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Affine Hecke algebras appeared as a tool in the theory of automorphic functions. Now they are indispensable in modern representation theory, combinatorics, geometry, harmonic analysis, mathematical physics, and the theory of special functions. The Langlands program, the theory of quantum groups, invariants of knots, the modern theory of the symmetric group and Weyl groups, Lusztig's modular conjectures, and applications to the theory of spherical and hypergeometric functions are well known examples.

Double Affine Hecke algebras, DAHA, are a fundamental development. They unify Affine Hecke algebras and Heisenberg-Weyl algebras. After they were employed to prove Macdonald’s conjectures including the celebrated constant term conjecture, they have attracted a lot of attention. There is recent progress in the following directions: harmonic analysis (Cherednik, Opdam, Delorme, Stokman), representations of DAHAs and K-theory (Vasserot), the theory of Calogero-Moser varieties (Etingof, Ginzburg), the applications of DAHAs in the so-called double arithmetic (Kapranov, Gaitsgory, Kazhdan), Schur algebras (Opdam, Rouquier, Varganolo, Vasserot), diagonal coinvariants (Gordon, Cherednik), quantization of certain algebraic varieties (Etingof, Oblomkov, Rains).

These developments, the progress in the theory of Affine Hecke algebras and relations to the Langlands program will be in the focus of this part of the meeting.

Affine and double affine Hecke algebras are directly connected with conformal field theory via the Knizhnik-Zamolodchikov equations, Quantum groups, Verlinde algebras. Their relations to the theory of matrix models are still not clarified; the recent q-theory of Gauss and Selberg-Mehta-Macdonald integrals, a part of a new difference harmonic analysis program are expected to be a natural link between the two theories. Establishing new connections to conformal field theory and matrix models, could be an important outcome of the meeting, as well as establishing connections of DAHAs with the p-adic Langlands program and Lusztig modular conjectures.


The Langlands Program has emerged in the late 1960s in the form of a series of far-reaching conjectures tying together seemingly unrelated objects in number theory, algebraic geometry, and the theory of automorphic forms (such as Galois representations, motives, and automorphic forms). In recent years it was realized that the Langlands conjectures (in the function field case) may be formulated geometrically, thereby allowing one to state them over an arbitrary field (e.g., the field of complex numbers). This approach has led to Drinfeld's proof of the Langlands conjecture for GL(2) in the function field case.

L. Lafforgue has proved the Langlands conjecture for GL(n) in the case of the field of functions on a curve over a finite field. Geometry of bundles on curves with additional structures (shtukas) plays an important role in his proof.

On the other hand, Beilinson and Drinfeld suggested a variant of the geometric Langlands correspondence over complex field. It relates Hecke eigensheaves on the moduli stack of G-bundles over a complex curve X and local systems for the Langlands dual group of G. They construct this correspondence via quantization of an integrable system on the cotangent bundle to the moduli space of G-bundles defined by Hitchin. Their work uses in an essential way methods from physics, especially, conformal field theory.

Recently, there was a significant development in this direction in the GL(n)-case by Gaitsgory, based on the work of E. Frenkel, Gaitsgory and Vilonen. It will be one of the key directions of the meeting together with important recent results by Laumon and Ngo on the Langlands-Shelstad Fundamental Lemma over a local field of characteristic p> 0. The latter has resisted all attempts to prove it for more than twenty years and is directly connected with the geometric Langlands program. The approach due to Bezrukavnikov to the Lusztig modular conjectures could be one of other direction.

There have been numerous other achievements in recent years, such as the proof of the local Langlands conjecture; Harris may talk about this. The ideas of local Langlands correspondence have been expanded to the realm of representations of loop groups and affine Kac-Moody algebras and D-modules on affine flag varieties by Feigin and Frenkel, and more recently by Frenkel and Gaitsgory (who may talk about this subject at the conference). This gives a new categorical version of the Langlands correspondence, involving categorifications of affine Hecke algebras and their representations.

Some important results have also been obtained in the study of higher dimensional analogues of the Langlands correspondence. These involve, in particular, the realization of the double affine Hecke algebras and its variants as Hecke algebras for reductive groups over two-dimensional local fields.

