Below, you will find past DRP projects listed by academic year. Links to the final presentations are provided where available. Prospective students are encouraged to browse recent projects for inspiration.
For projects prior to Spring 2022, see the old archive. The Rutgers DRP has been running since Summer 2005.
🌞 denotes Student Presentation Award.
🌝 denotes Honorable Mention.
A Complex-Analytic Proof of the Prime Number Theorem
Mentee Shawn Varghese
Mentor Riley Guyett
Abstract
We will cover preliminary results in complex analysis and detail a proof of the prime number theorem.
An Accessible Introduction to Category Theory via the Canonical Map V → V ∗∗
Mentee Jonathan Tan
Mentor Anna Rose Osofsky
Abstract
Students often hear that a finite-dimensional vector space V is “canonically” isomorphic to its double dual V ∗∗, but it is rarely explained what “canonical” really means. In this talk, I will use the familiar construction of the map V → V ∗∗ as a gentle introduction to the basic ideas of category theory: objects, morphisms, functors, and natural transformations. We will recast duals and double duals in this language and see how the maps V → V ∗∗ for different vector spaces V fit together into a single structure that behaves well with respect to all linear maps. This categorical viewpoint will make precise the sense in which the identification V ∼= V ∗∗ is distinguished from other isomorphisms, giving a concrete example of how category theory clarifies familiar constructions in linear algebra.
An Application of the Variational Method
Mentee Harry Haedrich
Mentor Mark Vaysiberg
Abstract
I will introduce the basic method of minimizing functionals, the so-called Direct Method, through application to a familiar physical example– the energy functional corresponding to the Schrödinger Equation. I will provide the necessary vocabulary where it comes up and explain in what ways this is a typical problem and in which ways it is not.
Bayesian Networks & Cancer Research Applications
Mentee Priya Rana
Mentor Aurora Hiveley
Abstract
A Bayesian network is a directed acyclic graph that represents probabilistic relationships among a set of variables. In the context of genomics, these variables correspond to genetic events such as mutations, which can be modeled as binary features across patient datasets. This presentation examines how Bayesian networks can be learned from genomic data using structure learning algorithms, with a focus on score-based methods such as Gobnilp. By modeling conditional dependencies between genes, these networks are able to capture complex patterns including co-occurrence and mutual exclusivity that are not apparent through pairwise analysis. We explore how these probabilistic graphical models provide insight into underlying biological mechanisms and enable the identification of distinct genetic subtypes of disease. The results demonstrate the effectiveness of Bayesian networks as an interpretable framework for analyzing high-dimensional genomic data.
Bayesian Networks: Examining The Effectiveness of Different Structure Learning Algorithms
Mentee Jash Prajapati
Mentor Aurora Hiveley
Abstract
A Bayesian Network is a directed acyclic graph with arcs that represent influence. Bayesian Networks encode conditional probabilities and improve computation when calculating probability. The presentation covers the implementation of different structure learning algorithms using multiple disease-related datasets. The graphs generated by each learning algorithm are then compared to a reference graph using various metrics.
Compactness Theorem and its analogy
Mentee Duke Bui
Mentor Violet Bianco
Abstract
This presentation will give a short introduction to the field of Model Theory. Next, I will introduce the statement of the Compactness Theorem and discuss some of its corollaries in other fields (graph theory, field theory, complex analysis, etc.).
Differential Equations and Dynamics
Mentee Aadithya Saravanan
Mentor Aprameya Girish Hebbar
Abstract
This presentation will begin by introducing flows and examples of dynamical systems and their behavior. Then, it will analyze the long-time behavior of systems using linearization.
Eigenvalues of Regular Graphs and Quasirandomness
Mentee Pranav Shankar
Mentor Milan Haiman
Abstract
This project explores regular graphs, their eigenvalues, and several criteria for quasirandomness of graphs. We then extend this framework to bipartite graphs, and conclude with the Alon-Boppana bound to make a regular graph as close to random as possible while making sure there are infinitely many graphs with similar characteristics.
Mentee Vaishnav Venkat
Mentor Dana Zilberberg
Abstract
Implemented Finite Element Method in one and two dimensions to approximate solutions to Schrodinger’s equation.
Mentee Dylan Patel
Mentor Caleb Fong
Abstract
Block designs are a combinatorial structure that show up across statistics, coding theory, and combinatorics. A particularly well-behaved class are balanced incomplete block designs (BIBDs), where every pair of elements appears together in exactly the same number of blocks. Many natural systems can be modeled as BIBDs, from tournament scheduling to error-correcting codes, making them a very interesting area of study in combinatorics. Fisher’s Inequality gives us a surprisingly strong constraint for these systems: any non-trivial BIBD must have at least as many blocks as elements. In this presentation we develop two proofs, a primary one using the rank of an incidence matrix and a secondary purely combinatorial proof, showing how two very different approaches lead to the same result.
Formalizing Codensity Transformations in Category Theory
Mentee Givi Tsvariani
Mentor Anna Rose Osofsky
Abstract
Category Theory can be utilized to formally prove validity of the logic and safety of compiler optimizations in programming languages like Haskell. Specifically, it is able to address the O(n2) → O(n) optimization achieved for Free Monads through a Codensity Transformation. By mapping the components of the Yoneda Lemma to a Kleisli Category and the "theoretical category" Hask that Haskell is built upon, we can demonstrate that the Codensity Monad is perfectly isomorphic to the original monad. Through this, Category Theory guarantees the logic and safety of the Condesity Transformations in Haskell.
From Smooth Manifolds to Fibre Bundles
Mentee Keshav Badri
Mentor Yiyang Liu
Abstract
Smooth manifolds are a platform for many interesting mathematical constructions, but one of particular interest to recent advancements in Physics is the fibre bundle. In this talk, I will review basic concepts and intuitions related to the fibre bundle construction, and introduce various objects that allow us to use it as a tool for describing smooth manifolds. I will then attempt to show an application of the construction in Physics which finds its foundations in fibre bundle theory.
From Smooth Manifolds to Moment Maps: A Journey Through Symplectic Geometry
Mentee Catherine Liu
Mentor António Agostinho Freitas Gouveia
Abstract
What is the right geometric structure for classical mechanics, and what does it reveal about symmetry? A smooth manifold is a space that locally looks like Rn but can be globally curved or closed. Equipped with a closed, non-degenerate 2-form ω, a symplectic form, it becomes the natural setting for Hamiltonian mechanics. The sphere S2, with its cylindrical coordinates (θ, h) and symplectic form ω = dθ ∧ dh, is our guiding example throughout, simple enough to draw and rich enough to carry the full story. On such a space, a Hamiltonian function H determines a vector field XH through the equation ιXH ω = dH, encoding the system’s dynamics in a completely coordinate-free way. When the system also has a continuous symmetry, a Lie group rotating or translating the space while preserving it, something remarkable happens: the symmetry forces the existence of a map that organizes all orbits of the action into a single, concrete geometric object. This talk traces that journey from first definitions to explicit computations, driven throughout by concrete examples and geometric pictures with no prior background in differential geometry assumed.
