Directed Reading Program

Department of Mathematics, UT-Austin

Fall 2020 Projects.

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


Video of online symposium


  1. Aaron Jackson
    • The Lamplighter Group
    • Abstract:
      • Ways to think about the lamplighter group and solving the word problem for the group
    • Prerequisites: Definition of a group and some group theory terminology, like kernel, generators etc.
    • Mentor: Jonathan Johnson

  2. Adarsh Hullahalli
    • Knots and Some Chess
    • Abstract:
      • An overview of basic knot theory concepts and application to diagrams created from chess games
    • Prerequisites: None
    • Mentor: Arun Debray

  3. Alexy Skoutnev
    • Support Vector Machine
    • Abstract:
      • Explain what support vector machine is, and how they are used in programming languages like R to classify datasets.
    • Prerequisites: Applied Statistics, some knowledge of R and computer programming fundamentals, and basic knowledge of Calculus
    • Mentor: Milad Eghtedari Naeini

  4. Andrew J Pease
    • An Introduction to Point Set Topology
    • Abstract:
      • Summary of point set topology, including continuous functions, metric spaces, compactness and connectedness, separation axioms, and typical topologies
    • Prerequisites: Elementary set theory
    • Mentor: Shiyu Liang

  5. Angela Cao
    • Titanic Survival Estimation via Naive Bayes
    • Abstract:
      • Using Naive Bayes Algorithm and Probability & Statistics knowledge to investigate the well-known Titanic Problem.
    • Prerequisites: Python programming (Pandas, Matplotlib libraries) and Probability & Statistics
    • Mentor: Shane McQuarrie

  6. Ashwin Devaraj
    • Riemannian Manifolds and Affine Connections
    • Abstract:
      • This writeup introduces the notion of a Riemannian manifold as a generalization of regular surfaces in R^3 and develops the notion of directional differentiation of vector fields via the affine connection. It concludes with the Fundamental Theorem of Riemannian Geometry.
    • Prerequisites: Basic linear algebra and point-set topology are required, and a basic familiarity with the geometry of curves and surfaces in R^3 is recommended.
    • Mentor: Dan Weser

  7. Carter Chu
    • Integer Partitions and the Rogers-Ramanujan Identities
    • Abstract:
      • An introduction to integer partitions and partition identities using bijections and generating functions. Afterward, we summarize a proof of the Rogers-Ramanujan identities using Gaussian polynomials and generating functions.
    • Prerequisites: Basic background in combinatorics and number theory.
    • Mentor: Erin Bevilacqua

  8. Ethan Roy
    • An Overview of the Simplicial and Singular Homology Groups
    • Abstract:
      • My project is about the simplicial and singular homology groups of topological spaces as well as one of their applications in the Euler Formulas
    • Prerequisites: Previous knowledge of kernels, images, homeomorphisms, and isomorphisms are needed for the project
    • Mentor: Yixian Wu

  9. Grant Kluber
    • Trotterization in QM Theory
    • Abstract:
      • If A and B are matrices that commute, then exp(A+B) = exp(A)exp(B). If A and B do not commute, the generalized limit exp(A+B) = [exp(A/n)exp(B/n)]^n where n goes to infinity does hold. This result is known as the Trotter Product Formula and has implications in both quantum spectral theory and numerical computing.
    • Prerequisites: Exposure to functional analysis and quantum mechanics is necessary to understand this project.
    • Mentor: Esteban Cardenas

  10. Himani Verma
    • Bipartite Graphs
    • Abstract:
      • In this paper we will discuss a special type of graph – the bipartite graph. We will introduce relevant definitions, work through examples, prove a theorem, and then finally culminate our definitions and examples in the discussion of the application of the theorem.
    • Prerequisites: Set theory
    • Mentor: Feride Ceren Kose

  11. Huiqian Wang
    • The Fast Fourier Transform
    • Abstract:
      • What is FFT and how FFT increases its efficiency compared to DFT
    • Prerequisites: Linear transformation
    • Mentor: Yiran Hu

  12. Jacob Wood
    • A Correspondence Between Vector Bundles and Modules
    • Abstract:
      • We introduce basic notions from algebraic geometry like the spectrum of a ring and talk about the correspondence between algebraic and geometric objects. To showcase this relationship, we show how a module over a ring can be used to construct a vector bundle over the spectrum of that ring.
    • Prerequisites: Point-set topology and algebra, particularly module and ring theory including the tensor product
    • Mentor: Jackson Van Dyke

  13. Jamie Pearce
    • The Wirtinger Presentation
    • Abstract:
      • A computation of the knot group for a knot embedded in 3-space.
    • Prerequisites: Topology & Algebra
    • Mentor: Kenny Schefers

