Tue, Feb 16  Wed, Feb 17  Thu, Feb 18  Fri, Feb 19  Sat, Feb 20  

8:009:00

8:009:20
Registration
9:309:45
Introduction

Registration


9:0010:00

9:4510:45
Vistoli

Edixhoven

Kramer

Mochizuki

ColliotThélène

10:0010:30

10:4511:15
Coffee

Coffee

Coffee

Coffee

Coffee

10:3011:30

11:1512:15
Chenevier

Jannsen

Geisser

Venjakob

Lazarsfeld

11:4512:45

Laza

Kim

Fantechi







14:30  15:30
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15:0015:30

Coffee

Registration/Coffee

Coffee

Coffee


15:3016:30

Ngô

Nicaise

Beilinson

Hida

16:00  Guided tour 
16:4517:45

Brosnan

Abramovich

Huybrechts




17:4518:45
Registration




19:00 h Banquet 
09:45  Angelo Vistoli  The essential dimension of a curve 
11:15  Gaëtan Chenevier  The infinite fern of Galois representations of unitary type 
15:30  Bao Châu Ngô  On the adelization of the trace formula 
16:45  Patrick Brosnan  Admissible normal functions and Hodge theory 
09:00  Bas Edixhoven  Computational aspects of 2dimensional Galois representations 
10:30  Uwe Jannsen  Strong resolution of singularities for twodimensional schemes 
11:45  Radu Laza  Moduli spaces birational to locally symmetric varieties of orthogonal or unitary type 
15:30  Johannes Nicaise  A proof of the motivic monodromy conjecture for abelian varieties 
16:45  Dan Abramovich  Varieties with a twist 
09:00  Jürg Kramer  An arithmetic RiemannRoch theorem for logsingular metrics on modular curves 
10:30  Thomas Geisser  Some remarks on Suslin homology 
11:45  Minhyong Kim  Diophantine Geometry and Galois Theory 
15:30  Alexander Beilinson  On the structure of the space of rational functions on a curve 
16:45  Daniel Huybrechts  Stability conditions on derived categories of K3 surfaces 
09:00  Takuro Mochizuki  Betti structure of holonomic Dmodules 
10:30  Otmar Venjakob  On the noncommutative Iwasawa Main Conjecture for CMelliptic curves 
11:45  Barbara Fantechi  Twists instead of logs 
15:30  Haruzo Hida  Characterization of abelian components of the 'big' Hecke algebra 
09:00  JeanLouis ColliotThélène  Dritte unverzweigte Kohomologie und ganzzahlige Hodge Vermutung 
10:30  Robert Lazarsfeld  Positivity of cycles on abelian varieties 
I will discuss a number of questions and results where the geometry and moduli of algebraic stacks help in understanding the geometry and moduli of algebraic varieties. In particular: work with Hassett gives an approach to understanding aspects of the boundary of moduli of polarized varieties; and questions of Campana which would shed light on the birational geometry of varieties with quotient singularities.
I will discuss a contractibility theorem conjectured by V. Drinfeld and proved recently by D. Gaitsgory.
Let H be a variation of pure Hodge structure of negative weight on a smooth complex algebraic variety S. Attached to H is a family J(H) over S of Griffiths intermediate Jacobians. Normal functions are certain sections of this family. The admissible normal functions are a subset of the normal functions formally introduced and studied by M. Saito in a 1996 Journal of Algebraic Geometry paper. Since normal functions are essentially analytic objects, it is clear that the zero locus of a normal function on S is an analytic subset of S. The main theorem of my talk will be a result recently obtained in joint work with Greg Pearlstein: that the zero locus is, in fact, algebraic. If time permits, I will also mention the independent approaches to this theorem by C. Schnell and by K. Kato, C. Nakayama and S. Usui.
Ganzzahlige Kohomologie Klassen von Grad 4 auf einer komplexen, glatten, projektiven Mannigfaltigkeit, die von Hodge Typ (2,2) sind, brauchen nicht unbedingt Bilder von algebraischen Zyklen zu sein, wie bekannte Beispiele von Atiyah und Hirzebruch, und andere Beispiele von Koll'ar, zeigen. In einer gemeinsamen Arbeit mit Claire Voisin werden solche Beispiele unter unirationalen Varietäten gefunden. Dafür wird eine Beziehung zwischen diesem Problem und dem Studium der dritten unverzweigten Kohomologie mit endichen Koeffizienten festgestellt. Dann können alte Beispiele von Ojanguren und dem Sprecher einberufen sein.
A deterministic algorithm that computes the twodimensional Galois representations associated to modular forms over finite fields, in time polynomial in the weight and in the cardinality of the field, will be described. Applications to computing Hecke operators T_p with p large, and to theta functions of lattices will be given. This is joint work with JM. Couveignes, R. de Jong, F. Merkl, J. Bosman, and P. Bruin.
Enumerative problems often require to compactify a moduli space while preserving the obstruction theory. E.g. Jun Li's proof of the degeneration formula for GW invariants uses a delicate analysis to do this (later simplified by Kim using log geometry). In this joint work with D. Abramovich we show how to use added stackiness (or twisting) instead. As an application, we extend the GW degeneration formula to orbifolds.
