I am a Hedrick Assistant Professor (i.e. postdoc) in the UCLA math department.
Until June 2018, I was a Ph.D. student at the University of Chicago. My advisor was Matthew Emerton.
Broadly, I am interested in algebraic number theory and Langlands program. I primarily study modular and automorphic forms, Shimura varieties and Galois representations, with an emphasis on the Taylor-Wiles-Kisin patching method and automorphy lifting theorems.
I am (jointly with Chi-Yun Hsu) organizing the UCLA number theory seminar. You can find a (partial) schedule of future talks here.
Math Sciences Building 6164
UCLA Mathematics Department
Los Angeles, CA 90095
This is an overview of the construction (due to Eichler and Shimura) of a two dimensional Galois representation associated to a weight two modular form (or more precisely, to a cuspform which is an eigenform for all of the Hecke operators). It is (a somewhat expanded version of) my topic proposal, which I wrote during the second year of my Ph.D. at UChicago.
This (current version: December 2019) is an expanded version of the appendix to my thesis. It's roughly my attempt to present the commutative algebra behind the Taylor-Wiles-Kisin patching argument (one of the main tools used in the proof of Fermat's Last Theorem) as cleanly and systematically as possible, using the "ultrapatching" approach introduced by Scholze. It is very much a work in progress, and I hope to eventually expand it to include some of the more sophisticated uses of patching (e.g. patching functors, Ihara avoidance, the Calegari-Geraghty method, derived deformation rings and Hecke algebra, etc.).