Mentee Sean LeClair
Mentor Danae Rupp
Abstract
Ramsey’s theorem tells us that for any finite edge coloring of an infinite graph there is a subgraph such that each edge is the same color. We ask how far can this theorem go?
Generating Functions and the Exponential Formula
Mentee Suhas Kondapalli
Mentor Michalis Lolis
Abstract
We will start with a review of ordinary and exponential generating functions and an introduction of the terminology of cards, decks, and hands in the context of connected labeled graphs. Then we will go over the exponential formula as an elegant and powerful tool for counting combinatorial structures. We will finally use this theory in order to solve a competitive programming problem with unexpected effectiveness.
Hidden Markov Models: Application in Market Regimes
Mentee Tristan Mihocko
Mentor Minhao Bai
Abstract
This presentation talks about Hidden Markov Models (HMMs), where the underlying state is unobservable but influences events/data that is observable through an emission model. We work through an application in finance by modeling market dynamics as a latent process with two regimes: bullish and bearish. By assuming Gaussian distributions for market returns and applying a recursive state-update algorithm at = B(Ot)AT at − 1, we demonstrate how a sequence of observable returns can be used to infer the most likely current market regime and inform investment decisions.
Independence of the Continuum Hypothesis
Mentee Akhil Vasagiri
Mentor Ben-Zion Weltsch
Abstract
The continuum hypothesis, a statement about the size of the real numbers, is known to be independent of ZFC. In this talk, we will introduce the method of forcing and show how it can be utilized to prove this independence.
Mentee Rahul Rajkumar
Mentor Minhao Bai
Abstract
We derive the first- and second-order sensitivities of the Black-Scholes pricing function, delta, gamma, theta, vega, and rho, and examine their analytic structure as partial derivatives of the solution to a parabolic PDE for option contracts.
Proving Existence Through Randomness using the Probabilistic Method
Mentee Akshat Singh
Mentor Milan Haiman
Abstract
This talk introduces the probabilistic method, a technique used to prove that certain mathematical objects exist by showing that a randomly chosen object has the desired property with positive probability. I will discuss the main idea behind the method through simple examples from combinatorics.
Reproducing Kernel Hilbert Spaces & Kernel Mean Embeddings with Applications to Hidden Markov Models
Mentee Pranav Tikkawar
Mentor Dennis Hou
Abstract
Reproducing kernel Hilbert spaces (RKHS) provide a geometric framework for studying functions via inner products and positive definite kernels, which encode similarity through Gram matrices. This talk gives an intuitive introduction to RKHS and kernel mean embeddings, which represent probability distributions as elements of an RKHS and enable nonparametric probabilistic reasoning. Building on these ideas, I will outline how hidden Markov models can be estimated nonparametrically by working with RKHS embeddings of state-dependent observation distributions instead of specifying parametric emission families, and briefly illustrate the resulting KernelHMM approach on simulated data.
Root Systems and the Weyl Group
Mentee Hamshika Rajkumar
Mentor Carlos Tapp Monfort
Abstract
This talk introduces root systems and their associated Weyl groups. We explore the geometric structure of root systems, how reflections across roots generate symmetries captured by the Weyl group, and some other important theorems. We conclude with a brief look at the classification of root systems and its role in Lie theory.
Schramm-Loewner Evolution and Random Curves
Mentee Ryan Wang
Mentor Ryan McGowan
Abstract
This talk introduces Schramm-Loewner Evolution (SLE) as a way to describe random curves. We begin with basic ideas from conformal maps and harmonic functions, and explain how they lead to the definition of SLE.
We then give examples of random curves and discuss why SLE provides a natural model for them. The talk concludes with a brief summary of the role of SLE in probability and related areas.
Seifert-Tait Knots and the Density of Alternating Knots
Mentee Jeyron Castillo
Mentor Timothy Bates
Abstract
In this talk, I will discuss a recent result of Huggett and Vdovina on the relationship between two graph constructions associated to knot diagrams: the Seifert graph, arising from the canonical Seifert surface, and the Tait graph, arising from a chessboard coloring of an alternating diagram. Although these constructions come from quite different geometric and combinatorial procedures, some alternating knots admit diagrams for which the Seifert and Tait graphs are isomorphic. Such knots are called Seifert-Tait knots.
After introducing alternating knots, Seifert graphs, Tait graphs, and the notion of density within families of knots, I will explain the main idea behind Huggett and Vdovina’s theorem: for any fixed genus g > 1, Seifert-Tait alternating knots dominate prime alternating knots of genus g as the crossing number tends to infinity. I will also describe the role of flat diagrams, whose Seifert and Tait graphs are isomorphic, and how earlier counting results for alternating knots by genus are used to prove this asymptotic dominance.
Simulating Stock Prices with Monte Carlo Methods
Mentee Krish Arora
Mentor Minhao Bai
Abstract
It is impossible to predict exact future stock prices due to the inherent noise of financial markets. Rather than attempting to forecast a single price, we can model a probability distribution of potential futures. This presentation builds a mathematical model starting from discrete random walks, to continuous Standard Brownian Motion, and finally Geometric Brownian Motion. Because it is difficult to get exact expected outcomes from these models, we can use Monte Carlo methods to calculate expected outcomes. By leveraging the Law of Large Numbers, we can simulate millions of random market paths to get a log-normal distribution of future prices. We can then apply these distributions to price European Call Options. Overall, using Monte Carlo allows investors to not only calculate the most likely expected returns but also understand unlikely outlier events.
Mentee Michael Fiore
Mentor Ben-Zion Weltsch
Abstract
When studying the real numbers, one must cope with the existence of several kinds of "ugly" sets: those which cannot be studied through the machinery of Lebesgue measure and Baire category. However, only one axiom of set theory seems truly culpable for the construction of such objects: choice. By weakening the axiom of choice and forcing our way to a "nicer" model of set theory, we can rid the real numbers of these pathological subsets.
The Gambler’s Ruin Problem
Mentee Joel Pulikkan
Mentor Nathan Jackson
Abstract
This presentation provides a derivation of the Gambler’s Ruin problem with Markov chains. We begin by phrasing the problem as a finite, absorbing Markov chain and defining the state space to be the current wealth, with ruin and success as the two absorbing states. From this, we derive the probability of success by solving a system of linear recurrence relations arising from the transition structure, forming exact closed-form solutions that capture the dependence on initial wealth, target fortune, and single-bet win probability. With this established, we highlight several applications and takeaways: the role of the Gambler’s Ruin in modeling sequential risk in finance, the implication that a negative win probability essentially guarantees bankruptcy as the target wealth increases, and the sensitivity of outcomes with small deviations from a fair game. We conclude by emphasizing the Gambler’s Ruin problem as not only a foundational result in applied probability, but also a natural entry point into the general theory of absorbing Markov chains.
