Related papers: Recent Advances in the Langlands Program

These are expanded notes of some lectures given by the author for a workshop held at the Indian Statistical Institute, Bangalore in June, 2010, giving an exposition on the modular representations of finite groups of Lie type and $p$-adic…

This paper can be viewed as a sequel to the author's long survey on the Zimmer program \cite{F11} published in 2011. The sequel focuses on recent rapid progress on certain aspects of the program particularly concerning rigidity of Anosov…

We prove the recent conjectures of Adams-Vogan and D. Prasad on the behavior of the local Langlands correspondence with respect to taking the contragredient of a representation. The proof holds for tempered representations of quasi-split…

Let G be a connected reductive group over a non-archimedean local field K, and assume that G splits over an unramified extension of K. We establish a local Langlands correspondence for irreducible unipotent representations of G. It comes as…

We study the variation of the local Langlands correspondence for ${\rm GL}_{n}$ in characteristic-zero families. We establish an existence and uniqueness theorem for a correspondence in families, as well as a recognition theorem for when a…

In the Proceedings of the AMS Boulder conference in 1965 Langlands states a combinatorial lemma involving families of characteristic functions attached to ordered partitions of an obtuse basis in a finite dimensional euclidean vector space.…

This paper describes my talk given to the 27th Johns Hopkins Workshop: Symmetries and Mysteries of M Theory, G\"oteborg, Sweden, 24-26 August, 2003. After a brief introduction to the lightcone worldsheet formalism for summing the planar…

After a short review of one of proposals to avoid complex stochastic processes in Complex Langevin studies, the recent progress in the former is reported. In particular, the new developments allow now to construct positive and normalizable…

In this article we briefly survey some developments in gauged linear sigma models (GLSMs). Specifically, we give an overview of progress on constructions of GLSMs for various geometries, GLSM-based computations of quantum cohomology,…

This is an expanded version of the notes by the second author of the lectures on Hitchin systems and their quantization given by the first author at the Beijing Summer Workshop in Mathematics and Mathematical Physics ``Integrable Systems…

I review recent developments in lattice QCD. I first give an overview of its formalism, and then discuss lattice discretizations of fermions. We then turn to a description of the quenched approximation and why it is disappearing as a…

This is a survey article on developments in modularity since the proof of Fermat's Last Theorem, with an emphasis on the historical development of the subject rather than any technical details.

Contents: * Community news: GGR activities, by Richard Price We hear that..., by Jorge Pullin Institute of Physics Gravitational Physics Group, by Elizabeth Winstanley Center for gravitational wave astronomy, by Mario Diaz * Research…

The Langlands correspondence for complex curves is traditionally formulated in terms of sheaves rather than functions. Recently, Langlands asked whether it is possible to construct a function-theoretic version. In this paper we use the…

A survey of three recent developments in algebraic combinatorics: (1) the Laurent phenomenon, (2) Gromov-Witten invariants and toric Schur functions, and (3) toric h-vectors and intersection cohomology. This paper is a continuation of…

The analytic Langlands correspondence describes the solution to the spectral problem for the quantised Hitchin Hamiltonians. It is related to the S-duality of $\cal{N}=4$ super Yang-Mills theory. We propose a one-parameter deformation of…

These are lecture notes from my talks at the "Current Developments in Mathematics" conference (Harvard, 2006). They cover a variety of topics involving symplectic cohomology. In particular, a discussion of (algorithmic) classification…

In this paper, we give a purely geometric approach to the local Jacquet-Langlands correspondence for GL(n) over a p-adic field, under the assumption that the invariant of the division algebra is 1/n. We use the l-adic etale cohomology of…

This is an extended version of the talk given by the author at the NATO Advanced Study Institute "Supersymmetry and Trace Formulae" (Isaac Newton Institute, Cambridge, UK, 8-19 Septemebr 1997). Contains the following sections: 1.…

The Langlands Program was launched in the late 60s with the goal of relating Galois representations and automorphic forms. In recent years a geometric version has been developed which leads to a mysterious duality between certain categories…