College of Science and Health > Academics > Mathematical Sciences > Research > 2022 NREUP

2022 Research Experience for Undergraduates

​​​​​​​​​​In Summer 2022, four DePaul University undergraduates will spend 8 weeks engaging in full-time mathematical research, thanks to funding from the Mathematical Association America's (MAA) National Research Experience for Undergraduates Program (NREUP). The 2022 NREUP at DePaul University is being run by Department of Mathematical Sciences faulty Karl Liechty and Emily Barnard.

The 2022 NREUP at DePaul University will focus on the relatively new field of  dynamical algebraic combinatorics. Broadly speaking, problems in dynamical algebraic combinatorics investigate enumerative questions about the orbit of a combinatorially defined map on some algebraic object. We will focus on two key maps on the symmetric group: the pop-stack sorting map and the kappa map. A typical question is: For an invertible map, is the average of "size" of each orbit the same? Our students will build on existing results about the behavior of these maps for the set of all permutations and for permutations which are 312-avoiding to study the maps on other pattern-avoiding permutations. Students will leave with a toolbox for exploring research questions in a variety of interconnected mathematical fields, including dynamical systems, extremal, algebraic and geometric combinatorics, and possibly topology and representation theory.

NREUP Seminar Schedule

As part of the 2022 NREUP, the department is hosting weekly seminars from speakers who will speak to the REU students about the type of mathematics they do and how they ended up pursuing and attaining a PhD in mathematics. The seminars are typically held Friday mornings at 11am.

Title: Pursuing Racial Equity within Schools

Abstract: Motivated by a desire to support the use of local data and multiple perspectives in policy development, I propose an adaptable, iterative process (CODAP) for data-driven decision making. CODAP refers to the key action steps of the process: Collect, Organize, Describe, Analyze, Prescribe. Taken in order, these action steps define a path of inquiry from observation (data) to informed response (policy). This talk will introduce the framework and provide an example of how the process is supporting racial equity work and policy analysis in a suburban school district. 

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: Equipartitions of Measures
or
Convex Geometry Meets Euclid
or
How to Write a Geometry Paper

Abstract: The talk highlights some recent work with S. Catoiu exploring how some classic results in convex set theory play out in the context of triangles and other polygons. More broadly, the talk will be about how there are many interesting unsolved problems in elementary Euclidean Geometry.

Title: Inverse Problems: Theory and Practice

Abstract: In this talk, we will discuss the concept of an inverse problem as well as its applications. In particular, we will focus on inverse problems arising from partial differential equations. These problems are to determine information about the mathematical model provided that one has measured values of the solution. The applications span the areas of medical imaging, on-destructive testing, cosmology, and geophysics. We will discuss how one can solve some inverse shape problems using qualitative (i.e. non-iterative) methods. These methods require little a priori information and are computationally simple to implement when one collects enough data.

Title: How do zebrafish get their stripes — or spots?

Abstract: Many natural and social systems involve individual agents coming together to create group dynamics, whether the agents are drivers in a traffic jam, voters in an election, or locusts in a swarm. Self-organization also occurs at much smaller scales in biology, and here I will focus on elucidating how brightly colored cells interact to form skin patterns in fish. Because they are surprisingly similar to humans genetically, we will investigate zebrafish, which are named for their dark and light stripes. Mutant zebrafish, on the other hand, have variable skin patterns, including spots and labyrinth curves. All these patterns form as the fish grow due to the interactions of tens of thousands of pigment cells. This leads to the question: how do mutations change cell behavior to create spotted zebrafish? In this talk, we will combine different modeling approaches (including agent-based and continuum) and topological data analysis to help shed light on this question. More broadly, we will explore how a combination of biological and mathematical approaches are being used to better understand how genes, cell behavior, and visible animal characteristics are related in fish.

Title: A brief introduction to the "KPZ" universality class

Abstract: The Kadar-Parisi-Zhang (KPZ) universality class, named after three physicists, is a collection of random growth models that share some limiting characteristics. We describe some of these characteristics and compare them to other simple probability models such a coin flipping. We give a description of a model in the KPZ class, called the asymmetric simple exclusion process (ASEP), and describe some current work. This talk is aimed at a wide audience and no background knowledge is assumed.

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

Abstract: The celebrated Mostow Rigidity Theorem provides 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. In this talk, we aim to understand the relevance of the question: how can this result be quantified? After providing an introduction to hyperbolic space, we will outline, time permitting, different math "ingredients" involved in joint work with Peter Shalen which investigates the question through the lens of imposing natural restrictions on the fundamental group of M.

Title: Panel discussion with three mathematicians named Emily

Abstract: The three Emilys (Emily Barnard - DePaul University, Emily Gundawan - University of Oklahoma, and Emily Meehan - Gallaudet University) will discuss their experiences applying and persevering in graduate school, and becoming and being mathematicians.

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