
Recent Developments in Algebraic and Arithmetic Geometry
Date: August 25th to 30th, 2014
Location: Alfréd Rényi Institute of Mathematics,
Budapest (Main Lecture Hall)
Main Speakers:
Francois Charles (CNRS, Orsay)
Finiteness results and the Tate conjecture (show abstract [here])
Finiteness results and the Tate conjecture (hide abstract [here])
This series of lectures will be devoted to explaining some new progress on
the Tate conjecture for divisors. In 1966, Tate described the ring of
endomorphisms of an abelian variety over a finite field by means of its
etale cohomology. This relied on finiteness results for families of abelian
varieties  which can be expressed through Zarhins' trick. The main theme
of these lectures will be the relationship between finiteness results and
construction of divisors on K3 surfaces. We will report on recent proofs of
the Tate conjecture for K3 surfaces over finite fields with an emphasis on
geometric proofs of finiteness statements and a version of Zarhin's trick.
Gaetan Chenevier (École Polytechnique, Paris)
An introduction to level 1 automorphic forms (show abstract [here])
An introduction to level 1 automorphic forms (hide abstract [here])
In this series of lectures, I will give an introduction to
automorphic forms and Langlands philosophy, with an emphasis on the "level
1" case over Q. This includes for instance Siegel modular forms for the
full Siegel modular group or functions on the space of even unimodular
lattices in the euclidean space of any given rank. Conjecturally, those
objects encode in particular the arithmetic of pure motives over Q with
good reduction everywhere. I hope to give some examples illustrating
Arthur's recent results in the case of "classical groups".
Duco van Straten (Mainz)
CalabiYau Motives and Differential Equations (show abstract [here])
CalabiYau Motives and Differential Equations (hide abstract [here])
Beyond the ordered world of elliptic curves and K3 surfaces
there is the wild zoo of geometrically and arithmetically interesting
CalabiYau threefolds. In the series of lecture I will first focus on
some classical examples and constructions of rigid CalabiYaus, then
go to the cases with 1dimensional moduli spaces and discuss examples
coming from quantum cohomology of Fanofourfolds. For them the theory
of classical modular forms looses its prominence, and the PicardFuchs
equations and their motivic local systems take centerstage. The question
arises how to recover geometric and arithmetic data from the differential
equation alone. We show by example how for certain threeterm equations
we come back to classical modular forms, generalising the seminal work
by Beukers and Stienstra. If time permits I will adress the problem of
contructing motivic local systems using the geometric Langlands approach
(work in progress with V. Golyshev.)
Angelo Vistoli (Scuola Normale Superiore Pisa)
The Nori fundamental group scheme (show abstract [here])
The Nori fundamental group scheme (hide abstract [here])
In my series of lectures I will treat the theory of the Nori
fundamental group scheme. In the first lecture I will give a brief account
of the theory of group schemes and torsors. In the second I will introduce
the tannakian formalism. In the third I will explain Nori's theory of the
fundamental group scheme, following the somewhat simplified approach due
to myself and Niels Borne. In the fourth I will explain about gerbes and
their tannakian interpretation, and how to use them to give a version of
Nori's fundamental group that does not depend on the choice of a base
point.
Balázs Szendröi (Oxford)
Motivic DT invariants of quantised CalabiYau quiver algebras (show abstract [here])
Motivic DT invariants of quantised CalabiYau quiver algebras (hide abstract [here])
After recalling the general definition CalabiYau quiver algebras, their geometric significance, and their motivic DT invariants, we calculate these invariants for some previously studied noncommutative examples such as quantised affine 3space, the affine cone over the Jordan plane, and some threedimensional Sklyanin algebras. Joint work with Brent Pym and Andrew Morrison.
Gergely Zábrádi (Eötvös University, Budapest)
Colmez's padic Langlands correspondence and generalizations (show abstract [here])
Colmez's padic Langlands correspondence and generalizations (hide abstract [here])
The first aim of this talk is to introduce the objects both on the Galois and on the automorphic side corresponding to each other in the padic local Langlands correspondence for GL_2(Q_p). I would like to outline some of Colmez's ideas how to construct such a correspondence. Finally, I will also sketch a generalization of one step in Colmez's proof to reductive groups other than GL_2(Q_p) (joint work with P. Schneider and M.F. Vigneras). For any Q_psplit reductive group G we construct a functor from padic Galois representations of Q_p to G(Q_p)equivariant sheaves of Q_pvector spaces on the flag variety G(Q_p)/P(Q_p) where P is a Borel subgroup of G.
Organizers:
Gavril Farkas (HU Berlin)
András Némethi (Rényi Budapest)
Gerard van der Geer (U Amsterdam)
András Stipsicz (Rényi Budapest)
Jürg Kramer (HU Berlin)
Tamás Szamuely (Rényi Budapest)
Everybody is welcome to participate.
In case of participation, please send an
email to Ms. Marion Thomma: thomma@math.huberlin.de
Schedule: [here]
Participants: [here]

