Presentations

Lax Equations and the Toda Lattice.  Student Analysis Seminar, University of Michigan, September 26th 2023.  [Notes] [Chalkboard]

This talk was given in the Student Analysis Seminar at the University of Michigan to an audience of 7 graduate students.  It was a summary of parts of a reading course that I took with several graduate students, led by Peter Miller.  The talk was ~55 minutes long.

Abstract: The Toda lattice is a system of ordinary differential equations describing the motion of particles on a line connected via springs.  This Toda lattice system is notable for its ability to be rewritten as a Lax equation, a certain differential equation of time-dependent linear operators.  In this talk, we will introduce the Lax equation and describe its connection to the Korteweg-de Vries equation, as well as establish the Toda lattice in the form of a Lax equation.  Time permitting, we will also discuss several methods for solving the Toda lattice system that use orthogonal polynomials, QR factorization, and a Riemann-Hilbert problem.

Residual Estimates for an Anisotropic Swift-Hohenberg Equation.  Student Analysis Seminar, University of Michigan, March 7th 2023.  [Notes] [Chalkboard]

This talk was given in the Student Analysis Seminar at the University of Michigan to an audience of 3 graduate students.  It described a senior writing project that I worked on in Spring 2021, during my time as an undergraduate at the University of Minnesota.  The talk was ~50 minutes long.

Abstract: The Swift-Hohenberg equation, originally derived to model convective instabilities, serves as a useful model for pattern formation.  Using a formal multiple scales expansion, we can derive an amplitude equation, known as the complex Ginzburg-Landau (CGL) equation, for an anisotropic Swift-Hohenberg (aSH) equation.  Solutions to (CGL) can be used to approximate solutions to (aSH) on a larger domain, with arbitrarily small residual estimates.  In this talk, we will demonstrate such estimates for an approximation in the space of bounded continuous functions on R^2.  We will then define uniformly local Sobolev spaces and establish similar estimates in these more suitable spaces.

Introduction to Fractional Differential Equations.  Student Analysis Seminar, University of Michigan, April 6th 2022.

This talk was given in the Student Analysis Seminar at the University of Michigan to an audience of 7 graduate students.  It served as a continuation of my "Fractional Derivatives" talk.  The talk was ~50 minutes long.

Abstract: Fractional calculus is an area of analysis that aims to generalize the derivative and integral operators to fractional order.  These fractional operators are collectively known as differintegrals, and they naturally lead to the idea of fractional differential equations.  Such equations have several applications in solving engineering and physics problems.  In this talk, we will introduce some basic fractional differential equations and discuss methods for solving them.  We will also use differintegrals to derive a formula for the heat flux for a certain diffusion problem.  Time permitting, we will show how differintegrals can be used to find solutions for ordinary differential equations.

Solutions and Asymptotics for the Phase-Diffusion Equation.  Undergraduate Mathematics Research Seminar (online), University of Minnesota, November 12th 2020.  [Slides]

This talk was given online in the Undergraduate Mathematics Research Seminar at the University of Minnesota to an audience of ≤ 10 undergraduate students.  It describes the research done as part of the University of Minnesota Complex Systems REU in Summer 2020, hosted by Arnd Scheel.  The talk was ~50 minutes long.

Abstract: The phase-diffusion equation is a partial differential equation, solutions of which are closely related to the formation of striped patterns.  In particular, on an expanding domain, one can introduce self-organizing patterns in order to study them further.  We will discuss both theoretical and numerical approaches to finding solutions to this equation.  We will also demonstrate the dependence of wavenumbers on wave speeds, and analyze the behavior of wavenumber asymptotics for large and small wave speeds.  This work was done as part of the 2020 Complex Systems REU at the University of Minnesota.

Abstract Calculus I.  Math Club, University of Michigan, October 5th 2023.  [Poster] [Notes] [Chalkboard]

This talk was given at the University of Michigan Math Club to an audience of about 35 people, most of whom were undergraduate students.  This talk discussed results presented in Non-Newtownian Calculus by Michael Grossman and Robert Katz, specifically the classical derivative, the geometric derivative, and the ⭑-derivative defined on an abstract arithmetic system.  The talk was ~55 minutes long.

Abstract: Consider the definition of the classical derivative f' of a function f.  The construction of this concept relies on the ideas of length and slope.  But if we change our perception of length and slope, then we can get some wacky new definitions of the derivative, such as the geometric derivative f* which is a key part of constructing calculus in an exponential sense.  We could go even further: create an abstract arithmetic, define derivatives in a more general abstract sense, and then observe the crazy consequences!  Let's explore what happens when we alter our foundation of arithmetic and try to adapt the familiar theorems from Calculus I into Abstract Calculus I!

Constructing the Real Numbers.  Math Club, University of Michigan, September 15th 2022.  [Poster] [Notes] [Chalkboard]

This talk was given at the University of Michigan Math Club to an audience of about 40 people, most of whom were undergraduate students.  This talk described a construction of the real numbers presented in Analysis I by Terence Tao.  The talk was ~60 minutes long.

