Three Honoured for Outstanding Research Achievements

CMS 2010 Krieger-Nelson Prize: Dr. Lia Bronsard (McMaster University)

The Krieger-Nelson Prize recognizes outstanding research by a female mathematician.

Lia Bronsard is one of Canada's leading mathematical analysts, whose interests lie in the field of partial differential equations and the calculus of variations. She specializes in the study of singular limits of solutions of partial differential equations. Her research brings rigorous methods of analysis to bear on problems arising in the physical sciences, and in particular those involving singular geometrical structures such as vortices, phase transition layers, and grain boundaries.

Bronsard was born in Québec in 1963 and received her Baccalauréat ès Sciences, in mathematiques from the Université de Montréal in 1983. She received her Ph.D. in 1988 from the Courant institute of Mathematical Sciences at New York University, working with R. V. Kohn on the De Giorgi conjecture connecting singularly perturbed reaction-diffusion equations and mean curvature flow. After her degree, she held positions at Brown University, the Institute for Advanced Study, and the Center for Nonlinear Analysis at Carnegie - Mellon University. In 1992, she moved to McMaster University, where she is now a Professor of Mathematics.

During the period after her thesis, Bronsard worked on energy driven pattern formation in collaboration with B. Stoth and others. Her paper with F. Reitich on the structure of triple-junction layers in grain boundaries, from her period at CMU, was the first mathematical analysis of these multiphase singular structures and has been highly influential.

In her current research, Bronsard studies the detailed structures of vortices in the phenomenon of Bose - Einstein condensation and in the Ginzburg - Landau models of superconductivity. In this area, her work, in collaboration with S. Alama, T. Giorgi, P. Mironescu, E. Sandier and colleague J. Berlinsky from Physics at McMaster University, sets a very high standard of quality, and is a model of interdisciplinary research.

CMS 2010 Jeffery-Williams Prize: Dr. Mikhail Lyubich (State University of New York at Stony Brook and the University of Toronto)

The Jeffery-Williams Prize recognizes mathematicians who have made outstanding contributions to mathematical research.

Mikhail Lyubich is a leader in the field of dynamical systems. He is one of the founders of modern real and complex one-dimensional dynamics, having in many ways shaped the development of the field.

Lyubich was born in 1959 in Kharkov, Ukraine, then a part of the Soviet Union. His interest in dynamics was influenced by his father Yuri Lyubich, who was a professor at Kharkov State University where Mikhail studied in the period 1975-80. Soviet political realities (in particular, tacit anti-Semitic policies) influenced Lyubich's early career. He was admitted to graduate school only in Tashkent, Uzbekistan, where he worked on holomorphic dynamics on his own. In his 1984 Ph.D. thesis, he proved fundamental results on ergodic theory and structural stability of rational maps; in particular, the existence of the measure of maximal entropy of a rational map, now known as Lyubich measure.

In 1989 Lyubich was able to leave the Soviet Union together with his family. He was invited by John Milnor to join the Institute for Mathematical Sciences at Stony Brook, founded at that time.

Lyubich received a Canada Research Chair at the University of Toronto in 2002, holding a joint appointment with Stony Brook. In 2007 he became Director of the Institute for Mathematical Sciences (Stony Brook). Lyubich is a sought-after speaker. He gave an invited address at the International Congress of Mathematicians in Zurich in 1994, as well as plenary talks at the Joint AMS Meeting in 2000 and the first CMS-SMF Mathematics Congress in 2003. He was awarded a Sloan Fellowship in 1991 and a Guggenheim Fellowship in 2002.

Among Lyubich's fundamental results in one-dimensional dynamics is his proof in the 1990s of hyperbolicity of renormalization for unimodal maps (conjectured by Feigenbaum and by Coullet and Tresser in the 1970s). Renormalization has been one of the main themes in low-dimensional dynamics for the past 40 years. Sullivan and later McMullen proved parts of the renormalization picture for unimodal maps, and Lyubich completed the proof of universality for bounded combinatorics. He later constructed a ``full hyperbolic horseshoe'' for the renormalization operator acting on real quadratic-like maps.

In earlier work on rigidity of quadratic polynomials, Lyubich resolved perhaps the most famous problem in dynamics on the real line by showing that hyperbolicity is dense in the real quadratic family. (This result was independently obtained by J. Graczyk and G. Świątek.)

One of the most fundamental problems in dynamics, for a parameterized family of maps, is to understand the asymptotic behaviour of almost every orbit for almost every value of the parameter. Even for the family of quadratic interval maps, this question had eluded experts for years. Lyubich's construction of the full renormalization horseshoe, together with his joint work with M. Martens and T. Nowicki, allowed him to obtain the definitive answer: almost every quadratic map is either regular or stochastic.

Lyubich's work was a major step towards the celebrated MLC (Mandelbrot set is locally connected) conjecture. A series of new breakthroughs has come in his recent results with J. Kahn, using the Kahn-Lyubich quasi-additivity law in conformal geometry.

CMS 2009 Coxeter-James Prize: Dr Patrick Brosnan (University of British Columbia)

The Coxeter-James Prize recognizes young mathematicians who have made outstanding contributions to mathematical research.

Patrick Brosnan is a young mathematician of unusual breadth, depth and scope; his work has had significant impact in several areas of mathematics, including motives, algebraic cycles, Hodge theory, algebraic groups, algebraic combinatorics, analytic number theory and mathematical physics.

Brosnan was born in Philadelphia, Pennsylvania in 1968 and grew up in Corpus Christi, Texas. He obtained a Bachelor of Arts degree from Princeton University in 1991 and a Ph.D. from The University of Chicago in 1998, studying algebraic cycles under the supervision of Spencer Bloch. Prior to joining the University of British Columbia, he held positions at Northwestern University, Max-Planck-Institut für Mathematik in Bonn, the University of California Irvine, the University of California Los Angeles, the State University of New York at Buffalo, and the Institute for Advanced Study in Princeton.

In a 2003 Duke Mathematical Journal paper with P. Belkale, Brosnan disproved the so-called “spanning tree” conjecture of the 1998 Fields medalist M. Kontsevich. The conjecture, which was motivated by research by the physicists D. Broadhurst and D. Kreimer into the number theoretical properties of Feynman amplitudes, was supported by a substantial body of empirical evidence. The work of Belkale and Brosnan was, consequently, entirely unexpected; and it has had a strong impact on the field.

Recently Brosnan has made important contributions to the theory of essential dimension. Brosnan's idea to extend the notion of essential dimension to the setting of algebraic stacks paved the way for wide-ranging applications of stack-theoretic methods which ultimately led to a number of striking developments. One of the applications, to appear in a joint Annals of Mathematics paper with Z. Reichstein and A. Vistoli, is an unexpectedly strong lower bound on the Pfister number of a quadratic form with trivial discriminant and Hasse-Witt invariant.

In a different direction, Brosnan and G. Pearlstein have recently made important contributions to Hodge theory. In another paper that will appear in the Annals, they show that a non-trivial admissible normal function on a curve can have only finitely many zeros. Normal functions are part of a conjectured program to prove the Hodge conjecture, one of the outstanding open problems in mathematics.