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.
In geometric group theory, Gromov's theorem on groups of polynomial growth, first proved by Mikhail Gromov, characterizes finitely generated groups of polynomial growth, as those groups which have nilpotent subgroups of finite index.
In mathematics, tropical geometry is the study of polynomials and their geometric properties when addition is replaced with minimization and multiplication is replaced with ordinary addition:
Homological mirror symmetry is a mathematical conjecture made by Maxim Kontsevich. It seeks a systematic mathematical explanation for a phenomenon called mirror symmetry first observed by physicists studying string theory.
In mathematics, Khovanov homology is an oriented link invariant that arises as the cohomology of a cochain complex. It may be regarded as a categorification of the Jones polynomial.
In mathematics, the Grothendieck–Katz p-curvature conjecture is a local-global principle for linear ordinary differential equations, related to differential Galois theory and in a loose sense analogous to the result in the Chebotarev density theorem considered as the polynomial case. It is a conjecture of Alexander Grothendieck from the late 1960s, and apparently not published by him in any form.
In mathematics, in particular algebraic topology, a p-compact group is a homotopical version of a compact Lie group, but with all the local structure concentrated at a single prime p. This concept was introduced in Dwyer & Wilkerson (1994), making precise earlier notions of a mod p finite loop space. A p-compact group has many Lie-like properties like maximal tori and Weyl groups, which are defined purely homotopically in terms of the classifying space, but with the important difference that the Weyl group, rather than being a finite reflection group over the integers, is now a finite p-adic reflection group. They admit a classification in terms of root data, which mirrors the classification of compact Lie groups, but with the integers replaced by the p-adic integers.
In algebraic geometry, a period is a number that can be expressed as an integral of an algebraic function over an algebraic domain. Sums and products of periods remain periods, such that the periods form a ring.
Daqing Wan is a Chinese mathematician working in the United States. He received his Ph.D. from the University of Washington in Seattle in 1991, under the direction of Neal Koblitz. Since 1997, he has been on the faculty of mathematics at the University of California at Irvine; he has also held visiting positions at the Institute for Advanced Study in Princeton, New Jersey, Pennsylvania State University, the University of Rennes, the Mathematical Sciences Research Institute in Berkeley, California, and the Chinese Academy of Sciences in Beijing.
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.
Aleksandr Sergeyevich Merkurjev is a Russian-American mathematician, who has made major contributions to the field of algebra. Currently Merkurjev is a professor at the University of California, Los Angeles.
In mathematics, specifically algebraic geometry, Donaldson–Thomas theory is the theory of Donaldson–Thomas invariants. Given a compact moduli space of sheaves on a Calabi–Yau threefold, its Donaldson–Thomas invariant is the virtual number of its points, i.e., the integral of the cohomology class 1 against the virtual fundamental class. The Donaldson–Thomas invariant is a holomorphic analogue of the Casson invariant. The invariants were introduced by Simon Donaldson and Richard Thomas (1998). Donaldson–Thomas invariants have close connections to Gromov–Witten invariants of algebraic three-folds and the theory of stable pairs due to Rahul Pandharipande and Thomas.
Christopher Deninger is a German mathematician at the University of Münster. Deninger's research focuses on arithmetic geometry, including applications to L-functions.
Alexander Nikolaevich Varchenko is a Soviet and Russian mathematician working in geometry, topology, combinatorics and mathematical physics.
Kari Kaleva Vilonen is a Finnish mathematician, specializing in geometric representation theory. He is currently a professor at the University of Melbourne.
Thomas Schick is a German mathematician, specializing in algebraic topology and differential geometry.
Amir Dembo is an Israeli-American mathematician, specializing in probability theory. He was elected a member of the National Academy of Sciences in 2022, and of the American Academy of Arts and Sciences in 2023.
Henri Moscovici is a Romanian-American mathematician, specializing in non-commutative geometry and global analysis.
Ivan Vadimovich Loseu is a Belarusian-American mathematician, specializing in representation theory, symplectic geometry, algebraic geometry, and combinatorial algebra.
