Laudable mathematics – The Fields Medal

Kicking off the International Congress of Mathematicians 2018 in Rio de Janeiro was this year’s Fields Medal awards ceremony, celebrating the brightest young minds in mathematics. The prize is awarded every four years to up to four mathematicians under the age of 40, and is viewed as one of the highest honours a mathematician can receive.

This year’s recipients come from diverse mathematical backgrounds, spanning the fields of algebraic geometry, number theory, and optimal transport. Honourees in 2018 are:

Caucher Birkar

For the proof of the boundedness of Fano varieties and for contributions to the minimal model program.

Alessio Figalli

For contributions to the theory of optimal transport and its applications in partial differential equations, metric geometry and probability.

Peter Scholze

For transforming arithmetic algebraic geometry over p-adic fields through his introduction of perfectoid spaces, with application to Galois representations, and for the development of new cohomology theories.

Akshay Venkatesh

For his synthesis of analytic number theory, homogeneous dynamics, topology, and representation theory, which has resolved long-standing problems in areas such as the equidistribution of arithmetic objects.

To celebrate the achievements of all of the winners, we’ve put together a reading list of free materials relating to the work that contributed to this honour.

A Quantitative Analysis of Metrics on Rn with Almost Constant Positive Scalar Curvature, with Applications to Fast Diffusion Flows, by Giulio Ciraolo, Alessio Figalli, and Francesco Maggi, published in International Mathematics Research Notices

The authors prove quantitative structure theorem for metrics on Rn that are conformal to the flat metric, have almost constant positive scalar curvature, and cannot concentrate more than one bubble.

The Langlands–Kottwitz approach for the modular curve, by Peter Scholze, published in International Mathematics Research Notices

Scholze shows how the Langlands–Kottwitz method can be used to determine the local factors of the Hasse–Weil zeta-function of the modular curve at places of bad reduction.

The Behavior of Random Reduced Bases, by Seungki Kim and Akshay Venkatesh, published in International Mathematics Research Notices

Kim and Venkatesh prove that the number of Siegel-reduced bases for a randomly chosen n -dimensional lattice becomes, for n→∞ , tightly concentrated around its mean, while also showing that most reduced bases behave as in the worst-case analysis of lattice reduction.

A Note on Sphere Packings in High Dimension, by Akshay Venkatesh, published in International Mathematics Research Notices

An improvement on the lower bounds for the optimal density of sphere packings. In all sufficiently large dimensions, the improvement is by a factor of at least 10,000.

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