Jay Hathaway

I am a 2nd-year graduate student in math at The University of Oregon. I am interested in categorical and geometric representation theory, and in their connections to low-dimensional topology and mathematical physics. My advisor is Ben Elias. Previously I was an undergraduate in physics and mathematics at The University of Texas where I did a senior thesis on the Springer correspondence, supervised by Sam Gunningham.

In the Winter 2019 quarter I will be a mentor for the directed reading program (DRP). I hope to pay forward the positive experiences I had as a mentee in the DRP at The University of Texas.

Motivation

In arXiv:1811.06188 B. Elias initiates a research program giving a Soergel bimodules analogue of D. Gaitsgory's construction of central perverse sheaves on affine flag varieties, which we call Gaitsgory's Central Sheaves (GCS). In short this program aims to associate a central complex of Soergel bimodules of (extended) affine type A to any finite dimensional representation of SL_n. This is achieved in all rank for the defining representation. We call these Gaitsgory's Central Complexes. Another outcome of this program is to give a Soergel bimodules analogue of the so-called flattening functor from the affine Hecke category to the finite Hecke category. This functor categorifies the flattening map from the cylindrical braid group to the ordinary braid group.

In a twist of fate, the flattening of the Gaitsgory's Central Complex associated to the defining representation of SL_n is sought after in work of E. Gorsky, A. Negut, and J. Rasmussen (GNR) conjecturing an equivalence between the Drinfeld center of the Hecke category of type A and coherent sheaves on the Hilbert scheme of n points in the plane. It is expected this complex corresponds to the tautological bundle on the Hilbert scheme.

Current Project

In view of (GNR), I am attempting to give a formula for the endomorphism X of the tautological bundle on the Hilbert scheme as a morphism of the Gaitsgory's Central Complex associated to the defining representation. (in writing)

Possible Future Projects

• Give an isomorphism between the endomorphism ring of Gaitsgory's Central Complex for the defining representation and the endomorphism ring of the tautological bundle on the Hilbert scheme.
• Compute the generating morphisms of SL_4 Webs as morphisms of Gaitsgory's Central Complexes.
• Write computer software to assist me in some of these calculations.

In Spring 2019 I am teaching Math 112 - Precalculus. The course website is here.

Teaching

Quarter	Class
Spring 2019	Math 112 - Precalculus - Course Website
Winter 2019	Math 111 - College Algebra
Fall 2018	Math 241 - Business Calculus (TA)
Spring 2018	Math 243 - Business Stats (TA)
Winter 2018	Math 111 - College Algebra
Fall 2017	Math 111 - College Algebra

Upcoming Travel

I will attend...

• The FRG workshop Hilbert schemes, categorification and combinatorics in Davis, California from June 19-23, 2019.

Past Travel

I attended...

• The Summer School on Geometric Representation Theory from July 9-13, 2018 in Vienna, Austria.
• The CIME School Geometric Representation Theory and Gauge Theory from June 25-29, 2018 in Cetraro, Italy.
• The Seminaire de mathematiques superieures (SMS) 2018 - Derived Geometry and Higher Categorical Structures in Geometry and Physics from June 11-22, 2018 in Toronto, Canada.
• The MSRI Summer Graduate School - Soergel Bimodules From June 26 - July 7, 2017 in Berkeley, California.

In addition, I have attended every WARTHOG since summer 2017, and I expect to attend every future one until I graduate from the University of Oregon. Please come to the next one! I am happy to show you around Eugene!

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