I am a Professor in the Geometry Research Group of the Mathematics Department, University of Texas, Austin.
NB:Much of this page has not been updated since 2016 - in progress! [Fall 2023]
See also links to notes and videos at MathOverflow
I am interested in the interface between representation theory, algebraic geometry and mathematical physics. More specifically, my research program is concerned with the interactions of geometric representation theory (in particular the geometric Langlands program) with powerful new tools from homotopical algebra (derived algebraic geometry) and organizing structures from physics (topological field theory and supersymmetric gauge theory). The main theme in my recent work is the interaction of representation theory and gauge theory. Representation theory seeks to classify and describe the possible realizations of symmetries, and to exploit symmetry by providing a tool to decompose symmetric structures into elementary constituents. Rep- resentation theory has been an essential tool in quantum physics almost from its inception, providing for example the structure of atomic orbitals. Gauge theories, quantum field theories built directly out of the structure of local symmetry, are the language of much of high energy physics, in partic- ular the standard model. Gauge theory in turn has had a tremendous impact on low dimensional topology and geometry. Recently the relationship between representation theory and gauge theory has been reversed. In a development tracing back to the work of Segal and reaching its modern state after the work of Kapustin-Witten and Lurie, the structure of gauge theory has emerged as a powerful organizing framework for representation theory in the abstract. In this paradigm the different kinds of repre- sentation theories correspond to the different kinds of gauge theories, while the correlation functions, observables and defects in gauge theory provide a radically new and uniform structure through which to understand the most sophisticated concerns of representation theorists. Kapustin and Witten demonstrated that the Geometric Langlands Program appears naturally from the study of electric-magnetic duality in supersymmetric gauge theory. More broadly, one can use electric-magnetic duality as a powerful paradigm for the entire Langlands program, with interesting consequences for number theory.
I completed my Ph.D. under the supervision of Edward Frenkel; check out our book Vertex Algebras and Algebraic Curves.
I've had longstanding collaborations with Tom Nevins, David Nadler, Andy Neitzke, Sam Gunningham and David Jordan. I'm currently engaged in three large-scale collaborations: with Yiannis Sakellaridis and Akshay Venkatesh; with Harrison Chen, David Helm and David Nadler; and with Sam Gunningham, David Jordan and Pavel Safronov. My other collaborators include Indranil Biswas, Matt Szczesny (my academic brother), Reimundo Heluani, John Francis, Anatoly Preygel, Adrien Brochier, Chris Beem, Tudor Dimofte and Mat Bullimore.
My current graduate students are
My slides and video from plenary talk ``Representation Theory as Gauge Theory" at the 2016 Clay Research Conference.
Slides for lecture at Luminy, July 2012
I miss the nuance of the old topos.
Sunrise on Io by David Ben-Zvi and Paul Burchard. Copyright 1993 by The Geometry Center, University of Minnesota. Used with permission.
More cool fractal pictures I helped make at the Geometry Center
Our Himalayan Adventure, David and friends in Himachal and Ladakh 2003
Click here for some old pictures.
I am an avid movie fan. Check out some lists of movies I've seen. [NB: hasn't been updated since..2003??]
Some quotes collected from my U.of Chicago Honors Calculus class.
A brief personal history
Image from Outside In:
[Copyright © Summer 1992 by The Geometry Center, Univerity of Minnesota. All rights reserved - see more info]
Department of Mathematics University of Texas, Austin Office 10.168 (512)471-8151 email@example.com