Thursday, November 20, 2014, 1:30 - 2:30 PM EST
Free Resolutions and Hilbert Functions, Part 11
Structured deterministic models applied to malaria and other endemic diseases Committee: Feng (Chair), Buzzard, Yip, Kribs
Thursday, November 20, 2014, 3:30 - 4:30 PM EST
On a Classic Ramanujan Equation and its Probabilistic ImplicationsAbstract: In the year 1911, Ramanujan published a one page note in the Journal of the Indian Mathematical Society stating what has now come to be known as Problem 294. Restated in the language of probability, it asks what fraction of the probability that a Poisson rv with mean n equals n should be added to the probability that it is less than n to make the sum exactly .5? Ramanujan conjectured that the answer, say theta_n, is between 1/3 and 1/2 for all n. In his first letter to Hardy, Ramanujan stated a finer conjecture. 103 years since Ramanujan first stated the problem, it is now known as "the ultimate famous problem of Ramanujan" - we do not know who first gave it that attribute. The conjecture has been proved (Szego, Watson, others). There is now a good deal of competent literature on Problem 294. In 1986, Herman Rubin and Jeesen Chen jointly conjectured that the median of a gamma rv with mean n+1 is between n + 1/3 and n + log2. This too has finally been proved. Problem 294 and the Rubin-Chen conjectures are related. The talk presents a new integral representation for Ramanujan's theta_n, which will be used to produce fixed n bounds on theta_n that are sharp, and an eight term asymptotic expansion for it. Some fixed n bounds are byproducts of the integral representation and properties of the convergents of a regular continued fraction expansion. Some other fixed n bounds are byproduucts of the integral representation and shape properties, convexity, monotonicity. The talk then connects these techniques to mega generalizations of the Rubin-Chen work. High order (very high) asymptotic expansions for a general quantile of a gamma distribution with mean n are given completely explicitly. The expansions imply that the standard practice of calculating a P-value of a chi-square test by using the CLT is basically wrong, as one will reject too soon.
Thursday, November 20, 2014, 3:30 - 4:30 PM EST
Blow up Phenomena for Shadow System of Gierer-Meinhardt ModelAbstract: We will present some results of blow-up phenomena for the shadow system obtained from the Gierer-Meinhardt model. Shadow system is formally derived by letting the diffusion coefficient of one of the components tend to infinity, leading to a coupled system of a diffusion and an ordinary differential equation. There is a huge discrepancy in terms of long time behaviors between the shadow and the original Gierer-Meinhardt systems. We will demonstrate this using integral estimates and a fixed point theorem. This is joint work with Fang Li.
Computational & Applied Mathematics Seminar, Professor Christopher Kribs, University of Texas at Arlington, REC 103
Thursday, November 20, 2014, 3:30 - 4:30 PM EST
Invasion Reproductive Numbers for Multistrain InfectionsAbstract: Reproductive numbers are well-known key threshold measures of an infection's ability to persist in a population, with measures developed to derive canonical expressions for the basic or control reproductive number. In many cases, however, multiple infections (or multiple strains of an infection) are cocirculating, with coinfections such as HIV+TB of great concern in some regions. A primary infection can make an individual more or less susceptible to secondary infections, entangling the transmission dynamics. Invasion reproductive numbers (IRNs) measure an infection's ability to invade a population where another infection is already resident. In this talk I will show how next-generation methods developed to derive R0 can be used to derive IRNs, and apply it to two examples--one where coinfections are advantaged and one where cross-immunity precludes them.
Thursday, November 20, 2014, 4:30 - 5:30 PM EST
The Feynman Propagator for the Wave EquationAbstract: For elliptic operators on compact manifolds, say without boundary, one has a good understanding of global functional analytic mapping properties. Concretely, one knows that such an operator is a Fredholm map between L^2 based Sobolev spaces of appropriate order. This is a fundamental and useful property of elliptic operators, as having a Fredholm map means, for example, that one can solve equations up to a finite dimensional obstruction. In this talk we will describe a systematic way to extend this framework to hyperbolic PDE, specifically to the wave operator on Minkowski space. We use the Feynman propagator (in place of the more familiar and commonly used advanced and retarded propagators), which allows us to directly relate the hyperbolic problem to an elliptic one via Wick rotation and microlocal propagation estimates. We will explain this and discuss related semilinear well-posedness results for the Feynman problem.
Thursday, November 20, 2014, 6:00 - 7:00 PM EST
A Tale of Two SeriesABSTRACT: This is an examination of what it means for a series to converge by looking at the strange stories of the harmonic and alternating harmonic series. Bernoulli, Euler, Dirichlet and Riemann come into our tale, with a guest appearance by Cauchy and some more modern lesser stars.
Epistemic Uncertainty Quantification in Scientific Committee: Xiu (Co-Chair), Dong (Co-Chair), Buzzard, Lin
Friday, November 21, 2014, 11:30 - 12:30 PM EST
Thresholding Algorithms for Curvature Driven FlowsAbstract: Curvature driven flows arise in a number of models from materials science, physics, and biology. The equations that model these flows are often geometric and hence highly nonlinear. In the early 90's, Bence-Merriman-Osher (BMO) developed a very simple "thresholding algorithm" for producing mean curvature motion for a hypersurface. Their ideas have been extended over the years to mean curvature motion for filaments in space, and fourth order flows such as Willmore flow and Surface Diffusion. In this talk I will discuss the thresholding algorithms for mean curvature and Willmore flows. In particular I will detail a new proof of the convergence of the original BMO algorithm to mean curvature flow, which does not use the maximum principle. This new technique seems promising for proving the convergence of the algorithm for filament motion, which involves vector valued functions.
