Gavin Brown Prize winners

On the range of lattice models in high dimensions, Probability Theory and Related Fields, 176, no. 3 (2020), 941–1009

In this paper the authors provide precise conditions that ensure that the range of lattice models converges to the range of the limiting superprocess. It is a very beautiful and inspiring presentation despite the difficulty of the subject and the delicate technicalities of the convergence, with several reviewers noting the careful explanations and the way in which new concepts are related to more familiar ones. In the words of the reviewers: “[…] this work not only solves some long-standing open problems, but also provides new methods that should have fruitful applications in the future […] it is a truly outstanding achievement in the area of interacting stochastic systems in high dimensions and is a landmark in our understanding of large space-time scaling.”

Dispersion relations, power laws, and energy loss for waves in the marginal ice zone, Journal of Geophysical Research: Oceans, 123 (2018), 3322–3335

The paper has had a significant impact on the waves-in-ice field and beyond. In particular, the discovery of power-law relationship has modified the approach to data analysis and theory, and set a new benchmark for theoretical contributions to numerical prediction models, such as the WAVEWATCH III third-generation spectral wave model and the CICE sea ice model. The paper was unanimously recommended for the Gavin Brown Prize by the assessors. In the words of one of the assessors: “It […] represents a milestone in the research of wave-ice interaction. I unequivocally support this paper for the Gavin Brown Best Paper Prize.”

Holomorphic curves in exploded manifolds: virtual fundamental class, Geometry & Topology 23 (2019), 1877–1960

The paper was unanimously recommended for the Gavin Brown Prize by the assessors.
In the words of one of the assessors: “I believe that Parker has done impressively original work and difficult work.
It shows mastery of many aspects of symplectic geometry.
Parker’s program is having significant impact on the field.
I can certainly recommend the nominated paper for the Gavin Brown prize in the strongest possible terms.”

Every flock generalised quadrangle has a hemisystem, Bull. London Math. Soc., Vol. 42, (2010) 795–810

This paper is considered a “genuine paradigm shift” in its field. The methods used in the paper are a clever mix of geometric and algebraic techniques. This work both settles the problem of whether every such generalised quadrangle has a hemisystem, and is an enormous step beyond what was known previously. The construction behind the main theorem also produces exponentially-many hemisystems, not just one per prime power q.

The assessors were uniform in their high praise for this paper, judging it to be “very influential in the field of Finite Geometry”, “innovative and excellent research which solves a significant problem in its field”. This paper “has unquestionably had a big impact in the field. So completely have the authors answered the question they tackled, that there have been almost no publications on the subject since the publication appeared. In other words, rather than opening up an area, this has closed one down.”

Kernel Density Estimation via Diffusion, The Annals of Statistics, Vol. 38, No. 5, (2010) 2916–2957

This work has been described as ‘stunning’, ‘exceptionally well written’ and ‘extraordinary’. Marrying ideas from mathematical analysis and applied statistics, the trio have been praised for their fast kernel density estimator, which also achieves excellent accuracy thanks to an ingenious bandwidth selection algorithm.
This work is yet another example of the unifying power of mathematics — our estimator uses the abstract concepts originally due to Fourier, Einstein and others, that describe phenomena ranging from the spread of heat in objects through to the spread of information in social networks.

The critical fugacity for surface adsorption of self-avoiding walks on the honeycomb lattice is 1+√2, Comm. Math. Phys. 326 (2014), 727–754

In 2012 H. Duminil-Copin and Fields Medalist S. Smirnov rigorously proved the longstanding conjecture of B. Nienhuis from 1982 on the connective constant for 2-dimensional self-avoiding walks on the hexagonal lattice. The paper by Beaton et al. gives the highly non-trivial extension of this result to self-avoiding walks on a hexagonal half-plane by analysing such walks subject to an attractive or repulsive boundary. The proof of this result beautifully extends the case of self-avoiding walks in the plane and draws from a wide range of sophisticated techniques and methods from statistical mechanics, combinatorics and probability theory.

