# Summer Undergraduate Applied Mathematics Institute

### May 30 - July 22, 2023

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 30 through July 22, 2023.

The goals of SUAMI are twofold: (1) to expose students to the nature, culture, and rigors of research 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 a more 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 2023, SUAMI will feature projects in number theory, partial differential equations, and discrete mathematics. More information about the projects and mentors will be posted soon.

We are particularly interested in including women and individuals from other groups underrepresented in higher mathematics. Your application should include a statement (at most two pages in length) that explains

- why you are interested in participating in SUAMI
- what research topic(s) you would be interested in working on
- a description of any obstacles or challenges that you have faced and overcome in your pursuit of studying mathematics

Applications should also include a transcript, CV, and contact information for two references.

Participating students will receive a $4,800 stipend, on-campus housing, a meal allowance, and reimbursement for domestic round-trip travel to and from Pittsburgh.

** The application deadline is February 15**. Late applications will be considered until all spots in the program are filled.

All applicants should apply online at MathPrograms.

##### ➤ Hidden number theoretic structures in analysis

**Mentors:** Theresa Anderson and Elisa Bellah

We will work in both number theory and analysis and not only become skilled at translating between these two areas of math, but perhaps make significant progress on a long-standing conjecture in harmonic analysis. Specifically, we will work on constructing an "exotic" measure that behaves differently on specific intervals on the real line than others, and relate this to the inherent number theory. Though the expectation is a write-up and presentation of our progress, previous REUs related to this theme have produced professional publications, so there is exciting potential for continuing the momentum. For an idea of the style of math that may come into to play, see this article [pdf] - particularly the first 2 sections and the last section (This is only to get a vague idea, you do not need to understand everything in this article -- all the previous groups have started with essentially no initial knowledge and made great progress, so you can too!)

**Prerequisites:** At least one course in proof-based analysis and at least one course in proof-based algebra. Elementary number theory preferred. Measure theory a plus but is not needed. Should be comfortable writing rigorous mathematical proofs.

##### ➤ Fractional and nonlocal calculus

**Mentor:** Hayley Olson

Life is not always differentiable. Classical differential equations are used to model a wide myriad of physical phenomena. However, they thrive in situations where the solution would be a nice, smooth function -- but life doesn't always look like that! Unlike their classical counterparts, fractional and nonlocal vector calculus operators can act on functions which are not necessarily differentiable nor continuous. This gives them applications in models such as swarming behavior of animals, fracture mechanics, image processing, and more. We'll dive into the history and development of fractional and nonlocal calculus, then analyze some systems and models using these operators. Methods may include: analyzing functions and systems of equations, estimates and inequalities, analysis tools, and computer simulation.

**Prerequisites:** Multidimensional calculus and at least one proof-based course. Some coding experience (e.g. Python or MATLAB), courses in (partial) differential equations, or courses in analysis are pluses, but not required.

##### ➤ Zero forcing in graph theory

**Mentor:** Ryan Moruzzi

Zero forcing is an iterative coloring process on a graph. Suppose a subset B of the vertices of a graph G are colored blue, and all vertices not in B are colored white. We can proceed with the following color change rule: a blue vertex v will force a white neighbor u blue if u is the only white neighbor of v. After carrying out this iterative color change process, either all the vertices in the vertex set of G will be colored blue, or there will be at least two white vertices stopping us from coloring the entire graph. The size of the smallest initial blue vertex set required to force all the vertices in G blue is called the zero forcing number of G. One can think about how to adapt this color change rule; for example, to be less restrictive than needing a unique white neighbor. Questions like the previous on variations of forcing have been adapted for different applications and research into these variations of forcing bring about exciting new directions which we will explore.

**Prerequisites:** At least one proof-based course. Some coding experience (e.g. Python or SAGE) are a plus, but not required.

##### ➤ Anti-Ramsey problems

**Mentor:** Juergen Kritschgau

A typical anti-Ramsey question will ask how many parts a large combinatorial structure should be partitioned into in order to guarantee the existence of a small substructure which is distributed across many parts. For example, one might partition the elements of $\mathbb Z/n\mathbb Z$ into enough color classes so that the equation $x+y=z$ has a solution $(a,b,c)$ where $a,b,c$ are in different parts. Another common context is to partition the edge set of a graph with the aim of finding a "rainbow" copy of a particular subgraph. On possible SUAMI project is to explore some anti-Ramsey question for equations in groups, or subgraphs in host graphs.

**Prerequisites:** Background in abstract algebra, graph theory, and combinatorics are a bonus, but not required.

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