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Mathematics & Statistics

# Colloquia — Spring 2010

## Friday, April 30, 2010

Title
Speaker

Time
Place

Zeros of Heine-Stieltjes polynomials and critical measures
Andrei Martínez-Finkelshtein
Almería, SPAIN
3:00pm-4:00pm
PHY 120
E. Rakhmanov

Abstract

Heine-Stieltjes polynomials are solutions of certain second order linear ODEs with polynomial coefficients. They generalize the families of classical orthogonal polynomials and arise in several areas of mathematics and physics. Their zeros are saddle points of certain discrete energy functionals, and the description of their asymptotics requires an extension of the standard notion of equilirium. We analyze some related potential theoretic concepts, as well as discuss briefly their connection with trajectories of quadratic differentials and extremal problems on the plane. This is a partial report of a joint work with E. Rakhmanov.

## Friday, April 23, 2010

Title
Speaker

Time
Place

Cycle spaces of graphs
Steven C. Locke
Florida Atlantic University
3:00pm-4:00pm
PHY 120
Greg McColm

Abstract

We consider a collection of theorems that arose from a single conjecture of Adrian Bondy.

The cycle space (over the field of two elements) of a graph $$G$$ can be considered as the collection of edge sets of even subgraphs of $$G$$. Under Dirac-like conditions, long cycles generate the cycle space. For Cayley graphs over abelian groups, The Hamilton cycles are (almost) enough.

Sketches of proofs for one or two of the easier results will be given.

## Friday, April 16, 2010

Title
Speaker

Time
Place

Minimum distance of linear codes and ideals generated by products of linear forms
Stefan Tohaneanu
University of Cincinnati
3:00pm-4:00pm
PHY 120
Boris Shekhtman

Abstract

If $$C$$ is an $$[n;k;d]_{i}$$ linear code, computing its minimum distance, $$d$$, leads to deciding if certain ideals I generated by products of linear forms are Artinian or not. In this talk we show that when these ideals are Artinian, then they must be powers of the maximal (irrelevant) ideal. We discuss some theoretical consequences of this result in connection to the projective minimal codewords.

## Friday, April 9, 2010

Title
Speaker

Time
Place

Asymptotic Dimension
Atish Mitra
Center for Advanced Studies in Mathematics
Ben Gurion University
Israel
3:00pm-4:00pm
PHY 120
Milé Krajčevski

Abstract

The notion of asymptotic dimension of metric spaces has been extensively studied since its introduction by M. Gromov, and since G. Yu showed that finitely generated groups with finite asymptotic dimension satisfy the Novikov conjecture. In the talk I will give a brief overview of the theory of asymptotic dimension, discuss recent results and applications to large scale geometry and geometric group theory. The talk will be accessible to students.

## Friday, April 2, 2010

Title
Speaker

Time
Place

Magnification relations and their violation
Arlie Petters
Duke University
3:00pm-4:00pm
PHY 120
Nagle Lecture Committee

Abstract

One of the central theorems in gravitational lensing yields universal magnification relations for multiple images. The talk will present the magnification-relation theorem and show examples where the theorem is obeyed in nature. Interestingly, there are observed cases where the theorem is violated. The lecture will overview how this violation points to the development of a mathematical theory of stochastic gravitational lensing and has applications to the cold dark matter theory on galactic scales. The talk is aimed at an audience of mathematicians, astronomers, and physicists.

