Benjamin Hoffman

Sixth year PhD candidate in mathematics. My advisor is Reyer Sjamaar.

Here is my CV.

201 Malott Hall
Cornell University
Ithaca, NY 14853
Office Hours: By appointment
Email: bsh68@cornell.edu

Research

I am interested in group actions on symplectic and Poisson manifolds. I have been working on two projects. A detailed description is in my research statement.

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 we conjecture are action-angle variables on generalizations of the Gelfand-Zeitlin integrable system (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 an upcoming work, 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).

Papers and Preprints

- (H1) Toric Symplectic Stacks (2019).

- (AHLL1) Concentration of symplectic volumes on Poisson homogeneous spaces, with A. Alekseev, J. Lane, and Y. Li (2018).

- (HS1) Stacky Hamiltonian Actions and Symplectic Reduction, with R. Sjamaar and C. Zhu (2018). Accepted for publication at 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).

- (ABHL1) Poisson Structures and Potentials, with A. Alekseev, A. Berenstein, and Y. Li (2017). In Lie Groups, Geometry, and Representation Theory.

Teaching

In Fall 2019 I am 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.