Mathematical Sciences

School of Natural Sciences & Mathematics

Geometry, Topology, Dynamical Systems Seminar, AY 13-14





DateSpeakerAffiliationTitleAbstract
May. 13 M. Dabkowski Mathematical Sciences, UTD SL(2C) character varieties II

May. 6 M. Dabkowski Mathematical Sciences, UTD SL(2C) character varieties I

May. 2 Oleg Makarenkov Mathematical Sciences, UTD A perturbation approach to study the dynamics of nonlinear differential equations II Some interesting examples will be presented.
Apr. 25 Aykut Satici Erik Jonsson School, UTD Swarming with Connectivity via Lagrange-Poincare Equations
One of the important goals of a multi-agent mobile network is coverage or surveillance of a given
area. This requires the agents to swarm or move in formation along a desired path/trajectory.
In other words, it is desired that the centroid of the formation move along a specified desired
trajectory. In addition when avoiding contact with the environment is an issue, we may also want
to specify a desired orientation trajectory of the multi-agent system.


While the swarming operation is under way, it is still desired to achieve and maintain a desired
connectivity measure whenever information sharing between agents in the network is required.
In this work, we propose a framework which nearly decouples the control design for these two
potentially conflicting goals by exploiting the inherent symmetries of mechanical systems.
Apr. 15 Oleg Makarenkov Mathematical Sciences, UTD A perturbation approach to study the dynamics of nonlinear differential equations I
One of the methods to access the dynamics of nonlinear differential equations is by deforming the differential equation to x’=0. I will discuss the connection between the solutions of x’=0 and those of the original differential equation. This will lead us to the so-called “bifurcation function”. A classical application of this (perturbation) approach will be discussed, where I will likely choose the periodic Van der Pol oscillator. My own contributions in this field will be mentioned just briefly.
Feb. 25 Gabriele La Nave Isotropic curvature, macroscopic dimension and fundamental group
I will discuss the proof a conjecture of Gromov’s to the effect that manifolds with uniformly positive
isotropic curvature (and bounded geometry) are macroscopically 1-dimensional on the scale of the isotropic curvature.
One of the main techniques involved is modeled on Donaldson’s version of H\”ormander technique to produce (almost) holomorphic sections. We use this to produce destabilizing sections of the restriction of the complexified tangent bundle of M to a stable embedded minimal disk.
As a consequence we prove that compact manifolds with positive isotropic curvature have virtually free fundamental groups.
Feb. 18 Derege H.Mussa Texas A&M University-Commerce Reconstruction of tetrahedron from Edge length

Dec. 10 T. Hagge Mathematical Sciences, UTD Representation theory IV

Representation theory III

Nov. 19 T. Hagge Mathematical Sciences, UTD Representation theory II

Nov. 12 T. Hagge Mathematical Sciences, UTD Representation theory I

Oct. 29 T. Hagge Mathematical Sciences, UTD An Introduction to the Character Theory of Representations II

Oct. 22 T. Hagge Mathematical Sciences, UTD An Introduction to the Character Theory of Representations I
A representation rho of a group G over a field F is a group homomorphism.
rho: G -> GL_n(F).
Representation theory has an enormous variety of applications. One can expect it to turn up in any situation in which
1) A group (or other algebraic structure) of symmetries is present, and
2) There is some applicable mathematics which can be reduced to linear algebra.

It is not hard to imagine that there are many such situations; representation theory turns up in number theory, the classification of finite simple groups, algebraic topology, combinatorics, differential equations, Fourier series analysis, and many other areas of mathematics. It appears ubiquitously in quantum mechanics; the “quanta” of quantum mechanics are in fact quantities derived from the representations of some group of symmetries.

As usual, one has the following questions: 1) (classification) or a given group G, what representations are possible? 2) (isomorphism detection) Given two representations of a group, how does one determine, both in theory and in practice, whether they are structurally the same?

It turns out that for some important classes of groups, including finite groups, the structure of a representation over the complex numbers C is determined entirely by the traces of the matrices rho(g). The study of these traces is known as Character Theory. (The story for representations over the reals R is essentially the same but with a modest additional complication). It bears some similarity to the study of transitive group actions, but has important structural differences.

Over the next several lectures, we will learn basic terminology and concepts of representation theory, discuss some basic examples, and develop the theory of characters. We will develop a structural understanding of how the trace is used as a tool in algebra and construct machinery sufficient to classify the irreducible representations of finite groups of small order.

The necessary background is an understanding of the first sentence of this abstract and an undergraduate course in linear algebra.

Sep 17-Oct. 15 Changsong Li Mathematical Sciences, UTD Braids and Temperley-Lieb algebra I-V