Embedded Files
DRP transitioned to an online format partway through 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 34 projects this semester.
Aidan O'Keeffe — Barebones Delay Differential Equations
Aidan O'Keeffe — Barebones Delay Differential Equations
Abstract: Delay differential equations
Prerequisites: ODEs and linear algebra
Mentor: María Soria Carro
Aislinn Smith — An overview of Coupled Vortex Dynamics and the Hamiltonian Description
Aislinn Smith — An overview of Coupled Vortex Dynamics and the Hamiltonian Description
Abstract: The math behind the equations of ideal fluid motion, specifically in the case of interacting irrotational point vortices.
Prerequisites: Differential and Integral Calculus, Differential Equations, some basic vector calculus and continuum mechanics.
Mentor: William Barham
Angela Cao — K-Means in Image Clustering
Angela Cao — K-Means in Image Clustering
Abstract: Exploring K-nearest neighbors and K-means with an application in pixel quantization.
Prerequisites: Some linear algebra and some Python
Mentor: Shane McQuarrie
Ashwin Devaraj — An Overview of Curvature
Ashwin Devaraj — An Overview of Curvature
Abstract: My project is a set of lecture notes that provide an introduction to differential geometry in Euclidean space, with an emphasis on the notions curvature of curves and regular surfaces. These notes culminate in the definitions of mean and Gaussian curvature, and finally the presentation of Gauss' Theorema Egregium.
Prerequisites: A strong background in multivariable calculus and linear algebra would be helpful. A full real analysis background isn't necessary, but it would be helpful to understand the definitions of open and closed sets in R^n.
Mentor: Daniel Weser
Binglin Zhang — Barycentric Coordinate and its Applications
Binglin Zhang — Barycentric Coordinate and its Applications
Abstract: Discussing the definition and construction of Barycentric Coordinate and its recent application on figure deformation.
Prerequisites: Basic concepts of Geometry and Algebra.
Mentor: Chuning Wang
Chuxuan Yang — A Proof of the Fundamental Theorem of Algebra
Chuxuan Yang — A Proof of the Fundamental Theorem of Algebra
Abstract: Using fundamental group of a circle
Prerequisites: Fundamental group
Mentor: Hunter Vallejos
CJ Izzo — Plane Algebraic Curves and Bezout's Theorem
CJ Izzo — Plane Algebraic Curves and Bezout's Theorem
Abstract: Defining affine algebraic curves, projective spaces and curves, and providing motivation for Bezout's theorem from the perspective of Plane Algebraic Curve theory
Prerequisites: A familiarity with Abstract Algebra
Mentor: Jon Johnson
Daisy Gomez — Points of finite order and applications to the Nagell-Lutz Theorem
Daisy Gomez — Points of finite order and applications to the Nagell-Lutz Theorem
Abstract: My project is a paper describing various properties of points with finite order on elliptic curves as well as the group law on these points in the projective plane. It includes a proof of a theorem about points of order 2 and 3 which motivates the Nagell-Lutz Theorem.
Prerequisites: Number Theory, Algebraic Structures, Geometry
Mentor: Yixian Wu
Daniel Naves — Local Theory of Plane and Space Curves
Daniel Naves — Local Theory of Plane and Space Curves
Abstract: My project explores the fundamental results regarding the theory of curves in Euclidean space
Prerequisites: Basic analysis and linear algebra
Mentor: Daniel Restrepo
Dylan Chase Alexander — Pointwise Ergodic Theorem
Dylan Chase Alexander — Pointwise Ergodic Theorem
Abstract: Proof and interpretation of a fundamental result regarding L1 functions on measure preserving dynamical systems
Prerequisites: A knowledge of measure theory and basic dynamical systems is required. An example from physics is provided but physics is not crucial for the reading of this article.
Mentor: Frank Lin
Ethan Jana — QM statistical mechanics and *-algebras
Ethan Jana — QM statistical mechanics and *-algebras
Abstract: How ideal gases are treated in Quantum Mechanics
Prerequisites: Basic quantum mechanics, in particular the QM harmonic osc.
Mentor: Michael Hott
Fornia Van — Binomial Model in the Stock Market
Fornia Van — Binomial Model in the Stock Market
Abstract: Using binomial model, martingale, markov, and conditional expectation to relate to random walk, and then relating random walk to the stock market.
