Albert Schwarz

Distinguished Professor
Mathematical physics and topology
C.Sc., 1958, D.Sc., 1961, Moscow University
Refereed publications: Via Math Reviews

Web Page:
Office: MSB 3105
Phone: 554-2815

Professor Albert Schwarz is a mathematician from the former Soviet Union who has contributed to many areas of mathematics, including topology, category theory, functional analysis, and the calculus of variations; in the second half of his long career, he has concentrated on quantum field theory and other topics in mathematical physics. On several occasions, he introduced concepts which were not fully understood at first, but which were later rediscovered and considered important. He is currently studying string theory, and especially the possibility that string theory will lead to a Theory of Everything.

Professor Schwarz's earlier papers gave certain calculations and definitions that are now part of standard introductions to topology and geometry. Some of his work from this initial period concerned counting the number of closed geodesics on a Riemannian manifold. Among other things, he noticed that the (unparametrized) loop space of a manifold is an orbifold and not itself a manifold, thereby correcting a fallacy in earlier work by Bott and Morse. In another paper, he established that the fundamental group of a negatively curved manifold has exponential growth, a result which was rediscovered by and is sometimes attributed to Milnor. And he had a series of papers defining and discussing a certain topological invariant of a fiber bundle called the genus [2]. The same notion was rediscovered by Smale, who applied it to the computational complexity of algorithmic problems, through the notion of topological complexity.

The thrust of Professor Schwarz's work in mathematical physics has been to apply topology to physics [5]. In this aim he anticipated some of the work of the physicist Edward Witten. His most important result, joint with Belavin, Polyakov, and Tyupkin, was the discovery of instantons, which are solutions to classical gauge field equations that are localized in four Euclidean dimensions [2]. Although they are Euclidean objects, instantons manifest themselves in 4-dimensional Minkowskian spacetime as avenues for quantum tunneling. In this guise, they were used by t'Hooft to arrive at one of the notorious predictions of a Grand Unified Theory of physics (excluding only gravity): that the proton must be unstable. Finally, instantons reappeared in pure mathematics in Donaldson theory, a body of ideas which applies gauge field theory techniques to demonstrate many surprising non-trivialities about smooth 4-manifolds.

Another notorious prediction of Grand Unified Theories due to t'Hooft is the existence of magnetic monopoles. They are examples of solitons, which are like instantons but localized in one fewer dimension. Professor Schwarz also noticed that magnetic monopoles must exist for topological rather than merely computational reasons [3]. His argument fortified the classical philosophical speculation that particles in nature could be akin to knots in a cord; they could be indivisible and indestructible because of topology. It is yet possible that elementary particles such as quarks and electrons are kinds of solitons.

In yet other work, Professor Schwarz noticed that quantum field theories can produce topological invariants of manifolds. Following Witten, such theories are now called topological quantum field theories. Drawing on this older result, he conjectured in 1987 that the Jones polynomial, the well-known invariant of knots and links, comes from Chern-Simons quantum field theory [4]. The next year, Witten independently cast and then spectacularly verified the same conjecture.

Dr. Schwarz has made several significant achievements while at UC Davis including:

    1. Mathematical theory of Batalin-Vilkovisky quantization procedure.

    2. Applications of noncommutative geometry to string/M-theory. Relation of Morita equivalence to T-duality.

    3. The analysis of supersymmetric deformations of SUSY Yang-Mills theory in the framework of homological algebra.

    4. Applications of arithmetic geometry to physics, in particular, to integrality theorems in the theory of topological strings.

Professor Schwarz has advised many graduate students over a period of five decades, both officially and unofficially. His advisees include Dmitry Fuchs, V. A. Fateev, A. A. Rosly, O. Zaboronsky, and A. Konechny.

Selected publications

    [1] "The genus of a fiber space. " Trudy Moskov Mat. Obsc., 11: 99-126, 1962.

    [2] "Pseudoparticle solutions to Yang-Mills equations" (with A. Belavin, A.Polyakov and Yu. Tyupkin), Phys. Lett. B, 59: 85-87, 1975.

    [3] "Magnetic monopoles in gauge theories, " Nuclear Phys. B, 112: 358-364, 1976.

    [4] "New topological invariants arising in the theory of quantized fields, " Abstracts of the International Topological Conference, Baku, Part II, 1987.

    [5] "Quantum Field Theory and Topology, " Springer 1993.

    [6] "Noncommutative Geometry and Matrix Theory: Compactification on Tori." (with Alain Connes and Michael R. Douglas), JHEP 9802:003, 1998, Math ArXiv 9711162 This paper (cited more than 1000 times) opened the way for applications of NC geometry to string theory.

    [7] "Integrality of instanton numbers and p-adic B-model" (with Maxim Kontsevich and Vadim Vologodsky), Phys.Lett. B, 637: 97-101, 2006, Math ArXiv 0603106.

Last updated: 2012-04-12