Vladimir Alexandrovich Voevodsky was a Russian-American mathematician. His work in developing a homotopy theory for algebraic varieties and formulating motivic cohomology led to the award of a Fields Medal in 2002. He is also known for the proof of the Milnor conjecture and motivic Bloch–Kato conjectures and for the univalent foundations of mathematics and homotopy type theory.
In mathematics, the Hodge conjecture is a major unsolved problem in algebraic geometry and complex geometry that relates the algebraic topology of a non-singular complex algebraic variety to its subvarieties.
John Torrence Tate Jr. was an American mathematician distinguished for many fundamental contributions in algebraic number theory, arithmetic geometry, and related areas in algebraic geometry. He was awarded the Abel Prize in 2010.
Pierre René, Viscount Deligne is a Belgian mathematician. He is best known for work on the Weil conjectures, leading to a complete proof in 1973. He is the winner of the 2013 Abel Prize, 2008 Wolf Prize, 1988 Crafoord Prize, and 1978 Fields Medal.
Motivic cohomology is an invariant of algebraic varieties and of more general schemes. It is a type of cohomology related to motives and includes the Chow ring of algebraic cycles as a special case. Some of the deepest problems in algebraic geometry and number theory are attempts to understand motivic cohomology.
In mathematics, an Euler system is a collection of compatible elements of Galois cohomology groups indexed by fields. They were introduced by Kolyvagin in his work on Heegner points on modular elliptic curves, which was motivated by his earlier paper Kolyvagin (1988) and the work of Thaine (1988). Euler systems are named after Leonhard Euler because the factors relating different elements of an Euler system resemble the Euler factors of an Euler product.
Kazuya Kato is a Japanese mathematician who works at the University of Chicago and specializes in number theory and arithmetic geometry.
Markus Rost is a German mathematician who works at the intersection of topology and algebra. He was an invited speaker at the International Congress of Mathematicians in 2002 in Beijing, China. He is a professor at the University of Bielefeld.
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.
Matthias Flach is a German mathematician, professor and former executive officer for mathematics at California Institute of Technology.
In mathematics, the study of special values of L-functions is a subfield of number theory devoted to generalising formulae such as the Leibniz formula for π, namely
In mathematics, the norm residue isomorphism theorem is a long-sought result relating Milnor K-theory and Galois cohomology. The result has a relatively elementary formulation and at the same time represents the key juncture in the proofs of many seemingly unrelated theorems from abstract algebra, theory of quadratic forms, algebraic K-theory and the theory of motives. The theorem asserts that a certain statement holds true for any prime and any natural number . John Milnor speculated that this theorem might be true for and all , and this question became known as Milnor's conjecture. The general case was conjectured by Spencer Bloch and Kazuya Kato and became known as the Bloch–Kato conjecture or the motivic Bloch–Kato conjecture to distinguish it from the Bloch–Kato conjecture on values of L-functions. The norm residue isomorphism theorem was proved by Vladimir Voevodsky using a number of highly innovative results of Markus Rost.
In mathematics, the Quillen–Lichtenbaum conjecture is a conjecture relating étale cohomology to algebraic K-theory introduced by Quillen, who was inspired by earlier conjectures of Lichtenbaum (1973). Kahn (1997) and Rognes & Weibel (2000) proved the Quillen–Lichtenbaum conjecture at the prime 2 for some number fields. Voevodsky, using some important results of Markus Rost, has proved the Bloch–Kato conjecture, which implies the Quillen–Lichtenbaum conjecture for all primes.
Christopher Deninger is a German mathematician at the University of Münster. Deninger's research focuses on arithmetic geometry, including applications to L-functions.
James S. Milne is a New Zealand mathematician working in arithmetic geometry.
Charles Alexander Weibel is an American mathematician working on algebraic K-theory, algebraic geometry and homological algebra.
Vasudevan Srinivas is an Indian mathematician working in algebraic geometry. He is a Distinguished Professor in the School of Mathematics Tata Institute of Fundamental Research, Mumbai. Srinivas is an elected Fellow of the Third World Academy of Sciences, American Mathematical Society, Indian National Science Academy, and the Indian Academy of Sciences.
Eric Mark Friedlander is an American mathematician who is working in algebraic topology, algebraic geometry, algebraic K-theory and representation theory.
Uwe Jannsen is a German mathematician, specializing in algebra, algebraic number theory, and algebraic geometry.
Mark Lee Green is an American mathematician, who does research in commutative algebra, algebraic geometry, Hodge theory, differential geometry, and the theory of several complex variables. He is known for Green's Conjecture on syzygies of canonical curves.
