Brian C. Hall
Professor
Education
Ph.D., Cornell University, 1993
Research Groups
Analysis and Partial Differential Equations, Differential Geometry, Mathematical Physics
Research Area
Mathematical Physics and Lie Groups
Bio
Research Interests
My research is in mathematical physics, specifically mathematical problems motivated by quantum mechanics and quantum field theory. My work involves several different types of mathematics, including functional analysis, Lie group theory, and probability theory.
One main theme of my work concerns generalizations of the SegalBargmann transform. The ordinary SegalBargmann transform was developed by I.E. Segal and V. Bargmann in the early 1960’s and provides a unitary transformation from the “position Hilbert space” to a new space called the SegalBargmann space. This space is a certain Hilbert space of holomorphic functions and using it as the quantum Hilbert space brings quantum mechanics closer to the underlying classical mechanics. The ordinary SegalBargmann transforms was for a quantum particle moving in Euclidean space; my work has been to generalize this to a quantum particle moving on a compact symmetric space. The generalized SegalBargmann transform that I have developed can also be understood as a system of generalized coherent states.
More recently, I have been interested a particular sort of quantum field theory known as twodimensional YangMills theory. In four spacetime dimensions, YangMills theory is a major ingredient in the standard model of particle physics. The twodimensional case may be seen as a warmup to the fourdimensional theory, but is also of interest in its own right, especially because of connections to string theory. I have been interested in the socalled "largeN limit," in which one considers the structure group U(N) and lets N tend to infinity. A key tool in analyzing this limit is the MakeenkoMigdal equation.
Selected Publications

B. K. Driver, F. Gabriel, B. C. Hall, and T. Kemp, The MakeenkoMigdal equation for YangMills theory on compact surfaces. Comm. Math. Phys. 352 (2017), 967–978.

B. K. Driver, B. C. Hall, and T. Kemp, Three proofs of the MakeenkoMigdal equation for YangMills theory on the plane. Comm. Math. Phys. 351 (2017), 741–774.

B. K. Driver, B. C. Hall, and T. Kemp, The largeN limit of the SegalBargmann transform on U(N). J. Funct. Anal. 265 (2013), 2585–2644.

B. C. Hall, Lie groups, Lie algebras, and representations. An elementary introduction. Second edition. Graduate Texts in Mathematics, 222. Springer, 2015.

B. C. Hall, Quantum theory for mathematicians. Graduate Texts in Mathematics, 267. Springer, 2013.
Email: bhall@nd.edu
Office: 110 HayesHealy Bldg