In representation theory and algebraic number theory, the Langlands program is a web of far-reaching and consequential conjectures about connections between number theory and geometry. Proposed by Robert Langlands, it seeks to relate Galois groups in algebraic number theory to automorphic forms and representation theory of algebraic groups over local fields and adeles. Widely seen as the single biggest project in modern mathematical research, the Langlands program has been described by Edward Frenkel as "a kind of grand unified theory of mathematics."
In mathematics, the Birch and Swinnerton-Dyer conjecture describes the set of rational solutions to equations defining an elliptic curve. It is an open problem in the field of number theory and is widely recognized as one of the most challenging mathematical problems. It is named after mathematicians Bryan John Birch and Peter Swinnerton-Dyer, who developed the conjecture during the first half of the 1960s with the help of machine computation. Only special cases of the conjecture have been proven.
In mathematics, more specifically in group theory, a group is said to be perfect if it equals its own commutator subgroup, or equivalently, if the group has no non-trivial abelian quotients. In symbols, a perfect group is one such that G(1) = G, or equivalently one such that Gab = {1}.
Geometric group theory is an area in mathematics devoted to the study of finitely generated groups via exploring the connections between algebraic properties of such groups and topological and geometric properties of spaces on which these groups can act non-trivially.
In mathematics, arithmetic geometry is roughly the application of techniques from algebraic geometry to problems in number theory. Arithmetic geometry is centered around Diophantine geometry, the study of rational points of algebraic varieties.
Daniel Segal is a British mathematician and a Professor of Mathematics at the University of Oxford. He specialises in algebra and group theory.
Brian Hayward Bowditch is a British mathematician known for his contributions to geometry and topology, particularly in the areas of geometric group theory and low-dimensional topology. He is also known for solving the angel problem. Bowditch holds a chaired Professor appointment in Mathematics at the University of Warwick.
Azriel Lévy is an Israeli mathematician, logician, and a professor emeritus at the Hebrew University of Jerusalem.
John Robert Stallings Jr. was a mathematician known for his seminal contributions to geometric group theory and 3-manifold topology. Stallings was a Professor Emeritus in the Department of Mathematics at the University of California at Berkeley where he had been a faculty member since 1967. He published over 50 papers, predominantly in the areas of geometric group theory and the topology of 3-manifolds. Stallings' most important contributions include a proof, in a 1960 paper, of the Poincaré Conjecture in dimensions greater than six and a proof, in a 1971 paper, of the Stallings theorem about ends of groups.
Zlil Sela is an Israeli mathematician working in the area of geometric group theory. He is a Professor of Mathematics at the Hebrew University of Jerusalem. Sela is known for the solution of the isomorphism problem for torsion-free word-hyperbolic groups and for the solution of the Tarski conjecture about equivalence of first-order theories of finitely generated non-abelian free groups.
In mathematics, the main conjecture of Iwasawa theory is a deep relationship between p-adic L-functions and ideal class groups of cyclotomic fields, proved by Kenkichi Iwasawa for primes satisfying the Kummer–Vandiver conjecture and proved for all primes by Mazur and Wiles. The Herbrand–Ribet theorem and the Gras conjecture are both easy consequences of the main conjecture. There are several generalizations of the main conjecture, to totally real fields, CM fields, elliptic curves, and so on.
Simon Brendle is a German-American mathematician working in differential geometry and nonlinear partial differential equations. He received his Dr. rer. nat. from Tübingen University under the supervision of Gerhard Huisken (2001). He was a professor at Stanford University (2005–2016), and is currently a professor at Columbia University. He has held visiting positions at MIT, ETH Zürich, Princeton University, and Cambridge University.
Arthur Bartels is a German mathematician.
Pham Huu Tiep is a Vietnamese American mathematician specializing in group theory and representation theory. He is currently a Joshua Barlaz Distinguished Professor of Mathematics at Rutgers University.
Shahar Mozes is an Israeli mathematician.
Francesco Damien "Frank" Calegari is a professor of mathematics at the University of Chicago working in number theory and the Langlands program.
Martin Liebeck is a professor of Pure Mathematics at Imperial College London whose research interests include group theory and algebraic combinatorics.
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.
Eamonn Anthony O'Brien is a professor of mathematics at the University of Auckland, New Zealand, known for his work in computational group theory and p-groups.