Prerequisites: Probability
Mentor: Yiran Hu
Grant Kluber — Spectral Theory in Quantum Mechanics
Grant Kluber — Spectral Theory in Quantum Mechanics
Abstract: This project is about the application of spectral theory to quantum mechanics, beginning with the proper definition of the spectrum for linear operators on Banach/Hilbert spaces and its three disjoint subsets: the point, continuous, and residual spectra. There is emphasis on examples, primarily since many things are hard to state exactly in spectral theory (like the spectral theorem), but are easy to behold/motivate. Intended as a blog post, I estimate a reading time of 10-20 minutes.
Prerequisites: Exposure to ideas/notation in quantum mechanics and real analysis.
Mentor: Logan F Stokols
Jacob Gutierrez — Number Fields and Class Number: A journey to understanding the Minkowski Bound
Jacob Gutierrez — Number Fields and Class Number: A journey to understanding the Minkowski Bound
Abstract: Number fields and class number how to compute them
Prerequisites: Linear algebra and number theory
Mentor: Tynan Ochse
Jamie Pearce — A Brief Introduction to the Cantor Set
Jamie Pearce — A Brief Introduction to the Cantor Set
Abstract: The Cantor set and a few of its interesting properties and uses in topology.
Prerequisites: Topology
Mentor: Kenny Schefers
Jordan Grant — Reconciling Vector Bundles
Jordan Grant — Reconciling Vector Bundles
Abstract: Equating the two most commonly used definitions of a vector bundle to one another.
Prerequisites: Classical Differential Geometry, Topology
Mentor: Riccardo Pedrotti
Joshua Wong — Representation Theory with a Perspective from Category Theory
Joshua Wong — Representation Theory with a Perspective from Category Theory
Abstract: A brief overview of representation theory leading up to a proof of Frobenius Reciprocity. Afterwards, we tackle Frobenius Reciprocity from the more general viewpoint of category theory.
Prerequisites: Linear algebra, group theory
Mentor: Saad Slaoui
Karthik Vaidyanathan — The WLLN and CLT
Karthik Vaidyanathan — The WLLN and CLT
Abstract: I give an introduction to measure-theoretic probability and describe the Weak Law of Large Numbers and the Central Limit Theorem with their proofs.
Prerequisites: Preferably some exposure to real analysis but the measure theory is explained in the presentation.
Mentor: Joe Jackson
Lincole Jiang — Finite Spaces Tweaks and Twerks
Lincole Jiang — Finite Spaces Tweaks and Twerks
Abstract: Separation axiom, homeomorphism, and connectivity about finite topological spaces
Prerequisites: Basic understanding of point-set topology
Mentor: Gillian Grindstaff
Matthew Huynh — A Brief Introduction to Riemann Surfaces
Matthew Huynh — A Brief Introduction to Riemann Surfaces
Abstract: Defining and giving examples of the Riemann surface associated to a function
Prerequisites: Some familiarity with complex analysis and topology. However I included a section introducing all required info to understand the theorem that I prove at the end.
Mentor: Sebastian Schulz
Michelle Brun — Basic Projective Geometry
Michelle Brun — Basic Projective Geometry
Abstract: My project is an introduction to projective space and some of its properties.
Prerequisites: Euclidean geometry
Mentor: Max Riestenberg
Neel Panging — The mathematical intuition of neural networks
Neel Panging — The mathematical intuition of neural networks
Abstract: My project explains the underlying mathematical concepts that allow neural networks to learn.
Prerequisites: A basic understanding of linear algebra and calculus would help
Mentor: Milad Naeini
Ning Kang — Intro to Machine Learning: Cross Validation
Ning Kang — Intro to Machine Learning: Cross Validation
Abstract: Cross-validation is a resampling procedure used to evaluate machine learning models on a limited data sample.
Prerequisites: Mean Squared Error and Root Mean Squared Error.
Mentor: Milad Eghtedari
Omar Aceval — The Fundamental Theorem of Galois Theory
Omar Aceval — The Fundamental Theorem of Galois Theory
Abstract: This project details some important implications of the fundamental theorem of Galois theory, most notably a detailed discussion about the problem of finding a general solution to fifth order polynomials.
Prerequisites: A decent understanding of fields and groups, and in particular normal subgroups. The most rigorous proof is not particularly detailed and so requires little background in abstract algebra.
Mentor: Natalie Hollenbaugh
Pratyush Potu — Plateau’s Problem and Minimal Surfaces
Pratyush Potu — Plateau’s Problem and Minimal Surfaces
Abstract: In this presentation, I give an introduction to the Calculus of Variations. I introduce the definition of a functional and the Euler-Lagrange Equation which I prove is a necessary and sufficient condition for finding extremal functions of functionals. Then, I talk about Plateau's Problem concerning minimal surfaces which are a particular application of the Calculus of Variations. I explain how the minimization of the surface area of a surface is a problem of minimizing a surface area functional, and how we can use the Euler-Lagrange Equation to find minimal surfaces. I conclude with some applications of minimal surfaces to other fields.
