Research

I am interested in Representation theory and its interactions with Algebraic geometry and Combinatorics. A big part of my research concerns the (geometric and combinatorial) representation theory of rational Cherednik and related Hecke algebras. You can see an introductory talk on rational Cherednik algebras I gave at ICMS Edinburgh in June 2019 here.  More generally, I am interested in the representation theory of quantized quiver varieties. In 2015, I was teaching assistant to Ivan Losev in a Masterclass on Quantized quiver varieties given at QGM Aarhus, Denmark. Here are videos of  the lectures. Lately, I have also been thinking of categorical link invariants, double affine Hecke algebras and quantum toroidal algebras.

Publications

1. Higher rank (q,t)-Catalan polynomials, affine Springer fibers and a finite shuffle theorem (joint with N. González and M. Vazirani). Trans. AMS, to appear.  arXiv. Extended Abstract in Sem. Loth. de Comb. 89B (2023), Article #84

2. Unitary representations of cyclotomic Hecke algebras at roots of unity: combinatorial classification and BGG resolutions. (joint with C. Bowman and E. Norton) J. Inst. Mat. Jussieu 23 (2), 2024, 557--608.

3. Harish-Chandra bimodules for type A rational Cherednik algebras. J. Algebra 642 (2024), 286--329. arXiv

4. Algebra and geometry of link homology. (joint with E. Gorsky and O. Kivinen) Bull. London Math. Soc. 55 (2), 2023, 537--591. These are lecture notes for a course E. Gorsky gave at IHÉS in July of 2021.

5. An elementary approach to the homological properties of constant-rank operators. (joint with A. Arroyo-Rabasa). C. R. Math. Acad. Sci. Paris 361 (2023), 45--63 

6. From representations of the rational Cherednik algebra to parabolic Hilbert schemes via the Dunkl-Opdam subalgebra. (joint wth E. Gorsky and M. Vazirani). Transf. Groups DOI 10.1007/s00031-022-09743-7 . arXiv. Video of a talk on this paper I gave at the Northeastern GPRT seminar, October 2020.

7. On BGG resolutions of unitary modules for quiver Hecke and Cherednik algebras. (joint with C. Bowman and E. Norton) Selecta Math New Ser. 28, 29 (2022)

8. Representations with minimal support for quantized Gieseker varieties. (joint with P. Etingof, V. Krylov and I. Losev) Mathematische Zeitschrift 298 (3) (2021) 1593--1621 arXiv. Slides from a talk on this (and other) papers I gave at the MIT Infinite-dimensional algebra seminar, February 2021.

9. On Harish-Chandra bimodules over rational Cherednik algebras. Advances in Mathematics 317 (2017) 299--349. arXiv

10. An alternative perspective on projectivity of modules. (joint with C. Holston, S. López-Permouth and J. Mastromatteo)  Glasgow Math. Journal 57 issue 1 (2015) 83--99. arXiv

11. Characterizing rings in terms of the extent of injectivity and projectivity of their modules.  (joint with S. López-Permouth). J. Algebra 362 (2012) 56--69. arXiv

Preprints

1. Cluster deep loci and mirror symmetry (joint with M. Castronovo, M. Gorsky and D. E. Speyer) arXiv.

2. Calibrated representations of the double Dyck path algebra (joint with N. González and E. Gorsky). arXiv. 

3. Cluster structures on braid varieties (joint with R. Casals, E. Gorsky, M. Gorsky, I. Le and L. Shen)  arXiv. Video of a talk on this paper I gave at CMO, September 2022. 

4. Positroid links and braid varieties. (joint with R. Casals, E. Gorsky and M. Gorsky). arXiv.

5. Algebraic weaves and braid varieties. (joint with R. Casals, E. Gorsky and M. Gorsky). Submitted.  arXiv. Video of a talk on this paper I gave at the RepNet Virtual Seminar, February 2021.

Other

1. Clusters and weaves (joint with M. Gorsky). Expository article to appear in Notices of the AMS.

2. Review of Mexican Mathematicians in the world: trends and recent contributions. (joint with A. Arroyo-Rabasa). Notices of the AMS 70 (8) 1274--1277