Using TDA for decoding motor imagery
Mentee Ankith Kunduru
Mentor Dennis Hou
Abstract
I will discuss the basics of homology, topological data analysis, the motivation behind classifying motor imagery, and connecting the two. We will discuss holes in each dimension, explaining H0, H1, and H2 visually. Then, we will discuss how to use persistent homology to find holes and therefore the shape of point cloud data. Finally, we will discuss different methods of embedding the time-series data from motor imagery datasets into point cloud data to be analyzed through TDA.
What the Central Binomial Coefficient Knows About Primes—Apostol’s Theorem on elementary bounds for π(n)
Mentee Alexander Montes
Mentor Michalis Lolis
Abstract
The prime number theorem says that the number of primes up to x satisfies π(x)~x/log(x). Before proving this full asymptotic, Apostol proves an elementary estimate showing that x/log(x) is already the correct order of magnitude. The main idea is surprisingly combinatorial: study the central binomial coefficient 2nCn = (2n)!/(n!)^2. Its prime factorization contains enough information to force explicit upper and lower bounds for π(n).
Why the Quintic Has No Formula in Radicals
Mentee Mehtab Bhangal
Mentor Timothy Bates
Abstract
Everyone is familiar with the quadratic formula, but few realize that no analogous formula exists for general polynomials of degree five. In fact, it is impossible to solve all quintic equations using only arithmetic operations and radicals. In this talk, I will explain why such a formula cannot exist, and how this problem leads naturally to the development of Galois theory. We will explore the connection between polynomial equations, field extensions, and group structure, and see how these seemingly distant ideas together determine when an equation is solvable by radicals.
Winning Games with Surreal Numbers
Mentee Arnav Kondagunta
Mentor Riley Guyett
Abstract
This presentation explores how to solve combinatorial games such as Hackenbush and Nim by introducing the surreal number system.
Analytic Number Theory: Modular Forms
Mentee Maria Baumgartner
Mentor Preston Walker
Abstract
This presentation will introduce some classical analytic number theory results and theory of modular and cusp forms, leading to a discussion of the functional equation for L-functions associated with Hecke eigenforms.
Mentee Alankar Sovani
Mentor Samanthak Thiagarajan
Abstract
Why isn't the unit ball compact in some infinite-dimensional spaces? We will examine this central question to motivate an introduction to weak topologies and weak compactness. Then we will explore how weaker notions of convergence partially resolve this issue.
Mentee Aadhithya Saravanan
Mentor Omar Aceval Garcia
Abstract
I begin with a brief introduction to simple and general continued fractions and their properties. I then move into a proof of a representation of e as a continued fraction.
Convex Regularized Portfolio Optimization
Mentee Ali Khan
Mentor Anupam Nayak
Abstract
Modern portfolio optimization is typically formulated as a mean–variance problem, balancing risk (variance) and return (expected value). However, classical mean–variance optimization (Markowitz 1952) can yield unstable or non-sparse portfolios, especially when the number of assets is large or covariance estimates are noisy. This project investigates how convex regularization techniques (ℓ_2, CVaR constraints) can improve the robustness and interpretability of optimized portfolios compared to classical models and heuristic baselines.
Mentee Jeremiah Davis
Mentor Aprameya Girish Hebbar
Abstract
We will discuss the core ideas of surface theory. First, I will define what a surface is. Then I will go into the first and second fundamental forms of surface theory. I will also use visual examples to further help the audience understand the applications of surface theory.
Durrett’s Probability: Theory and Examples (Ch. 1–3)
Mentee Artem Ivaniuk
Mentor Minhao Bai
Abstract
The semester was spent reading through Rick Durett's "Probability: Theory and Examples" textbook, diving into the depths of probabilistic measure theory, law of large numbers, and central limit theorems.
Equilibria in Non-Cooperative Games
Mentee Shawn Varghese
Mentor Riley Guyett
Abstract
The presentation will cover John Nash's Ph.D. thesis on non-cooperative games, following its outline by introducing the relevant definitions of n-person finite games, mixed strategies, pay-off functions, and equilibrium points. It will then detail the proof of the existence of equilibrium points using the Brouwer fixed-point theorem.
Mentee Jason Billings
Mentor António Gouveia
Abstract
An introduction to geodesics. We will discuss an intuitive model as well as introduce some definitions and show some examples and non-examples on S^2.
Mentee Khoi Vu
Mentor Pablo Blanco
Abstract
This project explores girth, the length of the shortest cycle in a graph, and its significance in understanding the structure and sparsity of graphs.
Mentee Sriraj Muddu
Mentor Ben-Zion Weltsch
Abstract
Löb's Theorem states that if a consistent formal system can prove a self-referential statement of the form, "If this statement is provable, then it is true," then the system must already be able to prove the statement outright. This illustrates the limits on a formal theory's ability to assert its own completeness or even conditional correctness.
Mentee Jeevan Shah
Mentor Adam Earnst
Abstract
I will be presenting one proof of quadratic reciprocity as well as discussing some important applications and history associated with quadratic reciprocity.
Mentee Dylan Patel
Mentor Caleb Fong
Abstract
The Ramsey (3, 4) problem is a simple question about coloring edges, yet it’s one of the first places where symmetry breaks and unexpected structure appears. It shows how quickly order can emerge from very minimal rules, and why even small Ramsey numbers point toward much deeper unanswered questions. This case offers a clear glimpse into the surprising behavior that makes Ramsey theory so challenging and fun to explore.
Mentee Frederick Chang
Mentor Joy Hamlin
Abstract
We introduce some definitions, Van Kampen's Theorem, and see some applications.
Mentee Hamshika Rajkumar
Mentor Timothy Bates
Abstract
Algebraic topology is the bridge that allows us to assign algebraic structure to topological spaces; we are able to do nice algebra and get some insight into the topological structure. In this talk, we will focus on the fundamental group by computing some examples and discussing some applications to other well-known theorems.
Mentee Akhil Vasagiri
Mentor Ben-Zion Weltsch
Abstract
In recursion theory, the halting problem is the problem of determining whether a computer program, given an input, will halt or run forever. In this talk, we will introduce a model of computation and discuss a method for encoding every possible program into a unique natural number. In doing so, we will be able to show that the halting problem is undecidable, meaning there is no single program that can solve the halting problem for all pairs of programs and inputs.
The Unsolvability of the Quintic Polynomial
Mentee Wentai Chen
Mentor Alexander Day
Abstract
The basic ideas of Galois theory: symmetries of the roots of a polynomial (the ways the roots can be permuted without changing algebraic relations) form a group, called the Galois group. This Galois group encodes essential information about how the roots are related and what algebraic expressions can be built from them. Through a deep correspondence between subgroups of the Galois group and intermediate fields of a field extension, we show a particular quintic polynomial cannot be solved by radicals.