## Friday, March 19, 2010

Title
Speaker

Time
Place

The Dirichlet problem for a cone and zeros of Jacobi polynomials
Herrmann Render
University College Dublin
Dublin, Ireland
3:00pm-4:00pm
PHY 120
Dmitry Khavinson

Abstract

We consider the polynomial or analytical solvability of the Dirichlet problem for polynomial or analytical data functions defined on a cone given by the equation $$\gamma^2\left(x^2+y^2\right)+\left(\gamma^2-1\right)\,z^2=0$$ for fixed $$\gamma$$ in $$(0,1)$$. Using Fischer decompositon methods, it follows that the polynomial solvability is equivalent to the statement that the Jacobi polynomials $$P_n^{k+(d-3)/2,k+(d-3)/2}(x)$$ do not vanish at the point $$\gamma$$. For rational $$\gamma=b/c$$ and coprime numbers $$b$$, $$c$$ it is shown that Jacobi polynomials $$P_n^{k+(d-3)/2,k+(d-3)/2}(x)$$ do not vanish in $$\sqrt{b/c}$$ for all $$k,n\in\mathbb{N}_0$$ provided that $$b\ne 1,3$$. The case of analytic solvability is related to the asymptotic behavior of the Jacobi polynomials. For rational numbers $$\gamma=b/c$$ we establish analytical solvability for $$b\ne 1,3$$.

Title
Speaker

Time
Place

An extremal problem for non-vanishing functions in the disc
Terry Sheil-Small
University of York
York, U.K.
4:00pm-5:00pm
PHY 120
Catherine Bénéteau

Abstract

We consider the problem of minimizing the $$A^{2}$$ area integral for non-vanishing functions whose first two Taylor coefficients are given. We show that, if $$f$$ is the extremal function, then, under some regularity conditions, there is a non-linear differential equation between $$f$$ and an associated analytic function $$K$$. This leads to some relationships between the area moments and the circle moments of $$|f|^{2}$$ and from these we can find $$f$$ and calculate the desired minimum value.

## Friday, March 5, 2010

Title
Speaker

Time
Place

A Variation of the Classical Turan Type Problem
Zi-Xia Song
University of Central Florida
3:00pm-4:00pm
PHY 120
Xiang-dong Hou

Abstract

Let $$D=\left(d_{1},d_{2},\dotsc,d_{n}\right)$$ be an integer sequence with $$d_{1}\ge d_{2}\ge\dotsb\ge d_{n}\ge 0$$. We say that $$D$$ is graphic if there is a graph $$G$$ with $$D$$ its degree sequence. In those circumstances, we say that $$G$$ is a realization of $$D$$. We say that $$D$$ is $$K_{k}$$-graphic if $$D$$ has a realization containing $$K_{k}$$ as a subgraph; and $$D$$ is $$K_{k}$$-free if no realization of $$D$$ contains $$K_{k}$$ as a subgraph. We consider an extremal problem for graphs as introduced by Erdös, Jacobson and Lehel in 1991. That is to find the minimum even integer $$t$$ such that every graphic sequence $$D=\left(d_{1},d_{2},\dotsc,d_{n}\right)$$ with $$\sum\limits_{i=1}^n d_{i}$$ at least $$t$$ is $$K_{k}$$-graphic. They conjectured that $$t=(k-2)(2n-k+1)+2$$. In this talk, we will survey the methods on solving this conjecture and recent results in this area on $$K_{k}$$-graphic sequences.

## Friday, February 26, 2010

Title
Speaker

Time
Place

Dynamics and Hydrodynamic Limits of the Dissipative Boltzmann Equation
Shui-Nee Chow
Georgia Tech
3:00pm-4:00pm
PHY 120
Yuncheng You

Abstract

We investigate the macroscopic description of a dilute, gas-like system of particles, which interact through binary collisions that conserve momentum and mass, but which dissipate energy, as in the case of granular media with inelastic collisions. Our starting point is on the mesoscopic level, through the Boltzmann equation. We deduce hydrodynamic equations for the macroscopic description that would reduce to the compressible Navier-Stokes equations if there were no energy dissipation. We do this in a regime where both the Knudsen number (the ratio of mesoscopic to macroscopic length scales) and the restitution deﬁcit (which measures the inelasticity) are small but non-zero. In this regime, we show that for small values of the Knudsen number and small inelasticity it is possible to relate the actual dynamics to a reduced dynamics on a “slow manifold”, which in the limit of zero inelasticity and zero Knudsen number is simply the “manifold” of local Maxwellians. Instead of expanding the Boltzmann equation itself, we expand this manifold in terms of these two small parameters. In this way, a number of ideas from the theory of dynamical systems, and especially geometric singular perturbation theory, enter our analysis. We discuss the resulting hydrodynamic equations, and compare them to those obtained by other researchers using other methods (suited to other regimes). As we explain here, the particular regime we investigate is especially interesting in the context of pattern formation in driven granular media.

