# Welcome to Niles's research page.

My research ranges from categorical algebra to computational topology. My main interests are in categorical and topological connections having to do with Morita theory, Brauer groups, Galois theory, Norm/Power operations, and homotopic descent. I've also worked with the UGA VIGRE Algebra group on two projects in Lie algebra cohomology, and developed packages of computer code for Weyl group and weight combinatorics relevant to those projects.

## Project Map

To find out more about my interest and work in a particular area, you can navigate to the relevant page using the links below. If you prefer, you can also jump directly to a list of my arXiv preprints. And if you'd like a broad overview of my research program, have a look at my research statement (pdf).

If you're particularly interested in something, I'd love to hear from you! Contact me by e-mail if you have any questions or comments.

## Research Projects, by Area

Use the links for more information about the projects in a particular area. Note that some items appear in more than one area!

### Categorical Algebra

- Morita Theory and Invertibility in Bicategories
- Azumaya Objects and Brauer Groups in Bicategories
- Azumaya and Brauer Theory for Tensor Triangulated Categories

### Algebraic Topology

- Norm and Power Operations for Ring Spectra
- Obstruction Theory for E_infty ring maps
- Azumaya Objects and Brauer Groups in (Triangulated) Bicategories
- Brauer Groups and Galois Theory for Commutative S-algebras
- Calculations for Complex-Oriented Cohomology Theories
- Ecological Niche Topology

### Representation Theory

- Low-Degree Lie Algebra Cohomology
- Weyl Group Linkage Computations
- Composition factors of Weyl modules

### Computer Calculations

- Computations for Complex-Oriented Cohomology Theories: Power Operations, Hopf Algebroids, Formal Group Laws
- Weyl Group Linkage Computations
- Composition factors of Weyl modules

## Undergraduate projects mentored ¶

### 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*.

### 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*.

#### Research Pages

#### Office Information

Hopewell 184

Newark Campus

Ohio State University

1179 University Drive

Newark, Ohio 43055

Tel: +1 740 755 7856

#### Office Hours

- Mondays 10:30 –12:00 (ECC, HP 84)
- Wednesdays 12:55 – 2:30 (ECC, HP 84)
- Thursdays by appointment

#### E-mail address

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For more information, see the

contact page.