METU Mathematics Seminars
Fall 2013


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


October 3
Midddle East Technical University
A Theorem of Wielandt on the Gamma Function
The purpose of this talk is to present a theorem of Wielandt which facilitates the introduction of the gamma function by providing swift if somewhat artificial demonstrations of its basic properties.

October 10
Hitit University
Arbitrarily Long Factorizations in Mapping Class Group
For a compact connected oriented surface of genus g with boundary, we consider positive factorizations of the Dehn twist about the boundary. We study the problem whether or not there is an upper bound for the number of positive Dehn twists in such a factorization. As time permits, we discuss the applications of our results to low dimensional topology.

October 24
Midddle East Technical University
What is tropical geometry?

October 31
Midddle East Technical University
A direct BEM solution to MHD pipe flow
The magnetohydrodynamic (MHD) flow of an incompressible, viscous, electrically conducting fluid in a pipe under an externally applied magnetic field is investigated. The flow through the pipe is driven by the current produced by a pressure gradient. A direct boundary element method (BEM) solution is obtained by using a fundamental solution which enables to treat the magnetohydrodynamic flow equations in their original coupled form with general wall conductivities. The method is applied to several test problems with different geometries and the results are presented in terms of equivelocity and induced magnetic field contours.

November 7
Hacettepe University
On some topological applications of Reidemeister torsion
In the present talk, we will give the definition and basic facts about the topological invariant, Reidemeister torsion. We will also present our topological applications of the torsion to Hitchin component and pleated surfaces.

November 14
Midddle East Technical University
Turkish Contributions to Mathematical Sciences
Although in a journey backwards in time the questions of ethnicity and scientific taxonomy tend to become inextricably complicated and possibly meaningless, it can be argued that there is at least a millenium of Turkish interaction with mathematical sciences, intermittent and feeble as it is. In the opinion of the speaker and on a more optimistic note, at the end of the relatively stable last ninety year interval of our history we may look forward to individual research of greater significance as well as to the establishment of mathematical schools of lasting impact.

November 28
Galatasaray University
Simple elliptic singularities and Generalization of Slodowy slices
Any simple elliptic singularity of type \tilde{D}_5 can be obtained by taking the intersection of the nilpotent variety and the 4-dimensional "good slices" in the semi-simple Lie algebra sl(2,\C) + sl(2,\C). We describe these new slices purely by the structure of the Lie algebra. We also construct the semi-universal deformation spaces of \tilde{D}_5-singularities by using the 4-dimensional "good slices".

December 5
Research Institute for Mathematical Sciences
Kyoto University
Non-displaceable Lagrangian submanifolds
Lagrangian submanifolds are important objects in symplectic geometry. They are not so easily displaced by Hamiltonian isotopies. Sometimes, no Hamiltonian isotopy can displace a Lagrangian submanifold. Such a Lagrangian manifold is called non-displaceable. I would like to discuss some criteria for non-displaceablity with examples.

December 12
University of Ljubljana
Topology and data
In this talk we would like to present some ideas on how methods from algebraic topology can help in the overwhelming task of understanding and analyzing data. We will first describe some basic topological models for reconstructing an object or a space from a sample of points in the form of a simplicial complex. Topological properties of this complex, for example the number of components and the number of holes of various types which are measured by homology groups, give insight into the properties of the underlying space. We will then introduce discrete Morse functions which can serve to analyze data representing the values of a function that is given or measured only at the sample points. A discrete Morse function on a simplicial complex is a discrete analogue of a smooth Morse function on a manifold. As in the classical smooth case, discrete Morse functions give insight into the overall behavior of function values as well as into the structure of the underlying space.

December 19 (special event)
Bilkent University
Polinom Cozumleri Uzerine
Special event on the 70th birthday of Prof. Dr. Mehpare Bilhan. Click here to see the poster.

December 26
Bilkent University
Lines generate the Picard group of a Fermat surface
It is easily seen that the Fermat surface $F_m$ given by $x^m+y^m+z^m=1$ contains $3m^2$ straight lines. We answer a question of T. Shioda and show that, if the degree $m$ is prime to six, any algebraic curve in $F_m$ is equivalent to a combination of these lines.

January 2
Bilkent University
New observations on Ramanujan's circular summation formula with applications to Ramanujan's partition congruences.
We present how constant term techniques can be used to evaluate Ramanujan's circular summation function at several moduli. As it turns out these evaluations can be used to give uniform proofs of Ramanujan's partitions congruences for the modulus 5, 7 and 11. This is a joint work with Alexander Berkovich and Frank Garvan from University of Florida. This talk will be accessible to a general audience.

January 9
Imperial College London
On Control and Random Dynamical Systems in Reproducing Kernel Hilbert Spaces
We introduce a data-based approach to estimating key quantities which arise in the study of nonlinear control systems and random nonlinear dynamical systems. Our approach hinges on the observation that much of the existing linear theory may be readily extended to nonlinear systems - with a reasonable expectation of success - once the nonlinear system has been mapped into a high or infinite dimensional Reproducing Kernel Hilbert Space. In particular, we develop computable, non-parametric estimators approximating controllability and observability energy functions for nonlinear systems, and study the ellipsoids they induce. It is then shown that the controllability energy estimator provides a key means for approximating the invariant measure of an ergodic, stochastically forced nonlinear system. We also apply this approach to the problem of model reduction of nonlinear control systems. In all cases the relevant quantities are estimated from simulated or observed data. These results collectively argue that there is a reasonable passage from linear dynamical systems theory to a data-based nonlinear dynamical systems theory through reproducing kernel Hilbert spaces. This is joint work with J. Bouvrie (MIT).