Time & Location: All talks are on Thursdays in Gibson 414 at 3:30 pm unless otherwise noted. Refreshments in Gibson 426 after the talk.
Comments indicating vacations, special lectures, or change in location or time are in green.
Organizer: Gustavo Didier
Numerical Methods for Hyperbolic Systems of PDEs with Uncertainties
Alina ChertockNorth carolina State University (Host Alexander Kurganov)
Many system of hyperbolic conservation and balance laws contain uncertainties in model parameters, initial or boundary data due to modeling or measurement errors. Quantifying these uncertainties is important for many applications since it helps to conduct sensitivity analysis and to provide guidance for improving the models. Among the most popular numerical methods for uncertainty quantification are stochastic spectral methods. Such methods decompose random quantities on suitable approximation bases. Their attractive feature is that they provide a complete probabilistic description of the uncertain solution.
A classical choice for the stochastic basis is the set of generalized Polynomial Chaos (gPC) spanned by random polynomials, continuous in the stochastic domain and truncated to some degree. It is well-known, however, that when applied to general nonlinear (non-symmetric) hyperbolic systems, such approximations result in systems for the gPC coefficients, which are not necessarily globally hyperbolic since their Jacobian matrices may contain complex eigenvalues. In this talk, I will present a splitting strategy that allows one to overcome this difficulty and demonstrate the performance of the proposed approach on a number of numerical examples including systems of shallow water and compressible Euler equations.
Ramanujan's Mock Theta Functions and Quantum Modular Forms
Holly SwisherOregon State UniversitY (Host: Victor Moll)
Nearly 100 years after his untimely death, Ramanujan's legacy is still intriguing mathematicians today. One of the last obsessions of Ramanujan were what he called mock theta functions. In this talk, we will begin by discussing Ramanujan's work on integer partitions and how they connect to objects called modular forms and mock theta functions. Then we will continue by exploring some recent work in this area, including the construction of a table of mock theta functions with some interesting properties, including what is called quantum modularity. Part of this work is joint with Sharon Garthwaite, Amanda Folsom, Soon-Yi Kang, and Stephanie Treneer. The rest is joint with Brian Diaz and Erin Ellefsen from their undergraduate REU project this summer.
Combinatorial Hopf Algebras and Antipode
Nantel BergeronYork University (Host Mahir Can)
Given a family of combinatorial objects we often have operations that allow us to combine them to create larger objects and/or ways to decompose them into smaller members of the family. In the best situation we have in fact an algebraic structure, i.e. a graded Hopf algebra. I will give example of such structure using graphs, trees, set partitions, etc.
The antipode is a map from the Hopf algebra into itself that is defined recursively, with a lot of cancelation and is difficult to compute in general.
I will motivate why we should care about the antipode and why we should aim to find a cancelation free formula.
An important example is the combinatorial Hopf algebra of graphs. In this case, a cancelation free formula of for its antipode is given by, Humpert and Martin. We will see that such formula gives a structural understanding of certain evaluations of the combinatorial invariants for graphs.
In particular we recover very nicely a classical theorem of Stanley for the evaluation of the chromatic polynomial at -1.
I will discuss some generalization of this example.
Speaker | institution
Early Sub-Exponential Epidemic Growth: Implications for Disease Forecasting and Estimation of the Reproduction Number.
Prof Gerardo ChowellGEORGia State University (Host: James Hyman)
There is a long tradition of using mathematical models to generate insights into the transmission dynamics of infectious diseases and assess the potential impact of different intervention strategies. The increasing use of mathematical models for epidemic forecasting has highlighted the importance of designing reliable models that capture the baseline transmission characteristics of specific pathogens and social contexts. More refined models are needed however, in particular to account for variation in the early growth dynamics of real epidemics and to gain a better understanding of the mechanisms at play. Here, we review recent progress on modeling and characterizing early epidemic growth patterns from infectious disease outbreak data, and survey the types of mathematical formulations that are most useful for capturing a diversity of early epidemic growth profiles, ranging from sub-exponential to exponential growth dynamics. Specifically, we review mathematical models that incorporate spatial details or realistic population mixing structures, including meta-population models, individual-based network models, and simple SIR-type models that incorporate the effects of reactive behavior changes or inhomogeneous mixing. In this process, we also analyze simulation data stemming from detailed large-scale agent-based models previously designed and calibrated to study how realistic social networks and disease transmission characteristics shape early epidemic growth patterns, general transmission dynamics, and control of international disease emergencies such as the 2009 A/H1N1 influenza pandemic and the 2014-2015 Ebola epidemic in West Africa.