Upper Bounds: What Incompleteness Means for the Mathematician and the Epistemologist (Accompanying Paper)
Mentee Kira Shouldis
Mentor Danae Rupp
Abstract
This paper will briefly present a proof of Gödel's Incompleteness Theorem, then provide summary of several different philosophical and mathematical attempts to reckon with its implications. It will recount an argument about mathematical ontology posited in an earlier article that I have written, explain how said argument was somewhat misguided, and address how the notion of Incompleteness can enrich the innovations attempted by said article.
An Introduction to Differential Geometry and Liouville's Theorem on Manifolds
Mentee Jacob Sze
Mentor Ryan Mc Gowan
Abstract
Geodesic fields and flows allow us to find the direction of all geodesic curves, which help us understand lengths on general surfaces, called manifolds. In this talk, we will explore how they are defined, and we will look at what one can say about volumes on a manifold using these ideas.
An Introduction to Differential Topology: Smooth Maps, Tangent Spaces, and Rank
Mentee Pranav Shankar
Mentor Timothy Bates
Abstract
This talk brings together differential calculus and topology by discussing smooth manifolds, the maps between them, and local coordinate representations of these maps. We then discuss tangent vectors on manifolds and the Rank Theorem for smooth maps.
Applications of Convex Functions in Analysis
Mentee Mahadevan Seetharaman
Mentor David Herrera
Abstract
We will illustrate, using some examples, how clever convexity/concavity arguments lead to proofs of inequalities and other theorems in real analysis.
Asymptotics of Generalized Derangements
Mentee Aden Pereira
Mentor Dennis Hou
Abstract
We first establish the symbolic method to construct generating functions. The symbolic method is then used to construct the generating function for generalized derangements. Finally, we use methods from complex analysis to determine asymptotics from the behavior of the singularities of the generating function.
First-Order Logic and Deductions
Mentee Calvin Peloso
Mentor Isabella Arcoleo
Abstract
An overview of first-order languages, wffs, and deductions as defined and described in A Mathematical Introduction to Logic by Herbert B. Enderton.
Foundations of Algebraic Geometry
Mentee Tijil Kiran
Mentor Jianing Zhang
Abstract
We will explore the foundational aspects of algebraic geometry, including algebraic curves, projective geometry, and a proof of the weak Nullstellensatz.
Mentee Alankar Sovani
Mentor Samanthak Thiagarajan
Abstract
Harmonic functions describe systems which are at a steady state. In this talk, we will go through a derivation of Laplace's equation, explore some remarkable properties of harmonic functions, and prove Dirichlet's Principle: equating minimizing energies to being harmonic.
Phase Transitions in the Ising Model
Mentee Keshav Badri
Mentor Qidong He
Abstract
The Ising Model is one of the simplest, yet important toy models in Physics. Its primary application is in modelling the magnetic properties of ferromagnetic materials using discrete spins. In this presentation, we will unpack the mathematical results from this model, primarily related to a finite parameter phase transition. We will then briefly discuss lattice models with continuous spins and the famous Mermin-Wagner theorem, using ideas from the Ising Model.
Separability and Metrization Properties of Topological Groups
Mentee Herbert Huachaca
Mentor Timothy Bates
Abstract
We go over basic properties of topological groups, and use these to show (complete) regularity of topological groups and metrizability of first-countable topological groups (Birkhoff-Kakutani). As a corollary we show that for first countable topological groups, second countability, Lindeloffness, and separability are equivalent.
Silver’s Theorem: Approaching GCH for Singular Cardinals
Mentee Sriraj Muddu
Mentor Ben-Zion Weltsch
Abstract
How close can we get to statements like CH and GCH within the constraints of ZFC? What restrictions do we need to prove that the powerset of a cardinal is its successor? I will be introducing some basic concepts of Set Theory and using them to prove Silver's Theorem.
Spectral Graph Theory Meets GNN: An Approach to Graph Coloring
Mentee Khoi Vu
Mentor Forrest Thurman
Abstract
This project explores the use of spectral graph theory to improve graph neural networks (GNNs) for the graph coloring problem. We incorporate spectral features, such as Laplacian eigenvalues and eigenvectors, into GNN inputs to provide global structural information. The approach is evaluated on small synthetic graphs due to time constraints. Early results suggest that adding spectral features can improve coloring accuracy and offer insights into graph structure.
The Class Equation and Corollaries
Mentee Harry Haedrich
Mentor Yuqiao Huang
Abstract
In this short presentation I will provide a survey of some basic facts required to describe the Class Equation, and with time, discuss some interesting corollaries.
Mentee Urvi Patel
Mentor Ryan Mc Gowan
Abstract
The Mittag-Leffler theorem establishes that, given a discrete set of poles in the complex plane and prescribed principal parts at each pole, there exists a meromorphic function realizing exactly those singularities. This result complements Weierstrass’ factorization theorem by providing a dual perspective: while Weierstrass controls zeros, Mittag-Leffler controls poles. In this talk, I will present the statement and proof strategy of the theorem, along with an overview of key topics in complex analysis that provide the necessary context.
Mentee William Zhang
Mentor Ben-Zion Weltsch
Abstract
The Paris-Harrington Theorem is one of the first natural examples of incompleteness in the wild, showing that a finite statement in Ramsey theory is true but not provable in Peano Arithmetic. This is an overview of the proof of the theorem.
The Unsolvability of the Quintic Polynomial
Mentee Aryan Navin
Mentor Alexander Day
Abstract
I will give an overview of why there is no general formula to solve quintic polynomials using radicals, utilizing techniques from Galois theory and group theory.
Mentee Jason Billings
Mentor Anupam Nayak
Abstract
A brief history and applications of the Ballot Theorem, along with a fun proof.
A Brief Introduction to Differential Geometry
Mentee Ahmadh Hassan
Mentor António Gouveia
Abstract
An overview of the basic notion of manifolds to the geometric tools—Riemannian metrics, geodesics, and curvature tensors—that let us measure shape and curvature.
Algebraic Number Theory and the Sum of Squares
Mentee Sophia Pan
Mentor Nicholas Backes
Abstract
This presentation discusses the ring of Gaussian integers as an introduction to Algebraic Number Theory. We explore some of its properties and provide a proof of Fermat’s theorem on sums of two squares.
An Introduction to Elasticity Theory
Mentee Mahadevan Seetharaman
Mentor Samuel Wallace
Abstract
Elasticity Theory is the branch of continuum mechanics dealing with the deformation of elastic bodies. It applies many branches of mathematics, including functional analysis, PDE theory, differential geometry, and more. In this talk, we will give the basic definitions of concepts in elasticity and see how they arise from physical considerations. We will also look at a problem in elasticity to illustrate these concepts.
An Introduction to the p-adic Numbers and Hensel’s Lemma
Mentee Hamshika Rajkumar
Mentor Omar Aceval Garcia
Abstract
What if 5 were closer to 130 than to 6? By redefining what it means for numbers to be “close”, the p-adic numbers create a number system with surprising and counterintuitive properties. In this talk, I will introduce the p-adic norm and explain how it gives rise to the p-adic numbers. We will build some intuition for their topology in relation to Q and prove Hensel’s Lemma.
