The mathematics department will be hosting an internal research experience for undergraduates and early graduate students in Summer 2023. If your idea of fun is thinking about mathematics, then this may be your chance for getting paid for having fun.

What's the job?

- You will be part of a group of researchers working on math problems, guided by the official mentor.
- You will be expected to be around most of the time, attend official functions, and be expected to work about 20 hours per week.
- Pay is $20/hr for 20 hours a week (that is, $400/week).

*Offers to undergraduate students will be made by late March*

## Summer 2023 Projects

**Group theory and formal group laws**

A formal group law is a power series F(x,y) in two variables x and y that satisfies certain properties akin to the properties of an abelian group. For example, F(x,F(y,z)) = F(F(x,y),z) for variables x,y,z, corresponding to associativity, while F(x,y)=F(y,x) corresponding to commutativity. Two examples are F(x,y) = x+y and F(x,y) = x+y+xy. Most examples are not this simple and in general F(x,y) is a true power series in the sense that it does not have a finite expansion in monomials x^ny^m. Just like groups, we can define homomorphisms between two formal group laws F(x,y) and G(x,y): A homomorphism is a power series f(X) such that f(F(x,y)) = G(f(x), f(y)). With this in hand, we can talk about the automorphisms of F(x,y), which are the invertible homomorphisms from F(x,y) to itself. The collection of all automorphisms from F(x,y) to itself forms a group, denoted Aut(F(x,y)). Indeed, if f(X) and g(X) are automorphisms, we can compose them to form a new automorphism f(g(X)) and one can check that this composition gives a group structure on Aut(F(x,y)). In this project, we will try to answer questions about the structure of the groups Aut(F(x,y)) for certain special choices of formal group laws F(x,y).

**Dates:** June 5 -- August 4

**Prerequisites:** MATH 3140

**Mentor:** Agnès Beaudry

**Graphs, quadratic forms, and period maps**

Period maps provide a method of parameterizing certain algebraic and geometric data arising from the Hodge theory of complex projective manifolds. In extending period maps to the boundary of the domains of definition, one is led to study cones of quadratic forms. Specifically, one wants to know whether a given monodromy cone is contained in a "standard" cone of quadratic forms. In certain cases, these monodromy cones can be described from the combinatorics of graphs. In this project, we will investigate this type of situation for the Prym map, where the monodromy cones arise from graphs, and where there are several open questions. From previous work there is an expectation as to what graphs will lead to monodromy cones that do not lie in a "standard" cone, and we would like to confirm this in further examples.

**Dates:** June 5 -- July 7

**Prerequisites:** Abstract Algebra 1 MATH 3140 and Abstract Algebra 2 MATH 4140 (although Linear Algebra MATH 2135 may be sufficient). Experience coding in Python may also be helpful.

**Mentor:** Sebastian Casalaina-Martin

**A novel direction in interacting systems, computational aspects, and applications**

The study of Schramm-Loewner Evolutions (SLE) in Probability theory started in 2000 when Schramm introduced a family of random fractal curves that describe the scaling limit of interfaces of many planar Statistical Physics models. Very recently, a version of the model, that is multiple SLE curves driven by Dyson Brownian Motion (DBM), was introduced. DBM is a system of interacting diffusion processes with many interesting properties. If you are interested in learning and applying Stochastic Analysis techniques useful as well in Mathematical Finance/ Random Matrix theory/ Computational techniques please contact me at my email address! The preferred direction can be chosen according to your interest. I am looking forward to discovering together this unexplored very recent territory between two different fields of Probability Theory: Random Matrices and SLE.

