The Summer Program in Analysis and Measure (SPAM) 2026 is a four-week program in which undergraduate and graduate students explore research problems in various fields of mathematics using techniques from analysis and measure theory.

Students participate in a 4-week summer research program funded by the National Science Foundation (award no. DMS-2532664) where they work in small groups supervised by a faculty mentor, learning advanced topics in analysis beyond the standard curriculum and exploring applications of measure theoretic techniques in partial differential equations (PDE), statistics and machine learning.

The program is intended to acquaint participants with mathematical thinking, the rigors and best practices in research, as well as working collaboratively to break down challenging problems. In particular, it is expected that the weekly interactions, training and mentoring will significantly enhance students' knowledge of methods used in applied analysis.

List of Projects

1. Conservation law models for traffic flow: Conservation laws appear in a wide range of applications, including gas dynamics, fluid mechanics, aerodynamics, geophysical flows, biological transport and traffic modeling. These equations describe the evolution of quantities such as mass, momentum, and energy and are typically formulated as hyperbolic partial differential equations. A fundamental difficulty in their analysis is the formation of discontinuities (shock waves) in finite time, even from smooth initial data, which necessitates the study of weak solutions. In this program, we will investigate a conservation law that models the flow of traffic, known as the Lighthill-Whitham-Richards (or LWR) traffic model.
   
   

2. Metric entropy in statistics and machine learning: Metric entropy (or ε-entropy) has been studied extensively in a variety of literature and disciplines. It plays a central role in various areas of information theory and statistics, such as in the study of rates of convergence of empirical minimization procedures, as well as optimal convergence rates in numerous convexity constrained function estimation problems. From a different perspective, metric entropy has been used to measure the set of solutions of nonlinear partial differential equations. In this program, we will explore its applications in performing average-smoothness analysis for efficiently learning real-valued functions on metric spaces.