Mentee Jessie Wang
Mentor Bojue Wang
Abstract
Using Burnside’s Lemma to count distinct ways to color the four vertices of a square using two colors. Combining group theory to introduce the formula and illustrate the counting process step by step. Then induce the general way to solve those symmetry-based coloring problems.
🌝Brownian Bridges in Stochastic Calculus
Mentee Artem Ivaniuk
Mentor Minhao Bai
Abstract
A presentation covering the foundations of Stochastic Calculus learned during the semester, culminating in the discussion of Brownian Bridges—Gaussian Processes or Brownian motions that are conditioned on some future value of the process, starting from an initial value.
Calculating Effective Volume for a Gaunt–Fisher Configuration
Mentee Derek Chan
Mentor Qidong He
Abstract
This topic explores how to rigorously define and calculate the effective volume associated with a given Gaunt–Fisher configuration—a defect in a close packing ground state.
Categories and the Yoneda Lemma
Mentee Winston Li
Mentor Bojue Wang
Abstract
The Yoneda Lemma is an important result in category theory. I will introduce some definitions and examples of categories. Then, I will go over the lemma, its proof, and a short application.
Mentee Alon Danai
Mentor Alexander Day
Abstract
Category theory is a nice way of abstracting various concepts in math, and it gives us a general language to describe many things. We will go over basic category theory as well as discuss limits and adjoints with examples.
Mentee Nuray Kutlu
Mentor Dennis Hou
Abstract
This presentation will introduce and provide some nice examples of Cayley graphs.
Mentee Yilan Liu
Mentor Joy Hamlin
Abstract
In this presentation, I will introduce the basic concepts of Topology and the key terms related to one theorem related to the compactness of Topology. The theorem is “The image of a compact space under a continuous map is compact.”
Complex Differentiation, Integration, and Cauchy’s Theorem
Mentee Alankar Sovani
Mentor Ryan Mc Gowan
Abstract
The theory of functions of a complex variable has been a substantial area of study for some of the greatest mathematicians since the 1700’s up to the modern day. In this talk, we will see how the concepts of differentiation and integration change when working with complex numbers, leading us to the ever-important Cauchy’s Integral Theorem.
🌝Conformal Maps and the Riemann Mapping Theorem
Mentee Jason Liu
Mentor Ryan Mc Gowan
Abstract
In this presentation, I give an introduction to conformal maps, some examples, and present the Riemann Mapping Theorem (RMT), which gives the existence of conformal maps under very general conditions. If time permits, I also discuss the Schwarz–Christoffel theorem, which gives explicit maps from the upper half plane onto polygons, as well as the Uniformization theorem, a generalization of the RMT.
Continuum Mechanics: Formulating the Harmonic Oscillator
Mentee Aditya Raman
Mentor Erik Bahnson
Abstract
Going over spectral theorem and introducing notions and properties of infinite dimensional Hilbert spaces. Then understanding spectral theorem and then applying it to the position operator. Then demonstrating self adjointness and demonstrating the position operators failure of self adjointness alongside the Riesz Representation Theorem and introducing notions of operator domains to obtain self-adjoint extensions. Finally, using all of the above to formulate the harmonic oscillator.
De Finetti’s Theorem: Exchangeability and Its Consequences
Mentee Luke Wernyj
Mentor Forrest Thurman
Abstract
This talk introduces the idea of exchangeability and presents De Finetti’s Theorem, which characterizes infinite sequences of exchangeable random variables as mixtures of i.i.d. variables. We will see how this result bridges symmetry to probability theory and forms a foundation for Bayesian inference.
Dive into the Cantor Space: A Visualization Using Trees
Mentee Jason Billings
Mentor Danae Rupp
Abstract
In this presentation we will be building an intuitive picture and model of how the Cantor space is built. Along the way, we’ll see how trees offer a natural way to describe the topology of the space, connect finite approximations to infinite objects, and visualize key properties like compactness.
Exploring Applications of Linear Optimization for the Maximum Flow Problem
Mentee Vikram Kirhsnaswamy
Mentor Anupam Nayak
Abstract
This presentation will explore linear optimization, its applications, and its uses in solving the maximum flow problem. Additionally, this presentation will go over the Ford–Fulkerson approach to solving the maximum flow problem.
Exploring Thresholds for Properties of Random Graphs
Mentee Jeffrey Xu
Mentor Milan Haiman
Abstract
In this presentation, we will introduce the concept of random graphs and threshold functions, before looking at some specific graph properties, such as connectivity or the existence of specific subgraphs.
Mentee Akhil Vasagiri
Mentor Ben-Zion Weltsch
Abstract
Consider the set of all infinite sequences of natural numbers. Intuitively, it seems there should be no “empty space”; between any two sequences there should always be a third that lies in between. However, when we impose a certain ordering and utilize uncountable numbers, unexpected gaps begin to appear. In this presentation, we will explore these gaps and their relation to the Continuum Hypothesis.
Mentee Nick Belov
Mentor Natasha Ter-Saakov
Abstract
I’ll start with a brief overview of algebra in the ring of formal power series. Then I’ll get into exponential families with cards, decks and hands and eventually build up to the exponential formula!
Group Extensions and Ext
Mentee Nadya Belova
Mentor Timothy Bates
Abstract
We will motivate and introduce basic notions in homological algebra by discussing their relationship to the problem of group extensions, and ultimately to the problem of classifying finite groups.
Kähler Geometry
Mentee Maxwell Goldberg
Mentor Riley Guyett
Abstract
I will be introducing the basics of Riemannian geometry and using that to introduce Kähler geometry.
Non-Sliding Hard-Core Lattice Particle Models—Proving Crystallization
Mentee Harry Haedrich
Mentor Qidong He
Abstract
I will introduce some basic concepts from the paper “High-Fugacity Expansion and Crystallization in Non-sliding Hard-Core Lattice Particle Models Without a Tiling Constraint” in an informal matter, go over the main result of the paper and what it says about particle systems that satisfy its assumptions.
High Girth and Chromatic Number
Mentee Frederick Chang
Mentor Caleb Fong
Abstract
The girth of a graph is the length of its shortest cycle, and its chromatic number is the smallest number of colors required to properly color its vertices. For every positive integer k, l, is there a graph with girth greater than l and chromatic number greater than k? It turns out there is!
Insurance Company Simulation Using ML
Mentee Tanesha More
Mentor Minhao Bai
Abstract
I am working on applying what I learned in my machine learning course and my actuarial courses to try and simulate how a basic insurance company must estimate losses incurred from a person, then trying to price the insurance premium that the person will need to pay.
Mentee Shreya Ghosh
Mentor Adam Earnst
Abstract
In this talk I will discuss basic topics of measure theory, the connection between measure theory and ergodicity, and the Ergodic Theorem.
Pricing a Perpetual American Put Option
Mentee Arnav Kondagunta
Mentor Daniela Elizondo
Abstract
This presentation will cover the binomial model for asset pricing in discrete time. We will introduce martingales and stopping times, then derive the price of a perpetual American put option.