  14. Jeremy Krill
    • Representation Theory of SU(2)
    • Abstract:
      • An overview of the representation theory of SU(2) and an introduction into its applications in quantum mechanics.
    • Prerequisites: Group Theory and Linear Algebra
    • Mentor: Rok Gregoric

  15. Jerry Villalobos
    • The Non-Measurable
    • Abstract:
      • I go over the construction and proof of existence of a non-measurable set.
    • Prerequisites: Some measure theory would be beneficial, but not necessary.
    • Mentor: Hunter Vallejos

  16. Jordan Grant
    • The Hodge Decomposition
    • Abstract:
      • This paper outlines a proof of the Hodge Decomposition, along with a very brief sneak peak of Poincare Duality.
    • Prerequisites: Smooth Manifold Theory, Algebraic Topology (needed to understand Poincare Duality, not the Hodge Decomposition)
    • Mentor: Riccardo Pedrotti

  17. Kaushik Katta

  18. Khue Tran

  19. Lincole Jiang
    • Kalman Filters in Epidemiological Studies
    • Abstract:
      • A brief description of Kalman filter (ensemble Kalman Filter, in particular) and its general application in epidemiological settings.
    • Prerequisites: A little statistics
    • Mentor: Harrison Waldon

  20. Maruth Goyal

  21. Mauricio Montes
    • Alice in the Riemannian Surface
    • Abstract:
      • We can reorganize the paths of complex polygons whose vertices are branch points and sides are branch cuts in such a way that we can always decompose our polygon into groups of consecutive corners with the same roots assigned to them.
    • Prerequisites: Complex Analysis, Algebra, Fundamental Groups
    • Mentor: Richard Wong

  22. Michael Keith
    • An Intuitive Guide to Lebesgue Measure
    • Abstract:
      • This project concerns the development of the fundamentals for Lebesgue measure and Lebesgue integration.
    • Prerequisites: Some knowledge of Real Analysis is helpful, though most of the relevant material is explained in the paper itself.
    • Mentor: Kenneth DeMason

  23. Neel Panging

  24. Noah Shimizu
    • Free Actions on Trees
    • Abstract:
      • In this talk, we prove a basic theorem, that if a group G acts on a tree freely and non-trivially, then G is a free group. We then use this theorem to prove a nice corollary about the subgroups of free groups.
    • Prerequisites: Free Groups, Group Actions, Graphs
    • Mentor: Teddy Weisman

  25. Oswaldo Ceballos
    • Applications of Fuzzy Set Theory: The Mamdani Controller
    • Abstract:
      • The project explored the conceptual idea of fuzzy logic. Using certain properties and the overall theory itself, an application was made showing how an app that would suggest users on whether they should go outside. Users can enter temperature, socialization and information about whether they have been exposed to covid or unsure. The fuzzy logic mentioned in the project is applied using a fuzzy module in python.
    • Prerequisites: A solid understanding of how boolean logic works is helpful to understand the project.
    • Mentor: Austin Alderete

  26. Porfirio Rodriguez
    • Representations of Finite Groups & Schur's Lemma
    • Abstract:
      • My project is a brief description of representations of finite groups, as well as Schur's Lemma and some theorems that follow from it. Over the course of the project, I progress towards a way to show that representations are irreducible.
    • Prerequisites: Some group theory, and linear algebra.
    • Mentor: Alberto San Miguel Malaney

  27. Qixin Wang

  28. Samuel Perales
    • Spectral Theorem for Bounded Self-Adjoint Operators with Projection-Valued Measures
    • Abstract:
      • We often care about eigenvector, eigenvalue pairs for a given operators and how we can use them to form a basis for our space. However, quantum operators can often fail to have true eigenvectors, so we need a notion of generalized eigenvectors to create a basis. This article surveys proofs regarding the spectral theory of bounded self-adjoint operators and gives some intuition as to how projection-valued measures can be used to state the Spectral Theorem for these operators and how they might be used in a quantum context.
    • Prerequisites: Some knowledge of operators, Hilbert spaces, and basic measure theory.
    • Mentor: Joseph Miller

  29. Tharit Tangkijwanichakul
    • Saddle Point Theorem for Convex Functionals
    • Abstract:
      • We stated the necessary and sufficient conditions for optimality of a minimization problem.
    • Prerequisites: Linear Algebra, Convex Optimization, Duality
    • Mentor: Tharathep Sangsawang

  30. Zac Schulwolf
    • Introduction to MCMC with applications in Generative Adversarial Networks
    • Abstract:
      • A short introduction to MCMC (Markov Chain Monte Carlo) and Markov Chains that focuses on the Metropolis-Hastings algorithm and some applications. In my presentation, I provide understanding of the Metropolis-Hastings algorithm and showcase an interesting application in the field of generative adversarial networks.
    • Prerequisites: Some probability and understanding of markov chains
    • Mentor: Joseph Jackson