|Geometry and Topology|
The geometry and topology group at the Mathematical Institute of Leiden University presently consists of the following people:
Peter Bruin Graduate Student email@example.com Bas Edixhoven Full Professor firstname.lastname@example.org Robin de Jong Assistant Professor email@example.com Martin Lübke Scientific Staff firstname.lastname@example.org Jacob Murre Emeritus Professor email@example.com Frans Oort Emeritus Professor (Utrecht) firstname.lastname@example.org Arjen Stolk Graduate Student email@example.com Lenny Taelman Postdoc firstname.lastname@example.org Antonius Van de Ven Emeritus Professor email@example.com
We are attached to the Rennes node of the Research Training Network of the European Union Arithmetic Algebraic Geometry.
In algebraic geometry the equations are given by polynomials. Classically, the coefficients and solutions were complex numbers. Number theorists consider integer or rational coefficients and solutions. The goal of arithmetic geometry is to understand the relations between algebraic geometry and number theory.
Three important notions in arithmetic geometry are ``algebraic variety'' (abstraction of system of polynomial equations), ``zeta function'' and ``cohomology''. Zeta functions associated to algebraic varieties are generating functions defined using the numbers of solutions in finite fields. Cohomology associates vector spaces equipped with certain structures to algebraic varieties. One important aim of arithmetic geometry is to understand the relations between the values of zeta functions at integers and properties of the set of rational solutions. Cohomology plays an important role here. Cohomology also provides representations of Galois groups, which is essential for Langlands's program (relations between such representations and ``automorphic'' representations of matrix groups). The most striking results obtained in this field are the proof of Weil's conjectures (Dwork, Grothendieck, Deligne), Faltings's proof of Mordell's conjecture, Fontaine's theory (comparison between certain cohomologies), Wiles's proof of Fermat's Last Theorem, Lafforgue's result on Langlands's conjectures, the proof of Serre's modularity conjecture (Khare, Wintenberger, Kisin....), and Taylor's proof of the Sato-Tate conjecture.
Apart from its numerous applications within mathematics, algebraic geometry over finite fields provides error correcting codes and crypto systems, both used in everyday life.