The modularity theorem states that elliptic curves over the field of rational numbers are related to modular forms. Andrew Wiles proved the modularity theorem for semistable elliptic curves, which was enough to imply Fermat's Last Theorem. Later, a series of papers by Wiles's former students Brian Conrad, Fred Diamond and Richard Taylor, culminating in a joint paper with Christophe Breuil, extended Wiles's techniques to prove the full modularity theorem in 2001.
Henri Paul Cartan was a French mathematician who made substantial contributions to algebraic topology.
In mathematics, Serre's modularity conjecture, introduced by Jean-Pierre Serre, states that an odd, irreducible, two-dimensional Galois representation over a finite field arises from a modular form. A stronger version of this conjecture specifies the weight and level of the modular form. The conjecture in the level 1 case was proved by Chandrashekhar Khare in 2005, and a proof of the full conjecture was completed jointly by Khare and Jean-Pierre Wintenberger in 2008.
Alexander A. Beilinson is the David and Mary Winton Green University professor at the University of Chicago and works on mathematics. His research has spanned representation theory, algebraic geometry and mathematical physics. In 1999, Beilinson was awarded the Ostrowski Prize with Helmut Hofer. In 2017, he was elected to the National Academy of Sciences. In 2018, he received the Wolf Prize in Mathematics and in 2020 the Shaw Prize in Mathematics.
Michel Pierre Talagrand is a French mathematician. Docteur ès sciences since 1977, he has been, since 1985, Directeur de Recherches at CNRS and a member of the Functional Analysis Team of the Institut de Mathématique of Paris. Talagrand was elected as correspondent of the Académie des sciences of Paris in March 1997, and then as a full member in November 2004, in the Mathematics section.
André Haefliger was a Swiss mathematician who worked primarily on topology.
In arithmetic geometry, the Tate–Shafarevich groupШ(A/K) of an abelian variety A (or more generally a group scheme) defined over a number field K consists of the elements of the Weil–Châtelet group , where is the absolute Galois group of K, that become trivial in all of the completions of K (i.e., the real and complex completions as well as the p-adic fields obtained from K by completing with respect to all its Archimedean and non Archimedean valuations v). Thus, in terms of Galois cohomology, Ш(A/K) can be defined as
Michael Rapoport is an Austrian mathematician.
In mathematics, Tate pairing is any of several closely related bilinear pairings involving elliptic curves or abelian varieties, usually over local or finite fields, based on the Tate duality pairings introduced by Tate and extended by Lichtenbaum (1969). Rück & Frey (1994) applied the Tate pairing over finite fields to cryptography.
In algebra, the Hochster–Roberts theorem, introduced by Melvin Hochster and Joel L. Roberts in 1974, states that rings of invariants of linearly reductive groups acting on regular rings are Cohen–Macaulay.
Prakash Belkale is an Indian-American mathematician, specializing in algebraic geometry and representation theory.
Jean-François Quint is a French mathematician, specializing in dynamical systems theory for homogeneous spaces.
Jun-Muk Hwang is a South Korean mathematician, specializing in algebraic geometry and complex differential geometry.
Shahar Mozes is an Israeli mathematician.
Sug Woo Shin is a professor of mathematics at the University of California, Berkeley working in number theory, automorphic forms, and the Langlands program.
Francesco Damien "Frank" Calegari is a professor of mathematics at the University of Chicago working in number theory and the Langlands program.
Bhargav Bhatt is a mathematician who is the Fernholz Joint Professor at the Institute for Advanced Study and Princeton University and works in arithmetic geometry and commutative algebra.
Thomas Hermann Geisser is a German mathematician working at Rikkyo University. He works in the field of arithmetic geometry, motivic cohomology and algebraic K-theory.
Tsachik Gelander is an Israeli mathematician working in the fields of Lie groups, topological groups, symmetric spaces, lattices and discrete subgroups. He is a professor in Northwestern University.
Ronen Eldan is an Israeli mathematician. Eldan is a professor at the Weizmann Institute of Science working on probability theory, mathematical analysis, theoretical computer science and the theory of machine learning. He received the 2018 Erdős Prize, the 2022 Blavatnik Award for Young Scientists and the 2023 New Horizons Breakthrough Prize in Mathematics. He was a speaker at the 2022 International Congress of Mathematicians.
