Fall 2021

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 

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 

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 

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 

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 

Abstract: We introduce principal components analysis and prove a theorem about the optimal encoding of low-dimensional data.

Mentor: Hunter Vallejos

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

Mentor: Hunter Vallejos 

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 

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

Mentor: Yiran Hu 

Abstract: The definition and uses of stochastic integration. 

Mentor: Enrique Leon 

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  

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 

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 

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 

Abstract: I would like to talk about measure theory, introduction, definition, and example about it and application (Lebesgue Measure/Lebesgue Integration).

Mentor: Daniel Weser 

Abstract: A priori estimates of elliptic partial differential equations. Interior and boundary regularity following the Schauder estimates.

Mentor: Jincheng Yang 

Abstract: We will look at why there is a distinction between sets and classes in set theory.

Mentor: Michael Hott 

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 

Abstract: The talk will explain the projective plane, Bezout's theorem, and give some examples of the theorem in practice. 

Mentor: Kyrylo Muliarchyk 

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 

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 

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 

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 

Abstract: Definition and intuition of free product. An intuition from Van Kampen's theorem. A brief proof about why it exists.

Mentor: Shiyu Liang 

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 

Abstract: My talk starts from the basics of graph theory by introducing the concept of a graph, vertices, edges, etc. Then the concept of a tree is explored, after which one particular algorithm for finding the minimum spanning tree of a graph will be covered.

Mentor: Desmond Coles

(not recorded)

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 

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 

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 

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 

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