*Abstracts of talks*

*Fedor Bogomolov (Courant Institute/Nottingham)*

*On the section conjecture in anabelian geometry*

The section conjecture plays a fundamental role in anabelian geometry. In my talk I will present recent ideas and results.

*Kevin Buzzard (ICL)*

*p-adic Langlands correspondences*

For a connected reductive group over the rational numbers, the classical Langlands philosophy predicts a relationship between, on the one side, automorphic representations of this group, and on the other, certain complex representations of the so-called global Langlands group, a group whose definition we do not know. This is not an ideal situation. One can formulate a precise conjecture if one restricts to algebraic automorphic representations and instead matches them with certain p-adic Galois representations. The introduction of the prime p into the picture on the Galois side introduces an asymmetry. However, and we are still missing a definition of a p-adic automorphic representation in full generality. Moreover one has to go back to basics and formulate a new local Langlands correspondence - the p-adic local Langlands correspondence - to ensure that it is meaningful to ask that the local and global correspondence coincide. I will give an overview of the state of the art and in particular of the state of the art of the p-adic local Langlands correspondence.

*Ted Chinburg* (*University of Pennsylvania)*

*Higher Chern classes in Iwasawa theory*

Many of the main conjectures of Iwasawa theory can be viewed as identifying the first Chern class associated to a complex of Galois modules with a class defined by p-adic L-series. In this talk I will discuss the problem of identifying the higher Chern classes of the natural complexes and Galois modules. Over imaginary quadratic fields, this leads to considering invariants in adelic K_2 groups defined by symbols associated to pairs of Katz p-adic L-functions. Joint work with F. Bleher, R. Greenberg, M. Kakde, G. Pappas, R. Sharifi and M. J. Taylor.

*Christopher Deninger (Münster)*

*Zeta functions and foliations*

We will explain and motivate a dictionary between arithmetical and foliation theoretical structures. This gives a potentially useful way to look at unsolved problems about Hasse-Weil zeta functions like the functional equation, the Riemann hypotheses and formulas for their values at integers.

*Edward Frenkel (Berkeley)*

*The Langlands programme and quantum dualities*

A geometric version of the Langlands programme has been linked, in the works of Witten and others, to the electromagnetic duality of four-dimensional quantum gauge theories. But where does that duality come from? It turns out that it has a natural geometric explanation from the point of view of a mysterious six-dimensional quantum field theory, whose existence is predicted in string theory and M-theory and which appears to have no Lagrangian description. Although they are still conjectural, various properties of this theory have been recently used to reveal surprising connections between 4D and 2D quantum field theories, some of which have now been rigorously proved. I will review some of these connections and their possible implications for the geometric Langlands programme.

*Dennis Gaitsgory (Harvard)*

*Picard-Lefschetz oscillators for Drinfeld-Lafforgue compactifications (after Simon Schieder)*

In their proof of the Langlands conjecture for functions fields, V. Drinfeld and L. Lafforgue used a compactification Bun_G of the diagonal morphism of the stack Bun_G of G-bundles on a curve X. The stack Bun_G is obtained by mapping the curve X to the stack-theoretic quotient of the Vinberg canonical semigroup attached to G, by means of GxG acting on by left and right translations.

In the talk we will focus on the case of the group G=SL_2. The stack Bun_G is highly singular, and we would like to understand the intersection cohomology sheaf. The answer turns out to be quite complex, but yet combinatorially tractable, and is obtained by applying the Picard-Lefschetz picture of vanishing cycles.

*Dominic Joyce (Oxford)*

*Derived symplectic geometry and categorification*

Pantev-Toen-Vaquie-Vezzosi defined a new geometric structure called "k-shifted symplectic structures" in derived algebraic geometry, for integers k. Ideas from ordinary symplectic geometry, such as Lagrangian submanifolds, have analogues in this "derived symplectic geometry". Ben-Bassat, Bussi, Brav and Joyce prove "Darboux Theorems" giving explicit local models for k-shifted symplectic derived schemes and stacks. The (somewhat mysterious) derived structure then tells us how these local models should be glued together globally.