Mentee Akash Dubey
Mentor Hong Chen
Abstract
This presentation explores the q-Binomial Theorem, highlighting its combinatorial and algebraic meanings. We introduce q-analogues, define q-binomial coefficients, and explain their role in weighted counting and generating functions.
Mentee Taha Rana
Mentor Yiyang Liu
Abstract
A discussion of regular surfaces in R³ and the three major criteria of regular surfaces.
Mentee Anusha Iyer
Mentor Brittany Gelb
Abstract
How can we use the “shapes” found in datasets to draw conclusions about them? This project explores the fundamental concepts behind topological data analysis while applying persistent homology techniques using Ripser to demonstrate these concepts visually.
The Mycielski Construction
Mentee Arya Swaminathan
Mentor Milan Haiman
Abstract
The Mycielski construction provides a method to construct a graph with a certain clique number and an infinitely high chromatic number.
Mentee Michael Fiore
Mentor Ben-Zion Weltsch
Abstract
This presentation aims to understand the Wold Decomposition Theorem by building up the required prerequisite knowledge and understanding the implications of the theorem and how it is influential in modeling.
Understanding the Wold Decomposition Theorem
Mentee Pranav Tikkawar
Mentor Forrest Thurman
Abstract
This presentation aims to understand the Wold Decomposition Theorem by building up the required prerequisite knowledge and understanding the implications of the theorem and how it is influential in modeling.
A Brief Overview of Point-Set Topology
Mentee Ahmadh Hassan
Mentor Joy Hamlin
Abstract
I will be doing a general presentation on some theorems in point set topology and examples of how it works.
A Look into Lie Theory: Exploring Lie Theory Through Matrix Groups
Mentee Maxwell Goldberg
Mentor Riley Guyett
Abstract
Lie groups are a very important structure in mathematics. By studying groups which are also smooth manifolds, we are able to study symmetries of a wide array of systems. In this talk, we will learn about some fundamental Lie groups, such as SO(n), as well as the connection between them and Lie algebras.
“Climability” of Aronszajn Trees: Cofinal Branches in Trees of Height ω and ω1
Mentee Harry Haedrich
Mentor Ben-Zion Weltsch
Abstract
When do trees have branches? For trees of countable height and finite levels, it can be easily shown that there must be some branch (a cofinal branch) which reaches all the way to the top—in plain words, you can “climb” them, one node at a time. However, when we move past the countable into the uncountable, things get tricky. Can Jack climb the ℵ1-Aronszajn beanstalk?
Combinatorial Game Theory: Toads and Frogs
Mentee Nuray Kutlu
Mentor Dennis Hou
Abstract
This presentation will introduce Combinatorial Game Theory notation, definitions, and game reduction techniques along with some Toads and Frogs results.
Elliptic Curves and Fermat's Last Theorem
Mentee Alon Danai
Mentor Preston Tranbarger
Abstract
We will sketch an outline of how Fermat’s Last Theorem was proven using the theory of elliptic curves and modular forms.
Energy Methods for the Wave Equation
Mentee Ish Shah
Mentor Anupam Nayak
Abstract
The wave equation is one of the most fundamental examples of a linear partial differential equation (PDE), solvable through elementary means. However, solving the PDE is not necessary for studying some qualitative properties of solutions. In this talk, we will explore physics-inspired energy methods that allow for the study of some qualitative properties, such as the uniqueness of solutions.
Mentee Jessie Wang
Mentor Bojue Wang
Abstract
The Gershgorin Disk Theorem helps locate the eigenvalues of a square matrix by showing they lie within specific disks in the complex plane. This presentation will explain the theorem, its proof, and its applications, with examples to make it clear.
Incompleteness and Undecidability
Mentee Michael Fiore
Mentor Ben-Zion Weltsch
Abstract
An introduction to undecidable problems and unprovable statements.
Introduction to Symmetric Polynomials
Mentee Kevin Guether
Mentor Hong Chen
Abstract
A polynomial is said to be symmetric if permuting the variables results in the original polynomial. Three examples of these would be the elementary symmetric polynomials, the complete symmetric polynomials, and power sums. Each of these come with their own generating function. We will be examining the relation between these three types of polynomials and their generating functions.
Lagrangian Mechanics, Manifolds, and Noether's Theorem
Mentee Keshav Badri
Mentor Yiyang Liu
Abstract
Newton’s laws provide us with a fundamental set of rules to analyze these systems but have a fundamental limitation when considering problems of higher complexity. Instead, physicists tend to prefer the Lagrangian method for solving these problems. The Lagrangian method has further advantage in being able to describe “motions” in Quantum Field Theory and Electromagnetism. This talk will discuss the Lagrangian method and a mathematical derivation for Lagrangian mechanics using manifolds and its relevant spaces. We will then use this manifold definition to prove an intuitive formula and definition for Noether’s Theorem, which describes the relation between symmetries and conserved quantities in a given system. We will then connect these mathematical concepts with physical systems to motivate our intuition behind this important result of Lagrangian mechanics.
🌞Obtaining Perfect Matchings of Bipartite Graphs Through Augmentation
Mentee Valentina Pappano
Mentor Natasha Ter-Saakov
Abstract
This talk will introduce the concepts of bipartite graphs, Hall's Theorem, and matching. We will describe how to utilize augmented paths to extend a partial matching on a bipartite graph to find a perfect matching.
Orientation and Poincaré Duality: Relating Algebraic Invariants of Orientable Manifolds
Mentee James Belov
Mentor Timothy Bates
Abstract
We will discuss homology and cohomology of topological spaces, and a helpful theorem relating them in a wide and interesting class of spaces.
🌝Rational Points on Cubic Curves
Mentee Shreya Ghosh
Mentor Adam Earnst
Abstract
This presentation explores the properties of rational points on cubic curves using Group Laws. It highlights Mordell’s Theorem, which states that the group of rational points on an elliptic curve is finitely generated.
Sperner's Theorem and Its Applications
Mentee Akshat Singh
Mentor Milan Haiman
Abstract
How Sperner’s theorem can be applied to problems such as the Littlewood–Offord problem.
The Unreasonable Effectiveness of the Calculus of Variations
Mentee Mahadevan Seetharaman
Mentor Samuel Wallace
Abstract
The calculus of variations is a general framework for finding the critical points of functionals (functions depending on functions, rather than finitely many real variables). Problems that can be tackled by variational techniques arise in many areas of mathematics and physics, and the subject is an active area of research. We will derive the Euler–Lagrange equation, a basic result that can be used to solve many variational problems, then talk about some examples with interesting solutions, including the brachistochrone problem and the isoperimetric problem.
The Weierstrass Factorization Theorem
Mentee Jason Liu
Mentor Ryan Mc Gowan
Abstract
In this presentation we examine the Weierstrass factorization theorem, which can be seen as a generalization of the Fundamental Theorem of Algebra, its proof, and, if time permits, some applications.
