Research

Probability and Related Fields

Wednesday, December 1, 2004

Title: Self-assembling DNA Model and related problems
Speaker: Ana Staninska
Time: 4:00pm–5:00pm
Place: PHY 118

Abstract

I will describe a theoretical model of self-assembly inspired by DNA nano-technology and DNA computing, and introduce related mathematical problems. This model consists of tiles that assemble into graph-like complexes, which assembled "properly" can represent a solution to a given problem. It can be shown that the computational power is equivalent to solving NP complete problems.

Wednesday, November 24, 2004

Title: Stopping Times and the Evolution of Random Structures
Speaker: Greg McColm
Time: 4:00pm–5:00pm
Place: PHY 118

Abstract

One increasingly popular area of applied probability to combinatorics is the evolution of random structures, especially of random graphs. Such “evolutions” can be used to study the behavior of assembly, accretion, and development. One of the fundamental questions is *when* an important threshhold is crossed. This is a stopping time problem. We look at some of the basic notions in this field.

Wednesday, November 17, 2004

Title: Products of Random Circulant Matrices, Part II
Speaker: Edgardo Cureg
Time: 4:00pm–5:00pm
Place: PHY 118

Wednesday, November 10, 2004

Title: Products of Random Circulant Matrices, Part I
Speaker: Edgardo Cureg
Time: 4:00pm–5:00pm
Place: PHY 118

Abstract

An $$n\times n$$ matrix of the form $$\begin{matrix} a(0) & a(1) & a(2) & \dotsm & a(n-1) \\ a(n-1) & a(0) & a(1) & \dotsm & a(n-2) \\ \dotsm & \dotsm & \dotsm & \dotsm & \dotsm \\ \dotsm & \dotsm & \dotsm & \dotsm & \dotsm \\ a(1) & a(2) & a(3) & \dotsm & a(0) \end{matrix}$$ is called a circulant matrix. Such matrices have been studied in the context of random walks, BCH codes, smoothing of data, analysis of random number generators, etc. (See P. Diaconis, Patterned Matrices, Proc. of Symposium of Appl. Math. 40, AMS, 37-58, 1990).

In this talk we discuss some basic properties of such matrices and consider the problem of convergence in distribution of products of i.i.d. circulants. Orthogonal matrices play a key role in our solution.

Wednesday, October 27, 2004

Title: Weak and weak$$^*$$-convergence, Part II
Speaker: Arunava Mukherjea
Time: 4:00pm–5:00pm
Place: PHY 118

Abstract

To continue last week's discussion, I'll prove my old result that in a non-compact group, the random walk escapes to infinity.

In other words, if $$G$$ is a locally compact Hausdorff non-compact group containing the support $$S(P)$$ of a probability measure $$P$$ such that no compact subgroup of $$G$$ contains $$S(P)$$, then for any compact subset $$K$$ of $$G$$, $$\Pr(Z(n)\in K$$) tends to zero as $$n$$ tends to infinity, where $$Z(n)$$ is the random walk induced by $$P$$.

Wednesday, October 20, 2004

Title: Weak and weak$$^*$$-convergence, Part I
Speaker: Arunava Mukherjea
Time: 4:00pm–5:00pm
Place: PHY 118

Abstract

This will be mostly an introductory talk. New results in this context will be presented by others later in the semester.

Wednesday, October 13, 2004

Title: Planar Graphs, Random Walks and Heat Content
Speaker: Patrick McDonald, New College at Sarasota
Time: 4:00pm–5:00pm
Place: PHY 118

Abstract

There is a well-known and well-studied relationship between Brownian motion, boundary value problems and the geometry of Euclidean domains. This relationship gives rise to discrete analogs relating random walks, problems for discrete difference operators and the geometry of graphs embedded in Euclidean spaces. In this talk we survey the discrete material, developing techniques for moving between categories and using these techniques to discuss recent results. In particular, we will construct a pair of isospectral graphs and prove that these graphs are distinguished by their heat content.

The talk is aimed at a general mathematical audience and is reasonably self-contained. In particular, we develop those probabilistic and geometric tools which we will require.

Wednesday, October 6, 2004

Title: Local Limit Theorems for Random Integer Partitions
Speaker: Ljuben Mutafchiev
Time: 4:00pm–5:00pm
Place: PHY 118

Abstract

Certain power series expansions will be used to prove a local limit theorem for the length of the side of a Durfee square in a random partition of a positive integer $$n$$ as $$n$$ tends to infinity.

Wednesday, September 29, 2004

Title: Hypergroups: Examples, Idempotent and Invariant Probability Measures, Part II
Speaker: Norbert Youmbi
Time: 4:00pm–5:00pm
Place: PHY 118

Wednesday, September 22, 2004

Title: Hypergroups: Examples, Idempotent and Invariant Probability Measures, Part I
Speaker: Norbert Youmbi
Time: 4:00pm–5:00pm
Place: PHY 118

Abstract

A semihypergroup (Hypergroup) is a locally compact space on which the vector space of finite regular Borel measures has a convolution structure preserving the probability measures. The class of semihypergroups (Hypergroups) includes the class of locally compact topological semigroups (Groups). Hypergroups generalizes in many aspects locally compsc groups. Many $$n$$-dimensional hypergroups are obtained from orthogonal polynomials on spaces on which no structure of a group could be defined. We will give some practical examples of hypergroups as well as presenting some results on invariants and idempotent probability measures on semihypergroups.

Wednesday, September 15, 2004

Title: Random Fibonacci Sequences
Speaker: Edgardo Cureg
Time: 4:00pm–5:00pm
Place: PHY 118

Abstract

Viswanath's determination of the rate of growth of $$(|x_n|)$$, where $$x_{n+1}=\pm\;x_n+x_{n-1}$$, $$n\ge 1$$, $$x_0=x_1=1$$, and the $$+$$ and $$-$$ signs each occur with probability $$1/2$$.

The techniques involved in the solution illustrate an interplay between the theory of random matrix products, the Stern-Brocot tree, fractal measures, and computer simulations. We also present some generalizations of the random Fibonacci sequence.

Wednesday, September 8, 2004

Title: When convergence in distribution of products of $$d\times d$$ i.i.d. matrices is determined essentially by their skeletons
Speaker: Arunava Mukherjea
Time: 4:00pm–5:00pm
Place: PHY 118

Abstract

Two nonnegative matrices $$A$$ and $$B$$ have the same skeleton if $$A(i,j)>0$$ whenever $$B(i,j)>0$$ and conversely.