My research interests are mostly within functional analysis, with the theory of Banach spaces as a starting point and then branching out to areas such as nonlinear functional analysis and operator spaces.
One frequent theme in my research so far is generalizations of well-known ideals of operators acting between Banach spaces to new situations. The study of such ideals of operators has historically provided tools for proving a myriad of interesting results whose utility goes beyond just Banach spaces, with applications in many other areas of analysis. For various reasons, it makes sense to try to generalize some of these ideas from Banach space theory to other contexts.
I have worked in two such situations: first the nonlinear functional analysis that arises when Banach spaces are replaced with general metric spaces, and then the operator space theory that concerns the case where Banach spaces are replaced with their noncommutative or quantized counterparts.
On a more applied note, I am interested in various aspects of signal processing. One of them is the recovery of low-rank matrices, which again has a noncommutative flavor. Another one is the theory of frames in Banach spaces.
Peer-reviewed journal articles
- Completely coarse maps are R-linear. With Bruno M. Braga. To appear in Proceedings of the American Mathematical Society. Preprint.
- Asymptotic dimension and coarse embeddings in the quantum setting. With Andrew Swift. Submitted for publication. Preprint.
- Isoperimetric and Sobolev inequalities for magnetic graphs. Preprint.
- Connectivity for quantum graphs. With Andrew Swift. Linear Algebra Appl. 608 (2021), 37–53. Preprint.
- Ando-Choi-Effros liftings for regular maps between Banach lattices. To appear in Positivity. Preprint.
- Operator p-compact mappings. With Verónica Dimant and Daniel Galicer. Journal of Functional Analysis, 277 (2019) 2865–2891. Preprint.
- Frame potential for finite-dimensional Banach spaces. With D. Freeman and K. Kornelson. Linear Algebra Appl. 578 (2019), 1–26. Preprint.
- An Ando-Choi-Effros lifting theorem respecting subspaces. J. London Math. Soc. (2) 100 (2019) 914–936. Preprint.
- Ideals of extendible Lipschitz maps. With A. Jiménez-Vargas. Submitted for publication. Preprint.
- p-converging operators and Dunford-Pettis Property of order p. With D. Chen and L. Li. J. Math. Anal. Appl. 461 (2018), no. 2, 1053–1066. Preprint.
- Pelczynski’s property (V∗) of order p and its quantification. With D. Chen and L. Li. Math. Nachr. 291 (2018), no. 2-3, 420–442. Preprint.
- Duality for ideals of Lipschitz maps. With M. G. Cabrera-Padilla, A. Jiménez-Vargas and M. Villegas-Vallecillos. Banach J. Math. Anal. 11, (2017), No. 1 108–129. Preprint.
- Maximal Banach ideals of Lipschitz maps. With M. G. Cabrera-Padilla, A. Jiménez-Vargas and M. Villegas-Vallecillos. Annals of Functional Analysis 7, (2016), no. 4, 593–608. Preprint.
- Some notions of transitivity for operator spaces. With Timur Oikhberg. Function spaces in analysis, 49–61, Contemp. Math., 645, Amer. Math. Soc., Providence, RI, 2015. Preprint.
- Lipschitz tensor product. With M. G. Cabrera-Padilla, A. Jiménez-Vargas and M. Villegas-Vallecillos. Khayyam Journal of Mathematics. 1, No. 2 (2017), 185–218. Preprint.
- Lipschitz p-convex and q-concave maps. Submitted for publication. Preprint.
- Stability of low-rank matrix recovery and its connections to Banach space geometry. With Denka Kutzarova. Journal of Mathematical Analysis and Applications 427 (2015), 320–335. Preprint.
- The Chevet-Saphar tensor norms for operator spaces. Houston Journal of Mathematics. 42, No. 2 (2016), 577–596. Preprint.
- Lipschitz factorization through subsets of Hilbert space. Journal of Mathematical Analysis and Applications, 418 (2014) 344–356. Preprint.
- Lipschitz (q,p)-mixing operators. Proceedings of the American Mathematical Society, 140 (2012), no. 9, 3101–3115. Preprint.
- Completely (q, p)-mixing maps. Illinois Journal of Mathematics, 56 (2012), no. 4, 1169-1183. Preprint.
- Duality for Lipschitz p-summing operators. Journal of Functional Analysis, 261 (2011), no. 2, 387–407. Preprint.
Undergraduate research advisees:
- Andrew Swift (2018-2021)