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Mathematics & Statistics

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Zeros of Heine-Stieltjes polynomials and critical measures

Andrei Martínez-Finkelshtein

Universidad de Almería

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.

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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.

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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.

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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.

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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.

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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 \(b\ne 1,3\).

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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.

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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.

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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.

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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.

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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.

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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.

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Beurling’s Theorem in the Bergman Space and Its Consequences

Rachel Weir

Allegheny College

3:00pm-4:00pm

PHY 120

Catherine Bénéteau

**Abstract**

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.