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Mathematical Physics
University of Rochester

The goal of physics as a natural science is to understand the structure and behavior of matter, from its smallest constituents to the entire universe. It was noticed right at the beginning of physics, in the movement of planets, that mathematics plays an important role in describing the laws of nature. As time went on, this connection has only become strengthened. Just to take an example, Calculus (analysis) and mechanics grew together, developed in tandem by the same great minds: Kepler, Galileo, Newton, Legendre, Lagrange, Laplace, Euler, Gauss, Jacobi, Hamilton, Poincare, and into modern times with Arnold, Faddeev, Sobolev, Moser and so on. Probability, which arose in the study of games, is at the foundation of thermodynamics and quantum mechanics. Geometry made an early appearance with Kepler's ellipses, but much deeper connections were discovered in our own time. Einstein found that Riemannian geometry was the natural language of his General Theory of Relativity; and the Yang-Mills theory of connections is the most fundamental theory of elementary particle physics we know today. Topology was born in Poincare's theory of dynamical systems, but grew up separately until the modern discoveries of Witten and Kontsevich placed it squarely in the middle of theoretical physics.

This close relationship has not been without its tensions. Often, mathematicians and physicists use very different languages to talk of the same thing. The value systems of the field are different. Physics is primarily an empirical science, with the ultimate test being experimental validity. In mathematics the test of validity is proof. There are several quite deep areas of mathematics which as yet have no application to physics. Still, from time to time deep connections are (re)discovered and both fields benefit from this interaction. We live in the middle of such an era of reconnection. Of the 29 Fields medals awarded since 1978, 12 have been for work related to physics in some way. Four out of six Abel prize-winners have also been very influential in theoretical physics.

Mathematical Physics at Rochester

As mathematics grows and evolves into the twentyfirst century, new connections are emerging. Quantum computing promises new approaches to the ancient problem of resolving numbers into products of primes; the physics of quantum observation leads to new kinds of stochastic differential equations. Random matrices, which arose in Nuclear Physics, now seem fundamentally related to the Riemann zeros of number theory.

Optical communications lead us to study solitons in nonlinear media. Contact geometry is being revived as a theory of thermodynamics and possibly also microeconomics. Metamaterials with remarkable new properties provoke deep questions in the theory of Partial Differential Equations: can we cloak a light source from far away observers? These are some of the questions being investigated at Rochester, at the departments of Mathematics, Optics and Physics.

Mathematics

Fred Cohen is primarily an algebraic topologist. His work has led to a computation of the cohomology of braid groups. He is currently involved in a collaboration with electrical engineers, applying the topology of graphs to communication networks.

Dan Geba studies the existence and regularity of solutions of Partial Differential Equations, many of which arise from physics. Phase space transforms and dispersive estimates are found to be useful in studying nonlinear wave equations. An ongoing project, with Rajeev, is to understand the time evolution of soliton solutions in a kind of wave map equation, the Skyrme model of nuclear physics.

Allan Greenleaf is an analyst with a special interest in Partial Differential Equations of mathematical physics. The inverse problems of electrostatics and scattering theory has made major advances recently. Surprsingly, mathematics shows that the interior of a system can be made invisible from its boundary: cloaking.

Rick Lavine is primarily interested in the scattering theory of quantum mechanics. The nature of time as an observable in quantum mechanics and the resolution of he quantum zeno paradox are among of the questions investigated.

Carl Mueller works on Stochastic Partial Differential Equations (SPDE). Most physical systems are described in terms of partial differential equations, and random noise influences most of these systems. How do random fluctuations in a medium affect the propagation speed of a signal? How do particle system converge to SPDE?

Sema Salur works in differential geometry, geometric analysis and mathematical physics. In particular, high dimensional complex geometry and singularity theory, manifolds with special holonomy and calibrated geometry, deformation and elliptic theory on noncompact manifolds, supersymmetry and Calabi-Yau manifolds, special Lagrangian submanifolds and the applications in mirror symmetry, associative and coassociative submanifolds of G_2 manifolds and the connections with M-Theory.

Shannon Starr does research on spin glasses: systems of spins with simple interactions among neighbors leading to a complex, disordered, large scale structure. Mean field theories of spin systems can now be treated with full mathematical rigor. He also works on quantum spin systems.

Optics

Govind Agrawal's research in quantum electronics, nonlinear optics and laser physics has led him to problems in the theory of nonlinear PDEs and stochastic differential equations. Propagation of solitons in noisy media was the subject of a collaboration with Rajeev. He is the author of seven standard textbooks in these areas.

Miguel Alonso is mainly interested in the propagation of waves. In physics, this leads to questions of interest in optics, accoustics and quantum mechanics. How to make estimates of wave fields in an arbitrary medium based on the knowledge of rays? On the mathematical side, he develops theories of integral transforms, phase space representations, and uncertainty relations.

Physics

Andrew Jordan works on several areas of theoretical physics that connect with mathematics. The eigenvalue statistics of chaotic quantum systems, spectral thoety of Ruelle resonances, theory of quantum noise, large deviation theory are among the topics covered. For a popular account of quantum measurement theory look here.

S. G. Rajeev does research on the mathematical aspects of High Energy Physics. Early work revived a radical idea of Skyrme, that fermions such as the proton can arise as solitons of bosonic fields. A mathematical derivation of this theory in the simpler two dimensional context was achieved later. Exact solutions of Yang-Mills theory on a cylinder and quantum gravity in two dimensions were also found . Diverse topics at the intersection of mathematics and physics are being studied now: contact geometry in thermodynamics, a theory of dissipative quantum systems, stochastic Partial Differential Equations of soliton propagation in random media, non-commutative differential geometry as a tool for studying Navier-Stokes equations. Recently, an old problem of classical physics was solved exactly: the spiral trajectory of a charged particle as it orbits a nucleus while emitting radiation.

Yonathan Shapir works at the intersection of condensed matter physics and statistical physics: critical phenomena in ordered and disordered systems.: spin glasses, fractal properties of percolation, kinetic models of growth and aggregation. The mathematical connections are to probability and stochastic differential equations.