Automorphic Forms and Representation Theory Seminar, Dr. Ori Parzanchevski, Institute for Advanced Study, REC 121
Friday, November 21, 2014, 1:30 - 2:30 PM EST
Combinatorics of Ramanujan ComplexesABSTRACT: Ramanujan graphs are sparse and highly connected graphs, with many remarkable and useful properties. They were first constructed by Lubotzky-Phillips-Sarnak and by Margulis, using the Ramanujan-Petersson conjecture for GL_2. Advances around the Ramanujan conjecture for GL_d have led several authors to study "Ramanujan Complexes", which are a natural high-dimensional analogue. I will talk on the spectral theory of Ramanujan complexes of dimension 2, and the combinatorial properties which can be inferred from it. Based on joint work with Konstantin Golubev.
DATE: Friday, November 21, 2014 — Note special Date and Time
Monday, November 24, 2014, 3:30 - 4:15 PM EST
Frozen Gaussian approximation and its applicationsAbstract: We propose the frozen Gaussian approximation for the computation of high frequency wave propagation. This method approximates the solution to the wave equation by an integral representation. It provides a highly efficient computational tool based on the asymptotic analysis on phase plane. Compared to geometric optics, it provides a valid solution around caustics. Compared to the Gaussian beam method, it overcomes the drawback of beam spreading. We will present numerical examples as well as preliminary application in seismology to show the performance of this method.
Monday, November 24, 2014, 3:30 - 4:30 PM EST
Transverse Weitzenbock Formulas on Totally Geodesic FoliationsAbstract: We will show a transverse Weitzenbock formula for the horizontal Laplacian of a totally geodesic foliation. This formula implies a optimal first eigenvalue lower bound, a sub-Riemannian Bonnet-Myers theorem and Li-Yau estimates for the horizontal kernel. This is joint work with B. Kim (Purdue University) and J. Wang (IMA).
Computational & Applied Mathematics Seminar, Professor Jinglai Li, Shanghai Jiaotong University, REC 122
Monday, November 24, 2014, 4:15 - 5:00 PM EST
High Dimensional Density Estimation with Optimal Transport MapsAbstract: Many machine learning problems such as Bayesian classifications require the estimation of density functions from data. In such problems, the dimensionality of the data can often pose a challenge for conventional density estimation approaches. In this talk we present a method for estimating high dimensional densities by constructing a transport map from a multivariate Gaussian random variable to the parameter of interest. In particular our method can ensure the monotonicity of the map. An efficient alternating minimization algorithm for solving the resulting optimization problem is provided and finally application examples are discussed.
Special Colloquium, Benedict (Ben) Williams, Postdoctoral Fellow, Department of Mathematics, University of British Columbia, REC 302
Monday, November 24, 2014, 4:30 - 5:30 PM EST
Azumaya Algebras and TopologyAbstract: Azumaya Algebras are a generalization of central simple algebras over fields, and have been studied since the 1950s. In this talk, I shall explain how comparing Azumaya Algebras over rings with topological PGLn-bundles can be used to answer questions in ring theory.
Monday, December 1, 2014, 3:30 - 4:30 PM EST
Reduced Order Modeling and Domain Decomposition Methods for Uncertainty QuantificationAbstract: Traditionally, terms in PDEs such as permeabilities, viscosities or boundary conditions have been treated as known deterministic quantities. However, these quantities are not always known with certainty, and there is much interest today in treating them as random fields. In this talk, I will present a reduced basis collocation method for efficiently solving PDEs with random coefficients, which is joint work with Howard Elman of University of Maryland. I will also present a domain-decomposed uncertainty quantification approach for complex systems, which is joint work with Karen Willcox of Massachusetts Institute of Technology.
PROBABILISTIC UNCERTAINTY QUANTIFICATION AND EXPERIMENT DESIGN FOR NONLINEAR MODELS: APPLICATIONS IN SYSTEMS BIOLOGY Committee: Buzzard (Chair), Rundell, Feng, Lin.
Wednesday, December 3, 2014, 4:30 - 5:30 PM EST
How Eigenvalues Are Actually ComputedAbstract: Eigenvalues are a standard topic in undergraduate and graduate Linear Algebra courses, but the algorithms presented in such classes, while of great theoretical importance, are far from the ones used in most science and engineering applications. The most notable difference is that they are much slower. In this talk I introduce some of the important ideas behind fast eigenvalue solvers such as compressed storage, fast matrix-vector products, and parallelism. As an example, I discuss my recent work with my advisor Jianlin Xia, the first near-linear complexity eigenvalue solver for general symmetric rank-structured matrices. I also discuss some important applications of fast eigenvalue solvers in science and engineering. Numerical results and examples will be given to motivate theory and support claims.
Friday, December 5, 2014, 11:30 - 12:30 PM EST