# METU Mathematics Seminars

Fall 2016

Previous Seminars: Spring 2016, Fall 2015, Spring 2015, Fall 2014, Spring 2014, Fall 2013, Spring 2013, Fall 2012, Spring 2012, Fall 2011, Spring 2011.

The seminars will be held in Gündüz İkeda Room at 15:40 unless otherwise stated.

October 13

Rutgers University

How difficult is it to classify the minimal homeomorphisms of the Cantor set?

In this talk, we will analyze the topological conjugacy problem for minimal
homeomorphisms of the Cantor space from the point of view of descriptive set theory. We shall
not assume any prior knowledge on the topic and the first half of the talk will be devoted to the
mathematical framework provided by descriptive set theory to analyze the relative complexity of
classification problems. In the second half of the talk, we shall focus on the topological conjugacy
relation on the space of minimal homeomorphisms of the Cantor space and show that the Friedman-Stanley
jump of the identity relation on reals is a lower bound for the Borel complexity of this relation.

October 20

METU

The Bebutov-Kakutani Representation Theorem

Famously investigated by M. V. Bebutov, there is a natural flow on
the set of real valued functions on the real line with the metric
of uniform convergence on compact sets. This flow is universal in the
sense that each flow on a compact metric space can be imbedded as its
subflow. A weaker version of this fact was first proven by V. V.
Nemytskii in a paper which remained mostly unnoticed until S. Kakutani
provided a brief and natural demonstration.

October 27

Bilkent University

Lines in quartic surfaces

November 3

İstanbul University

On extended Legendrian dualities and their applications

November 10

Marmara University

All Separable Infinite Dimensional FrechetSpaces are Homeomorphic

Abstract

November 17

METU Department of Chemical Engineering

The dynamics of the rise and fall of empires

November 24

METU Department of Physics

The right-hand side of Einstein's equation

It is well-known that Einstein's equation has two sides.
About these two sides, Einstein wrote "it is similar to a building, one wing of which is made of fine marble(left part of the equation),
but the other wing of which is built of low grade wood (right side of equation).
What he called "marble" is the geometry side and "wood" the matter side. The wood part is really still problematic.
I will explain this and also talk about the recent observation of binary black hole mergers and the sound they produced.

December 1

Bilkent University

Orthogonal polynomials for continuous singular measures

First we recall some results about asymptotic behaviour
of orthogonal polynomials for finite gap sets and then discuss
the case of continuous singular measures.

December 8

Atılım University

Disconjugacy via Lyapunov and Vallee-Poussin type inequalities for forced differential equations

In this talk, in the case of oscillatory potentials, we present some
new Lyapunov and Vall´ee-Poussin type inequalities for second-order forced
differential equations. No sign restriction is imposed on the forcing term. The
obtained inequalities generalize and complement the existing results in the
literature.
This is joint work with Ravi P. Agarwal.

December 15

METU

Intermediate Growth in Finitely Presented Algebras

Let A be an (not necessarily associative) algebra over a field k
generated by a finite set X. The growth function of A with respect to
X is defined as the dimension of the subspace of A spanned by all
monomials on X of length at most n. The asymptotic behavior of this
function does not depend on the generating set X and it is called the
growth of A. The growth rate is a widely studied invariant for
finitely generated algebraic structures such as algebras, groups,
semigroups. This talk will be a survey of history, open problems and
some new results around this notion. The main focus will be on
finitely presented algebras whose growth is intermediate between
polynomial and exponential.

December 22

Doguş University

Title

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 group F of
fixed points of A is an STI-subgroup of G, then F is solvable if and only if G is solvable.

December 29

Bilkent University Department of Physics

Topological Effects in Non-relativistic Quantum Mechanics:An overview of the 2016 Nobel Prize

The 2016 Nobel Prize in Physics has been awarded to J. Michael Kosterlitz, Duncan Haldane, and David J. Thouless, for
“theoretical discoveries of topological phase transitions and topological phases of matter”.
In this talk I will try to explain why Topology, a branch of geometry which studies properties which do not change with continuous changes in size or shape, becomes important in determining the physical properties of a system and phase transitions between different states of matter. Without presenting any
mathematical details, I will motivate the order parameter concept and its connection to topological defects. I will try to review some of the concepts regarding Kosterlitz Thouless phase transitions, Thouless charge pumps, Thouless-Kohmoto-Nightingale-den Nijs conductance formula, Haldane gap in
spin chains and the Haldane model. Finally I would like to review current status of the field, including our recent work[1].
The talk is aimed at a broad audience with minimal background in mathematics and physics, and should be accessible to undergraduate students.
[1]F. Nur Ünal, Erich J. Mueller, and M. Ö. Oktel PHYSICAL REVIEW A 94, 053604 (2016) “Nonequilibrium fractional Hall response after a topological quench.”

January 5

Galatasaray University

Arithmetic of the étale co-site of the modular curve

The central question of studying the absolute Galois group (together with the theorem of Belyi) has been the key
ingredients of (so-called) geometric Galois actions which was outlined in a research proposal entitled "Esquisse d’un Program” by Grothendieck.
This theory, although focusses on the étale site of the modular curve, can be used for "non-étale" covers, too. We will exemplify related
arithmetic questions by discussing covers that arise from annuli, which apparently were quite well-known to Gauss! If time permits, we explain
a generalisation that arises by replacing the modular group by Hecke groups. This is joint work with M.Uludag.
This research is funded by TUBITAK 1001 grant 114R073.