Welcome to Niles's research page.

[picture of Niles]

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

[diagram of projects I work on]

The projects are organized roughly by area, but there is considerable overlap among these areas and the placement of each project is intentionally imprecise. Notes in blue indicate completed work on parts of various projects, while notes in maroon indicate current work in progress. It's ok if these are unreadable, since the precise status is constantly in flux! For more details on a particular item, see the links below.

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

During the summer of 2011 I mentored two undergraduates through independent summer projects. Their work was exciting and encouraging—I'm glad I had the chance to support them!

Jack Farnsworth: Cross polytope numbers

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

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.

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