Applied Math Collective

Applied Math Collective was initiated by my advisors and Fernando Guevara Vasquez. The aim is to provide an informal platform where the speaker discusses general-interest "SIAM review"-style applied math papers, led by either faculty or graduate student. We meet Thursdays at 4pm in LCB 323, when the Department Colloquium does not have a speaker. Please contact me if you would like to attend or give a talk so that I can add you to the mailing list.

➜ Summer 2018

June 14
Speaker: Liz Fedak
Title: Avoidable bistability: Analyzing and establishing novel bifurcations
Abstract: Bistability and hysteresis are widely used mechanisms in biological modeling. While systems classically feature a single switch between one low and one high equilibrium branch, a recently constructed biological system features two hysteretic switches on either end of a collapsing high equilibrium. Increasing the bifurcation parameter too quickly causes the system to bypass the bistable region, inspiring the term ``avoidable bistability.'' More surprisingly, the two switches interact with each other through two previously unstudied bifurcations: a codimension-1 bifurcation characterized by a one-dimensional locus of saddle-node bifurcations, and a codimension-2 bifurcation characterized by a one-dimensional locus of Takens-Bogdanov bifurcations. This talk will focus on the ongoing process of establishing the theoretical framework for both bifurcations. We construct generic polynomial representations with the desired behavior, use perturbation theory to determine Hopf bifurcation criticality and estimate homoclinic orbit location, and discuss derivation of the bifurcation conditions required to prove whether the generic representations are indeed normal forms. In short, listeners will be taken on a journey starting with cancer biology and ending with topological equivalence proofs.

June 21
Speaker: Kyle Steffen
Title: The Difference Potentials Method
Abstract: The Difference Potentials Method (DPM) is a framework for highly accurate and efficient numerical approximation of partial differential equations. In this talk, I will review the underlying mathematics, introduce the DPM, and discuss recent work.

June 28
Speaker: Chris Miles
Title: The wiggles: diffusive search for a diffusive target
Abstract: Suppose you're a particle diffusing in space, looking for a tiny ($\varepsilon$ wide) target that lives on some surface. Add the following twist: the target is also diffusing along the surface. Does this extra wiggling help you find your target faster or make (the already grim) matter worse? In this talk, I'll start by motivating this scenario with a classical result in biophysics. I'll then discuss how Sean Lawley and I added the target diffusion by formulating a PDE with random boundary conditions, resulting in a stochastic PDE (but not the Martin Hairer kind!). To study it, we'll borrow some results from dusty old electrostatics books involving Green's functions of the Laplacian to constructed a matched asymptotics solution in $\varepsilon$.

July 5
Speaker: Chee Han Tan
Title: Schauder Fixed Point Theorem
Abstract: Fixed point theorems are some of the most general yet powerful results in mathematics. I will discuss Brouwer fixed point theorem and prove two generalisations: Schauder and Schaefer's fixed point theorem. Time permitting, I will demonstrate how one uses Schauder's fixed point theorem to establish the existence of weak solutions of semilinear elliptic PDEs.

July 12
Speaker: Rebecca Hardenbrook
Title: Padé Approximation: In Theory and In Practice
Abstract: Inspired by the work of Frobenius, Henri Padé introduced a technique in the late 1800s by which a function's power series could be approximated by rational functions. This method has been shown to produce more rigorous bounds than the well known method of Taylor series truncation, even providing approximations in the case that the Taylor series of the given function does not converge. In this talk, we will introduce Padé approximants and some theory surrounding them, including theory pertaining to Stieltjes series and representations. We will conclude with a discussion on current research utilizing this approximation technique for the homogenized advection-diffusion equation.

July 19
Speaker: Huy Dinh
Title: Factorization of Functions, an overview
Abstract: Meromorphic functions, which are analytic away from isolated poles, were studied by Weiestrass' and his student Mittag-Leffler. I plan to discuss their work on infinite product representations as another tool to understand functions. Illustrations of infinite products with their resulting functions, significant theorems from Weierstrass and Mittag-Leffler and an application will be presented.

July 26
Speaker: China Mauck
Title: The Mathematical Foundation of Radar
Abstract: Radar was originally developed within the engineering community but is now the subject of many open problems in mathematics. In this talk, I will walk through the mathematics underlying basic radar systems and methods, providing an intuitive understanding of the basic principles of radar. This talk is based on material from the book "Fundamentals of Radar Imaging" by Margaret Cheney and Brett Borden.