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

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.

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

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.

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

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

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.

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

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 \(P\).

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

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

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

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.

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

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.

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

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

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.

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

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.

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

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