2009
The AHP Prize 2009 was awarded to D. Dolgopyat and B. Fayad for the paper
This paper concerns outer billiards, a dynamical system similar to the conventional (inner) billiards. Their study was put forward by J. Moser in the 1970s and provides an interesting example of an area preserving two- dimensional mapping with an explicit geometrical description. In particular, Moser posed the problem whether the orbits of the outer billiards can escape to infinity. The motivation for this question was that if the boundary of the outer billiard table is strictly convex and sufficiently smooth, then KAM-type arguments prove that all orbits stay bounded.
Surprisingly enough, the methods of this paper are essentially of the KAM type, although the authors prove an “anti-KAM” kind of result; these methods are likely to be applicable to similar problems, and their theorem opens the door for further study.
To summarize, this paper solves an old problem in an unexpected way, and its method can certainly be applicable to a bunch of new models, too.
2008
The AHP Prize 2008 was awarded jointly to
P. Bálint and I. P. Tóth for the paper
Billiards with some hyperbolicity have played a key role in the development of dynamical systems, since they represent a highly nontrivial natural example of chaotic dynamics. The nominated paper is very well-written and accessible to non billiard experts. It settles a long standing conjecture (modulo an additional assumption which is most likely generic) and clarifies our understanding of ergodicity and mixing properties of billiards.
and to
L. Parnovski for the paper
The Bethe-Sommerfeld conjecture concerns a basic property of an operator with wide application in physics and was considered a challenging problem in spectral theory in the last decades. In the case of rational lattices and in all dimensions the proof has achieved by Skriganov (1984) and Scrikanov & Sobolev (2006). The definitive result has been obtained by Parnowski in this paper, which proves the conjecture for any periodicity lattice, in all dimensions greater than two and with an arbitrary smooth potential.
2007
The AHP Prize 2007 is attributed to Fabien Vignes-Tourneret for the paper "Renormalization of the Orientable Non-commutative Gross–Neveu Model"
This paper introduces new methods in noncommutative field theory and solves several non-trivial and difficult mathematical issues. It opens a new category of quantum field theories to renormalization, namely non-commutative Fermionic quantum field theories. This is an important step for physics as well as for mathematics, as the condensed matter version of such theories, although still to be developed, should be the relevant framework for a future deeper understanding of the physics of the quantum Hall effect.
2006
For 2006, the AHP Prize laureates are Giuseppe Benfatto, Alessandro Giuliani and Vieri Mastropietro for their paper entitled "Fermi Liquid Behavior in the 2D Hubbard Model at Low Temperatures” in which they prove that the weak coupling 2D Hubbard model away from half filling is a Landau Fermi liquid up to exponentially small temperatures.
On the picture: Vincent Rivasseau, Chief Editor of the Journal Annales Henri Poincaré, poses with the winners of the AHP Prize during the award ceremony at the Annual Meeting of the Swiss Physical Society (March 27 2008, Genève). From left to right: Marc Herbstritt (Birkhäuser Verlag), Vieri Mastropietro (University Roma "Tor Vergata", Italy, AHP Prize Winner 2006), Alessandro Giuliani (University Roma Tre, Italy, AHP Prize Winner 2006), Giuseppe Benfatto (University Roma "Tor Vergata", Italy, AHP Prize Winner 2006), Alexander Sobolev (University College London, U.K., AHP Prize Winner 2005) and Vincent Rivasseau (Chief Editor Annales Henri Poincaré).
2005
For 2005, Alexander V. Sobolev receives the AHP Prize for his paper entitled "Integrated Density of States for the Periodic Schrödinger Operator in Dimension Two" in which he provides a rigorous and insightful investigation on the high energy asymptotics of the density of states for the Schrödinger operator L2.