# METU Mathematics Seminars

Spring 2018

Previous Seminars: Fall 2017, Spring 2017, Fall 2016, 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.

February 22

Bilkent University

Why cannot you untie a knot or unlink a link?

I will give a gentle introduction to the classical theory of knots
and links in the 3-dimensional space. I will explain why a trivial
knot can always be untied algorithmically but, nevertheless, this
algorithm can never conclude that a given knot is nontrivial.
Therefore, to distinguish knots and links, one needs invariants,
i.e., certain quantities that, on the one hand, are known to take
the same value on equivalent knots and, on the other hand, are easy
to compare. I will describe a naive diagram based approach to the
construction of link invariants and illustrate this approach on a
few examples, starting from the classical signature and Alexander
polynomial and culminating in the new invariant, called slope,
discovered recently in collaboration with Vincent Florens and Ana
G. Lecuona. If time permits, I will also discuss briefly the basic
properties of slopes and their relation to the Alexander polynomial
and signature.

March 1

Boğaziçi University

Revisiting torsion and curvature

Modern differential geometry is based on two fundamental theories. The
first emerged around 1850 with the revolutionary ideas of Bernhard Riemann
and later evolved into what is now called Riemannian geometry. The second
emerged around 1870 with the revolutionary ideas of Sophus Lie and is now
called Lie theory. In this talk, we will explain that the concept of covariant
differentiation, in particular the concepts of torsion and curvature, which originated
in Riemannian geometry, emerge also from Lie theory, but with quite
different geometric interpretations of torsion and curvature. If time permits,
we will indicate some surprising consequences of this new derivation.

This talk is a summary of Part I of the book "An Alternative Approach to
Lie Groups and Geometric Structures" to be published by OUP in July.

March 8

Bilkent University

Cohomology of small categories and gluing problems

Given a small category C and a commutative ring R,
an RC-module is defined as a functor from C to the category of R-modules.
The category of RC-modules has enough projectives to allow us to
define the cohomology of the category C. I will describe some gluing problems
from representation theory where the obstruction group for the solution of the
gluing problem can be expressed in terms of the cohomology of a certain category.
I will also present some recent results, joint with Olcay Coşkun, on a gluing problem
for biset functors.

March 15

Bilkent University

Time Delays and Dynamics of Networks

This talk will consider dynamics on networks that evolve under a discrete Laplacian operator and in the presence of time delays. The spatial structure naturally invites notions from graph theory, and time delays lead to coupled functional differential equations on this structure. We discuss the effects of structure and delays on the dynamics, for both undirected and directed networks, and for discrete and distributed delays. Time permitting, we will indicate extensions to networks with time-varying connections, as well as to signed graphs, where the links can have positive or negative connection weights, which is the case for brain networks and social interactions.

March 22

Bilkent University

Dessins d'Enfants

I will present "Dessins d'Enfants", which are interesting objects due to
the fact that they, on the one hand, admit a topological characterization
and on the other hand, admit a combinatorial characterization.
Consequently, they provide an equivalence between a certain
topological/geometric setup and a certain combinatorial/algebraic setup.
After establishing this equivalence, I will mention examples where it is
used to conclude meaningful results.

March 29

Atılım University

Automorphisms of homogeneous symmetric groups

April 5

Atılım University

Waring-Goldbach Problem with Sparse Subsets of Primes

April 12

Sabancı University

On the Normalization Procedure of Baouendi-Huang

I will explain how Baouendi-Huang's normalization procedure applies in order to study the classification problem for formal holomorphic embeddings between Shilov Boundaries of Bounded and Symmetric Domains.

April 19

Tosun Terzioğlu Remembrance Day

METU Department of Physics

Thermodynamics of Geometry and Geometry of Thermodynamics

In the last 50 years, black hole thermodynamics has been a
major research area and it seems that Einstein's equations can be
formulated as
equations of thermodynamics. So, Riemannian geometry as it appears in
gravity could be the thermodynamic limit of a more fundamental quantum
theory of gravity. On the other-hand, since the early work of Constantin
Carathéodory, thermodynamics is known to be described by contact geometry.
I hope to give an introduction to these topics.

April 26

Bilkent University

Operads and Homotopy Theory

In this talk we will discuss operads and their applications in homotopical algebra. We will start by giving basic definitions, facts, and standard examples, all with an emphasis on algebraic operads. Given two isomorphic chain complexes, one with some extra algebraic structure one can transfer this algebraic structure to the second one by using an isomorphism between these chain complexes. Moreover, the transferred algebraic structures on the second chain complex would be isomorphic to the first one. In this talk, we will discuss the case where the two chain complexes are only quasiisomorphic. Here the main questions are:

1. What kind of algebraic structures can be transferred?

2. Is the transferred structure equivalent to the original one in some way?

The answers to these questions are given by homotopy transfer theorem, or HTT for short. Finally we will talk about applications of HTT in a wide variety of subjects.

May 3

METU

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May 10

Boğaziçi University

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May 17

Institute

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May 24

Institute

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