Directed Reading Program

Department of Mathematics, UT-Austin

Spring 2020 Projects.

Note: Some projects with larger video files may take a little while to load.


  1. Aidan O'Keeffe

  2. Aislinn Smith

  3. Angela Cao
    • K-Means in Image Clustering
    • Abstract:
      • Exploring K-nearest neighbors and K-means with an application in pixel quantization.
    • Prerequisites: Some linear algebra and some Python
    • Mentor: Shane McQuarrie

  4. Ashwin Devaraj
    • An Overview of Curvature
    • Abstract:
      • My project is a set of lecture notes that provide an introduction to differential geometry in Euclidean space, with an emphasis on the notions curvature of curves and regular surfaces. These notes culminate in the definitions of mean and Gaussian curvature, and finally the presentation of Gauss' Theorema Egregium.
    • Prerequisites: A strong background in multivariable calculus and linear algebra would be helpful. A full real analysis background isn't necessary, but it would be helpful to understand the definitions of open and closed sets in R^n.
    • Mentor: Daniel Weser

  5. Binglin Zhang
    • Barycentric Coordinate and its Applications
    • Abstract:
      • Discussing the definition and construction of Barycentric Coordinate and its recent application on figure deformation.
    • Prerequisites: Basic concepts of Geometry and Algebra.
    • Mentor: Chuning Wang

  6. Chuxuan Yang

  7. CJ Izzo
    • Plane Algebraic Curves and Bezout's Theorem
    • Abstract:
      • Defining affine algebraic curves, projective spaces and curves, and providing motivation for Bezout's theorem from the perspective of Plane Algebraic Curve theory
    • Prerequisites: A familiarity with Abstract Algebra
    • Mentor: Jon Johnson

  8. Daisy Gomez
    • Points of finite order and applications to the Nagell-Lutz Theorem
    • Abstract:
      • My project is a paper describing various properties of points with finite order on elliptic curves as well as the group law on these points in the projective plane. It includes a proof of a theorem about points of order 2 and 3 which motivates the Nagell-Lutz Theorem.
    • Prerequisites: Number Theory, Algebraic Structures, Geometry
    • Mentor: Yixian Wu

  9. Daniel Naves
    • Local Theory of Plane and Space Curves
    • Abstract:
      • My project explores the fundamental results regarding the theory of curves in Euclidean space
    • Prerequisites: Basic analysis and linear algebra
    • Mentor: Daniel Restrepo

  10. Dylan Chase Alexander
    • Pointwise Ergodic Theorem
    • Abstract:
      • Proof and interpretation of a fundamental result regarding L1 functions on measure preserving dynamical systems
    • Prerequisites: A knowledge of measure theory and basic dynamical systems is required. An example from physics is provided but physics is not crucial for the reading of this article.
    • Mentor: Frank Lin

  11. Ethan Jana

  12. Fornia Van
    • Binomial Model in the Stock Market
    • Abstract:
      • Using binomial model, martingale, markov, and conditional expectation to relate to random walk, and then relating random walk to the stock market
    • Prerequisites: Probability
    • Mentor: Yiran Hu

  13. Grant Kluber
    • Spectral Theory in Quantum Mechanics (QM)
    • Abstract:
      • This project is about the application of spectral theory to quantum mechanics, beginning with the proper definition of the spectrum for linear operators on Banach/Hilbert spaces and its three disjoint subsets: the point, continuous, and residual spectra. There is emphasis on examples, primarily since many things are hard to state exactly in spectral theory (like the spectral theorem), but are easy to behold/motivate. Intended as a blog post, I estimate a reading time of 10-20 minutes.
    • Prerequisites: Exposure to ideas/notation in quantum mechanics and real analysis.
    • Mentor: Logan F Stokols

  14. Jacob Gutierrez

  15. Jamie Pearce

  16. Jordan Grant
    • Reconciling Vector Bundles
    • Abstract:
      • Equating the two most commonly used definitions of a vector bundle to one another.
    • Prerequisites: Classical Differential Geometry, Topology
    • Mentor: Riccardo Pedrotti

  17. Joshua Wong
    • Representation Theory with a Perspective from Category Theory
    • Abstract:
      • A brief overview of representation theory leading up to a proof of Frobenius Reciprocity. Afterwards, we tackle Frobenius Reciprocity from the more general viewpoint of category theory.
    • Prerequisites: Linear algebra, group theory
    • Mentor: Saad Slaoui

  18. Karthik Vaidyanathan
    • The WLLN and CLT
    • Abstract:
      • I give an introduction to measure-theoretic probability and describe the WLLN and CLT with their proofs.
    • Prerequisites: Preferably some exposure to real analysis but the measure theory is explained in the presentation.
    • Mentor: Joe Jackson

  19. Lincole Jiang
    • Finite Spaces Tweaks and Twerks
    • Abstract:
      • Separation axiom, homeomorphism, and connectivity about finite topological spaces
    • Prerequisites: Basic understanding of point-set topology
    • Mentor: Gillian Grindstaff

  20. Matthew Huynh
    • A Brief Introduction to Riemann Surfaces
    • Abstract:
      • Defining and giving examples of the Riemann surface associated to a function
    • Prerequisites: Some familiarity with complex analysis and topology. However I included a section introducing all required info to understand the theorem that I prove at the end
    • Mentor: Sebastian Schulz

