Fall 2021 Symposium.
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Speakers in order of appearance

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

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

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

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

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

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 lowdimensional data.

 Mentor: Hunter Vallejos
Speakers in order of appearance

Yijie Lian
 Markov Chain

Abstract:
 Property, examples about Markov chain and random walks of graphs.
 Mentor: Hunter Vallejos

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, sigmaalgebras, and the like.
 Mentor: Joseph Jackson

William Yan
 Ito's Integral

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

Jacob Way
 Stochastic Calculus

Abstract:
 The definition and uses of stochastic integration.

Mentor: Enrique Leon

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

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
Speakers in order of appearance

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

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

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

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

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
Speakers in order of appearance

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

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

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

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

John Teague
 QuasiCoherent Sheavs and the Proj Construction

Abstract:
 I will give a brief introduction to the basics of scheme theory, quasicoherent 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
Speakers in order of appearance

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 nonunique prime factorizations in a ring of integers will be examined using the class number.
 Mentor: Tynan Ochse

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

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

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
Speakers in order of appearance

Tyler Dean
 QuaternionsWhat 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

Michael Updike
 Lie Algebra and Quantization

Abstract:
 Quantum mechanics stipulates that observables don’t commute. The simplest encapsulation of this noncommutative 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

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

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