# 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.

Past AMC: [Fall 2017] | [Spring 2017] | [Fall 2016]

### ➜ Spring 2018

**January 25**

Speaker: Braxton Osting

**Title: **Extremal Spectral Gaps for Periodic Schrödinger Operator

**Abstract: ** The spectrum of a Schrödinger operator with periodic potential generally consists of bands and gaps.
I will discuss the problem of finding a potential that gives maximum length spectral gaps. This is joint work with
C.-Y. Kao.

**Feburary 8**

Speaker: Dong Wang

**Title: **Recent Advances on the Threshold Dynamics Method

**Abstract: ** The threshold dynamics method developed by Merriman, Bence, and Osher (MBO) is an efficient method for
simulating the mean curvature flow. In this talk, I will discuss some recent advances on the threshold dynamics method.

**Feburary 15**

Speaker: Chee Han Tan

**Title: **Extremal Problems for Laplacian Eigenvalues

**Abstract: ** Historically, the first spectral optimisation problem is due to Lord Rayleigh in 1877, where he conjectured
that the disk should minimise the first Laplace-Dirichlet eigenvalue among all planar domains of equal area. Almost 30 years
later G. Faber and E. Krahn proved this result independently and the conjecture holds in higher dimension as well. Much
progress has been made over the past few decades due to significant development in variational analysis and optimisation
methods. We review existing results for the constrained problem of finding an optimal domain that minimises or maximises the
$k$-th eigenvalue of the Laplacian operator with boundary conditions such as Dirichlet, Neumann or Steklov. We will then
discuss the proof of the Faber-Krahn inequality for the first Laplace-Dirichlet eigenvalue and the Hersch-Payne-Schiffer
inequality for the second Laplace-Steklov eigenvalue, the latter leads to the assertion that the disk maximises the second
Laplace-Steklov eigenvalue among plane open sets of given area.

**Feburary 22**

Speaker: Huy Dinh

**Title: **Fractional Brownian Motion: Formulation and Applications

**Abstract: ** Fractional derivatives are derivatives where the order of differentiation may be any number, real in this
presentation. While they were first discussed by Leibniz in 1695 during the formulation of calculus, their lack of a
transparent physical and geometric meaning deterred their development and application. Now, they have found numerous,
naturally motivated applications. We will present the history of the Riemann-Liouville (RL) Fractional Derivative to motivate
its formulation. We will then discuss how RL fractional differential equations can naturally arise by presenting examples. Two
other formulations of fractional derivatives, Cauchy and Caputo, will be compared to the RL formulation. Applications to time series and signal processing will be discussed.

**March 1**

Speaker: Nathan Willis

**Title: **Sloshing and the Two-Dimensional Ice-Fishing Problem

**Abstract: ** Physically, sloshing is the problem of a fluid moving in a container which can lead to significant
consequences. These consequences can be either everyday occurrences such as spilling your cup of coffee or more severe examples
such as an overturned milk truck or oil tankers. Mathematically, the sloshing problem is an eigenvalue problem where the
eigenvalue appears in the boundary condition. In this talk I will consider a special case of the sloshing problem known as the
ice-fishing problem. I will give a brief history and derivation of the problem. Then I will present the work done to determine
several of the smallest eigenvalues. Much of the talk will be based on the work done by Peter Henrici, B. A. Troesch and Luc
Wuytack in their 1970 paper
"Sloshing Frequencies for a Half-space with Circular or Strip-like Aperture".

**April 19 **Note the time 4:30pm****

Speaker: Hyunjoong Kim

**Title: **The Chapman Kolmogorov Equation: A Universal Tool for Math Biology

**Abstract: ** Most of the biological phenomena in a cellular level can be described by Stochastic Differential Equations
(SDE). The Kolmogorov forward equation translates SDE to PDE and yields easier for "mathematician" to analyze the problem.
I will discuss a derivation of the forward equation and first moment equation of switching environment as an application. This talk is based on

[1] J. Keener. Lecture notes
on stochastic processes in physiology.

[2] P. C. Bressloff and S. D. Lawley. Moment equations for a piecewise deterministic PDE, Journal of Physics A: Mathematical and Theoretical, 48 105001 (2015).

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