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A Fields Medal reading list

One of the highest points of the International Congress of Mathematicians, currently underway in Seoul, Korea, is the announcement of the Fields Medal prize winners. 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 sees the first ever female recipient of the Fields Medal, Maryam Mirzakhani, recognised for her highly original contributions to geometry and dynamical systems. Her work bridges several mathematic disciplines – hyperbolic geometry, complex analysis, topology, and dynamics – and influences them in return.

We’re absolutely delighted for Professor Mirzakhani, who serves on the editorial board for International Mathematics Research Notices. To celebrate the achievements of all of the winners, we’ve put together a reading list of free materials relating to their work and to fellow speakers at the International Congress of Mathematicians.

Ergodic Theory of the Earthquake Flow” by Maryam Mirzakhani, published in International Mathematics Research Notices

Noted by the International Mathematical Union as work contributing to Mirzakhani’s achievement, this paper investigates the dynamics of the earthquake flow defined by Thurston on the bundle PMg of geodesic measured laminations.

Ergodic Theory of the Space of Measured Laminations” by Elon Lindenstrauss and Maryam Mirzakhani, published in International Mathematics Research Notices

A classification of locally finite invariant measures and orbit closure for the action of the mapping class group on the space of measured laminations on a surface.

Mass Forumlae for Extensions of Local Fields, and Conjectures on the Density of Number Field Discriminants” by Majul Bhargava, published in International Mathematics Research Notices

Manjul Bhargava joins Maryam Mirzakhani amongst this year’s winners of the Fields Medal. Here he uses Serre’s mass formula for totally ramified extensions to derive a mass formula that counts all étale algebra extentions of a local field F having a given degree n.

Model theory of operator algebras” by Ilijas Farah, Bradd Hart, and David Sherman, published in International Mathematics Research Notices

Several authors, some of whom speaking at the International Congress of Mathematicians, have considered whether the ultrapower and the relative commutant of a C*-algebra or II1 factor depend on the choice of the ultrafilter.

Small gaps between products of two primes” by D. A. Goldston, S. W. Graham, J. Pintz, and C. Y. Yildrim, published in Proceedings of the London Mathematical Society

Speaking on the subject at the International Congress, Dan Goldston and colleagues prove several results relating to the representation of numbers with exactly two prime factors by linear forms.

On Waring’s problem: some consequences of Golubeva’s method” by Trevor D. Wooley, published in the Journal of the London Mathematical Society

Wooley’s paper, as well as his talk at the congress, investigates sums of mixed powers involving two squares, two cubes, and various higher powers concentrating on situations inaccessible to the Hardy-Littlewood method.

 

Image credit: (1) Inner life of human mind and maths, © agsandrew, via iStock Photo. (2) Maryam Mirzakhani 2014. Photo by International Mathematical Union. Public Domain via Wikimedia Commons.

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