I recently got my PhD in mathematics from Cornell University. My PhD advisor was Reyer Sjamaar.
I'm currently a teaching associate at Cornell. I'm on the market for jobs at the intersection of conservation and math/computation, broadly.
I make a calendar of the cats of Fall Creek neighborhood.
Here is my CV.
Cornell University
Ithaca, NY 14853
Office Hours: see course webpage
Email: bsh68@cornell.edu
I am interested in group actions on symplectic and Poisson manifolds. I have been working on the following projects. A detailed description is in my research statement, which is now a bit out of date.
Collective integrable systems and toric degenerations of affine varieties (with J. Lane):
In (HL1), we give a construction of a collective integrable system for every multiplicity-free Hamiltonian K-manifold, for K a compact Lie group. This can be viewed as a generalization of the famous Gelfand-Zeitlin integrable system on u(n)*, constructed by Guillemin-Sternberg.
Tropicalization of Poisson-Lie groups (with A. Alekseev, A. Berenstein, J. Lane, and Y. Li):
The goal of this project is to study coadjoint orbits of compact groups using tools from the theory of total positivity and geometric crystals. We built a collection of Poisson manifolds called "partial tropicalizations," which are meant to describe the behavior at infinity of a 1-parameter family of Ginzburg-Weinstein isomorphisms (ABHL1). The symplectic leaves of these manifolds have the representation-theoretic properties one hopes for (ABHL2). We used partial tropicalizations to prove a new result about the concentration of symplectic volume under a 1-parameter family of symplectic forms on the flag manifold K/T (AHLL1). In (AHLL2), we apply this theory to construct large Darboux charts on multiplicity free K-manifolds, and derive new bounds on the Gromov width of regular coadjoint orbits.
Symplectic Stacks (with R. Sjamaar):
Given a symplectic manifold with a locally free Hamiltonian action of a Lie group, the reduced spaces will in general fail to be manifolds, but will be differentiable stacks. With R. Sjamaar, we built a framework for Hamiltonian actions and symplectic reduction in a stacky setting (HS1). This led to the definition of toric symplectic stacks, and a classification theorem generalizing a celebrated result of Delzant (H1).
- (HL1) Canonical bases and collective integrable systems, with J. Lane (2020). Slides of a talk.
- (AHLL2) Action-angle coordinates on coadjoint orbits and multiplicity free spaces from partial tropicalization, with A. Alekseev, J. Lane, and Y. Li (2020).
- (H1) Toric Symplectic Stacks (2019). In Advances in Mathematics.
- (AHLL1) Concentration of symplectic volumes on Poisson homogeneous spaces, with A. Alekseev, J. Lane, and Y. Li (2018). In Journal of Symplectic Geometry.
- (HS1) Stacky Hamiltonian Actions and Symplectic Reduction, with R. Sjamaar and C. Zhu (2018). In International Mathematics Research Notices.
- (ABHL2) Langlands duality and Poisson-Lie duality via cluster theory and tropicalization, with A. Alekseev, A. Berenstein, and Y. Li (2018). In Selecta Mathematica.
- (ABHL1) Poisson Structures and Potentials, with A. Alekseev, A. Berenstein, and Y. Li (2017). In Lie Groups, Geometry, and Representation Theory.
In Spring 2020 I was a grader for Math 4540, Differential Geometry. I also co-taught a course called Game Theory at Cayuga Correctional Facility with the Cornell Prison Education Program.
In Fall 2019 I was a TA for Math 2930, Differential Equations for Engineers.
In Fall 2018 I was an instructor for Math 1120, Calculus II.
In Summer 2018 I was an instructor for Math 1110, Calculus I.
In Fall 2017 I taught intermediate algebra at Cayuga Correctional Facility with the Cornell Prison Education Program. I also graded for Math 4420, Combinatorics II.
In the academic year 2016-2017, I attended the Master Class in Geometry, Topology, and Physics at the University of Geneva in Switzerland.
In Spring 2016 I was an instructor for Math 1110, Calculus I.
In Fall 2015 I was a recitation TA for Math 2210, Linear Algebra.