Mentee Liza Ter-Saakov
Mentor Nicholas Backes
Abstract
What sort of properties are true for groups with topologies on them? In this talk, we will introduce topological groups and discuss some interesting theorems that shed light on their structure and significance, which then leads us to locally compact groups and the Haar measure on them.
Understanding Singular Value Decomposition
Mentee Aiman Koli
Mentor Nilava Metya
Abstract
My presentation will go over Singular Value Decomposition and its applications in Machine Learning that include Dimensionality Reduction, Recommendation Systems, and reconstructions. I will also address how I have implemented this computationally in a real-life project and show the limitations of the implementation. The next part will go over root finding techniques and will build to problems in convex optimization such as quadratic approximations and least squares.
Mentee Sophia Pan
Mentor Sam Spiro
Abstract
An introduction to combinatorial game theory, the study of sequential games with perfect information. This presentation focuses on 2-player games and introduces Nim, nimbers, impartial games, value of games, and concludes with the Sprague–Grundy Theorem.
An Overview on Park and Pham’s Proof of the Kahn–Kalai Conjecture
Mentee Rui Zhang
Mentor Charles Kenney
Abstract
I will provide a brief review of Jinyoung Park’s proof of the Kahn–Kalai Conjecture, which demonstrates the close relationship between expectation thresholds and actual thresholds in random structures.
Birkhoff ’s Curve Shortening Process
Mentee Eric Shim
Mentor Dongyeong Ko
Abstract
An introduction to geodesics, the Birkhoff curve shortening process, and some applications for chords of manifolds with boundaries.
Brownian Motion and the Dirichlet Problem
Mentee Andrew Xie
Mentor Anupam Nayak
Abstract
Theory of stochastic processes with selected applications to potential theory and the Dirichlet problem such as representation of solutions and properties of harmonic functions.
Coproducts in Algebraic Categories
Mentee Kyan Valencik
Mentor Alexander Day
Abstract
An overview of coproducts in various categories (Set, Group, Vect, etc.). Based on the readings of Algebra: Chapter 0 by Aluffi.
Demystifying Sheaves: A Basic Introduction to Sheaves
Mentee Maxwell Goldberg
Mentor Riley Guyett
Abstract
For my talk, I will be explaining what sheaves are and attempting to develop intuition for understanding them and what they do.
Differential Forms: Theory and Applications
Mentee Mahadevan Seetharaman
Mentor Ishaan Shah
Abstract
We will explain what manifolds are and construct differential forms on manifolds. We will briefly discuss applications.
∞-Categories with a View Toward Simplicial Homotopy Theory
Mentee Qichang Huangfu
Mentor Jishen Du
Abstract
Introduction to ∞-categories and simplicial homotopy theory.
Introduction to Stochastic Calculus and Its Applications in Finance
Mentee Pranav Tikkawar
Mentor Forrest Thurman
Abstract
An introduction to stochastic calculus and its applications in finance.
Mentee Nick Belov
Mentor Maxwell Aires
Abstract
A solution to the following problem: How do you partition the nodes of cactus into pairs so that you maximize the sum of the shortest distances between the nodes in the pairs?
Quantum Groups and Knot Invariants
Mentee Trisha Kothavale
Mentor Dennis Hou
Abstract
An introduction of the theory of quantum groups and how representations of quantum groups can be used to derive the Jones polynomial for a framed link.
Smooth Manifolds & Symplectic Manifolds
Mentee Nilay Tripathi
Mentor Forrest Thurman
Abstract
We start by giving a brief overview of topological manifolds, smooth structures on manifolds, and differential forms. We then discuss the basic concepts and definitions regarding symplectic manifolds and some brief applications in Hamiltonian mechanics.
Mentee James Belov
Mentor Timothy Bates
Abstract
An overview of one of the most efficient methods for computing fundamental groups.
A Brief Introduction to Differential Topology
Mentee Iris You
Mentor Bernardo Do Prado Rivas
Abstract
This presentation offers an introduction to the foundations of differential topology. Beginning with smooth maps, we illustrate how one obtains smooth manifolds and discuss their properties. Then tangent spaces are introduced as tools for studying the infinitesimal behavior of manifolds. Finally, we introduce tensors as a way to encode geometric relationships on curved spaces.
Mentee Ramesh Balaji
Mentor Rashmika Goswami
Abstract
This presentation provides an overview of what error-correcting codes are, what differentiates low-density parity-check (LDPC) codes from other kinds of error-correcting codes, and an overview of different decoding algorithms for LDPC codes.
An Overview of the Prime Number Theorem
Mentee Ish Shah
Mentor Nicholas Backes
Abstract
In this presentation, the prime number theorem will be stated. Key ingredients used in the proof, such as the Riemann zeta function, will be discussed. Finally, the presentation culminates in an overview of the proof of the prime number theorem.
Mentee Hanbo Xie
Mentor Forrest Thurman
Abstract
Basic introduction to strongly regular graphs and their connection with association schemes.
🌞Braid Groups and the Burau Representation
Mentee Ava Ostrem
Mentor Dennis Hou
Abstract
This presentation will cover the definition of braid groups and their basic properties. Then we will discuss linear groups and the linearity of the braid groups.
Mentee Trisha Kothavale
Mentor Bojue Wang
Abstract
An overview of the multilinear construction of the Clifford algebra and some of its basic properties, and some examples.
Complex Analysis: The Beautiful Implications of Being Holomorphic
Mentee Maxwell Goldberg
Mentor Riley Guyett
Abstract
Complex analysis isn't always an area of math focused on in a Rutgers undergraduate curriculum. However, it has many amazing results and consequences that don't appear in real analysis. In my presentation, I will give a highlight reel of complex analysis to showcase its beauty and convince you to learn it as well.
Dirichlet Characters, L-functions, and Applications
Mentee Liza Ter-Saakov
Mentor Charles Kenney
Abstract
I will give an introduction to characters and Dirichlet characters, then L-functions and applications. Prerequisites are basic group theory.
Mentee Rui Zhang
Mentor Ben-Zion Weltsch
Abstract
Filters are powerful mathematical structures with diverse applications in set theory, topology, and beyond. This presentation explores the fundamental concepts and properties of filters in the context of set theory.
Financial Mathematics and Personal Financial Decisions
Mentee Elvina Abzalimova
Mentor Minhao Bai
Abstract
The application of mathematical methods to financial problems, in particular personal finance. A look into mortgage and retirement calculations.
Optimized Portfolio Allocation
Mentee Artem Ivaniuk
Mentor Minhao Bai
Abstract
An overview of the Brownian Motion assumption in the stock market, its applications through the Geometric Brownian Motion (GBM) Model, which yield Markowitz’s Modern Portfolio Theory (MPT), and most recent research on optimizing the MPT through risk-averse 1/n into each n Brownian motions strategy.
Poset Dimensions and Scrambling Sets
Mentee Eric Yang
Mentor Milan Haiman
Abstract
The order dimension of partially ordered sets on the Boolean lattice is closely related to the notion of minimal scrambling sets, which describe the minimum number of permutations of n elements needed to place any one element above any k others. Exact values are not known for small values of k compared to n, but a variety of combinatorial and probabilistic techniques have narrowed the bounds on this value.
