# Research

## Graduate Seminar Series (Leader: Nathan Hayford <nhayford (at) usf.edu>document.write('<a href="mai' + 'lto:' + 'nhayford' + '&#64;' + 'usf.edu' + '">Nathan Hayford</a>');)

### Friday, December 3, 2021

Title: An Introduction to Coding Theory
Speaker: Austin Dukes
Time: 2:00pm–3:00pm
Place: Zoom Meeting

#### Abstract

In this talk we will discuss the basic notions of coding theory. A (linear) $$[n,k,d]_q$$ code $$C$$ forms a vector space over $$\mathbb{F}_q$$, the finite field with q elements, so we will begin by providing a brief overview of finite fields. Then we will shed some light on the relationships among the parameters $$n, k, d$$ by observing the singleton bound. Finally, if time permits, we will consider some fundamental constructions of maximum distance separable (MDS) codes, i.e., codes whose parameters achieve equality with respect to the singleton bound. Audiences of all backgrounds are welcome.

### Friday, November 12, 2021

Title: A Brief Introduction to Lamplighter Groups and Diestel-Leader Graphs
Speaker: Austin Gaffglione
Time: 2:00pm–3:00pm
Place: Zoom Meeting

#### Abstract

In 1990, W. Woess proposed the following question: Is every “nice” infinite graph quasi-isometric to the Cayley graph of some finitely generated group? In 2001, R. Diestel and I. Leader conjectured that some of graphs in the family they defined were not quasi-isometric to the Cayley graph of any finitely generated group. This was later proven to be true in a 2012 paper by A. Eskin, D. Fisher, and K. Whyte. We will examine the general construction of a Lamplighter group and observe some of its algebraic properties. We will also develop the notion of a horocyclic product of two trees, which will define a subfamily of the Diestel-Leader graphs. In particular, we will focus on the graph DL(2) and show that it is isomorphic to the Cayley graph of the Lamplighter Group L2 with respect to a certain choice for the generating set.

### Friday, November 5, 2021

Title: Enhanced diffusivity and skewness of a diffusing tracer in the presence of an oscillating wall
Speaker: Lingyun Ding
University of North Carolina at Chapel Hill
Time: 2:00pm–3:00pm
Place: Zoom Meeting

#### Abstract

We develop a theory of enhanced diffusivity and skewness of the longitudinal distribution of a diffusing tracer advected by a periodic time-varying shear flow in a straight channel. Although applicable to any type of solute and fluid flow, we restrict the examples of our theory to the tracer advected by flows which are induced by a periodically oscillating wall in a Newtonian fluid between two infinite parallel plates as well as flow in an infinitely long duct. These wall motions produce the well-known Stokes layer shear solutions which are exact solutions of the Navier–Stokes equations. With these, we first calculate the second Aris moment for all time and its long-time limiting effective diffusivity as a function of the geometrical parameters, frequency, viscosity, and diffusivity. Using a new formalism based upon the Helmholtz operator, we establish a new single series formula for the variance valid for all time. We show that the viscous dominated limit results in a linear shear layer for which the effective diffusivity is bounded with upper bound $$\kappa (1+A^2/(2L^2))$$, where $$\kappa$$ is the tracer diffusivity, $$A$$ is the amplitude of oscillation, and $$L$$ is the gap thickness. Alternatively, for finite viscosities, we show that the enhanced diffusion is unbounded, diverging in the high-frequency limit. Non-dimensionalization and physical arguments are given to explain these striking differences. Asymptotics for the high-frequency behavior as well as the low viscosity limit are computed. We present a study of the effective diffusivity surface as a function of the non-dimensional parameters which shows how a maximum can exists for various parameter sweeps. Physical experiments are performed in water using particle tracking velocimetry to quantitatively measure the fluid flow. Using fluorescein dye as the passive tracer, we document that the theory is quantitatively accurate. Specifically, image analysis suggests that the distribution variance be measured using the full width at half maximum statistic which is robust to noise. Further, we show that the scalar skewness is zero for linear shear flows at all times, whereas for the nonlinear Stokes layer, exact analysis shows that the skewness sign can be controlled through the phase of the oscillating wall. Further, for single-frequency wall modes, we establish that the long-time skewness decays at the faster rate of $$t^{-3/2}$$ as compared with steady shear scalar skewness which decays at rate $$t^{-1/2}$$. These results are confirmed using Monte-Carlo simulations.

### Friday, October 29, 2021

Title: COVID-19 Spreading: An Eigenvalue Viewpoint
Speaker: Haowei (Alice) Chen
Time: 2:00pm–3:00pm
Place: Zoom Meeting

#### Abstract

How will a virus propagate? How long does it take to disinfect given infection rate and virus death rate? What is the best node to immunize? We answer these questions by developing a nonlinear dynamical system that models viral propagation in any arbitrary network and also a general epidemic threshold is proposed: the inverse of the largest eigenvalue of the adjacency matrix.