The case k = -1 is relevant to Lagrangian intersections and Calabi-Yau 3-fold moduli spaces. The BBBJ Darboux theorems tell us that a -1-shifted symplectic derived scheme (or stack) is Zariski locally (or smooth locally) modelled on the critical locus of a regular function f on a smooth scheme U. There are several important constructions associated to critical loci: the Milnor fibration, the perverse sheaves/D-modules/mixed Hodge modules of vanishing cycles, motivic Milnor fibres, matrix factorization categories. So we can ask: given a -1-shifted symplectic derived scheme/stack, which is locally modelled on critical loci, can we glue these constructions on local models to get an interesting global object? For example, we prove that given a -1-shifted symplectic derived scheme or stack X with an "orientation", we can glue the local models for perverse sheaves of vanishing cycles to get a canonical global perverse sheaf on X.

I discuss applications of this to define a "Fukaya category" of complex or algebraic Lagrangians in a complex or algebraic symplectic manifold, and to defining "Cohomological Hall Algebras" of 3-Calabi-Yau categories, categorifying Donaldson-Thomas theory of Calabi-Yau 3-folds.

*Mikhail Kapranov (Yale)*

*Lie algebras and E_n-algebras associated to secondary polytopes*

The secondary polytope Sigma(A) of a finite set of points A in R^n is a convex polytope whose vertices correspond to certain triangulations of the polytope Conv(A). It has a remarkable factorization property: each face of Sigma(A) is itself a product of several secondary polytopes Sigma(B) for a subset B of A. This leads to a construction of factorization algebras (and therefore E_n-algebras and L_\infty-algebras) of explicit combinatorial nature, by using (co)chain complexes of secondary polytopes (possibly with coefficients in appropriate sheaves). For n=2 these algebras appear naturally in the problem of deformation of triangulated categories with exceptional collections. This suggests possible higher-categorical interpretation for n>2. Joint work with M. Kontsevich and Y. Soibelman.

* Minhyong Kim (Oxford)*

* Non-abelian reciprocity laws and Diophantine geometry*

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* *A generalisation of Artin's reciprocity law is formulated with coefficients in a wide class of varieties.

Some Diophantine finiteness results and a non-abelian conjecture of Birch and Swinnerton-Dyer type will be among the consequences.

* Robert Langlands *(*IAS*)

*Problems in the theory of automorphic forms: 45 years later (Video Lecture)*

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The text of the lecture will be made available to the participants of the conference at the end of June. Questions about it can be emailed to Ivan, and some of them may be answered by the author in the final version of the text which will be put on the IAS www-page.

*François Loeser (Univ. Paris 6-7)*

*Motivic integration and representation theory*

We shall present a broad overview of some recent developments of motivic integration in relation with representation theory. We shall start by explaining how the definable character of many objects occurring in p-adic representation theory implies strong uniformity properties. This allows to transfer results from characteristic p to characteristic zero (the Fundamental Lemma is a typical example by work with Cluckers and Hales) or from characteristic zero to characteristic p (integrability of characters in positive characteristic, work of Cluckers, Gordon, and Halpuczok). In a second part we shall focus on more global aspects by presenting a motivic version of the Poisson formula due to Hrushovski and Kazhdan which has already several applications: to Fourier transforms of matching test functions (Hrushovski and Kazhdan) and to geometric analogues of Manin's conjecture on points of bounded height (work with Chambert-Loir).

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*Ralf Meyer (Univ. Göttingen)*

*Groupoids and higher groupoids*

Many interesting geometric objects are moduli spaces or quotient spaces. When these quotients are too singular, they may profitably be replaced by groupoids. The same construction is possible on a higher level: replace a quotient group(oid) by a 2-group(oid). 2-groupoids arise naturally as symmetries of groupoids. I will explain how 2-groupoids act on groupoids by partial equivalences and give some examples. A nice example concerns the quotient of the circle group by a dense cyclic subgroup, which naturally gives a 2-group T//Z. The algebra of functions on this quotient space (forget the group structure) is a noncommutative torus, which is suggested by Manin as a real analogue of an elliptic curve. The noncommutative torus carries no trace of a group structure, however. The analogue of the multiplication action of a group on itself may be encoded as an action of the 2-group T//Z on the noncommutative torus. Groupoids may be turned into a category using functors; these are the appropriate morphisms if we view groupoids as generalised spaces. When we view groupoids as generalised groups, then a less-known type of morphism is more appropriate; these do not induce a map on the orbit space but induce a morphism between groupoid C*-algebras; they are sometimes called comorphisms or Zakrzewski morphisms, I prefer to call them actors because they cooperate well with groupoid actions. Both functors and actors may be further combined with equivalence of groupoids.

