Research

My research interests lie in the fields of operator algebras, operator theory, and functional analysis with connections to dynamical systems and geometric group theory. In particular, I am interested in finite representation theory of operator algebras and graph products in operator algebras.

My research in finite representation theory of operator algebras considers representations of C*-algebras and von Neumann algebras in II1-factor von Neumann algebras with the goal of relating properties of these maps to properties of the domains and codomains. This research has connections to amenability, ultraproduct analysis of von Neumann algebras, and the Connes Embedding Problem.

The notion of graph products simultaneously generalizes free and direct/tensor products. I like to consider the interplay between the operation of taking a graph product and certain operator algebraic/theoretic notions including completely positive maps, positive definite functions, unitary dilation, von Neumann's inequality, dynamical systems, and approximation properties of groups and C*-algebras.

I am an active participant in the Fractal Research Group, Mathematical Physics & Dynamical Systems, and Operator Algebras seminars at UCR. I organized the UCR Operator Algebras seminar during the 2020 Winter Quarter.

MathSciNet

Publications

Peer Reviewed

  • (with S. Kunnawalkam Elayavalli) On ultraproduct embeddings and amenability for tracial von Neumann algebras. Int. Math. Res. Not. IMRN to appear. preprint 2019. arXiv:1907.03359

  • Some results on tracial stability and graph products. Indiana Univ. Math. J. to appear. preprint 2018. arXiv:1808.04664

  • On graph products of multipliers and the Haagerup property for C*-dynamical systems. Ergodic Theory Dynam. Systems 40 (2020), no. 12, 3188-3216.. arXiv:1803.01881

  • Graph products of completely positive maps. J. Operator Theory 81 (2019), no. 1, 133-156. arXiv:1706.07389

  • Minimal faces and Schur's lemma for embeddings into R^U. Indiana Univ. Math. J. 67 (2018), no. 4, 1327-1340. arXiv:1608.08189

  • (with C. Ramsey) Unitary dilation of freely independent contractions. Proc. Amer. Math. Soc. 145 (2017), no.4, 1729-1737. arXiv:1601.00613

  • Convex sets associated to C*-algebras. J. Funct. Anal. 271 (2016), no. 6, 1604-1651. arXiv:1509.00822

Preprints

Talk slides

  • Amenability via ultraproduct embeddings for II_1 factors. UCR Fractal Research Group Seminar, May 7 and 14, 2020. (Expository version with expanded background) SLIDES (PDF)

  • Amenability via ultraproduct embeddings for II_1 factors. UK Operator Algebras Virtual Seminar, April 30, 2020. (Expository version) SLIDES (PDF)

  • Amenability via ultrapower embeddings for tracial von Neumann algebras. Wabash Modern Analysis Seminar (virtual edition), April 25, 2020. SLIDES (PDF)

  • Ultraproduct embeddings and amenability for tracial von Neumann algebras. BIRS Workshop on Classification in von Neumann algebras, October 4, 2019. SLIDES (PDF) VIDEO (LINK)

  • Unitary dilation. University of Tennessee Chattanooga Summer Colloquium Series, July 12, 2019. SLIDES (PDF)

  • A selective version of Lin's theorem. Virginia Operator Theory and Complex Analysis Meeting, October 27, 2018. SLIDES (PDF)

  • Some recent results on graph products. Great Plains Operator Algebras Symposium, May 29, 2018. SLIDES (PDF)

  • Graph products of completely positive maps. East Coast Operator Algebras Symposium, October 7, 2017. SLIDES (PDF)

  • Minimal faces and Schur's lemma for embeddings into R^U. Barcelona Conference on C*-algebras: Structure, Classification, and Dynamics, June 19, 2017. SLIDES (PDF)

  • Unitary dilation of freely independent contractions. Young Mathematicians in C*-algebras, July 27, 2016. SLIDES (PDF)

  • Convex sets associated to C*-algebras. JMM Special Session on Classification Problems in Operator Algebras, January 8, 2016. SLIDES (PDF)