## Friday, February 19, 2010

Title
Speaker

Time
Place

Distributed algorithms and graph theory
Andrzej Czygrinow
Arizona State University
3:00pm-4:00pm
PHY 120
Brendan Nagle

Abstract

In recent years there has been a growing interest in applying graph-theoretic methods to design eﬃcient distributed algorithms. In this talk, we will discuss distributed complexity of some of the classical problems in graph theory like, for example, the maximum matching problem or the minimum dominating set problem. We will overview known facts for general graphs and show that much more efficient solutions can be obtained for special classes of graphs. In addition to the algorithmic results we will show that some of the purely graph-theoretic facts lead to interesting consequences for the theory for distributed algorithms.

## Friday, February 5, 2010

Title
Speaker

Time
Place

Scrambling and Supercoiling in Ciliate DNA: a relationship between topology and recombination
Mark Daley
University of Western Ontario
3:00pm-4:00pm
PHY 120
Masahiko Saito and Nataša Jonoska

Abstract

The ciliated protozoa are a group of unicellular eukaryotes which maintain two types of nuclei: functional macronuclei, and germline micronuclei. Following conjugation, in which two ciliates exchange haploid micronuclear genomes, ciliates regenerate their macronuclei from the new micronuclei. This process involves extensive genome processing and, in the case of the stichotrichous ciliates, some micronuclear genes actually have to be descrambled (via nontrivial block permutations of substrings) to form functional macronuclear genes.

In this talk we give an introduction to the underlying biology of this process and a high-level overview of a template-guided recombination formal model for gene descrambling along with a short exposition of computability results. Although the template model is theoretically intriguing, and some experimental evidence now exists to support it, it suffers from certain thermodynamic liabilities. In order to address these liabilities, we have introduced an extension which uses the topological conformation of DNA to drive the recombination process. In this new model we describe an iterative process in which recombination drives topological change which, in turn, facilitates further recombination.

We will conclude the talk with a summary of our efforts to chase down biological evidence for this model including exciting excursions into electron microscopy, molecular genetics and the computational simulation of DNA topology via Brownian dynamics.

## Friday, January 29, 2010

Title
Speaker

Time
Place

Virtual endomorphisms and self-similar groups
Zoran Šunić
Texas A&M University
3:00pm-4:00pm
PHY 120
Milé Krajčevski

Abstract

The concept of self-similar group of automorphisms of a regular rooted tree is, in one rather natural interpretation, coming from considering symbolic dynamics on a rooted tree (as opposed to a line). Starting from here, we arrive at the concept of virtual endomorphism, which is a purely algebraic framework for the study of self-similar groups.

We use virtual endomorphisms to provide self-similar actions of several well known groups (for instance, $$\mathrm{PSL}_{2}$$(Z)) and describe all ﬁnite $$p$$-groups that have self-similar actions on roted $$p$$-ary trees with abelian level stabilizers.

## Friday, January 22, 2010

Title
Speaker

Time
Place
In 1949, Beurling showed that every closed $$z$$-invariant subspace of the Hardy space is generated by a classical inner function. Aleman, Richter and Sundberg established a result which can be viewed as a Bergman space version of Beurling’s theorem in 1996. Given the more complicated nature of the invariant subspaces of the Bergman space, this was a significant breakthrough. Shimorin continued this progress in 2000 and 2001 when he extended this result to certain weighted Bergman spaces, using a different approach. We will describe these developments and their consequences for both the weighted and unweighted Bergman spaces. In addition, we will outline several open problems and some recent related work.