Lucas Mason-Brown


MIT Department of Mathematics
Office: 
Simons Building (Building 2)
Room 2-233A
Email: lmbrown@mit.edu

I am a fifth year graduate student in the mathematics department at MIT. I am primarily interested in representation theory, the study of abstract mathematical symmetries. Most of my research is inspired by the Orbit Method, a mysterious and conjectural correspondence between co-adjoint orbits (geometric objects related to classical mechanics) and unitary representations (analytic objects related to quantum mechanics). I am working under the supervision of David Vogan. Next year, I will be a Titchmarsh research fellow at Oxford Universtiy.

My CV is here. My research statement is here.

Research

Gluing Sheaves Over Codimension 1 (joint with James Tao, link to PDF )

Let X be a smooth variety which decmposes under the action of an algebraic group G into an open orbit U and a closed orbit Z of codimension 1. We provide an algebraic description of the category of G-equivariant vector bundles on X. The resulting theory has numerous applications to the representation theory of real reductive groups.

Unipotent Representations Attached to the Principal Nilpotent Orbit (arXiv)

I construct and classify the (special) unipotent representations of a real reductive group attached to the principal nilpotent orbit.

Upper Triangularity for Unipotent Representations (arXiv)

There is evidence to suggest that every unitary representation (of a real reductive group) can be built (through several types of induction) from unipotent representations. I show that under favorable conditions, every unipotent representation can be built (through a more general kind of induction) from a smaller set of representations, namely unipotent representations attached to birationally rigid orbits.

Unipotent Representations and Microlocalization (arXiv)

I develop a theory of microlocalization for Harish-Chandra modules, inspired by the work of Ivan Losev on primitive ideals. This theory gives rise to a nice formula for the K-types of a unipotent representation attached to a nilpotent orbit with small boundary, proving (in a large family of cases) an old conjecture of Vogan.

Natural Operations in Differential Geometry (masters thesis, link to PDF)

What do the Lie bracket (of vector fields), the wedge product (of differential forms), and the exterior derivative (of differential forms) have in common? They are all natural operations (i.e. independent of local coordinates). In this thesis, I use the representation theory of GL(n) to classify all such operations on differential forms and alternating multi vector fields.

Decoding Roger Williams: The Lost Essay of Rhode Island's Founding Father (link to Amazon)

As a college student, I used statistical techniques to decipher an unpublished essay of Rhode Island's founding father. In this book, the text of this essay is revealed and placed in historical context.

Speaking

Upper Triangularity for Unipotent Representations (7/27/2020, `Unipotent Representations and Associated Varieties, Beijing Center for Mathematical Research, Beijing, China)

Unipotent Representations Attached to Principal Nilpotent Orbits (2/3/2020, Yale University Seminar on Geometry, Symmetry, and Physics, New Haven, CT)

Unipotent Representations Attached to Principal Nilpotent Orbits (1/17/2020, AMS-MAA-SIAM Joint Mathematics Meeting, Denver, CO)

Coherent Sheaves on Fastened Chains (10/23/19, UMD Lie Groups and Representation Theory Seminar, College Park, MD)

Unipotent Representations and Microlocalization (6/26/19, Representation Theory XVI, Dubrovnik, Croatia)

The K-types of Unipotent Representations (4/3/19 and 4/10/19, MIT Lie Groups Seminar, Cambridge, MA)

Seminar

I organized a graduate student seminar at MIT on the Local Langlands Correspondence for real reductive groups during the 2018-2019 school year. You can find our syllabus here.