The 2018 Fields Medal Candidates and Their Notable Contributions

The Fields Medal is awarded every four years to mathematicians under 40 years of age, recognizing their outstanding achievements in the field. In the lead-up to the 2018 International Congress of Mathematicians in Rio de Janeiro, several promising candidates were widely discussed as possible recipients. This article delves into the work of four of these candidates and their significant contributions to mathematics:

1. Peter Scholze

Known for his work in arithmetic algebraic geometry and his development of perfectoid spaces, Peter Scholze is a leading candidate due to his groundbreaking contributions. Scholze's research has transformed the landscape of arithmetic algebraic geometry over p-adic fields. His introduction of perfectoid spaces has opened new avenues for studying Galois representations and has led to the development of new cohomology theories. Scholze's work holds immense potential for shaping the future of mathematics, making him a key figure in the mathematical community.

2. Akshay Venkatesh

Akshay Venkatesh is a prominent figure in analytic number theory. His work spans various areas, including harmonic analysis, representation theory, and homogeneous dynamics. Venkatesh's ability to bridge different fields and synthesize complex concepts makes him a strong contender for the Fields Medal. His contributions to these areas, particularly his resolution of long-standing problems in analytic number theory and homogeneous dynamics, demonstrate his profound understanding of the field and his potential for future contributions.

3. Nikhil Srivastava

Nikhil Srivastava is recognized for his contributions to random matrix theory and the Kadison-Singer problem. His work has significant implications in both pure and applied mathematics. The Kadison-Singer problem, which he helped resolve, has profound implications for a wide range of disciplines, including computer science and engineering. Srivastava's ability to tackle complex problems with innovative solutions positions him as a notable candidate for the Fields Medal.

4. June Huh

June Huh is a rising star in combinatorial geometry. His work on the Hodge theory of matroids and his insights into algebraic combinatorics have positioned him as a noteworthy candidate. Huh's innovative approach to combinatorial geometry, which draws on techniques from algebraic geometry, has opened new possibilities for the field. His groundbreaking research in understanding the structure of combinatorial objects through the lens of algebraic geometry marks him as a promising potential recipient of the Fields Medal.

5. Caucher Birkar

Caucher Birkar is recognized for his contributions to algebraic geometry, particularly in relation to minimal models and the theory of Fano varieties. His work has transformed the way mathematicians approach these areas, leading to significant advancements in the field. Birkar's proof of the boundedness of Fano varieties and his contributions to the minimal model program have further solidified his position as a strong candidate for the Fields Medal.

The actual recipients were announced during the International Congress of Mathematicians in Rio de Janeiro, Brazil, in August 2018. Here are the recipients and their contributions:

Alessio Figalli

Alessio Figalli was awarded for his contributions to the theory of optimal transport and its applications in partial differential equations, metric geometry, and probability.

Caucher Birkar

Caucher Birkar received the award for his proof of the boundedness of Fano varieties and his contributions to the minimal model program.

Peter Scholze

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

Akshay Venkatesh

Akshay Venkatesh was recognized 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.

These mathematicians were recognized not only for their individual achievements but also for their potential to shape the future of mathematics. The 2018 Fields Medal recipients continue to inspire and push the boundaries of mathematical research.