I've had the good fortune to do independent reading and research work with undergraduate students on a few occasions. Here are some brief descriptions of their projects.

Micah Darrell: Topological Hochschild Homology and p-adic Hodge Theory (Summer 2017)

Abstract:

This is the final report from a summer project guided by the wonderful Niles Johnson. The goal was to try to develop the necessary background to understand the anticipated paper of Bhargav Bhatt, Matthew Morrow, and Peter Scholze in which the results of their paper ’Integral p-adic Hodge Theory’ will be explained using Topological Hochschild homology. The focus of our project was to try to understand topological Hochschild homology, and begin to understand the connection to p-adic cohomology theories in anticipation of this paper. In this paper an outline of the new construction of THH from [10] is given, and a very rough sketch of the connection of THH to crystalline cohomology is given.

Johann Miller: Universal Elements (Autumn 2014)

Introduction:

In algebra and topology, there are sometimes very general questions. How do I create a new commutative ring from a given commutative ring K with an indeterminate element x? Given an integral domain D, how do I construct a field containing D? Given a set of topological spaces Xi for i in I , how do I determine a natural topology on the Cartesian product of the underlying sets Xi? These questions all have answers which fit very nicely...However, there is more than one solution to each...There is a reason why the first answers are very desirable, as they are the "most efficient" solution to the given problem. We make this notion rigorous by defining universal elements.

Jack Farnsworth: Cross polytope numbers (Summer 2011)

Abstract:

This study on recursive definitions of Cross Polytope numbers was motivated by finding a formula for the number of lines spanned by all paths of length n from a fixed origin along the ZZ^d lattice. This led to the equivalent problem of finding the maximum number of non-adjacent vertices a distance n-1 along the same lattice. These are precisely the Cross Polytope numbers and help to justify certain recursion formulas used to define these numbers.

Eddie Beck: On Calculations of p-Typical Formal Group Laws (Summer 2011)

Abstract:

Formal group law theory provides computational tools with which to explore algebraic topology and homotopy theory. This paper studies the formal sum and the cyclic power operation for p-typical formal group laws, specifically to reduce prohibitive computation times through algorithm and time complexity analysis. We provide a combinatorial algorithm that directly computes terms of arbitrary degree using Mahler partitions. We also provide an online algorithm for computing the cyclic power operation, meaning that the precision of the calculations can be increased without restarting the computations. We measured the time complexity by counting the number of monomial multiplications required. These algorithms are at worst sub-exponential on the degree of the precision. Our algorithm substantially reduced previous computation times and shows that the McClure formula on MU_p is non-zero for p ≤ 61.

tags: fun-topics