Uwe Jannsen (born 11 March 1954) [1] is a German mathematician, specializing in algebra, algebraic number theory, and algebraic geometry.
Uwe Jannsen (born 11 March 1954) [1] is a German mathematician, specializing in algebra, algebraic number theory, and algebraic geometry.
Born in Meddewade, Jannsen studied mathematics and physics at the University of Hamburg with Diplom in mathematics in 1978 and with Promotion (PhD) in 1980 under Helmut Brückner and Jürgen Neukirch with thesis Über Galoisgruppen lokaler Körper (On Galois groups of local fields). [2] In the academic year 1983–1984 he was a postdoc at Harvard University. From 1980 to 1989 he was an assistant and then docent at the University of Regensburg, where he received in 1988 his habilitation. From 1989 to 1991 he held a research professorship at the Max-Planck-Institut für Mathematik in Bonn. In 1991 he became a full professor at the University of Cologne and since 1999 he has been a professor at the University of Regensburg.
Jannsen's research deals with, among other topics, the Galois theory of algebraic number fields, the theory of motives in algebraic geometry, the Hasse principle (local–global principle), and resolution of singularities. In particular, he has done research on a cohomology theory for algebraic varieties, involving their extension in mixed motives as a development of research by Pierre Deligne, and a motivic cohomology as a development of research by Vladimir Voevodsky. In the 1980s with Kay Wingberg he completely described the absolute Galois group of p-adic number fields, i.e. in the local case. [3]
In 1994 he was an Invited Speaker with talk Mixed motives, motivic cohomology and Ext-groups at the International Congress of Mathematicians in Zürich. [4]
He was elected in 2009 a full member of the Bayerische Akademie der Wissenschaften and in 2011 a full member of the Academia Europaea.
His doctoral students include Moritz Kerz. [5]
Jean-Pierre Serre is a French mathematician who has made contributions to algebraic topology, algebraic geometry and algebraic number theory. He was awarded the Fields Medal in 1954, the Wolf Prize in 2000 and the inaugural Abel Prize in 2003.
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.
In number theory, Iwasawa theory is the study of objects of arithmetic interest over infinite towers of number fields. It began as a Galois module theory of ideal class groups, initiated by Kenkichi Iwasawa, as part of the theory of cyclotomic fields. In the early 1970s, Barry Mazur considered generalizations of Iwasawa theory to abelian varieties. More recently, Ralph Greenberg has proposed an Iwasawa theory for motives.
In algebraic geometry, motives is a theory proposed by Alexander Grothendieck in the 1960s to unify the vast array of similarly behaved cohomology theories such as singular cohomology, de Rham cohomology, etale cohomology, and crystalline cohomology. Philosophically, a "motif" is the "cohomology essence" of a variety.
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 arithmetic geometry, the Weil–Châtelet group or WC-group of an algebraic group such as an abelian variety A defined over a field K is the abelian group of principal homogeneous spaces for A, defined over K. John Tate named it for François Châtelet who introduced it for elliptic curves, and André Weil, who introduced it for more general groups. It plays a basic role in the arithmetic of abelian varieties, in particular for elliptic curves, because of its connection with infinite descent.
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.
In mathematics, the absolute Galois groupGK of a field K is the Galois group of Ksep over K, where Ksep is a separable closure of K. Alternatively it is the group of all automorphisms of the algebraic closure of K that fix K. The absolute Galois group is well-defined up to inner automorphism. It is a profinite group.
In number theory and algebraic geometry, the Tate conjecture is a 1963 conjecture of John Tate that would describe the algebraic cycles on a variety in terms of a more computable invariant, the Galois representation on étale cohomology. The conjecture is a central problem in the theory of algebraic cycles. It can be considered an arithmetic analog of the Hodge conjecture.
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
In mathematics, p-adic Hodge theory is a theory that provides a way to classify and study p-adic Galois representations of characteristic 0 local fields with residual characteristic p. The theory has its beginnings in Jean-Pierre Serre and John Tate's study of Tate modules of abelian varieties and the notion of Hodge–Tate representation. Hodge–Tate representations are related to certain decompositions of p-adic cohomology theories analogous to the Hodge decomposition, hence the name p-adic Hodge theory. Further developments were inspired by properties of p-adic Galois representations arising from the étale cohomology of varieties. Jean-Marc Fontaine introduced many of the basic concepts of the field.
Jean-Marc Fontaine was a French mathematician. He was one of the founders of p-adic Hodge theory. He was a professor at Paris-Sud 11 University from 1988 to his death.
Pierre Colmez is a French mathematician and directeur de recherche at the CNRS (IMJ-PRG) known for his work in number theory and p-adic analysis.
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.
Christopher Deninger is a German mathematician at the University of Münster. Deninger's research focuses on arithmetic geometry, including applications to L-functions.
Haruzo Hida is a Japanese mathematician, known for his research in number theory, algebraic geometry, and modular forms.
Reinhardt Kiehl is a German mathematician.
Peter Bernd Schneider is a German mathematician, specializing in the p-adic aspects of algebraic number theory, arithmetic algebraic geometry, and representation theory.
Bhargav Bhatt is an Indian-American mathematician who is the Fernholz Joint Professor at the Institute for Advanced Study and Princeton University and works in arithmetic geometry and commutative algebra.
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