Besides various technically different definitions of topological groupoids, there are Lie groupoids and groupoids in algebraic geometry. The theory of groupoids may be developed uniformly for all these cases, by studying groupoids in a category with pretopology. In this setting, we may define groupoid principal bundles, equivalence of groupoids, functors, actors.

* Matthew Morrow (Bonn)*

* On the deformation theory of algebraic cycles*

I will survey some very recent developments in the deformation theory of algebraic cycles in families, in the style of variational Hodge and Tate conjectures. In particular, I will discuss the formulation of a crystalline variational Tate conjecture in characteristic p and its proof for line bundles, based on continuity results in topological cyclic homology due to B. Dundas and myself. Similar ideas in characteristic 0 lead earlier to the formulation and proof of an infinitesimal Hodge conjecture, which was independently considered by S. Bloch, H. Esnault, and M. Kerz.

*Sergey Oblezin (Nottingham)*

*Whittaker functions, mirror symmetry and the Langlands correspondence*

In 2009 jointly with A. Gerasimov and D. Lebedev, a new class of integral representations of Whittaker functions and automorphic L-functions for GL(N,R) in the framework of 2-dimensional topological field theories was proposed. In this setting the archimedean Langlands correspondence is identified with the mirror symmetry of the underlying TFTs. My talk will outline the main constructions of this approach and explain the new geometric interpretation of the archimedean Whittaker functions and L-factors as equivariant symplectic volumes of certain Kaehler U(N)-manifolds. I will present applications of these results including the archimedean analogue of the Langlands-Shintani formula and discuss further directions.

*Urs Schreiber (Nijmegen)*

*Correspondences of cohesive linear homotopy types and quantization*

Geometric quantization (the orbit method) is really the process of forming the push-forward in twisted equivariant K-theory of the prequantum line bundle on phase space. Incidentally, this is also the definition of D-brane charge in string theory. In this talk I first explain how all the ingredients that go into these statements have a characterization in Goodwillie-tangent spaces to those infinity-toposes which I call "cohesive". A prequantized Lagrangian correspondence here is a kind of pure motive of cohesive higher stacks equipped with cocycles in bivariant twisted differential generalized cohomology. I then discuss how transfer through such correspondences by twisted Umkehr maps serves as a path integral quantization for topological field theories and close by indicating some examples.

*Christophe Soulé (IHES)*

*A singular arithmetic Riemann-Roch theorem*

We present several variants of the Riemann-Roch theorem in Arakelov geometry, including a recent one with Henri Gillet, valid for varieties over the integers with arbitrary singular special fibres. Next, we speculate on a (yet to be defined) adelic geometry.

*Masatoshi Suzuki (Tokyo Institute of Technology)*

*Translation invariant subspaces and GRH for zeta functions*

I will describe new approaches to study analytic properties of zeta functions, such as analytic continuations, functional equations and distributions of poles or zeros, from the viewpoint of translation invariant subspaces of suitable function spaces. Two kinds of invariant subspaces come up in this story. If time permit, we will discussed their possible relationship that assumed Connes' zeta operator a cardinal point.

One is used to describe a correspondence between zeta functions and mean-periodic functions which originated in work of Fesenko. In this correspondence, meromorphic continuations and functional equations of zeta functions are explained as mean-periodicity of corresponding functions in some function space. Here, the main interest is in mean-periodicity in relation to arithmetic zeta functions.

Another is used to study zeros of zeta functions in terms of canonical systems of the first order differential equations. In this direction, the RH is reduced to a construction of expected canonical systems, which is quite difficult because it is a kind of inverse spectral problem. In this talk, we try to avoid such difficulty by considering some integral operators endowed with kernels consisting of quotients of zeta functions and using Euler products.

*Constantin Teleman (Berkeley/Oxford)*

*Categorical representations and Langlands duality*

We outline a theory of topological representations of a compact Lie group G on categories ("locally trivial representations") in terms of a mirror-symmetric construction which involves the Langlands dual group. A distinguished class of representations, symplectically induced from Levi subgroups, is described on the mirror side in terms of the Toda integrable system. The description is supported by the computation of the mirrors of generalized flag varieties, which recovers the construction by K. Rietsch. The governing ideas come from 2- and 3-dimensional topological field theory.

*Yuri Tschinkel (Courant Institute)*

*Introduction to almost abelian anabelian geometry*

I will survey recent developments in the birational anabelian geometry program aimed at the reconstruction of function fields of algebraic varieties over algebraically closed fields from pieces of their absolute Galois groups.