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DRP was entirely online this semester. In lieu of the DRP Symposium, students were asked to submit a project in one of the following formats:
Short paper (handwritten or typed)
Slideshow that might have accompanied a talk
Recorded 12-15 minute talk
Blog post
A clear explanation of a proof
Projects are sorted alphabetically by the speakers' first names. There were 30 projects this semester.
Adarsh Hullahalli — Knots and Some Chess
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
Alexy Skoutnev — Support Vector Machine
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
Andrew J Pease — An Introduction to Point Set Topology
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
Angela Cao — Titanic Survival Estimation via Naive Bayes
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
Ashwin Devaraj — Riemannian Manifolds and Affine Connections
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
Carter Chu — Integer Partitions and the Rogers-Ramanujan Identities
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
Ethan Roy — An Overview of the Simplicial and Singular Homology Groups
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
Grant Kluber — Trotterization in QM Theory
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
Himani Verma — Bipartite Graphs
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
Huiqian Wang — The Fast Fourier Transform
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
Jacob Wood — A Correspondence Between Vector Bundles and Modules
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
Jamie Pearce — The Wirtinger Presentation
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
Jeremy Krill — Representation Theory of SU(2)
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
Jordan Grant — The Hodge Decomposition
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
Kaushik Katta — Pricing American Options through Backwards Induction
Kaushik Katta — Pricing American Options through Backwards Induction
Abstract: The project is about what American options are and how they can be priced in a binomial model by using backwards induction.
Prerequisites: Probability
Mentor: Anastasiya Tanana
Khue Tran — Introduction to Dynamical Systems and Applications in Neuroscience
Khue Tran — Introduction to Dynamical Systems and Applications in Neuroscience
Abstract: An introduction to concepts used in dynamical systems and applying them to study the dynamics of action potentials in neuroscience.
Prerequisites: Differential Equations
Mentor: William Warner
Lincole Jiang —Kalman Filters in Epidemiological Studies
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
Maruth Goyal — Differential Privacy: How to not say too much, but just enough
Maruth Goyal — Differential Privacy: How to not say too much, but just enough
Abstract: Differential Privacy: a mathematically rigorous foundation for studying privacy
Prerequisites: Basic understanding of probability
Mentor: Hunter Vallejos
Mauricio Montes — Alice in the Riemannian Surface
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
Michael Keith — An Intuitive Guide to Lebesgue Measure
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
Neel Panging — Stochastic Processes and Discrete-time Markov chains
Neel Panging — Stochastic Processes and Discrete-time Markov chains
Abstract: The definition and the different types of stochastic processes with a focus on discrete-time Markov chains and some of their properties.
Prerequisites: Basic probability
Mentor: Milad Eghtedari Naeini
Noah Shimizu — Free Actions on Trees
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
Oswaldo Ceballos — Applications of Fuzzy Set Theory: The Mamdani Controller
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
Porfirio Rodriguez — Representations of Finite Groups & Schur's Lemma
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
Qixin Wang — The benefit of Residual Network
Qixin Wang — The benefit of Residual Network
Samuel Perales — Spectral Theorem for Bounded Self-Adjoint Operators with Projection-Valued Measures
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
Tharit Tangkijwanichakul — Saddle Point Theorem for Convex Functionals
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
Zac Schulwolf — Introduction to MCMC with applications in Generative Adversarial Networks
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
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