The p-adic Langlands Program requires special comments. Congruences between Fourier coefficients of modular forms have been directly involved in many of the important advances in number theory in the past few decades. For example, congruences played its role in the dramatic proof of modularity of elliptic curves over Q, a few years ago. The striking fact is that classical modular forms (technically: eigenforms for the Hecke operators and of finite slope) are intimately connected by a very important and remarkably intensive web of congruences. This has developed into a promising field, where especially in recent years, the power of the theory automorphic forms (for a large collection of reductive groups) has been enhanced by considering p-adic analytic spaces parameterizing automorphic forms and their p-adic interpolations; these parameterizing spaces are sometimes called eigenvarieties.

What is in progress, here, is a recasting of the Langlands program in the language of eigenvarieties, to incorporate all this congruence data and its rich arithmetic applications. For this, one needs a p-adic-valued theory of automorphic representations as well. This direction, including recent developments, will be well represented at the conference. Through the Tate-Mumford p-adic uniformization, this direction is hopefully connected with the new theory of q-functions based on DAHAs; the meeting could be a good place to try to understand this connection. There are new important lines of the interaction between LP and physics to be discussed at the conference such as the recent results by Kapustin and Witten, by Pioline and others.


The focus of the CFT-MM part of the meeting will be the theories of random matrix models and their applications in conformal field theory, 2D quantum gravity and string theory (c = 1, superstrings, and branes), and statistical mechanics, and the relations to Calabi-Yau manifolds and the AdS/CFT correspondence. The CFT itself will be well represented too, including recent results in the boundary and perturbed conformal theories.

The connections between matrix models, integrable systems and topological string theory on Calabi-Yau manifolds (Vafa and collaborators), and understanding the Dirichlet branes in low dimensional string theory sparked by the construction of boundary states in the Liouville theory (Fateev, the Zamolodchikovs, Teschner) are among the highlights of recent work in matrix models and string theory.

The long-standing problem of finding a matrix model realization of the twodimensional black hole using the matrix models of the topological string theory attracts a lot of attention, including computations of the black hole entropies, a generalization to other classes of Calabi-Yau manifolds and integrable structures in the AdS/CFT correspondence. The results of Alexandrov, Kazakov, Kostov and Kutasov on a matrix model approach to the theory of 2D dilatonic black hole will be reported at the meeting.

While the field of matrix models and conformal field theory in condensed matter theory is very broad, one area which we intend to cover in depth is the relation between matrix models and conformal field theory; it includes growth processes and Schramm-Loewner evolutions, SLE, and various models of statistical mechanics, for instance, dimers on two-dimensional lattices or spin models on dynamical random lattices. Bernard and Boyer obtained exciting results connecting CFT to the stochastic Loewner equation, to be discussed at the meeting.

Another quickly developing subject to be represented is the integrable structure in the perturbative N = 4 superconformal Yang-Mills theory and in the superstring on the AdS5 ×S5 space for the Anti-de Sitter space AdS5, two sides of the famous AdS/CFT duality. This topic will be discussed in the lectures of Beisert and Kazakov; it represents a completely new application of the old integrability tools, such as Lax and Zakharav-Shabat equations, finite zone solutions, algebraic curves and Bethe ansatz, in the context of string and field theories.

There is a hope that an exact and virtually complete solution of the first nontrivial 4D gauge theory can be found in this way.

The following remarks reflect our expectations concerning some other possible topics of the meeting.

Other possible topics

Seiberg and coauthors recently proposed a matrix model description of the N = 0 noncritical superstring theory and managed to describe all D-branes there. The results beautifully agree with the Super Liouville CFT.

Al. Zamolodchikov's new results on the non-critical strings via the Liouville CFT will be reported; they discovered with A. Zamolodchikov a new type of Dbranes, that appeared (recently) connected with matrix models. A. Zamolodchikov and Lukyanov also found a new class of 2D boundary conformal field theories corresponding to some specific branes called "paperclip" and "pillow". Eguchi obtained an interesting CFT description of the N = 2 Super Liouville CFT.

Maldacena and Lunin found a new supergravity solution describing a strong coupling limit of N=4 super Yang Mills (SYM) theory with "1/2" of supersymmetry preserved; they used the Toda equations to describe the classical limit of the string theory on the corresponding gravitational background.

Kutasov obtained interesting results on the string theory and quantum gravity on AdS3; there are two different phases in this theory, depending on the radius of AdS3.

Klebanov with coauthors discovered new RG flows for certain conformal SYM theories; they described the RG cascades in the dual quiver gauge theories.

Marino with coauthors studied topological open string amplitudes on orientifolds without fixed planes; they determined the contributions of the untwisted and twisted sectors and the BPS structure of the amplitudes.