Deterministic and Stochastic Reduced Order Modeling of Microscopic Organism Motility Mechanisms
Prof. Sorin MitranUniversity of North Carolina at Chapel Hill (Host: Lisa Fauci)
Microscopic organisms exhibit various modes of propulsion: ciliary or flagellar beating, lamellipodium protrusion followed by attachment/detachment to a substrate, taking over control of actin production in a host cell. High-throughput computational simulation can provide detailed description of specific motility aspects, but their cost and complexity is an impediment to furthering biological understanding of organism propulsion. This talk presents work on the transformation of detailed computational simulation into tractable reduced-order models. Two specific cases are considered: (1) ciliary propulsion to illustrate model reduction of a deterministic system, and (2) propulsion of Listeria monocytogenes to illustrate aspects of stochastic model reduction. In ciliary propulsion, molecular dynamics level computation is used to furnish a detailed description of mechanical behavior of the microtubule constituents of a cilium. The data is used to construct a finite element model that is markedly different from the Euler-Bernoulli beam models typically used in cilia studies. L. monocytogenes moves by taking over the production of actin within a host cell. Stochastic modeling of the growth of the host cytoskeleton catalyzed by L. monocytogenes is used to construct a statistical model of the flight/forage behavior that can be used to infer infection virulence. Model reduction in this case involves consideration of the differential geometry of probability distributions, a field of study known as information geometry. The model reduction procedures are presented at a conceptual level, avoiding technical details, concentrating on the goal of arriving at correct models of direct utility to biology.
Limit Theorems for Composition of Functions
Michael AnshelevichTexas A&M (host: Mahir Can)
Limit theorems for sums of independent random variables (or, equivalently, for convolutions of measures) are a cornerstone of classical probability theory. Distributions arising as limits in these theorems are called infinitely divisible.
We will discuss limit theorems for repeated composition of functions on the upper half-plane. Note that unlike addition or convolution, composition is a non-commutative operation. What are the limit theorems? Which functions arise as limits? We will see both parallels and differences from the usual setting. This is joint work with John D. Williams.
On invariant measures for Hamiltonian PDEs
Geordie RichardsUtah state university (Host: Nathan Glatt-holtz)
Abstract: We will survey some recent results on the construction and proof of invariance for certain canonical measures, such as the Gibbs measure, under the flow of dispersive Hamiltonian PDEs. Proving the invariance of these measures is often nontrivial due to the low regularity of functions belonging to their support. Focus will be placed on the generalized Korteweg-de Vries (gKdV) equations; Bourgain proved invariance of the Gibbs measure for KdV and mKdV, which have quadratic and cubic nonlinearities, respectively. Previously, we proved invariance of the Gibbs measure for the quartic gKdV by exploiting a nonlinear smoothing induced by initial data randomization. More recently, in joint work with Tadahiro Oh (Edinburgh) and Laurent Thomann (Nantes), we have established this invariance for gKdV with any odd power (defocusing) nonlinearity using a probabilistic construction of solutions.
Efficient Discretization Methods for Magnetohydrodynamic Flow Simulationc
Prof. Leo RebholzClemson University (Host Kun Zhao)
After an introduction to magnetohydrodynamics (MHD), which describes the flow of electrically conducting fluids, we will discuss the major difficulties associated with computing MHD solutions. We will then present two novel approaches to circumventing these difficulties. The first is using the Elsasser change of variable approach, which allows for an unconditional energy-stable decoupling of the different physical processes at each time step in a temporal discretization. A second approach is based on an algebraic decoupling scheme (build the matrix, then split) that also allows for an accurate decoupling but with a mild CFL condition. High level details of the analysis and derivation of these methods will be discussed, and several open problems will be presented.
Spectral and Nonlinear Stability of Viscous Detonation Waves
Gregory LyngUniversity of Wyoming (HOST: Vincent Martinez)
In this talk we give an overview of a body of results pertaining to the stability of detonation waves. These are particular, dramatic solutions to systems modeling mixtures of reacting gases. They are known to have delicate stability properties. On the mathematical side, the centerpiece of the program is the Evans function. This is a spectral determinant whose zeros agree in location and multiplicity with the eigenvalues of the linearized operator about the wave; it enters the analysis at both the nonlinear and linear/spectral levels. We discuss both theoretical aspects of the Evans function and also issues related to its practical computation. On the physical side, much of the novelty of this body of work stems from the inclusion of oft-neglected diffusive effects (e.g., viscosity, heat conductivity, species diffusion) in the analysis. Indeed, this modeling choice sometimes leads to surprising results.