# Summer Undergraduate Applied Mathematics Institute

### May 28 - July 20, 2024

The Summer Undergraduate Applied Mathematics Institute (SUAMI) is an eight-week summer research program for undergraduate students. The program will be held on the campus of Carnegie Mellon University from May 28 through July 20, 2024.

The goals of SUAMI are twofold: (1) to expose students to the nature, culture, and rigors of research in pure and applied mathematics by working on a research project in collaboration with other program participants with faculty and postdoctoral mentors; (2) to give the students a taste of the graduate school experience to help them make an informed decision on whether they should attend graduate school, as well as inform them about other possible career paths in mathematics. SUAMI is part of the *Summer Scholars Program* run by the Mellon College of Science at Carnegie Mellon, and SUAMI students will have opportunities to participate in the academic, cultural, and social programs of the larger MCS summer community.

In 2024, SUAMI will feature projects in number theory, combinatorics, financial mathematics, and applied mathematics.

Participating students will receive a stipend, on-campus housing, a meal allowance, and reimbursement for domestic round-trip travel to and from Pittsburgh.

We are particularly interested in including women and individuals from other groups underrepresented in higher mathematics.

**The application deadline is February 15.**

SUAMI welcomes applications from all undergraduate students, regardless of citizenship.

##### ➤ Turán-type problems on partially ordered sets

**Primary Advisor:** Shanise Walker

**Abstract:** The goal of this project is to work on problems tied to Turán theory. Turán theory problems study the largest size of a subset of a combinatorial object (graph, grid, lattice, etc.) that does not contain some forbidden substructure. We investigate Turán-type problems for partially ordered sets (posets), which are sets with a binary relation that is reflexive, antisymmetric, and transitive. Given a poset $P$, we study the maximum size of a family of subsets in the $n$-dimensional Boolean lattice poset that does not contain $P$ when $P$ is a weak or strong subposet of the Boolean lattice. As a small example, the image below shows the 3-dimensional Boolean lattice and a poset of interest outlined in blue.

##### ➤ Optimization Algorithms for Problems Related to Partial Differential Equations

**Primary advisor:** Lucas Bouck

**Abstract:** Many problems in engineering lead to nonconvex optimization problems in high dimensions. The goal of this project is to study and implement various optimization algorithms for such problems and potentially develop new algorithms. The focus will be on global optimization for nonconvex problems, as well as techniques to accelerate gradient-based methods. These methods will then be applied to the computation of numerical solutions for calculus of variations problems arising in non-Euclidean elasticity. Students will learn techniques to analyze algorithms for optimization and implement numerical codes in Python.

**Prerequisites** for this project include a background in programming, calculus, and linear algebra. Background knowledge in numerical methods is recommended but not essential.

##### ➤ Markoff Triples and Periods of Linear Recurrence Sequences

**Primary Advisor:** Elisa Bellah

**Abstract:** The Markoff Equation is defined by $X^2+Y^2+Z^2=3XYZ$, and the integer solutions to this equation (such as (1, 1, 1) and (1, 2, 5)) are called Markoff triples. These triples were first introduced by Andrei Markoff in the late 1800s in the context of Diophantine approximation and quadratic forms, and over time have found applications in many different areas of mathematics (such as the study of free groups on two generators and the index of certain 4-dimensional manifolds). More recently, work has been done to study the mod p solutions to the Markoff equation, particularly due to their cryptographic applications.

In this project, we will investigate the number theoretic properties of Markoff mod p points by studying the path structure of certain associated graphs. We’ll leverage a new observation (made in https://arxiv.org/pdf/2311.11468.pdf) that the cycle structure of these graphs can be understood by studying the periods and intersections of certain associated linear recurrence sequences.

**Prerequisites:** a first course in elementary number theory and familiarity with some of the basic topics from a first course in abstract algebra (including groups, rings and fields). Coding experience is preferred, but not required.

##### ➤ Parameter estimation from data

**Primary advisors:** Jerry Wang, Rachel Kurchin

**Abstract:** A ubiquitous challenge in science and engineering is inferring model parameters from data, whether generated via simulations or experiments. This challenge is exacerbated by the presence of noise in that data and by potential errors in the form of model assumed. In this project, we will tackle the characterization of statistical error for a particular class of random walks that occurs in a wide range of nanoscale engineering applications. Students engaging in this project will sharpen their skills at the intersection of applied mathematics, statistics, physics, and scientific computing.

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