Brown University Center for Statistical Sciences Seminar
Refreshments beginning at 3:15pm |
Abstract: Epidemic models are used to describe infectious spread in a population. Many epidemic models exhibit drastically different behavior depending on whether a critical parameter, the reproductive number, is below or above the critical value of one. In general, major epidemic is only possible above the critical value. We will describe this threshold phenomena for a class of simple infectious disease models, when only temporary immunity is possible. We study the final outcome of an epidemic, or the total number of infections, in different regions of the critical parameter space.
Department of Mathematics Distinguished Lecture Series
Abstract: A quasi-isometry is a ap between two metric spaces which is bilipshitz up to an additive constant. In this lecture I will attempt to argue that a quasi-isometry, even though it has no local structure, can be usefully thought of as a sort of differentiable map.
Joint Stochastic Systems and PDE Seminar
Abstract: We will develop several results on hitting probabilities of random fields which highlight the role of the dimension of the parameter space. This yields upper and lower bounds in terms of Hausdor measure and Bessel-Riesz capacity, respectively. Then, we will apply these results to systems of stochastic wave equations in spatial dimension $k\geq 1$, driven by an additive, d-dimensional, spatially homogeneous Gaussian noise that is white in time and colored in space. Finally, we will report on the method, based on Malli- avin calculus, to extend these results to multiplicative noise. This work is part of a joint project with Robert C. Dalang, EPFL, Switzerland.
Scientific Computing Seminar
Abstract: I will give a transverse introduction to how Boundary Integral techniques (or their numerical version, the Boundary Element Method) have been and are being used for scattering problems. While in the beginning the field of boundary integral equations dealt mainly with simple exterior problems, the study of the scattering of waves by penetrable obstacles offers a set of more realistic problems where BEM techniques have often to be coupled with better known methods for PDEs, such as the Finite Element Method. We will see how a method-oriented approach can be valid for wave problems in the frequency and time domains, and how the Laplace transform plays a surprising role for numerical discretization in the time domain.
Special Stochastic Systems Seminar
Abstract: Max-plus methods were originally developed for deterministic control problems, based on the max-plus linearity of the associated semigroup (dynamic programming operator). While investigating an application in UAV tasking, it was noticed that a max-plus curse-of-dimensionality-free method could be used. It was soon realized that this was a general property, i.e., that such curse- of-dimensionality-free methods were possible for stochastic control problems. One simply uses the max-plus associative property. In fact, the key to the curse-of-dimensionality-free methods is that the set of quadratic functions maps into itself under the semigroup operator; max-plus linearity is not necessary. There are, however, difficult complexity-growth attenuation problems which must be handled in the stochastic case. At each step of these algorithms, the solutions are represented as max-plus (or min-plus) sums of simple functions. Our problem becomes: Given an approximate solution representation as a max-plus sum of M functions, find the best approximation as a max-plus sum of N functions (with $N < M$). We find that for certain classes of problems, the optimal reduced-complexity representation is comprised of a subset of the original set of functions. Some comments on game problems, max-plus convexity and min-max semiring expansions may also be included.
PDE Seminar
Abstract: We discuss recent work to prove global existence of solutions to the Cauchy Problem for the Relativistic Boltzmann equation with near Vacuum data. Our proof generalizes the work of Glassey (2006) to some restricted bounded cross sections which need not decay at infinity. Our estimates are independent of the speed of light globally in time. We will discuss applications to the Newtonian limit, c \to \infty if time permits.
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