAM Student Chapter. The speaker's visit to Turkey is supported by Tübitak.

On Algebraic Curves Over Finite Fields

Bounding the number of points, with coordinates in a finite field, on algebraic curves has attracted much attention, especially after the discovery of V.D. Goppa of good linear codes from such algebraic curves. We are going to survey on Curves with Many Points, specially the so-called maximal curves; i.e., the ones attaining Hasse-Weil upper bound (equivalent to the validity of Riemann Hypothesis in this context).

March 1

University of Waterloo

Efficient Arithmetic and Compact Representations in Finite Fields

The efficiency of many cryptographic protocols relies on fast arithmetic and compact representation in the underlying group. I will present my recent work on obtaining compressed representations for finite field elements and speeding up certain finite field operations. I will compare these techniques with the previously-known methods, discuss some applications to pairing-based cryptography, and conclude by stating some open problems and directions for future research.

March 8

Middle East Technical University

Dehn twists, commutators, and bounded cohomology

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 the relation between them for the mapping class group of a surface.

March 15

Bilkent University

Carmichael Meets Chebotarev

We show the existence of infinitely many Carmichael numbers whose prime factors behave in a specified manner when factored over a given number field. This result implies that for every natural number $n$ there are infinitely many Carmichael numbers of the form $a^2+ n b^2$.

March 29

Middle East Technical University

Strongly normal cones

Strong normality of cones in ordered topological vector spaces is discussed. Several results on positive operators in Banach spaces with strong normal positive cones are presented.

April 5

Middle East Technical University
Physics Department

On some old and new relations between mathematics and physics

 

April 12

Ahmet Irfan Seven

Middle East Technical University

Associahedra and Grassmannians

 

April 19

Institut Galilée Université PARIS 13

Local equivalences between finite Lie groups

Abstract

April 26

Bilkent University

Entanglement and Classical Realism

May 3

Middle East Technical University

Hyperbolicity of Geodesic Flows

Each Riemannian manifold hosts a natural flow in its unit tangent bundle, the so called geodesic flow. A celebrated theorem of D. V. Anosov vouchsafes that the geodesic flow of a Riemannian manifold of everywhere negative sectional curvature is hyperbolic, a property that heralds complicated dynamical behaviour on compact manifolds. Of this phenomenon, I will elaborate the well-known instance of the Poincare half-plane and try to summarize some recent work of mine, if time permits. The exact conditions under which the geodesic flow of a Riemannian manifold is hyperbolic are unknown.

May 10

Bilkent University

Group Categories and Their Blocks

A group category has finite groups as objects, linear combinations of permutation bisets as morphisms. The direct sum of the hom-sets is an algebra, whose modules, called group functors, arise frequently in representation theory and in the study of group actions in topology. When the structures are based only on induction and restriction maps, the category of group functors is semisimple in characteristic zero. But, in many contexts of application, inflation maps are present too. We shall conclude with a discussion of some joint work with Robert Boltje concerning blocks of categories of group functors in such cases.

May 17

McMaster University

Symmetry in Geometric Topology

This will be a survey talk about connections between the topology of manifolds and the structure of their finite symmetry groups. I will describe some work on this theme, both old and new, and present some open problems for further research.

May 24

Sabanci University

Elliptic Curves, Drinfeld Modules and Curves over Finite Fields

In this talk I will start by introducing Elliptic Curves and their characteristic p analogues, Drinfeld Modules. I will talk about modular curves and varieties, which parametrize these objects and show how they can be used in the construction of curves over finite fields with many rational points.