  21. Michelle Brun
    • Basic Projective Geometry
    • Abstract:
      • My project is an introduction to projective space and some of its properties.
    • Prerequisites: Euclidean geometry
    • Mentor: Max Riestenberg

  22. Neel Panging
    • The mathematical intuition of neural networks
    • Abstract:
      • My project explains the underlying mathematical concepts that allow neural networks to learn.
    • Prerequisites: A basic understanding of linear algebra and calculus would help
    • Mentor: Milad Naeini

  23. Ning Kang
    • Intro to Machine Learning: Cross Validation
    • Abstract:
      • Cross-validation is a resampling procedure used to evaluate machine learning models on a limited data sample.
    • Prerequisites: Mean Squared Error and Root Mean Squared Error.
    • Mentor: Milad Eghtedari

  24. Omar Aceval
    • The Fundamental Theorem of Galois Theory
    • Abstract:
      • This project details some important implications of the fundamental theorem of Galois theory, most notably a detailed discussion about the problem of finding a general solution to fifth order polynomials.
    • Prerequisites: A decent understanding of fields and groups, and in particular normal subgroups. The most rigorous proof is not particularly detailed and so requires little background in abstract algebra.
    • Mentor: Natalie Hollenbaugh

  25. Pratyush Potu
    • Plateau’s Problem and Minimal Surfaces
    • Abstract:
      • In this presentation, I give an introduction to the Calculus of Variations. I introduce the definition of a functional and the Euler-Lagrange Equation which I prove is a necessary and sufficient condition for finding extremal functions of functionals. Then, I talk about Plateau's Problem concerning minimal surfaces which are a particular application of the Calculus of Variations. I explain how the minimization of the surface area of a surface is a problem of minimizing a surface area functional, and how we can use the Euler-Lagrange Equation to find minimal surfaces. I conclude with some applications of minimal surfaces to other fields.
    • Prerequisites: Calculus up to the level of M408D.
    • Mentor: Erica De La Canal

  26. Rachel Thornton
    • Foundational Ideas in Topology
    • Abstract:
      • Definition of topology, open sets, limits, and closures, and theorem: A set is open if and only if its compliment is closed.
    • Prerequisites: Strong background in proofs, no topology needed
    • Mentor: Casandra Monroe

  27. Rithvik Reddy Golamari
    • Bousfield Localization of Spectra
    • Abstract:
      • The project discusses some introductory stable homotopy theory and proceeds to construct the Bousfield localization of spectra with respect to a homology theory. A small, non-technical application is also discussed in the form of the Thick Subcategory Theorem. The project is in the form of a blog-post so I've attached a link to the post below.
    • Prerequisites: Algberaic topology, category theory and some homotopy theory
    • Mentor: Richard Wong

  28. Ruben Dimas

  29. Ryan Chatterjee
    • Lie Groups and their Algebras
    • Abstract:
      • Explores the representation theory of Lie groups, by defining the associated Lie algebra, and explaining how the Lie algebra describes structure of the associated group via Baker-Campbell-Hausdorff.
    • Prerequisites: Some topology (especially differential), linear algebra, abstract algebra
    • Mentor: Tom Gannon

  30. Shannon Scofield
    • An Introduction to Representation Theory
    • Abstract:
      • My project is a talk explaining some basics of representation theory and identifying the irreps of S_3.
    • Prerequisites: Group Theory, Linear Algebra would be useful
    • Mentor: Tom Gannon

  31. Sonali Singh
    • Stochastic Calculus for Finance: Random Walk
    • Abstract:
      • "This project focuses on properties concerning Random Walk. Random walk is a stochastic (random) process of discrete steps on a fixed length. The steps do not depend on previous steps. Symmetric random walk is when the direction of each step (up or down, left or right) is equally likely, meaning p=½. In asymmetric random walk p≠½. However, both types of random walk have the same set of possible paths. In two examples, we further explore the applied value of the theory. In the first, we explore properties of random walk between dimensions (2D and 3D). In the second, we explore the reflection principle, a theory that explores how the probability of reflected paths can be utilized."
    • Prerequisites: Probability and Intro to Financial Math
    • Mentor: Luhao Zhang

  32. Yangxinyu Xie
    • Knots, Links and Spatial Graphs
    • Abstract:
      • In 1983, John Conway and Cameron Gordon published a well-known paper Knots and links in spatial graph [CMG83] that stimulated a wide range of interests in the study of spatial graph theory. In this note, we review this classical paper and some of the earliest results in intrinsic properties of graphs.
    • Prerequisites: Basic Knot Theory
    • Mentor: Alexandra Embry

  33. Yongqi Pang
    • Statistics and Data Analysis
    • Abstract:
      • This project is about using python to approach to data analysis. The main part of this project is to manipulate, process, clean and crunch data in python. We focus on learning several libraries such as NumPy, Pandas, Matplotlib and Seaborn. Besides, the topic we focus on is about the current trend—Covid-19 and we learn about the details of SIR model. At last, we try to construct a simple SIR model using python.
    • Prerequisites: Knowledge of Python programming, Statistics, and SIR model
    • Mentor: Jincheng Yang

  34. Zhecheng Wang
    • Hyperbolic Geometry in Parameterization of Surfaces
    • Abstract:
      • An idea to achieve conformal mapping of Genus 2 surface with Hyperbolic Geometry.
    • Prerequisites: Discrete Math, some basic topology definitions and a solid understanding of Euclid's Postulates.
    • Mentor: Neža Žager Korenjak