Suslin and Voevodsky proved that Suslin homology of a variety over an algebraically closed field, with coefficients prime to the characteristic of the field, is dual to etale cohomology. In the first half of the talk we discuss properties of Suslin homology with rational coefficients, and coefficients a power of the characteristic. In the second half we focus on finite base fields. We define a candidate for a wellbehaved integral homology theory, which is related to tamely ramified class field theory (even for singular schemes).
In the first 20 minutes, we try to explain why such a characterization is important. Then, we state a (conjectural) example of such characterization, and at the end, we explain a proof of one of such statements.
For varieties over fields of characteristic zero there are very strong results (by Hironaka and others) on resolution of singularities: a canonical and to a certain extent also functorial procedure by socalled permissible blowups. Over fields of positive characteristic or for schemes with mixed characteristic very little is known, and only in small dimension  more precisely in dimensions 2 and 3. Even in dimension 2, where one has several methods (e.g., a procedure by Lipman using alternately normalization and blowups in points), there were until recently no results on strong resolution as described above. I will report on joint work with V. Cossart and S. Saito, in which we fill this gap. We can even treat nonirreducible or nonreducible schemes, as well as 'boundaries' on them.
In his manuscripts from the 1980's Grothendieck proposed ideas that have been interpreted variously as embedding the theory of schemes into either group theory and higherdimensional generalizations ; or homotopy theory. It was suggested, moreover, that such a framework would have profound implications for the study of Diophantine problems. In this talk, we will discuss mostly the little bit of progress made on this last point using some mildly nonabelian motives associated to hyperbolic curves.
In our joint work with J. Burgos and U. Kühn we generalized arithmetic intersection theory originally introduced by H. Gillet and C. Soule to the extent that hermitian vector bundles equipped with logarithmically singular metrics can be taken into account. This is of particular interest when applying methods of arithmetic intersection theory to automorphic vector bundles equipped with their natural invariant metric on Shimura varieties of noncompact type. The last key element missing in the development of our theory is the establishing of an arithmetic RiemannRoch theorem. In order to state such a theorem the definition of a direct image of the hermitian vector bundles under consideration for proper morphisms between arithmetic varieties is needed. In our talk we will present joint work with T. Hahn in which first results in this direction have been established for the Hodge bundle on modular curves equipped with the hyperbolic metric. Using different techniques a similar result has been obtained by G. Freixas.
The cones of divisors and curves defined by various positivity conditions on a smooth projective variety have been the subject of a great deal of work in algebraic geometry, and by now they are quite well understood. However the analogous cones for cycles of higher codimension have started to come into focus only recently. I will discuss a couple of computations on abelian varieties where one can work out the picture fairly completely  already here one sees some nonclassical phenomena. I will also discuss at some length some of the many open problems that present themselves around this circle of ideas. (This is joint work in progress with Olivier Debarre, Lawrence Ein and Claire Voisin.)
We would like to give a report on the rather recent progress in the study of holonomic Dmodules. Then, we will describe an attempt to define an appropriate notion of ``Betti structure'' for holonomic Dmodules with nice functoriality.
We formulate a global form of Denef and Loeser's motivic monodromy conjecture, and we prove it for tamely ramified abelian varieties A over a discretely valued field. More precisely, we show that the motivic zeta function of A has a unique pole, which coincides with Chai's base change conductor c(A), and that this pole corresponds to a monodromy eigenvalue on the tame elladic cohomology of A. This is joint work with Lars Halvard Halle (Hannover).
We discuss under which assumptions the (commutative) 2variable Main Conjecture for CMelliptic curves (due to Rubin, Yager, Katz etc.) implies the noncommutative Main Conjecture as formulated together with Coates, Fukaya, Kato and Sujatha. This is joint work with Thanasis Bouganis.
Let k be a base field of characteristic 0. The essential dimension of a projective curve C over an extension K of k is the largest transcendence degree over k of a field of definition of C. Fix a genus g; a reasonable question to ask is: what is the least upper bound for the essential dimension of a smoth curve C of genus g over some extension of k? In other words, how many parameters are needed to write a smooth curve of genus g? The answer to this question is known, due to work of Brosnan, Reichstein and myself. I will discuss this result, and some recent extensions of it to singular curves.