Directed Reading Program

Department of Mathematics, UT-Austin

Fall 2021 Symposium.

Note: Some rooms with many speakers have large video files that may take longer to load.

Video of Room 1

Speakers in order of appearance
  1. Veronica King
    • The Vigenere Cipher (and how to try to crack it)
    • Abstract: 
      • In this talk, I will provide an introduction to an encryption method called the Vigenere Cipher and use Python code to demonstrate some cryptography concepts. I will explain how to encrypt and decrypt using the Vigenere Cipher, and then demonstrate a method that can be used to crack a ciphertext encrypted by the Vigenere Cipher.
    • Mentor: Allie Embry
  2. Tara Roshan
    • Attacks on RSA Encryption
    • Abstract:
      • We'll be giving a brief introduction to RSA encryption & go over an example. Then we'll discuss how the encryption can be cracked and have the audience figure out some secret messages that were encrypted using the RSA protocol.
    • Mentor: Addie Duncan
  3. Marilyn Lionts
    • Introduction to Topological Data Analysis
    • Abstract:
      • Topological data analysis (TDA) allows us to apply topological tools to data sets in order to analyze its intrinsic properties. This talk will give an overview of a typical workflow in the TDA process.
    • Mentor: Erin Bevilacqua
  4. Lohit Jagarapu
    • Perceptrons: the building blocks of Machine Learning
    • Abstract:
      • Perceptrons, which are single neuron neural networks, are the first and most fundamental neural networks created. This talk will focus on the theoretical model of a perceptron and the formulas that govern it. No machine learning familiarity is required.
    • Mentor: Ziheng Chen 
  5. Edward Shao
    • Introduction to Neural Network
    • Abstract:
      • The talk will give an introduction on neural networks and the process of training it. The talk will also give a very brief view of the neural network reading inputted images of handwriting.
    • Mentor: Jacky Chong
  1. Kyle Cox
    • An Introduction to Principal Components Analysis and Applications
    • Abstract:
      • We introduce principal components analysis and prove a theorem about the optimal encoding of low-dimensional data.
    • Mentor: Hunter Vallejos

Video of Room 2

Speakers in order of appearance
  1. Yijie Lian
    • Markov Chain
    • Abstract:
      • Property, examples about Markov chain and random walks of graphs. 
    • Mentor: Hunter Vallejos

  2. Utkarsh Nigam
    • Probability: A Measure Theoretic Perspective 
    • Abstract:
      • Most students learn basic probability. However, it turns out even basic probability is fascinatingly difficult to formalize -- but is a jewel of a subject when expressed with measures, measurable functions, sigma-algebras, and the like.
    • Mentor: Joseph Jackson

  3. William Yan
    • Ito's Integral
    • Abstract:
      • Introduce brownian motion, simple process, and the definition of Ito's Integral. 
    • Mentor: Yiran Hu

  4. Jacob Way
    • Stochastic Calculus
    • Abstract:
      • The definition and uses of stochastic integration. 
    • Mentor: Enrique Leon
  5. Wenxuan Jiang
    • Stochastic Calculus for Finance
    • Abstract:
      • How do we set a price to trade a derivative security at the moment given possible future payoff? We investigated the binomial pricing model, martingale, Markov processes, and how they were deployed for such purposes. 
    • Mentor: Luhao Zhang
  6. Maximillian DeMarr 
    • A Probabilistic Method for Proofs
    • Abstract:
      •  I'll be introducing a unique approach for proofs, utilizing the Probabilistic Method. This applies the idea of probability in various areas which wouldn't be obvious otherwise.
    • Mentor: Jayden Wang

Video of Room 3

Speakers in order of appearance
  1. Samuel Perales
    • Can you hear the shape of a drum?
    • Abstract:
      • If I have two drums, can I always tell them apart based only on the sound? What does this problem really mean in the context of math and how can we use computers to help understand a difficult to prove answer to this question?
    • Mentor: Ziheng Chen

  2. Wenting Lu
    • The Metric Topology
    • Abstract:
      • I am going to give an introduction about what is a basis of topology and what is a metric topology is like. I will give examples of these concepts. 
    • Mentor: Ben Nativi

  3. Doan Ngyuyen 
    • The Measure Thepry
    • Abstract:
      • I would like to talk about measure theory, introduction, definition, and example about it and application (Lebesgue Measure/Lebesgue Integration).
    • Mentor: Daniel Weser

