Lamia Belhaji Title: The ``two-scale'' contact process Particle systems are usually defined on a homogeneous graph, the interaction neighborhoods of sites of the graph are linked each other by translation. In this work, we study the contact process on a non homogeneous graph designed to model the interactions within metapopulations. The $d$-dimensional lattice is turned into a ``chessboard'' through the superposition of a mesoscopic lattice on the usual microscopic lattice. Each site of the mesoscopic lattice is the center of a square of the chessboard.Interactions occur at both site (microscopic) level and square (mesoscopic) level. The superposition of two interaction levels induces two birth rates, called microscopic and mesoscopic birth rates. Similarly, deaths occur at both levels: individual deaths at microscopic level, and mass extinctions (destruction of all the particles contained in a given square) at mesoscopic level. Our ``two-scale'' contact process can be viewed as a metapopulation model describing the evolution of a set of interacting local populations. We study the effect of coarseness, defined as the ratio of the mesoscopic scale over the microscopic scale, on the survival probability of the particle system. We find that, in the absence of mass extinctions, particles are more likely to spread out as coarseness increases, even if the mesoscopic birth rate decreases significantly with the square size. In the presence of mass extinctions, coarseness has only a limited effect. Dov Chelst Title: Statistical Physics and Mobile Ad Hoc Networks. Abstract: We examined wireless multi-hop networks where each node transmits at the same frequency and power level P. Our numerical study calculated the relationship between the power level P and various measures of network connectivity. This was conducted for networks of different sizes and models. A percolation-type critical power level was noted among other features regarding thermodynamic behavior. Jean-Marc Derrien Title: Duality between conductances and resistances for the study of random walks on electric networks. Abstract: The explicit asymptotic variance of one-dimensional random walks on electric networks lets appear a symmetry between conductances and their inverses. We explore new aspects of this duality. Rohini Kumar Title: Space-Time Current Process for Independent Random Walks in One-Dimension Abstract: In a system made up of independent random walks, the hydrodynamic limit of particle distribution is the solution of a transport pde. Fluctuations of order n^{1/4} from the hydrodynamic limit, where n is the scaling parameter, come from particle current across characteristics of the transport pde. We show that a two-parameter space-time particle current process converges to a two-parameter Gaussian process. Tom La Gatta Title: Global Length-Minimizing Curves of Random Riemannian Metrics I will discuss the existence of global length-minimizing geodesics on random Riemannian metrics, and make the connection to discrete first passage percolation models. Siu Tang Leung Title: Optimal Exercising Rules for Employee Stock Options Employee stock options (ESOs) have become an important component of compensation. They are subject to suboptimal exercises of the holders, which arise from risk aversion, trading and hedging constraints, and job termination risk. We consider the valuation of ESOs as a stochastic control problem with voluntary and involuntary stopping. In Markovian framework, this leads to the study of a chain of nonlinear free-boundary problems of reaction-diffusion type. We provide a characterization of the optimal exercise times, and examine the behavior of the optimal exercise boundaries. Trevis Litherland Title: Longest Increasing Subsequences for Finite and Infinite Alphabets Using an elementary formulation of the problem, we investigate the asymptotics of the length of the Longest Increasing Subsequence for general iid finite alphabets, and extend some of these results to infinite alphabets. Elizabeth Meckes Title: Exchangeable pairs and multivariate normal approximation Abstract: Stein's method of exchangeable pairs is a powerful method for proving probabilistic limit theorems with convergence rates. I will review the method in general and its implementation in the case of normal approximation. I will then discuss a version for multivariate normal approximation (joint with Sourav Chatterjee), and give several examples of the method in action. Mark Meckes Title: The norm of a random Toeplitz matrix. Abstract: Bryc-Dembo-Jiang and Hammond-Miller independently recently showed that properly scaled random symmetric Toeplitz matrices possess a limiting spectral distribution. The limiting distributions have unbounded support, and Bryc-Dembo-Jiang raise the question of the asymptotic behavior of the spectral norms of such random matrices. We address this and related questions using tools from the theory of stochastic processes. Pitor Milos Title: Occupation time fluctuations of Poisson of certain branching processes. For a branching system one can invastigate occupation time process and its fluctuations. In certain cases (eg. branching system consisting of partices undergoing alpha-stable motion with finite branching) a kind of CLT-like theorem can be obtained. Soumik Pal Title: Brownian motions interacting through ranks and a phase transition phenomenon. Abstract: Consider n positive diffusions whose logarithms are Brownian motions with the following interaction among themselves. At any time point, one orders the Brownian particles in increasing ranks, and the drift at any time is a function of their ranks. Hence, as time progresses, their ranks and drifts both change in an interacting way. There are necessary and sufficient conditions on the drift function so that the difference between the maximum and the minimum particle remains stable. Under such conditions, as n grows to infinity, a curious phenomenon occurs. Look at the positive diffusions divided by their sum, that is to scale them such that they add up to one. Under very weak conditions, one of three things can happen to the scaled values: either they all go to zero, or the maximum grows to one while the rest go to zero, or they stabilize and converge in law to a Poisson-Dirichlet point process. The last convergence is delicate, and the parameters of the process need to be in a critical strip. We also discuss how starting with infinite particles is very different from starting with n particles and taking n to infinity. This is based on separate joint works with Sourav Chatterjee and Jim Pitman. Lea Popovic Title: Degenerate Diffusion Limits in Gene Duplication We consider two processes that have been used to study gene duplication, Watterson's double recessive null model, and Lynch and Force's subfunctionalization model. Though the state spaces of these diffusions are 2 and 6 dimensional respectively, we show in each case that the diffusion stays close to a curve. Using ideas of Katzenberger we show that one dimensional projections converge to diffusion processes, and we obtain asymptotics for the the time to loss of one gene copy. Amber Puha Title: A Fluid Limit for a Shortest Remaining Processing Time Queue Consider a GI/GI/1 queue operating under shortest remaining processing time with preemption. To describe the evolution of this system, we use a measure valued process that keeps track of the residual service times of all jobs in the system at any given time. Of particular interest is the waiting time for large jobs, which can be tracked using the frontier process, the largest service time of any job that has ever been in service. We propose a fluid model and present a functional limit theorem justifying it as an approximation of this system. The fluid model state descriptor is a measure valued function for which the left edge of the support is the fluid analog for the frontier process. Under mild assumptions, we prove existence and uniqueness of fluid model solutions. Furthermore, we are able to characterize the left edge of fluid model solutions as the right continuous inverse of a simple functional of the initial condition, arrival rate, and service time distribution. When applied to various examples, this characterization reveals the dependence on service time distribution of the rate at which the left edge of the fluid model increases. Julia Reffy Title: Large deviations and potential theory In large deviation theory of empirical eigenvalue distribution of random matrices the weighted logarithmic energy functions play important role. Finding the equilibrium measure for the functionals helps to find the limit distribution. This talk will give an overview about the results concerning the diffrent types of random matrices. Ramon van Handel Title: Robustness and approximations in nonlinear filtering Brief description: when nonlinear filters are approximated, one often observes that the approximation error is bounded uniformly in time--a highly desirable property from the point of view of applications. I will describe some recent progress in understanding this phenomenon in the continuous time setting. Hua Xu Title: Concentration of spectral measure for random matrices with infinitely divisible entries. Abstract: We study the real symmetric or Hermitian random matrices whose on and above diagonal entries form an infinitely divisible vector, or more in particular an alpha-Stable one. The concentraion of spectral measure and the largest eigenvalues are given.