**Dates:** May 15 -- June 30

**Prerequisites:**** **Good computational/programming skills, and basic knowledge of Calculus and Probability.

**Mentor:** Vlad Margarint

**Topological Data Analysis, Computational Algebraic Geometry and Applications in the Sciences**

Two of the main challenges of modern data analysis are how to separate noise and outliers from significant parts of data samples and the high dimensionality of data. One of the most recent and powerful methods to attack these problems comes from topology. It is the goal of this REU to study methods from topological data analysis (TDA), computational algebraic geometry and apply them to examine data sets coming from the sciences. It is in particular planned to consider the so-called 'variety learning' problem which means to find varieties (i.e. zero sets of polynomials) and their singularities from which a particular data set is sampled from. We will then apply the methods to energy landscape as they appear in physics and chemistry and see in how far TDA and computational geometry can help to better understand them. During the REU a review of simplicial complexes and their homology will be given, afterwards we will study methods from computational algebraic geometry. We will also do some hands-on experimental mathematics using software tools for TDA and computational homology. Part of the REU will also be to learn the software/computational tools we need (from MATLAB, python, topoology toolkit etc...). In the end we hopefully can apply all this to some more realistic data scenarios coming from physics and chemistry.

**Dates:** May 11 -- June 19

**Prerequisites:** Knowledge of linear algebra and willingness

to learn some abstract math and do computer assisted computations.

**Mentor:** Markus Pflaum

**Apollonian Circle Packings**

An integral Apollonian circle packing is a certain fractal collection of circles, all disjoint or mutually tangent, for which all of the (infinitely many) circles have integer curvatures (curvature = 1/radius). There is a long history of studying this collection of curvatures from the point of view of number theory. Many questions begin by asking about the set of curvatures for a fixed packing. It is also possible to ask questions about the packings which contain a fixed curvature. Or about the relationships between pairs of curvatures. We will investigate some of these questions, using tools from number theory and geometry which we will introduce at the beginning of the summer, including integral binary quadratic forms and hyperbolic geometry. There's a natural programming component to this project also.

**Dates:** May 15 - June 30

**Prerequisites: **Math 3110 (Introduction to Number Theory) and Math 3140 (Algebra 1) preferred. Programming skills are a plus.

**Mentor:** James Rickards and Kate Stange

**Why are those two things related?**

Recent insights in the land of combinatorial Hopf algebras have suggested an algebraic foundation for surprising connections between families of combinatorial gadgets (graphs, integer partitions, Dyck paths, etc.). These kind of results lead to unexpected combinatorial consequences: for example, graphs to integer partitions gives insights on graph coloring problems. This project seeks to work out more of these examples with an eye towards a more general theory. Depending on the interest of the researchers this project could skew more algebraic (representation theory) or combinatorial.

**Dates:** May 15 - June 30

**Prerequisites:** Definitely a good comfort level with linear algebra. Math 3140 or Math 3170 are a bonus.

**Mentor:** Nat Thiem

## How do I apply?

Applications for the above Summer 2023 REU projects are open and are due March 3, 2023. Offers to students will be made by late March. If you have any questions, please contact Nat Thiem.

Women and other underrepresented groups are encouraged to apply. This program is not funded with federal money, so all students are welcome to apply regardless of eligibility for federal support or immigration status.

A completed application includes:

- This completed Google form (MUST be signed in to Google with your CU identikey)
- A .pdf copy of your (un)official transcript (note that you can't have graduated before Summer 2023)
- A ranking of the projects you'd be interested in participating in
- A statement describing why you would like to do mathematics research, and if applicable why your favorite project is especially compelling to you. Other things you might add to your statement include:
- If your experience does not meet the prerequisites of a project, but you nevertheless feel prepared, then please explain why
- If you unsuccessfully applied to this REU in previous years, please let us know (in general this will positively affect your application)

The University of Colorado Boulder is committed to building a culturally diverse community of faculty, staff, and students dedicated to contributing to an inclusive campus environment. We are an Equal Opportunity employer. Human diversity includes, but is not limited to ethnicity, race, gender, age, socio-economic status, sexual orientation, religion, disability, political viewpoints, veteran status, gender identity or expression, and health status.