Abstract: Real numbers are so prominent in math today that it's easy to take them for granted.  How are we so sure that real numbers exist?  Are they even formed on some basis of rigorous mathematical logic? Or maybe we just made them up, and everything we've done so far just magically worked out... There is in fact a way to rigorously construct the real numbers!  In this talk, we'll discuss one construction to convince ourselves that the real numbers are actually... well, real!  But in order to do that, we'll need to rigorously construct the integers and the rational numbers first, and that requires more strange symbols and careful details than you may think.  We'll even reveal the bare minimum that must be assumed about natural numbers in order to rigorously form the rest of our mathematical universe!

Fractional Derivatives.  Math Club, University of Michigan, February 3rd 2022.  [Poster] [Notes] [Chalkboard] [Desmos]

This talk was given at the University of Michigan Math Club to an audience of about 25 people, most of whom were undergraduate students.  My motivation for studying this subject matter was to learn how to compute a "half derivative" of a function, and then to present what I learned to a suitable audience.  The talk was ~50 minutes long.

Abstract: After one semester of calculus, we can compute as many derivatives of a function as we want: 1st derivatives, 2nd derivatives, 100th derivatives, and so on.  But is it possible to define a "-1st derivative" or a "1/2 derivative"? These "fractional derivative" operators, also called "differintegrals," do in fact exist!  In this talk, we'll introduce how to take the derivative of a function of order q, where q is an arbitrary real number, using only first-year calculus (and some help from the gamma function).  We'll compute differintegrals of some basic functions, discuss their properties, observe the graphs of differintegrals, and explore one or two applications of fractional derivatives.

REU Panel.  Math Club (online), University of Minnesota, November 19th 2020.

This talk was given at the University of Minnesota Math Club to an audience of ≤ 10 undergraduate students.  It was a panel consisting of me and 3 other undergraduate students who had participated in a Research Experience for Undergraduates (REU), with the intent to educate other undergraduate students about our REU experiences and encourage them to apply.  The panel was ~50 minutes long.

Abstract: If you haven't heard about REUs before, they are summer programs for college students where you are paid to do research at a university for 8-10 weeks.  Come hear our great panelists talk about their experiences with applying for and attending REUs in mathematics!

MREG Presentation: Parabolic PDEs.  Parabolic PDEs MREG (online), University of Michigan, July 1st 2022.  [Slides]

This talk was given online at the University of Michigan.  It describes material studied in the as part of the Michigan Research Experience for Graduates (MREG) on Parabolic PDEs in Summer 2022, organized by University of Michigan graduate students and led by Neel Patel.  The talk was presented with the other participants of the Parabolic PDEs MREG, including Abhishek Adimurthi (Indiana University Bloomington), Santiago Cordero-Misteli (Stony Brook University), Zhengjun (Jasper) Liang (University of Michigan Ann Arbor), Ning Tang (University of California Berkeley), and Lizhe Wan (University of Wisconsin Madison).  This talk was given to the other undergraduate participants and graduate mentors of the REU that worked on separate projects.  The talk was ~35 minutes long.

Topics include using the Fourier transform to solve parabolic PDEs, deriving decay estimates for a linearization of the Muskat problem, and deriving and analyzing solutions to a nonlinear PDE modeling the distribution of roots for a polynomial under differentiation.

Arcsine Solution to a Nonlocal Transport Equation.  Parabolic PDEs MREG (online), University of Michigan, June 27th 2022.  [Slides]

This talk was given online at the University of Michigan.  It describes material studied in the as part of the Michigan Research Experience for Graduates (MREG) on Parabolic PDEs in Summer 2022, organized by University of Michigan graduate students and led by Neel Patel.  This talk was presented by myself to the other 6 participants of the Parabolic PDEs MREG.  Here, I describe in detail an arcsine solution to a nonlinear PDE modeling the distribution of roots for a polynomial under differentiation, including the derivation and the motivation for the solution.  The talk was ~20 minutes long.

Distributional Derivatives of Discontinuous Functions.  DRP Final Project Event (online), University of Minnesota, December 3rd 2020.  [Slides]

This talk was given online at the University of Minnesota as a final project for the Directed Reading Program in Fall 2020, mentored by graduate student Cole Jeznach, where we studied functional analysis.  We read Chapters 1-3 of Introductory Functional Analysis with Applications by Erwin Kreyszig and Chapters 1-2 of Lecture notes: harmonic analysis by Russell Brown.  My motivation for participating in this project was to understand precisely what the "delta function" is using mathematical rigor, and connecting its definition to ideas in functional analysis.  These slides provide my interpretation of the connection between functionals and the "delta function."  The talk was ~15 minutes long and was presented to 2 other Directed Reading Program participants.

Quenching in the Phase-Diffusion Equation.  Complex Systems REU (online), University of Minnesota, July 22nd 2020.  [Slides]

This talk was given online in the Undergraduate Mathematics Research Seminar at the University of Minnesota to an audience of ≤ 10 undergraduate students.  It describes the research done as part of the University of Minnesota Complex Systems REU in Summer 2020, hosted by Arnd Scheel.  This talk was presented with Kelly Chen (Massachusetts Institute of Technology), my partner in the REU project, and was presented to the other 4 undergraduate participants and 3 graduate mentors of the REU that worked on separate projects.  The talk was ~20 minutes long.

Abstract: We study traveling-wave solutions to the two-dimensional phase-diffusion equation on the half-plane with nonlinear boundary condition, and describe a connection between the solution with a stripe solution of the Swift-Hohenberg equation.  We use both theoretical and numerical methods to analyze the dependence of wavenumbers on wave speeds and compute limiting values.