  4. Kyle Alkire
    • Regularity of Elliptic Partial Differential Equations
    • Abstract:
      • A priori estimates of elliptic partial differential equations. Interior and boundary regularity following the Schauder estimates..
    • Mentor: Jincheng Yang
  5. Joan Antonio Artero Calvo
    • Sets and Classes
    • Abstract:
      • We will look at why there is a distinction between sets and classes in set theory.
    • Mentor: Michael Hott

Video of Room 4

Speakers in order of appearance
  1. Michael Panner
    • Introduction to Algebraic Geometry
    • Abstract:
      • A basic introduction to algebraic geometry including a discussion of vanishing sets, the ideal of a set of points, and the Hilbert Nullstellensatz.
    • Mentor: Amy Bradford

  2. Luis Kim
    • Bezout's Theorem
    • Abstract:
      • The talk will explain the projective plane, Bezout's theorem, and give some examples of the theorem in practice. 
    • Mentor: Kyrylo Muliarchyk

  3. Snighda Pakala
    • The Group Law on Elliptic Curves
    • Abstract:
      • I will be discussing basic group theory and then a little introduction to what elliptic curves are and their purpose, leading up to the argument/ proof that elliptic curves are groups. 
    • Mentor: Isaac Martin

  4. Simon Xiang
    • The Yoneda Lemma
    • Abstract:
      • In the talk, I plan to give a brief overview of basic category theory and a proof of the Yoneda lemma.
    • Mentor: Rok Gregoric
  5. John Teague
    • Quasi-Coherent Sheavs and the Proj Construction
    • Abstract:
      • I will give a brief introduction to the basics of scheme theory, quasi-coherent sheaves, and the Proj construction, as well as touch on the twisting sheaf of Serre in recovering algebraic information of sheaves of graded modules (especially on Proj). We will go over some common examples, cautions, and theorems relating these ideas.
    • Mentor: Alberto San Miguel Malaney

Video of Room 5

Speakers in order of appearance
  1. Matthew Allen
    • Number Fields and Prime Factorization
    • Abstract:
      • We will introduce number fields and their corresponding ring of integers and class groups and how the class group relates to factorization in the ring of integers with the class number. In particular, the behavior of non-unique prime factorizations in a ring of integers will be examined using the class number.
    • Mentor: Tynan Ochse

  2. Ruiqi Zou
    • Free Product Exists
    • Abstract:
      • Definition and intuition of free product. An intuition from Van Kampen's theorem. A brief proof about why it exists.
    • Mentor: Shiyu Liang

  3. Andrew Pease
    • The Lamplighter Group
    • Abstract:
      • I will define the lamplighter group in several ways and compute the word length. If time permits, I will end with showing the lamplighter group has dead end elements of arbitrary depth.
    • Mentor: Teddy Weisman

  4. Ethan Sollenberger 
    • Snowflakes Among Tropical Trees
    • Abstract:
      • We describe two geometric objects: the tropical Grassmannian and the Stiefel image contained within it. We show that there exists an element of the tropical Grassmannian which is not contained within the Stiefel image, by first associating points in the tropical Grassmannian tropGr(2,n) to phylogenetic trees, and then by providing a necessary and sufficient condition for a phylogenetic tree to be in the Stiefel image.
    • Mentor: Austin Alderete

Video of Room 6

Speakers in order of appearance
  1. Tyler Dean
    • Quaternions-What are they and how to use them
    • Abstract:
      • The first half of the talk will be dedicated to defining what a quaternion is and how it functions. Meanwhile, the second part of the talk will focus more on applications of quaternions over issues like gimble lock and how to turn rotations in R3 to quaternions and back..
    • Mentor: Colin Walker

  2. Michael Updike 
    • Lie Algebra and Quantization
    • Abstract:
      • Quantum mechanics stipulates that observables don’t commute. The simplest encapsulation of this non-commutative structure is the Lie algebra. Starting with a few examples, I aim to show how Lie algebras lead naturally to quantum theory.
    • Mentor: Jackson Van Dyke
  3. Abigail Perryman
    • Translation Symmetry of the Hamiltonian
    • Abstract:
      • The discrete translational symmetry of the Hamiltonian provides insight into its physical properties. Examining the discrete translation group representation and its eigenstates provides a path to solve for the eigenstates of the Hamiltonian. This is one example of the usefulness of symmetry groups in studying physical systems, and other symmetries also have profound applications..
    • Mentor: William Stewart
  4. Ryan McWhorter
    • Noether's Theorem and Symplectic Geometry
    • Abstract:
      •  We provide a brief overview of symplectic geometry, and use it to develop an understanding of Noether’s theorem. Beyond this, we explain the possibility of applications beyond purely mathematical physics.
    • Mentor: Riccardo Pedrotti