Research
Research Interests

Broadly speaking, I am interested in functional analysis and operator algebras. I primarily study C*-algebras built from locally compact groups and groupoids, including crossed products and other related constructions. I have also recently begun to work with C*-algebras associated to directed graphs and inverse semigroups. These sorts of constructions provide operator algebraists with interesting examples of C*-algebras, whose properties are often determined by the underlying algebraic object and/or dynamics. Most of my work revolves around crossed products of C*-algebras by locally compact groupoids. I have specifically worked on determining when such C*-algebras are nuclear or exact, and I have also spent some time investigating exact groupoids. I am currently interested in extending these sorts of results to Fell bundles, which subsume many other C*-algebraic constructions involving groupoids.

I also occasionally engage in research projects outside the realm of functional analysis. Recently I have worked on some problems in pattern avoidance, dynamical systems, and combinatorial group theory.

More information can be found in the current version of my research statement.



Publications

The following is a current list of publications, including some from my days as an undergraduate/applied mathematician.

  1. Some consequences of the stabilization theorem for Fell bundles over exact groupoids, in progress.
  2. Condition (K) for inverse semigroups and the ideal structure of their C*-algebras (with David Milan and Jamie Scott), in progress.
  3. On some permanence properties of exact groupoids, submitted, May 2017. arXiv:1703.05190 [math.OA]
  4. Amenability and uniqueness for groupoids associated with inverse semigroups (with David Milan), to appear in Semigroup Forum, 2016. arXiv:1511.01517 [math.OA]
  5. Allowed patterns of symmetric tent maps via commuter functions (with Kassie Archer), SIAM J. Discrete Math., 31 (2017), no. 1, 317–334. arXiv:1606.01317 [math.DS]
  6. Equivalence and exact groupoids, Houston J. Math, 42 (2016), no. 4, 1267–1290. arXiv:1411.1027 [math.OA]
  7. Nuclearity and exactness for groupoid crossed products, J. Operator Theory, 74 (2015), no. 1, 213–245. arXiv:1406.1749 [math.OA]
  8. Understanding the effects of polymer extrusion filter layering configurations using derivative-free optimization (with K.R. Fowler and E.W. Jenkins), Optimization and Engineering, 11(2), 339-354, June 2010.
  9. A simulation-based optimization approach to polymer extrusion filter design (with K.R. Fowler, E.W. Jenkins, and C.L. Cox), Filtration, 9(3), 224--230, July 2009.
  10. A simulation-based optimization approach to polymer extrusion filter design (with K.R. Fowler, E.W. Jenkins, and C.L. Cox), Proceedings of the American Filtration and Separations Society Annual Conference, (May 19-22) Valley Forge, PA 2008.


Talks and Posters

Here are some Mathematica demonstrations from a talk I gave on Fourier analysis.


Slides from an undergraduate-level talk I gave to prospective graduate students about my research. (The talk contains animations – for best results, open in Acrobat Reader.)

Invited Talks
Fell bundles: Unifying C*-algebras associated to groupoids
Linear Analysis Seminar, Texas A&M University. September 29, 2017.

Some recent progress on C*-algebras associated to groupoids
Analysis Seminar, University of Houston. April 11, 2017.

A brief overview of groupoid C*-algebras
Analysis Seminar, University of Houston. April 10, 2017.

Groupoid dynamical systems, C*-algebras, and crossed products
Mathematics Colloquium, Sam Houston State University. March 8, 2017.

Permanence properties of exact groupoids
Analysis Seminar, University of Houston. September 29, 2016.

Equivalence and exact groupoids.
Workshop in Analysis and Probability Seminar, Texas A&M University. July 31, 2014.

Nuclearity and exactness for groupoid crossed product C*-algebras.
Thesis defense, Dartmouth College. May 5, 2015.

Equivalence theorems for groupoids and their C*-algebras.
University of Texas at Tyler. March 5, 2014.
Mathematics Colloquium, Dartmouth College. February 20, 2014.

Results on nuclearity and exactness for groupoid crossed product C*-algebras.
Special Session on Classification Problems in Operator Algebras, Joint Mathematics Meetings, Baltimore, MD. January 16, 2014.

Groupoid equivalence and exactness.
Special Session on Groupoids in Analysis and Geometry, AMS Fall Central Sectional Meeting, Washington University in St. Louis. October 20, 2013.


Other Research Presentations
Permanence properties of exact groupoids
Great Plains Operator Theory Symposium, Texas Christian University, May 23, 2017.

Permanence properties of exact groupoids
AMS Contributed Paper Session on Operator Theory, Joint Mathematics Meetings, Atlanta, GA. January 5, 2017.

Amenability and uniqueness for groupoids associated with inverse semigroups.
Great Plains Operator Theory Symposium, University of Illinois at Urbana-Champaign, May 23, 2016.

Equivalence and exact groupoids.
AMS Session on Operator Theory, Joint Mathematics Meetings, San Antonio, TX. January 10, 2015.

Nuclearity and exactness of groupoid crossed products.
Great Plains Operator Theory Symposium, University of California, Berkeley. May 22, 2013.

Exact groupoids and crossed products.
New Hampshire Operator Theory Symposium, University of New Hampshire. May 4, 2013.

Properties of groupoid dynamical systems and their crossed products (Poster).
Dartmouth Graduate Poster Session. April 10, 2013.

Properties of groupoid dynamical systems and associated crossed product C*-algebras (Poster).
YMN-Project NExT Poster Session at the Joint Mathematics Meetings, San Diego, CA. January 9, 2013.

Nuclearity of groupoid crossed products.
New Hampshire Operator Theory Symposium, Dartmouth College. October 27, 2012.


Educational and Student-Oriented Talks
Projects in an introductory abstract algebra course.
MAA Contributed Paper Session on Teaching or Learning Advanced Mathematics, Joint Mathematics Meetings, San Antonio, TX, January 12, 2015.

Stokes and the surveyor's shoelaces
Math Club, University of Texas at Tyler. February 15, 2017.

Fun with Fourier analysis.
Math Club, University of Texas at Tyler. April 15, 2015.
SIAM Chapter Meeting, Dartmouth College. May 6, 2014.
Mathematical Adventures & Diversions, Lafayette College. February 12, 2014.


Functional Analysis Seminar, Dartmouth College
Graduate Student Seminar, Dartmouth College