Related papers: Recent progress on Langlands reciprocity for $\mat…

This expository article is an expanded version of talks given at the "Current Developments in Mathematics, 2002" conference. It gives an introduction to the (generalized) conjecture of Rapoport and Goresky-MacPherson which identifies the…

In this paper we develop representation theory of the two fold central extension of SL(2) over the field of 2-adic numbers. As a consequence, we obtain a local Shimura correspondence between the two fold central extension of SL(2) and the…

We use the theory of trianguline $(\varphi,\Gamma)$-modules over pseudorigid spaces to prove a modularity lifting theorem for certain Galois representations which are trianguline at $p$, including those with characteristic $p$ coefficients.…

We study branching laws for a classical group $G$ and a symmetric subgroup $H$. Our approach is through the {\it branching algebra}, the algebra of covariants for $H$ in the regular functions on the natural torus bundle over the flag…

We study the supersingular locus of a reduction at an inert prime of the Shimura variety attached to $\mathrm{GU}(2,n-2)$. More concretely, we decompose the supersingular locus into a disjoint union of iterated fibrations over (classical)…

In an earlier paper of the authors, a general family of instances of the relative Langlands duality of Ben-Zvi-Sakellaridis-Venkatesh [BZSV] were proposed and studied in the setting of branching problems for smooth representations. In this…

We prove in this paper a classicality result for overconvergent modular forms on PEL Shimura varieties of type (A) or (C) associated to an unramified reductive group on $\mathbb{Q}_p$. To get this result, we use the analytic continuation…

We prove a modularity lifting theorem for minimally ramified deformations of two-dimensional odd Galois representations, over an arbitrary number field. The main ingredient is a generalization of the Taylor-Wiles method in which we patch…

In this paper we use automorph class theory formalism to construct a lifting of similitudes of quadratic Z-modules of arbitrary ternary nondegenerate quadratic forms to morphisms between certain subrings of associated Clifford algebras. The…

We study universal localisations, in the sense of Cohn and Schofield, for finite dimensional algebras and classify them by certain subcategories of our initial module category. A complete classification is presented in the hereditary case…

We give an elementary introduction to Classical Invariant Theory and its modern extension "Transcending Classical Invariant Theory", commonly known as the theory of local theta correspondence. We explain the two fundamental assertions of…

We generalize the $\Lambda$-adic Shintani lifting for $GL_2(Q)$ to indefinite quaternion algebras over $Q$.

We propose a formulation of the Equivariant Tamagawa Number Conjecture for modular motives with coefficients in universal deformation rings and Hecke algebras; something which seems to have been heretofore missing because the complexes of…

This paper is the second in a series of five that together prove the geometric Langlands conjecture. Our goals are two-fold: (1) Formulate and prove the Fundamental Local Equivalence (FLE) at the critical level; (2) Study the interaction…

We describe a topological field theory that studies the moduli space of solutions of the symplectic vortex equations. It contains as special cases the topological sigma-model and topological Yang-Mills over Kahler surfaces. The correlation…

In this paper, we discuss Galilean relativistic Maxwell theory in detail. We first provide a set of mapping relations, derived systematically, that connect the covariant and contravariant vectors in the Lorentz relativistic and Galilean…

The lagrangian-based Standard-Model Extension framework offers a broad description of possible gravitational effects from local Lorentz violation. In this talk, I review the status of the theoretical and phenomenological work in this area.…

We propose a duality in the relative Langlands program. This duality pairs a Hamiltonian space for a group $G$ with a Hamiltonian space under its dual group $\check{G}$, and recovers at a numerical level the relationship between a period on…

We prove a new converse theorem for Borcherds' multiplicative theta lift which improves the previously known results. To this end we develop a newform theory for vector valued modular forms for the Weil representation, which might be of…

Let $f$ be a newform of weight $k\geq 2$, level $N$ with coefficients in a number field $K$, and $A$ the adjoint motive of the motive $M$ associated to $f$. We carefully discuss the construction of the realisations of $M$ and $A$, as well…