Writing

Stats 525 Final Project.  April 28th 2024.  [PDF]

This paper was written as a final project for Stats 525: Probability Theory, taken in the Winter 2024 semester at the University of Michigan, taught by Hai Le.

In this final project for Stats 525, we introduce basic definitions in probability from a measure theory perspective, including properties involving random variables and expectation.  We state and sketch a proof of Kolmogorov's 0-1 law as an application of measure theory in probability.  We then define conditional expectation, first in a natural sense for discrete random variables, and then extending the definition to general random variables, and give basic properties thereof.  We then introduce martingales and give applications to random walks and betting strategies.  We end with the supermartingale convergence theorem and build the foundation for its proof.

Notes on Abstract Derivatives.  September 1st 2023.  [PDF]

These notes were written for a Math Club talk at the University of Michigan, given in October 2023.  It summarizes chosen sections of Multiplicative calculus and its applications by A. Bashirov, E. Kurpınar, and A. Ozyapıcı, and Non-Newtonian Calculus by M. Grossman and R. Katz.

In these notes, we establish various constructions of derivatives by defining them through functions of uniform growth.  We review classical derivatives and their familiar properties, and compare them to those of geometric derivatives.  We prove coexistence of classical and geometric derivatives, giving formulas for each in terms of the other, in addition to stating geometric analogues of various classical differentiation theorems.  We briefly discuss anageometric and bigeometric differentiation, and prove coexistence of all four derivative definitions.  We then investigate the abstract arithmetic and generalize the derivative to this altered system, relating it to all the aforementioned derivative constructions.

Math 656 Final Project.  December 10th 2022.  [PDF]

This paper was written as a final project for Math 656: Introduction to Partial Differential Equations, taken in the Fall 2022 semester at the University of Michigan, taught by Zaher Hani.

This final project for Math 656 covers Sections 4.2 and 4.4 of Partial Differential Equations by Lawrence C. Evans. More specifically, topics include: traveling wave solutions of the Korteweg-de Vries equation, stability curves for the reaction-diffusion equation, similarity solutions under scaling for the porous medium equation, the Cole-Hopf transformation for a quasilinear parabolic equation, and potential functions for Euler’s equations.

Notes on Fractional Calculus.  March 12th 2022.  [PDF]

These notes were written for a Math Club talk at the University of Michigan, given in February 2022, and then expanded upon for a Student Analysis Seminar talk at the University of Michigan, given in April 2022.  It summarizes chosen sections of The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order by Keith B. Oldham and Jerome Spanier.  The motivation of studying this topic was to rigorously define and make sense of a "fractional derivative," also known as a "differintegral."  The idea of extending derivatives to arbitrary order is known as "fractional calculus."  These notes are intended to be an introduction to fractional calculus, emphasizing computations and examples rather than abstract theory and proofs.  These notes are intended for those who have interest in learning about fractional derivatives and have taken Calculus I, although some topics from Calculus II and ordinary/partial differential equations appear.

In these notes, we investigate derivatives and integrals of arbitrary order by deriving the Riemann-Liouville integral formula that gives the q-differintegral of a function for real q.  We discuss the composition rule for differintegrals and analyze specific cases to show that interchanging differintegrals does not always work.  We work towards deriving a formula for the differintegral of a power function, and use it to give expressions for differintegrals of other functions using Taylor series.  We give examples of applications of fractional calculus, including the evaluation of definite integrals, solving the tautochrone problem, discussing solutions to Legendre’s and Bessel’s equations, and analyzing certain diffusion problems.  We also give a brief introduction to extraordinary differential equations, including semidifferential equations and applications to the heat equation.

Residual estimates for an anisotropic Swift-Hohenberg equation.  May 10th 2021.  [PDF]

This paper was written for a senior writing project at the University of Minnesota, mentored by Arnd Scheel in Spring 2021.  It is based off of work done in two textbooks: Pattern Formation: An Introduction to Methods by Rebecca Hoyle, and Nonlinear PDEs: A Dynamical Systems Approach by Guido Schneider and Hannes Uecker.  In those books, Hoyle derives an amplitude equation (the Ginzburg-Landau equation) for an anisotropic Swift-Hohenberg equation in one dimension, and Schneider and Uecker establish residual estimates for the approximation in uniformly local Sobolev spaces.  

In this paper, we replicate the work done by these authors for an anisotropic Swift-Hohenberg equation in two dimensions.  We derive an amplitude equation for an anisotropic Swift-Hohenberg equation via multiple scales expansion by establishing an approximation using a complex Ginzburg-Landau equation.  We demonstrate estimates for this approximation by making the residual arbitrarily small in the space of bounded continuous functions.  We also make progress toward estimating the approximation in uniformly local Sobolev spaces, while developing the framework necessary to do so.

Complex Analysis.  December 11th 2020.  [PDF]

This paper is a write-up of various topics in complex analysis, written in the style of lecture notes.  Along with stating definitions and theorems, the paper emphasizes iconic examples and connections rather than abstract theory and proofs.  It is intended for anyone who may be interested in learning basic results in complex analysis through detailed examples. 

This write-up contains the following topics.  Constructions of the complex numbers, complex arithmetic and algebra, sequences and series, complex differentiation, Cauchy-Riemann equations, Cauchy-Goursat theorem, Cauchy's integral formula and corollaries.  I initially intended to keep updating this write-up over time, but it seems unlikely that I will be able to do that...

Quenching in the phase-diffusion equation.  July 22nd 2020.  [PDF]

This paper was written with Kelly Chen (Massachusetts Institute of Technology) for the University of Minnesota Complex Systems REU hosted by Arnd Scheel in Summer 2020.  The research done for this paper was funded by the University of Minnesota’s Undergraduate Research Opportunities Program and the National Science Foundation.  The work that was done for this report was adapted into a publication.

In this paper, 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.

Algebraic Topology for Everyone.  May 1st 2020.  [PDF]

This paper was written as a final project for the University of Minnesota Directed Reading Program, mentored by graduate student Tobias Ramaswamy in Spring 2020, where we studied algebraic topology.  It summarizes Chapter 1 of Algebraic Topology by Allen Hatcher, covering the basics of the fundamental group, the van Kampen theorem, and covering spaces.  The paper is written for both mathematical and non-mathematical audiences, with the intent that anybody can read it and understand the material.  To accomplish this, along with giving pictures and simple examples, we write each definition and theorem in two parts: a mathematically rigorous description, and an "idea" which intuitively describes that rigorous description.