Commensurated Subgroups of Arithmetic Groups, Totally Disconnected Groups and Adelic Rigidity, Geometric and Functional Analysis, Vol. 23, 1631–1683 (2013)

The paper is notable for a number of reasons. The authors answer the conjecture of Margulis and Zimmer for a broad class of groups, roughly speaking for groups commensurable with G(O), where G is a Chevalley group over a global field K of characteristic zero, and O is the ring of integers of K. The methods they use are novel, imposing a topology in such a way that they are able to draw on results from the study of general totally disconnected groups. Finally, they provide a unified framework for considering a number of results and conjectures in the rigidity theory of arithmetic groups. A number of experts consider that this final contribution may perhaps become the strongest legacy of the paper.

*Yehuda Shalom is not eligible to receive the Gavin Brown Prize, as he has not been a member of AustMS during the last 10 years.

Ergodic billiards that are not quantum unique ergodic, Annals of Mathematics 171:2 (2010), 605–619.

In this paper, Andrew Hassell makes a breakthrough in the theory of quantum ergodicity, by producing the first example of a planar domain for which the billiard flow is quantum ergodic, but not quantum unique ergodic (QUE). The study of quantum ergodicity is a young and active field, which gained prominence when Elon Lindenstrauss won a 2010 Fields medal in part for showing QUE holds on arithmetic surfaces. The domain on which Hassell worked is a Bunimovich stadium (shown below).
The study of ergodicity of dynamical systems is an important subject in mathematics. Quantum ergodicity, replaces the natural billiard flow on a domain, by the consideration of the distribution of eigenfunctions of the Dirichlet problem for the Laplacian on the domain. In a sense, the billiard flow is the classical limit of the quantum system.

On a Bunimovich stadium, a billiard’s flow is ergodic, but not uniquely ergodic. This is because as it can bounce back and forth periodically from between flat sides, albeit with probability zero. The flow was also known to be quantum ergodic, but it was an open question as to whether it was QUE, meaning that eigenvectors of the Laplacian cannot concentrate on a proper subset. Hassell has shown that they can indeed so concentrate. Thus, QUE typically fails on such a domain.
Terry Tao discussed the problem of QUE on such a domain in his 2007 blog¹. The next year², he presented Hassell’s result on this problem, describing it as “one of his favorite problems”. The result was also discussed in Peter Sarnak’s survey article³.

References

¹ http://terrytao.wordpress.com/2007/03/28/open-question-scarring-forthe-bunimovich-stadium/. This blog also appeared in Tao’s book Structure and Randomness, published by the AMS in 2008.
² http://terrytao.wordpress.com/2008/07/07/hassells-proof-of-scarringfor-the-bunimovich-stadium/, “Hassell’s proof of scarring for the Bunimovich stadium”, was republished by the AMS in Tao’s book Poincaré’s Legacies, Part II, 2009.
³ Peter Sarnak, “Recent progress on the quantum unique ergodicity conjecture”, Bulletin of the AMS, vol 48 (2011), 211–228 (http://www.ams.org/journals/bull/2011-48-02/S0273-0979-2011-01323-4/home.html)

Proof of the fundamental gap conjecture, J. Amer. Math. Soc. 24 (2011), 899–916.

In this paper the authors employ entirely new methods to resolve a long-standing conjecture in mathematics and physics concerning the minimum gap between the first and second eigenvalues of the Dirichlet Laplacian. The conjecture had been highlighted as a major open problem at many conferences over a long period and was considered a central problem in geometric spectral theory. Their achievement in its resolution has already been acclaimed at conferences, including Banff and Oberwolfach workshops where discussion of the result was given a central place.

Regularity of Potential Functions of the Optimal Transportation Problem
Archive for Rational Mechanics and Analysis, 177 (2005), no. 2, 151–183.

The paper unlocked a problem mentioned in the Fields medallist Cedric Villani’s 2003 AMS book on optimal transport as “the most important” remaining to be understood in the area of smoothness of optimal transport, namely the regularity in a geometric (Riemannian) setting. The problem seemed to be monstrously difficult and Caffarelli apparently thought it to be intractable. Then arrived Ma, Trudinger and Wang, and their discovery (by brute analytic force) of what is now called the ‘Ma–Trudinger–Wang tensor’. The paper was like a lightning strike, and was the start of a new direction of research stimulating many papers and it was their contribution that made all this possible, a remarkable insight both in the theory of optimal transport and in differential geometry.

*Xi-Nan Ma is not eligible to receive the Gavin Brown Prize, as he has not been a member of AustMS during the last 10 years.