The Canadian Mathematical Society (CMS) has selected Penny Haxell (Waterloo) as the recipient of the 2006 Krieger-Nelson Prize, Andrew Granville (Montréal) as the recipient of the 2006 Jeffery-Williams Prize, Robert McCann (Toronto) as the winner of the 2005 Coxeter-James Prize, and Vasilisa Shramchenko (Concordia) as the winner of the 2005 CMS Doctoral Prize.

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

Dr. Penny E. Haxell works in Combinatorics and Graph Theory, focussing on extremal problems. She has obtained highly interesting results using combinatorial, probabilistic and, more recently, topological tools in a very fascinating manner. In all her work she exhibits impressive capability, originality and technical ability, and her pioneering work is well known internationally.

Her work in 1995 with Kohayakawa and Luczak, led to a profound study of Szemeredi's lemma in a sparse setting, and their methods are still being developed fruitfully by others.

Shortly thereafter, she gave an ingenious proof of a conjecture of Aharoni. This led to a collaboration that culminated in a beautiful and celebrated paper applying topological ideas to give a simple sufficient condition for a system of distinct representatives in a hypergraph family. The work has already found manifold applications.

The Haxell-Rödl Theorem, published in 2001, asserts that an optimal fractional packing of one graph into another can be converted into an actual packing that is more or less as good. It can facilitate the proof of a stability, or structural, result without the need for a pre-existing extremal theorem, from which it may then be possible to recover the extremal theorem itself. This line of research is very new, with great potential and prospects.

One of her latest papers gives a proof that the strong chromatic number of a graph is at most three times the maximum degree. This is a direction of research in which Alon's ten-year-old result was the previous best but, whereas Alon's approach was probabilistic, Haxell's is a lovely demonstration of traditional graph theory.

Dr. Haxell received a B. Math.Honours from the University of Waterloo in 1988 and was awarded the University of Waterloo Alumni Association Gold Medal for highest academic achievement in the Faculty of Mathematics. She received her Ph.D., supervised by Bela Bollobas, from the University of Cambridge in 1993. She has been a member of the Department of Combinatorics and Optimization, University of Waterloo, since 1993, rising to rank of Full Professor in 2004. In 2002, she was a Visiting Professor at Bell Laboratories, Lucent Technologies, New Jersey.

She received an NSERC Women's Award from 1993 to 1998, and an Ontario Premier's Research Excellence Award from 2001 to 2006. Dr. Haxell is a managing editor of the Journal of Combinatorial Theory (Series B), widely regarded as the best in the subject. She has been active in the organization of international conferences and seminars. From 1997 to 2001, she served on the Board of Directors of the Canadian Mathematical Society.

Dr. Penny Haxell will present the 2006 Krieger-Nelson Prize Lecture at the CMS 2006 Summer Meeting hosted by the University of Calgary in June 2006.

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

Dr. Andrew Granville is recognized world-wide as a leading analytic number theorist. He has contributed to various areas of number theory, and in each one, he has left his mark with striking results and solutions to long standing problems.

A sample of his first rate work has to include his proof with Alford and Pomerance of the infinitude of Carmichael numbers with the important practical consequence that many commercially available primality tests falsely certified many composite numbers as prime. One should also mention his papers with Friedlander, and very recently with Soundararajan, on the irregularities of the distribution of primes, and more generally, of arithmetic sequences in arithmetic progressions; his work with Bombieri and Pintz on squares in arithmetic progressions; his series of papers with Soundararajan on the distributions of character and exponential sums; and his interesting paper with Stark on the abc-Conjecture and Siegel zeros of L-functions of imaginary quadratic fields.

The above papers contain major breakthroughs and constitute fundamental advances in analytic number theory. Accordingly, they have appeared in the top tier of mathematical journals. Dr. Granville's contributions have been recognized by an invitation to speak at the International Congress of Mathematicians in Zurich 1994, as well as numerous prizes and honours.

An excellent expositor, Dr. Granville is highly sought after as a plenary speaker or to explain sophisticated mathematical concepts and facts in an intriguing and understandable way. A typical example of his mastery of the art of exposition is his very recent comprehensive article "It is easy to determine whether a given integer is prime" in the Bulletin of the American Mathematical Society.

The Canadian mathematical community is extremely fortunate that Dr. Granville returned to Canada in 2002, where he earlier undertook graduate studies and a postdoctoral fellowship. While not in Canada, he maintained strong ties through the organization of sessions, service to NSERC, the Fields Institute, as well as to the Canadian Number Theory Association.

Dr. Andrew Granville received a Bachelor of Arts (Honours) degree in 1983 and a Certificate of Advanced Studies (Distinction) in 1984, both at Trinity College, Cambridge University. He went Queen's University where, under the supervision of Paulo Ribenboim, he completed his Ph. D. in 1987 (which included what were then some of the best results known on Fermat's Last Theorem).

