➤ Convex Sets and Probability

Advisor: Konstantin Tikhomirov

Convex sets naturally appear in data science and optimization as domains for high-dimensional data points. Probabilistic methods can be very efficient in understanding geometry of convex sets: volume distribution, the shape of sections and projections, arrangement of supporting hyperplanes. The research area is very active but there are still many interesting open problems to consider. The goal of this project is to get familiar with fundamental results in high-dimensional convex geometry and to apply probability to study domains given by linear and quadratic constraints.

Prerequisites: An introductory course on Linear Algebra/Matrix Theory (such as 21-241, 21-242 or equivalent) and Probability (21-325 or equivalent).

June 3 - July 26

➤ Counting Quantifiers in Topology

Advisor: Riley Thornton

We can measure the complexity of a sentence by counting the number (and type) of quantifiers needed to express it. For instance, saying that a relation is an equivalence relation only requires universal quantifiers. But, saying that an order has a maximum element requires an existential quantifier followed by a universal quantifier. (Of course, proving that you really need these quantifiers takes some ingenuity.) In this project we will measure the complexity of some notions from topology.

Prerequisites: Concepts of Mathematics, and one of Basic Logic, Real Analysis, or Topology (or an equivalent).

May 8 - July 3

➤ Enhancing Generative Diffusion Models with Statistical Estimation for Computational Chemistry Applications

Advisor: Maria Chikina (University of Pittsburgh)

Neural networks are now recognized for their potential to solve complex problems in probability, such as high-dimensional density estimation and generating samples from complex distributions (e.g., DALL·E ). Despite these advances, significant challenges remain. Different probabilistic problems demand bespoke solutions, particularly as current leading sample-generating models lack the capability to provide density estimates.

This project seeks to bridge the gap between the generation of high-quality data samples, a hallmark of diffusion models, and the accurate estimation of data densities, leveraging the statistical efficiencies of methods like noise contrastive estimation. The ultimate goal is to apply this in the context of computational chemistry where density normalizing constants have physical interpretations that can be leveraged in molecular design.

Prerequisites: This is an interdisciplinary project combining statistical physics, statistics, and machine learning and we encourage students of all backgrounds to apply. Programming ability is required. Probability courses (21-325) and Linear Algebra (21-241, 21-341 or 21-344) are useful but not required.

May 13 - July 5

➤ Problems in Anti-Ramsey Theory

Advisor: Michael Young

A set is considered to be rainbow if each element of the set is assigned a distinct color. Anti-Ramsey problems ask the question: How many colors must appear in the coloring of a set to guarantee that some preidentified subset is rainbow. For example, we may color the integers in the set $[n]$ with $r$ colors and want to guarantee that no matter how we assign the colors, some set $\{x, y, z\}$ that satisfies the equation $x + y = z$ is rainbow. We could also color the edges of a complete graph and want to guarantee a rainbow cycle. In this project, we will explore some anti-Ramsey questions.

Prerequisites: Concepts of Mathematics (21-127 or 21-128) and Discrete Mathematics (21-228).

May 28 - July 19

➤ One-dimensional Calculus of Variations

Advisor: Reinaldo Resende

In this project we will focus on the examination of single-variable functions, their trajectories, and the comprehension of variational concepts in one dimension. The objective is to compute Euler-Lagrange equations to investigate critical points of certain functionals. Examples that may be investigated include the brachistochrone problem, minimal surfaces of revolution, and the cycloid (the latter representing the path that minimizes the time from top to bottom in the .gif below). Further exploration may include (in one-dimension): examples of non-differentiable critical points and the correlation between convexity and the existence of minimizers. These sorts of techniques are applicable to, and inspired by, problems in physics, engineering, and economics, showcasing the versatility and power of variational techniques.

Prerequisites: 21-235 Principles of Real Analysis or Math Studies: Analysis I.

June 24 - August 16

➤ Mathematics of Deep Learning Architectures

Advisor: Junichi Koganemaru

Deep learning is an umbrella term for a popular class of algorithms that have seen major applications in recent years. Aside from its many practical applications in data science, they have also been used as a useful tool for helping researchers prove rigorous mathematical theorems. For example, Physics-informed neural networks (PINNs) were recently employed by teams of researchers to identify potential blow-up solutions to fluid equations.

The main goals of this project are three-fold.

1. Investigate and understand the mathematics underlying prototypical deep learning architectures and architectures proposed in the literature.
2. Learn how to use deep learning tools to solve complicated problems in practice
3. Learn how to write mathematical papers.

Prerequisites: Since this is an interdisciplinary project, we encourage students of all backgrounds to apply, but some familiarity with Linear algebra (21-240 or similar), Probability (21-325 or similar) and Real analysis (21-355 or similar) is highly recommended. While programming experience is not required, it can also be beneficial for the project.

May 13 - July 5