$227,184; NSF Launching Early-Career Academic Pathways in the Mathematical and Physical Sciences (LEAPS-MPS)

Sharp Inequalities in Probability and Analysis
Principal Investigator: Phanuel Mariano, assistant professor of mathematics

Inequalities, a classical topic of study in mathematics, describe the relationships between quantities which may not be known precisely, but can be estimated or approximated. Through the study of inequalities, PI Mariano will investigate the connections between the mathematical fields of probability, analysis, and partial differential equations (PDEs), which are all fundamental tools in understanding physical phenomena. This research is relevant to our rapidly changing society, where data and randomness are becoming more and more present in our daily lives.

PI Mariano’s project will focus on two main research directions: 1) proving sharp inequalities involving the expected lifetime of a diffusion started inside a domain and the principal Dirichlet eigenvalue; and 2) addressing functional inequalities for degenerate diffusions. Additionally, a third research focus on the theory of explicit limit theorems for the products of random singular matrices will be explored with the undergraduate researchers involved in this project. Through his research, PI Mariano will discover new probabilistic tools that will lead to further exploration of the connection between analysis, geometry, and probability, as well as the connection between probability and other fields.

Through this project, PI Mariano aims to increase the representation and achievement of groups underrepresented in mathematics and probability. To achieve this goal, the PI will focus recruitment efforts on and encouraging undergraduates from groups underrepresented in STEM to work on this research. Furthermore, PI Mariano will host a summer conference on the recent progress in discrete continuous probability where he will create a welcoming environment for early career mathematicians, postdocs, and graduate students from all backgrounds to learn from and network with leading researchers in the field and to give short presentations of their own work. As part of this conference, PI Mariano will host lunch panel for early career mathematicians that will cover topics such as obtaining a postdoctoral and permanent position, how to network and initiate research collaborations, as well as information on careers outside of academia.

The PI also aims to increase the public’s scientific literacy of and engagement in STEM by hosting a distinguished lecture series on interdisciplinary applications of probability that would of interest to the general public, scholars, and students of political science, computer science, philosophy, physics, and engineering. Examples of these topics include gerrymandering, polling, and the impacts machine learning and data science in our daily lives.