Prerequisites: Calculus up to the level of M408D.
Mentor: Erica De La Canal
Rachel Thornton — Foundational Ideas in Topology
Rachel Thornton — Foundational Ideas in Topology
Abstract: Definition of topology, open sets, limits, and closures, and theorem: A set is open if and only if its compliment is closed.
Prerequisites: Strong background in proofs, no topology needed
Mentor: Casandra Monroe
Rithvik Reddy Golamari — Bousfield Localization of Spectra
Rithvik Reddy Golamari — Bousfield Localization of Spectra
Abstract: The project discusses some introductory stable homotopy theory and proceeds to construct the Bousfield localization of spectra with respect to a homology theory. A small, non-technical application is also discussed in the form of the Thick Subcategory Theorem. The project is in the form of a blog-post so I've attached a link to the post below. [link is no longer active]
Prerequisites: Algebraic topology, category theory and some homotopy theory
Mentor: Richard Wong
Ruben Dimas — Studying Subgroups of Free Groups with Folding
Ruben Dimas — Studying Subgroups of Free Groups with Folding
Abstract: Showing how to find the rank of a subgroup of F2 using directed graphs and folding
Prerequisites: Definition of a free group
Mentor: Teddy Weisman
Ryan Chatterjee — Lie Groups and their Algebras
Ryan Chatterjee — Lie Groups and their Algebras
Abstract: Explores the representation theory of Lie groups, by defining the associated Lie algebra, and explaining how the Lie algebra describes structure of the associated group via Baker-Campbell-Hausdorff.
Prerequisites: Some topology (especially differential), linear algebra, abstract algebra
Mentor: Tom Gannon
Shannon Scofield — An Introduction to Representation Theory
Shannon Scofield — An Introduction to Representation Theory
Abstract: My project is a talk explaining some basics of representation theory and identifying the irreps of S3 .
Prerequisites: Group Theory, Linear Algebra would be useful
Mentor: Tom Gannon
Sonali Singh — Stochastic Calculus for Finance: Random Walk
Sonali Singh — Stochastic Calculus for Finance: Random Walk
Abstract: This project focuses on properties concerning Random Walk. Random walk is a stochastic (random) process of discrete steps on a fixed length. The steps do not depend on previous steps. Symmetric random walk is when the direction of each step (up or down, left or right) is equally likely, meaning p=½. In asymmetric random walk p≠½. However, both types of random walk have the same set of possible paths. In two examples, we further explore the applied value of the theory. In the first, we explore properties of random walk between dimensions (2D and 3D). In the second, we explore the reflection principle, a theory that explores how the probability of reflected paths can be utilized.
Prerequisites: Probability and Intro to Financial Math
Mentor: Luhao Zhang
Yangxinyu Xie — Knots, Links and Spatial Graphs
Yangxinyu Xie — Knots, Links and Spatial Graphs
Abstract: In 1983, John Conway and Cameron Gordon published a well-known paper Knots and links in spatial graph [CMG83] that stimulated a wide range of interests in the study of spatial graph theory. In this note, we review this classical paper and some of the earliest results in intrinsic properties of graphs.
Prerequisites: Basic Knot Theory
Mentor: Alexandra Embry
Yongqi Pang — Statistics and Data Analysis
Yongqi Pang — Statistics and Data Analysis
Abstract: This project is about using python to approach to data analysis. The main part of this project is to manipulate, process, clean and crunch data in python. We focus on learning several libraries such as NumPy, Pandas, Matplotlib and Seaborn. Besides, the topic we focus on is about the current trend—Covid-19 and we learn about the details of SIR model. At last, we try to construct a simple SIR model using python.
Prerequisites: Knowledge of Python programming, Statistics, and SIR model
Mentor: Jincheng Yang
Zhecheng Wang — Hyperbolic Geometry in Parameterization of Surfaces
Zhecheng Wang — Hyperbolic Geometry in Parameterization of Surfaces
Abstract: An idea to achieve conformal mapping of Genus 2 surface with Hyperbolic Geometry.
Prerequisites: Discrete Math, some basic topology definitions and a solid understanding of Euclid's Postulates.
Mentor: Neža Žager Korenjak
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