# Undergraduate projects mentored.

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.

This document is available on Micah's homepage.

### 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