Results in Analysis and General Topology
Mentee Mahadevan Seetharaman
Mentor Ishaan Shah
Abstract
Point-set topology and mathematical analysis form a significant subset of the foundation for more advanced mathematics and have interesting applications in their own right. In the first half of this talk, we will discuss and prove some results about topological spaces and continuous functions. In the latter half of this talk, we will discuss three theorems from analysis: the Contraction Mapping Principle, the Inverse Function Theorem, and the Implicit Function Theorem. We will prove these theorems and talk about some applications.
Schemes from the Viewpoint of “Functor of Points”
Mentee Qichang Huangfu
Mentor Theodore Gonzales
Abstract
Considering the functor of points of a scheme can make it easier to study some problems in algebraic geometry such as the theory of algebraic groups. I will mainly discuss a theorem about a condition for a functor to be representable by a scheme.
The Addressing Problem for Loop Switching
Mentee Nuray Kutlu
Mentor Natasha Ter-Saakov
Abstract
This presentation talks about the math behind creating short and efficient addresses for data blocks in networks that transmit data between loops.
Mentee Zachary Roth
Mentor Ben-Zion Weltsch
Abstract
A history of choice and its controversy and some independence results.
🌝The Undecidability of Hilbert's Tenth Problem: Applications of Recursion Theory
Mentee Sean LeClair
Mentor Fanxin Wu
Abstract
Is there an algorithm that can decide if a Diophantine equation has an integer solution? No!
Untangling Knot Theory: Seifert Surfaces and Knot Genus
Mentee Jeffrey Tang
Mentor Timothy Bates
Abstract
A quick overview on knots, knot diagrams, classification of knots, and the notion of knot equivalence. The main focus will be on Seifert surfaces and Seifert's algorithm as a method for knot classification. Additionally, describing the genus of a knot using a Seifert surface and its properties and applications.
Algebraic Geometry: An Introduction with Hilbert's Nullstellensatz
Mentee Maxwell Goldberg
Mentor Riley Guyett
🌞An Overview of Hermite Polynomials
Mentee Nilay Tripathi
Mentor Forrest Thurman
Mentee Liza Ter-Saakov
Mentor Charles Kenney
Groups and Representations in Quantum Mechanics: Representations of Finite-Dimensional Groups
Mentee Mahadevan Seetharaman
Mentor Ishaan Shah
Mentee Daniel Elwell
Mentor Ben-Zion Weltsch
Representation Theory and Symmetric Functions
Mentee Emily Howlett
Mentor Hong Chen
Mentee Nuray Kutlu
Mentor Pablo Blanco
🌝ZFC and the Continuum Hypothesis
Mentee Ava Ostrem
Mentor Dennis Hou
Banach Fixed-Point Theorem and Application
Mentee Hana Huber
Mentor Karuna Sangam
Fourier Transform with Its Application
Mentee Xinchen Hua
Mentor Weihao Zheng
Gröbner Bases: A Sledgehammer for Solving Systems of Polynomials
Mentee Maxwell Goldberg
Mentor Bojue Wang
Introduction to Computational Homology
Mentee Varshini Gopalakrishnan
Mentor Brittany Gelb
Minesweeper Complexity: Is Minesweeper NP-complete?
Mentee Raj Limbasia
Mentor Robert Dougherty-Bliss
Representation Theory with Representations of S3 as an Example
Mentee Qichang Huangfu
Mentor Tae Young Lee
Shannon's Noisy Channel Coding Theorem over a Binary Symmetric Channel
Mentee Ramesh Balaji
Mentor Adarsh Srinivasan
Sylow Subgroups and Wilson's Theorem
Mentee Sakshi Koul
Mentor Dennis Hou
The Basics of Fourier Analysis
Mentee Tijil Kiran
Mentor Nicholas Backes
Mentee Alexander Valentino
Mentor Rashmika Goswami
An Overview of Kaplansky's Set Theory and Metric Spaces
Mentee Hasan Kunukcu
Mentor Fanxin Wu
Complex Analysis: An Introduction
Mentee Anish Suresh
Mentor Devin Bristow
Elliptic Curves and Cryptographic Applications: The Discrete Log Problem and Diffie-Hellman
Mentee Brian Zhang
Mentor Forrest Thurman
High Frequency Trading in a Limit Order Book
Mentee Akhil Sharma
Mentor Minhao Bai
Introduction to the Theory of Computation
Mentee Jean Paul Sadia
Mentor Devin Bristow
Pendulums and ODEs
Mentee Varshini Gopalakrishnan
Mentor Soham Chanda
Streaming Model and Misra–Gries Algorithm
Mentee Rohit Rao
Mentor Aditi Dudeja
Mentee Emily Howlett
Mentor Hong Chen
Mentee Jonathan Rasnitsyn
Mentor Caleb Fong
Mentee Yebai Zhao
Mentor Bojue Wang
A Probabilistic Perspective on the Open Question of When to Stop
Mentee Akhil Sharma
Mentor Minhao Bai
An Extremely Quick Look at the Sensitivity Conjecture
Mentee Ravi D'Elia
Mentor Sam Spiro
Complexity Theory
Mentee Daniel Baumgartner
Mentor Caleb Fong
Elliptic Curves over ℂ
Mentee Francisco del Campo
Mentor Nicholas Backes
Introduction to Knot Theory
Mentee Varshini Gopalakrishnan
Mentor Karuna Sangam
Lagrange's Theorem
Mentee Yakov Burton
Mentor Bojue Wang
Numeric Algorithms and RSA Encryption
Mentee Devin Holmes
Mentor Quentin Dubroff
The Lebesgue Integral
Mentee Vincent Caputo
Mentor Brittany Gelb
Complexity Theory
Mentee Gloria Liu
Mentor Maxwell Aires
Equivariant Maps and the Equivariant Rank Theorem
Mentee Daniel Bernstein
Mentor Lawrence Frolov
Mentee Ethan Kwok
Mentor Mitchell Bast
Mentee Mark Vaysiberg
Mentor Bojue Wang
Lie Groups, Homogeneous Spaces, and Model Geometries
Mentee Emily Howlett
Mentor Dennis Hou
Polynomial Factorization over Finite Fields, a Computational Approach
Mentee Alex Valentino
Mentor Louis Gaudet
Riemannian Geometry
Mentee Cormac Grindall
Mentor Dongyeong Ko
The Cauchy Integral Formula, Liouville's Theorem, and the Fundamental Theorem of Algebra
Mentee Yakov Burton
Mentor Sriram Raghunath
Mentee Akhil Sharma
Mentor Minhao Bai
Differential Topology: A Brief Journey
Mentee Maxwell Goldberg
Mentor Soham Chanda
Stochastic Processes and Time Series
Mentee Akhil Sharma
Mentor Minhao Bai