Iwasawa Theory and Motivic L-functions

We illustrate the use of Iwasawa theory in proving cases of the (equivariant) Tamagawa number conjecture

Mirror Symmetry, Hodge Theory and Differential Equations

The following is the report on the Oberwolfach workshop Mirror Symmetry, Hodge Theory and Differential Equations (April 2015), which brought together researchers from various areas such as quantum cohomology, complex algebraic geometry, Hodge theory representation theory etc

Families of toric varieties

The goal of this paper is to define families of toric varieties and to study their properties. These families are locally trivial fibrations over some base, whose fibres are isomorphic to a fixed complete toric variety. The study is motivated by the occurence of such families in the context of gauged sigma models for toric varieties.Comment: 25 pages, LaTe

Birational geometry of cluster algebras

We give a geometric interpretation of cluster varieties in terms of blowups of toric varieties. This enables us to provide, among other results, an elementary geometric proof of the Laurent phenomenon for cluster algebras (of geometric type), extend Speyer's example of an upper cluster algebra which is not finitely generated, and show that the Fock-Goncharov dual basis conjecture is usually false.Comment: 50 pages, to appear in Algebraic Geometr