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➤ Randomly weighted minimum spanning trees

Advisor: Tomasz Tkocz

Suppose every edge of a complete graph on $n$ vertices is assigned a weight, independently uniformly at random from $[0,1]$. Let $W_n$ be the total weight of a minimum spanning tree in the graph. Frieze's celebrated $\zeta(3)$-result says that the expected value ${\bf E} W_n$ goes to $\zeta(3)$ as $n\to\infty$. The goal of this project is two-folded. First, we will learn techniques leading to such results. Second, we will explore several constrained version of such problems, where the edges are also assigned independent random costs and we seek the minimal combinatorial structure satisfying a cost constraint.

Prerequisites: Probability course (21-325 or equivalent), Combinatorics course (21-301 or equivalent).

June 15 - August 25

➤ Gaussian assignment problem

Advisor: Tomasz Tkocz

Suppose every edge of a complete bipartite graph $K_{n,n}$ is assigned a weight, independently at random according to the uniform distribution on $[0,1]$. Let $M_n$ be the total weight of a minimum perfect matching in $K_{n,n}$. What is the expected value ${\bf E} M_n$? Aldous showed that it goes to $\zeta(2)$ as $n \to \infty$. When the distribution of the weights is exponential, a non-asymptotic result is known (Parisi's formula). The goal of this project is to learn more about such results and explore a possibility of extending them to other probability distributions, notably the Gaussian distribution, ubiquitous in probability theory.

Prerequisites: Probability course (21-325 or equivalent), Combinatorics course (21-301 or equivalent).

June 15 - August 25

➤ Optimization and Model Selection

Advisor: Hayden Schaeffer

Learning equations from data is an important task in model selection and parameter estimation. The goal is to construct data-based algorithms to uncover hidden processes that govern the data, for example, learning Kepler’s law from images of celestial bodies. For most datasets, the form and structure of the governing equations are rarely known a priori; however, based on the sparsity-of-effect principle one may assume that the number of candidate functions needed to represent the data is relatively small. Thus the learning problem is tractable. Students will develop optimization-based algorithms for fitting equations to observations. This project consists of modeling, computation, and theory.

Requirements: If you are interested in programming and applications, then knowledge of Python and MATLAB are required. If you are interested in the theoretical aspects, then successful completion of analysis and linear algebra is required.

June 15 - August 10

➤ Dynamical Systems and Neural Networks

Advisor: Hayden Schaeffer

Neural networks are a popular approach for various image processing tasks. However, their use in scientific modeling is limited. In this project, students will work on one application of neural networks for problems in computational physics. The goal is to develop approaches for integrating mathematical models and physical data using structured neural network architectures that respect physical laws. This project consists of modeling and computation.

Requirements: The main requirement is knowledge and/or experience in Python (knowledge in PyTorch or TensorFlow would be helpful). The mathematical requirement is successful completion of a course in ordinary differential equations.

June 15 - August 10

➤ Research topics in Discrete Mathematics

Advisor: Seyed Kaave Hosseini

We will introduce a few research projects related to Additive Combinatorics and Ramsey Theory that have applications in Number Theory and Computer Science. After introducing these projects and discussing them in depth, depending on the interests of the students, we will focus heavily on one or two of them.

Prerequisites: 21-241 and 21-301 (or very strong performance in 21-228).

May 25 - July 18

➤ Background and reading projects for undergraduates planning to take graduate level courses in mathematical logic

Advisor: Ernest Schimmerling

Students who have done very well in 21-300 (undergraduate logic) and 21-329 (undergraduate set theory) are prepared to try PhD level courses in mathematical logic such as 21-602 (graduate set theory). In the past, some strong undergraduate students filled in missing background and gained relevant intuition on their own by doing recommended reading before taking graduate mathematical logic courses. This summer, a more structured eight-week option, directed by Prof. Schimmerling, will be available. Study plans will be tailored to individual students. There will be reading assignments and exercises, student presentations, and discussions via Zoom with Prof. Schimmerling, sometimes in groups, sometimes one-on-one. The content will overlap 21-300 and 21-329 but other reading material could be included depending on each student's background and interests.

May 25 - July 18

➤ Problem-solving sessions in analysis

Advisor: Giovanni Leoni

The problem-solving session is a chance to work on challenging analysis problems in a casual environment.

I will start the session by reviewing the material. After this review, I will assign some problems, and students will work together on solving them. At the end of the sessions, we will go over the solutions.

Topics will include inequalities; sequences and series of real numbers and functions; limits, liminf, and limsup; continuity, partial derivatives, differentiability, higher-order derivatives; Taylor's formula; optimization and constraint optimization, Lagrange multipliers; the implicit function theorem and the inverse function theorem; multiple integrals, repeated integration, and change of variables.

Prerequisites: These sessions are intended for students with a strong background in analysis. These include students who have successfully passed 21-269 Vector Analysis or (21-355 Principles of Real Analysis I and 21-356 Principles of Real Analysis II). Students who have only taken 21-268 Multidimensional calculus might be considered.

Number of students: no more than 25.

June 1 - July 25