The 2018 Fields Medalists honored

Every four years the Fields Institute of Toronto, Canada announces its Fields Medal recipients. This year’s recipients are Caucher Birkar, Alessio Figali, Akshay Venkatesh and Peter Scholze.

The Fields Institute announces its awardees at the every-four-years International Congress of Mathematicians (ICM) meeting, which this year is being held in Rio de Janeiro. The awards, which are made to a maximum of four exceptional mathematicians under the age of 40, are often considered the “Nobel Prize” of Mathematics.

Here is some information on this year’s awardees and their work:

  1. Caucher Birkar. Birkar was born and raised in a very poor farming village in the Kurdish region of Iran. But eventually he managed to enroll at the University of Tehran. He recalls seeing photos of various Fields medalists on the walls, wondering “Will I ever meet one of these people.” Little did he know…

    In his senior year, while traveling in England, Birkar sought and received political asylum. He subsequently enrolled in the University of Nottingham, where he excelled in his mathematical studies. He currently is on the faculty at the University of Cambridge.

    The work for which Birkar was recognized is in the area of “algebraic geometry,” namely algebraically equivalent ways of viewing geometric objects. The set of solutions common to a given set of equations is known as an “algebraic variety.” Birkar used the theory of “birational transformations” to show that a class of geometric objects known as “Fans varieties” form neat, orderly families in every dimension.

    For more details on Birkar and his work, see this article by Kevin Hartnett.

  2. Alessio Figalli. Figalli was born in Rome. Although he liked mathematics from an early age, he didn’t seriously study the subject until his third year of high school. He was admitted into the Scuola Normale Superiore of Pisa, a university devoted to mathematically and scientifically gifted students, but struggled at first because other students were more advanced. But his talent quickly became apparent, as he caught up with and then exceeded the other students. He currently is on the faculty of ETH Zurich.

    Figalli became interested in the “optimal transport problem,” namely to find the most efficient path and means to move materials from one location to another. This is closely related to problems of “minimal surfaces,” which have long been studied by Italian mathematicians.

    In one key paper published in 2010, Figalli, together with his colleagues Franceseco Maggi and Aldo Pratelli, presented a proof of the stability of energy-minimizing shapes like crystals and soap bubbles. The proof takes the form of a precise inequality, which implies that if one increases the energy of a system such as a bubble or crystal by a certain amount, the resulting shape will deviate from the original shape by no more than a related amount. They further showed that this inequality is optimal in a certain precise sense.

    For more details on Figalli and his work, see this article, also by Kevin Hartnett.

  3. Akshay Venkatesh. Venkatesh was born in New Delhi, India, but grew up in Perth, Australia. Although he describes his childhood as a “fairly normal suburban experience,” he did win some international physics and math Olympiad competitions at ages 11 and 12. He later attended Princeton University, and worked under the well-known number theorist Peter Sarnak.

    Venkatesh has always viewed himself somewhat harshly. He described his own doctoral dissertation as “mediocre,” and had even considered leaving the field once, working for his uncle on a machine learning startup one summer. But then he was offered a C.L.E. Moore faculty position at MIT, and he accepted. He later wondered why his advisor (Sarnak) had written such a glowing recommendation. A fellow mathematician explained, “Sometimes, people see things in you that you don’t see.”

    One of Venkatesh’s papers dealt with generalizations of the well-known Riemann zeta function hypothesis. Venkatesh focused on L-functions. He employed ideas from the theory of dynamical systems to establish subconvexity estimates for a huge family of L-functions.

    For more details on Venkatesh and his work, see this article by Erica Klarreich.

  4. Peter Scholze. Scholze started teaching himself college-level mathematics at the age of 14, while attending a Berlin high school that specialized in mathematics and science. He recalls being fascinated by Andrew Wiles’ 1997 proof of Fermat’s Last Theorem. He confessed that at first although he was eager to understood the proof, he had much to learn. So for the next few years he studied some of the background theory, including modular forms and elliptic curves, together with numerous other techniques from number theory, algebra, geometry and analysis.

    Scholze began his research in arithmetic geometry, namely the usage of geometry-connected tools to study integer solutions to multivariate equations, e.g., x y2 + 3 y = 5. He found, as others had earlier, that it was useful to study these problems using the p-adic numbers, which are like the real numbers but where closeness is defined as a difference that is divisible numerous times by the prime p.

    Scholze ultimately proved a deep result known as the weight-monodromy conjecture. Scholze’s results expanded the scope of reciprocity laws, which govern the behavior of polynomials that use simple modular arithmetic (e.g., 5 + 10 mod 12 = 3). These reciprocity laws are generalizations of the quadratic reciprocity laws that were a favorite of Gauss. 20th century mathematicians found surprising links between these laws an hyperbolic geometry, which links have been elaborated more recently as the “Langland’s program,” an instance of which was the key to the proof of Fermat’s Last Theorem. Scholze has shown how to extend the Langland’s program to a much wider range of structures in hyperbolic spaces.

    For more details on Scholze and his work, see this article, also by Erica Klarreich.

Congratulations to all of this year’s winners!

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