He was a postdoctoral fellow at the University of Toronto (1987-1989) and a member of the Institute for Advanced Study, Princeton (1989-1991). In 1991, he took up a position as Assistant Professor at the University of Georgia, where he attained the rank of Associate Professor in 1993 and became Full Professor and holder of the David C. Barrow Chair of Mathematics in 1995. In 2002, he returned to Canada, accepting a Canada Research Chair at the Université de Montréal.

His numerous honours include an Alfred P. Sloan Research Fellowship (1992-1995), a Presidential Faculty Fellowship (awarded by President Clinton) from 1994 to 1999, the 1995 Hasse Prize of the Mathematical Association of America, and the 1999 Ribenboim Prize of the Canadian Number Theory Association.

He served on NSERC's Grant Selection Committee from 1995 to 1998, on the NSERC Membership Subcommittee from 1996 to 1997, and the Computation Subcommittee from 1997 to 1998. For the United States National Science Foundation (NSF), he was a member of the CAREER Panel in 1996, the POWRE Panel in 2000, and the "Committee of Visitors" in 2001.

Dr. Granville has served on the AMS Editorial Boards Committee (1996-1999), the AMS Conference Program Selection Committee (1998-2000), serving as Chair from 1999 to 2000, and is currently a member of the CMS Research Committee. He has also served on the Editorial Boards of close to a dozen other journals. He has supervised many graduate students and postdoctoral fellows.

Dr. Granville will present the 2006 Jeffery-Williams Prize Lecture at the CMS Summer Meeting, hosted by the University of Calgary in June 2006.

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

The referees describe Robert McCann as a "creative, deep and dynamic mathematician". In the ten years since he graduated from Princeton, he has become one of the leading figures in the area of optimal transportation and its many applications. His work balances very "pure" and rigorous contributions to deep mathematics with the discovery of new applications to image recognition, atmospheric circulation patterns, and the kinetic theory of granular media. He is considered to be one of the most innovative geometric analysts of his generation.

In his 1994 Princeton dissertation, Dr. McCann introduced an extremely original interpolation technique into the calculus of variations, based upon Y. Brenier's polar factorization of vector fields. In recent years this breakthrough has motivated a lot of research into the fundamental inequalities of mathematical physics and geometry.

In recent years he has been at the forefront of the progress in Monge-Kantorovich mass transfer theory. Dr. McCann's joint work with Caffarelli and Feldman uses geometric measure theory and analysis to contribute to the solution of Monge's original transportation problem, an unsolved problem for over 200 years.

Over the past years, Dr. McCann has also collaborated with many others on a wide variety of related problems, motivated by applied problems in image processing, mathematical economics, and meteorology. An extensive survey documenting his central contributions is contained in C. Villani's recent book on Optimal Transportation.

Dr. McCann graduated from Queen's University in 1989 with a B.Sc. degree and obtained his Ph.D. from Princeton University in 1994. From 1994 to 1998, he was the Tamarkin Assistant Professor at Brown University and held an NSERC Postdoctoral Fellowship from 1994 to 1996 and an American Mathematical Society Centennial Fellowship from 1996 to 1998.

In 1998, he joined the University of Toronto as an Associate Professor and was promoted to Full Professor in 2004.

He was awarded the Monroe H. Martin Prize in Applied Mathematics in 2001 and, together with Caffarelli, Evans, Feldman, and Gangbo, has held two National Science Foundation Focused Research Group Grant (2000 - 2007).

Dr. Robert McCann will present the 2005 Coxeter-James Prize Lecture at the CMS Winter Meeting, hosted by the University of Victoria in December 2005.

The CMS Doctoral Prize recognizes outstanding performance by a doctoral student who graduated from a Canadian university.

The Canadian Mathematical Society is delighted to award the 2005 Doctoral Prize to Dr. Vasilisa Shramchenko for her work on Frobenius manifolds associated with Hurwitz spaces of branched covers. The work is situated at the crossroads of many areas and disciplines, requiring deep and broad knowledge that she masterfully commands.

The reviewers refer to her work as "a great surprise" and declare that the impact and originality of her contribution is "outstanding". She uses an analytical approach to objects of classical algebraic geometry that stems from classical theories and is enhanced by contemporary constructions. Besides providing a wealth of results, she poses important questions and problems, answers to which would further extend her "major breakthrough" in the area of Frobenius manifolds and would greatly enhance our understanding. She has already published several papers, with more accepted for publication. She clearly has an impressive career ahead of her.

Dr. Vasilisa Shramchenko received her Honours Diploma as a Mathematician and Mathematics teacher in 2000 from St. Petersburg State University in the Department of Probability and Statistics. She pursued graduate studies in the Department of Mathematics and Statistics at Concordia University and completed her Ph.D. under the direction of Professor Dmitry Korotkin in October 2004. Her thesis was entitled "Frobenius Structures, Integrable Systems, and Hurwitz Spaces".

Dr. Vasilisa Shramchenko will present the 2005 Doctoral Prize Lecture at the CMS Winter Meeting, hosted by the University of Victoria in December 2005.

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