2023 Research Experience for Undergraduates

Title: The chromatic polynomial and the chromatic symmetric function

Abstract: The study of graph colorings arose in the mid 1800's when Francis Guthrie conjectured that four colors would suffice to color a map so that adjacent countries are distinct colors. In the modern version of the problem, we study the colorings of vertices so that adjacent vertices are distinct colors. In this talk, we will look at chromatic polynomials and chromatic symmetric functions, two algebraic ways to encode all graph colorings. We will discuss some applications of graph colorings and long-standing questions. We will also discuss the results from an Undergraduate Student Research Awards (USRA) project that studied the e-positivity of trees (joint work with Adrian She and Stephanie van Willigenburg).

Title: Random growth and complex integrals

Abstract: I will introduce two classes of random growth models: The Gaussian Universality class and the KPZ Universality class. For some simpler models, I’ll describe how to use complex integrals to analyze the model in the long-time limit. I will also try to show a lot of pictures and videos.

Title: WHAT IF…?: Assessing Alternatives to Current Policy

Abstract: This talk concerns the use of alternative scenarios to assess a particular, long-standing grading policy at a high school. Like most high schools, students enrolled in honors-level courses have an opportunity to earn more credit towards their grade point average (GPA), which is reflected in the student’s weighted GPA.

However, some school districts have broad discretion in how to compute weighted GPA, and this district’s approach, known informally as API*, was fundamentally different from the policies used in neighboring and peer districts given the importance of GPA as an indicator of academic achievement, and known differences in honors enrollment patterns, the district wanted to understand how the policy was working for all students.

Using grade data and other background information, the talk will discuss the API approach, how it differs from conventional grading policies, and demonstrate some implications of the approach for computing GPAs for students who earn honors credit vs those who don’t. The talk will also summarize recent honors enrollment trends in the district and estimate the impact of a modified policy on student outcomes.

*API stands for Academic Program Index and serves as an indicator of academic rigor across a student’s cumulative academic record.

Bio: Lincoln advises organizational leaders on operations strategy, program evaluation, and data use, with an emphasis on public-­private partnerships and cross-­functional teams. Prior to launching his own practice, he was a consultant with McKinsey & Company, and part of the management team of the Civic Consulting Alliance. Lincoln earned Masters and Ph.D. degrees in applied mathematics from the MIT Operations Research Center, and he is also a graduate of Florida A&M University, earning his bachelor’s degree in Computer and Information Sciences. A native and resident of Chicago's West Side, Lincoln is a founding board member of the Chicago Center for Arts and Technology (CHICAT), and he currently serves as the chair of the Programs and Evaluation Committee. Past Board appointments include the Education Pioneers Alumni Board (chair), and Theater Momentum (Artistic Associate).

Title: The Mathematics of Political Polarization (and how I got here)

Abstract: The present political climate of the United States suggests that political partisanship has reached an all-time high. How can this be quantified? In this talk I will give a brief overview of several different methods used to study this topic using mathematics, including a research project I have been involved in with undergraduate collaborators, investigating polarization in the U.S. Senate. I will also discuss what led me to this work, from my mathematical beginnings to today.

Title: In a world where friends of your friends are also your friends, Möbius came to invert it all.

Abstract: We will partially talk about orders (or perhaps talk about partial orders?) and explore how a function that we can define on orders relates to other areas of mathematics.

I will be speaking about posets, the Möbius function, and connections of these two with other areas. I will not assume previous knowledge of any of these topics since I am planning to explain it all.

Title: An introduction to hyperbolic 3-manifolds and ideas of quantitative Mostow Rigidity

Abstract: What makes hyperbolic 3-manifolds and related results interesting? We will gain an understanding of hyperbolicity via introductory material, providing a framework for a celebrated result in the study of manifolds, The Mostow Rigidity Theorem, which gives a remarkable bridge between the geometry and topology of complete, finite-volume hyperbolic n-manifolds in dimension at least 3. In particular, the topological type of a closed, orientable, hyperbolic 3-manifold M completely determines its geometry. We aim to understand the relevance of the question: How can this result be quantified? We will then explore different math "ingredients" involved in joint work with Peter Shalen which investigates the question through the lens of imposing natural restrictions on a mathematical object, the fundamental group of M, and provides new lower bounds on volume that improve manifold classification.

Title: Marlon Bundo Does Math

Abstract: Marlon Bundo is a charming childhood tale of a bunny determined to accomplish a goal and the stinkbug who gets in his way. Although Marlin Bundo's journey is fictitious, it is an unfortunate fact of life that stinkbugs are both real and rampant. In this talk, I will describe some of the stinkbugs I've encountered on my mathematical journey and the ways in which I've dealt with them. In addition to learning some stinkbug fighting techniques, observers of the talk should be assured they will also hear about some interesting mathematical problems.