3. Postdoctoral Program - MSRI

Your Name: Tristram Bogart
Year of Ph.D: 2007
Institution of Ph.D.: University of Washington
Ph.D. advisor: Rekha R. Thomas
MSRI FELLOWSHIP FINAL REPORT
TRISTRAM BOGART
1. Basic Information
Institution prior to obtaining the MSRI PD fellowship: Queen’s University, Canada
Position at that institution: postdoctoral fellow
Mentor: Gregory G. Smith
Institution where you held your MSRI PD fellowship: San Francisco State University (SFSU)
Mentor at that institution: Federico Ardila
Institution where you are going next year: Universidad de los Andes, Colombia
Position: assistant professor
Anticipated length: tenure-track
2. Research
I worked on three research projects during my year at SFSU, each of which I briefly describe
below. The three projects are not directly related, but they share a common theme of connecting
algebra or algebraic geometry to the theory of convex polytopes. I completed the first project, am
currently writing up the second and plan to submit it in summer 2011, and am in the early stages
of the third.
(1) Obstructions to lifting tropical curves in hypersurfaces: I began this joint project with fellow
MSRI postdoc Eric Katz during the tropical geometry semester in fall 2009 and we submitted
our paper [BK] in January 2011. In very general terms, tropicalization is a procedure that
turns an algebraic variety into a polyhedral complex, which is a piecewise-linear object that
can be studied by computational and combinatorial methods. The tropical lifting problem
asks for conditions under which the procedure can be reversed. The problem is quite difficult
and obstructions have been found in several cases, including linear spaces and curves. In our
paper we develop a new local obstruction to lifting pairs of tropical varieties, one embedded
in another.
(2) Mapping polytopes: In this project with SFSU professor Joseph Gubeladze and former
student Mark Contois, we study the construction of mapping polytopes. This construction
is key to developing a categorical theory of polytopes that includes homomorphims (affine
maps), direct products, tensor products, and more. Given full-dimensional polytopes P ⊆
R n and Q ⊆ R m , the mapping polytope is simply the set of affine linear maps Φ : R n → R m
satisfying the property that Φ(P) ⊆ Q. This set is itself a polytope of dimension (n + 1)m
Date: May 31, 2011.
1