Douglas pioneered a new, statistical, approach to the search for admissible physical vacua of the superstring theory, that could eventually lead to a "description" of our universe.

A new approach to the calculation of exact effective superpotentials in the N = 1 SYM theories based on the matrix models due to Dijkgraaf and Vafa is an important development. Their recent interpretation of the string partition function as some sort of "wave-function of the universe" also attracts a lot of attention.

Wiegmann and coauthors extended the stochastic Loewner evolution to the case of some critical systems with continuous symmetries and with domain walls that carry an additional spin=1/2 degree of freedom; the stochastic evolution results in the Knizhnik-Zamolodchikov equation.

Mathematical aspects

Many of these areas have already attracted mathematicians and interactions with the participants of the other sections of the meeting can be definitely expected. Links between the geometric Langlands correspondence and conformal field theory seem a very promising direction of future cooperation. Almost all topics of the CFT-MM program of the meeting heavily involve integrable systems in a fundamental way; it opens a wide range of possibilities for cooperation with experts in the representation theory, including Kac-Moody algebras, super Lie algebras, AHA-DAHA, and with mathematicians from other fields.
As for the mathematical aspects of CFT-MM to be discussed during the conference, we would like to mention here Kac's results on the classification of the Virasoro-type algebras and the results by Okounkov and Reshetikhin on the dimer models and beyond; Okounkov may talk about the Gromov-Witten and Donaldson-Thomas correspondence. Di Francesco and Zinn-Justin recently (partially) proved the Razumov-Stroganov conjecture. They and Zuber found new determinant formulas for some tiling problems and applied them to the, so-called, fully packed loops system.

Pioline may talk about new connections between the black hole micro-states and automorphic forms of exceptional Lie groups, a relation of the Langlands program to the gravitational physics. There will be other topics too.


The areas of AHA, LP, and CFT-MM are certainly among the most active areas of current research in mathematics and physics. They are so deeply interconnected that experts in one of these fields need to monitor permanently the progress in the others; our conference will provide an opportunity to do so.

Let us briefly discuss some of these connections.

First of all, there is an important conceptual connection between LP and CFT. Indeed, the main object of the geometric Langlands program is the double quotient G(O)\G(A)/G(K), where G is a reductive algebraic group, K the field of rational functions k(X) on a smooth projective curve X over the ground field k, A is the ring of adeles of K, and O the ring of integers in A. If k is the field of complex numbers, then G(A) is essentially a product of affine Kac-Moody groups, and the Langlands program in this case is closely related to the representation theory of affine Kac-Moody algebras, vertex algebras, and the Wess-Zumino-Witten model. The double coset space G(O)\G(A)/G(K) is nothing but the moduli stack of principal Gbundles on X in this case, which is the main geometric object in conformal field theories with G-symmetry.

There are well known connections between AHA and LP. One of the central directions of the LP is the study of representations of p-adic groups, where a very important topic is the representation theory of AHA. As it was shown by Kapranov, the representation theory of DAHA may play the same fundamental role in the representation theory of algebraic groups over 2-dimensional local fields, an important step toward the 2-dimensional Langlands program. This theory is now being developed by D. Gaitsgory and D. Kazhdan.

There are direct connections between AHA and CFT. For example, the Knizhnik-Zamolodchikov-Cherednik equations are of fundamental importance for the theory of AHA and DAHA with exciting applications in the harmonic analysis on symmetric spaces and combinatoris. On the other hand, the Knizhnik-Zamolodchikov equations are the main tool in the CFT and in the theory of the Wess-Zumino-Witten model.

It is very important that AHA, LP and CFT are deeply related to matrix models. For instance, partition functions for matrix models may be written in terms ofmultivariable orthogonal polynomials (such as Jack polynomials), which are closely related to AHA and DAHA (through works of Heckman, Opdam, Cherednik, and others). As for CFT and MM, they are now two essential and closely connected ingredients of modern string theory.

Finally, a very important role in string theory and quantum field theory is played by S-dualities, in which one side involves a compact Lie group G, and the other involves its Langlands dual. This provides a promising analogy with LP.

Quite a few mathematicians will participate in the CFT-MM part of meeting. Our special concern is to prepare the program that could be understandable to them and to young mathematicians and physicists. It is a difficult task, but we hope that the organizing committee of the conference and the speakers we invite can manage it.

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