### Friday, October 22, 2021

Title: Zeros of Harmonic Polynomials and Related Applications
Speaker: Azizah Alrajhi
Time: 2:00pm–3:00pm
Place: Zoom Meeting

#### Abstract

In this thesis, we study topics related to harmonic functions, where we are interested in the maximum number of solutions of a harmonic polynomial equation and how it is related to gravitational lensing. In Chapter 2, we studied the conditions that we have to have on the real or complex coeffcients of a polynomial $$p$$ to get the maximum number of distinct solutions for the equation $$p(z)-\bar{z}^2=0$$, where $$\deg p=2$$. In Chapter 3, we discuss the lens equation when the lens is an ellipse, a limaçon, or a Neumann Oval. Also, we discuss a counterexample for a conjecture by C. Bénéteau and N. Hudson in [1]. We also, in particular, discuss estimates related to the maximum number of solutions for the lens equations for the Neumann Oval.

##### Reference
1. C. Bénéteau and N. Hudson. A survey on the maximal number of solutions of equations related to gravitational lensing. Complex Analysis and Dynamical System, Trends Math. Cham: Birkhäuser/Springer, 23–38.

### Friday, October 15, 2021

Title: The Continuum Limit of Theta Functions
Speaker: Fudong Wang
University of Central Florida
Time: 2:00pm–3:00pm
Place: Zoom Meeting

#### Abstract

The purpose of the talk is to introduce Venakides' 1989 CPAM XLII paper on the continuum limit of theta functions. The problem is closely related to the semiclassical limit (or small dispersion, or nonlinear WKB analysis) for the famous nonlinear PDE: Korteweg-de Vries equation. Theta function can be considered as high dimensional Fourier series. The continuum limit of theta function we will discuss is the limit when the size of the period matrix of the theta function goes infinite in a certain way (or one can say the genus of the corresponding Riemann surface goes infinite). And we will associate the leading behavior with a minimizing problem (variational problem).

### Friday, October 8, 2021

Title: Introduction to Itô Calculus
Speaker: Nathan Hayford
Time: 2:00pm–3:00pm
Place: Zoom Meeting

#### Abstract

The Itô calculus is at the foundation of the theory of stochastic PDEs. In this talk, I will introduce the calculus, and prove Itô’s lemma. I will then demonstrate how the theory of martingales $$+$$ Itô’s lemma can be applied to derive deterministic equations for certain quantities appearing in the stochastic calculus. In particular, I will prove the conformal invariance of Brownian motion and the harmonic measure. If time permits, I will also start a discussion about Schramm-Loewner evolutions (SLEs).

### Friday, October 1, 2021

Title: Martingales and Stopping Times, II
Speaker: Zachary Forrest
Time: 2:00pm–3:00pm
Place: Zoom Meeting

### Friday, September 24, 2021

Title: Martingales and Stopping Times
Speaker: Zachary Forrest
Time: 2:00pm–3:00pm
Place: Zoom Meeting

#### Abstract

In Stochastic Calculus and elsewhere, the notion of Martingales and Stopping Times are invaluable for characterizing random processes whose mean behavior is invariant but whose data changes over time. In this talk we will clearly define the notions of Martingales, Stopping Times, and any necessary analytical objects; introduce basic properties of these objects, including Doob's “Optimal Sampling” Theorem; and briefly touch upon some applications of Martingales and Stopping Times in research.

### Friday, September 17, 2021

Title: Drifting through random walks and Brownian motions
Speaker: Nathan Hayford
Time: 2:00pm–3:00pm
Place: Zoom Meeting

#### Abstract

In this talk, we will leisurely wander through the area of study of random walks and Brownian motions. We will begin by studying many of the important properties and realizations of random walks. In particular, we will examine the return probabilities of random walks in 1, 2, and 3 dimensions; as more eloquently stated by S. Kakutani, “a drunk man will find his way home, but a drunk bird will be lost forever”. We will also study the scaling limit of a random walk, the so-called Brownian motion. Many of the properties of random walks will be reflected by Brownian motions, and we will see that they are the first example of a very important class of random processes known as martingales.

### Friday, September 10, 2021

Title: Introduction to Probability, Part II
Speaker: Louis Arenas
Time: 2:00pm–3:00pm
Place: Zoom Meeting

### Friday, September 3, 2021

Title: Introduction to Probability, Part I
Speaker: Louis Arenas
Time: 2:00pm–3:00pm
Place: Zoom Meeting

#### Abstract

In this talk, we reinterpret the concepts from measure theory under the lens of probability theory. Using concrete examples, the aim is to introduce important concepts in probability theory and interpret their statistical meaning and connection to measure theory. This first talk aims to build an intuitive idea of the machinery used in probability theory to prove weak and strong law of large numbers, a theorem often taken for granted in applied statistics. In the second part we hope to motivate central limit theorem, and give a proof using characteristic functions. The target audience are students who have limited exposure to measure theory (statistics students), and students who are currently studying real analysis.