CU Boulder is offering a research opportunity for Summer 2023 in Mathematics. The research project is in **abstract algebra**. The deadline to apply is** ****February 1, 2023**.

The opportunity is a 10-week summer research internship for a rising junior or senior through the Summer Multicultural Access to Research Training (SMART) program. For 2023, SMART begins Monday, June 5th and ends Friday, August 11th. You will be part of a community of students doing research in different departments at CU Boulder.

The SMART Program pays program costs. Interns receive:

- Tuition for upper-division undergraduate credit in independent study at the University of Colorado at Boulder
- Room and board for the 10-week program
- Transportation to and from Boulder, Colorado
- A stipend of $5,000 to be paid in 3 monthly increments of $1,666.67
- Living arrangements: Interns live together in University of Colorado housing arranged by program staff. Meals are provided each day.

For more information on how to apply, visit: https://www.colorado.edu/initiative/cdi/undergraduate-stem-research/smart-program-information/how-apply-smart

**Project description:**

__Abstract algebra: 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)). The automorphisms of F(x,y) 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 can check that this composition gives a group structure on Aut(F(x,y)). In this project, we will explore questions about the structure of the groups Aut(F(x,y)) for certain special choices of formal group laws F(x,y).

Pre-requisite: An undergrad course in abstract algebra

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If this particular project is not for you, note that there are other projects in different fields (not just math) at CU Boulder. You can find a list of all CU Boulder projects here:

There are also many math opportunities you can access through the Leadership Alliance

https://theleadershipalliance.org/summer-research-early-identification-program