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We construct a monomorphism of the De Rham complex of scalar multivalued meromorphic forms on the projective line, holomorphic on the complement to a finite set of points, to the chain complex of the Lie algebra of $sl_2$-valued algebraic…

The local $L^2$-mapping property of Fourier integral operators has been established in H\"ormander \cite{H} and in Eskin \cite{E}. In this paper, we treat the global $L^2$-boundedness for a class of operators that appears naturally in many…

Let R and S be differential graded algebras. In this paper we give a characterisation of when a differential graded R-S-bimodule M induces a full embedding of derived categories M\otimes - :D(S)--> D(R). In particular, this characterisation…

In the setting of the geometric Langlands conjecture, we argue that the phenomenon of divergence at infinity on Bun_G (that is, the difference between $!$-extensions and $*$-extensions) is controlled, Langlands-dually, by the locus of…

Point singularities of solutions to the classical Lane-Emden-Serrin equation have a polyhomogeneous asymptotic expansion whose logarithmic corrections are determined by a first order ODE. Surprisingly, we are able to discover such an ODE…

We investigate the integrability anomalies arising in the self-dual sectors of gravity and Yang-Mills theory, focusing on their connection to both the chiral anomaly and the trace anomaly. The anomalies in the self-dual sectors generate the…

The present paper is devoted to the description of local and 2-local automorphisms on Cayley algebras over an arbitrary field $\mathbb{F}$. Given a Cayley algebra $\mathcal{C}$ with norm $n$, let $O(\mathcal{C},n)$ be the corresponding…

In this paper, we prove that a $\mathrm{GL}(2n)$-eigenvariety is \'etale over the (pure) weight space at non-critical Shalika points, and construct multi-variable $p$-adic $L$-functions varying over the resulting Shalika components. Our…

Let $\pi$ be an irreducible admissible (complex) representation of $GL(2)$ over a non-archimedean characteristic zero local field with odd residual characteristic. In this paper we prove the equality between the local symmetric square…

Let $f $ be a holomorphic Hecke eigenform or a Hecke-Maass cusp form for the full modular group $ SL(2, \mathbb{Z})$. In this paper we shall use circle method to prove the Weyl exponent for $GL(2)$ $L$-functions. We shall prove that \[ L…

We construct and study various dual pairs between finite dimensional classical Lie groups and infinite dimensional Lie algebras in some Fock representations. The infinite dimensional Lie algebras here can be either a completed infinite rank…

We consider the centre of the affine vertex algebra at the critical level associated with the orthosymplectic Lie superalgebra. It is well-known that the centre is a commutative superalgebra, and we construct a family of its elements in an…

The extension problem asks whether positive semi-definite functions on a symmetric unital subset of a discrete group can be extended to positive semi-definite functions on the whole group. It has been known at least since the work of Rudin…

We determine local test vectors for Waldspurger functionals for GL(2), in the case where both the representation of GL(2) and the character of the degree two extension are ramified, with certain restrictions. We use this to obtain an…

We conjecture an explicit bound on the prime characteristic of a field, under which the Weyl modules of affine $\mathfrak{sl}_2$ and the minimal series modules of Virasoro algebra remain irreducible, and Goddard-Kent-Olive coset…

It has been observed that certain localizations of the spectrum of topological modular forms are self-dual (Mahowald-Rezk, Gross-Hopkins). We provide an integral explanation of these results that is internal to the geometry of the…

Two extension problems are solved. First, the class of locally matricial algebras over an arbitrary field is closed under extensions. Second, the class of locally finite dimensional semisimple algebras over a fixed field is closed under…

This is the second of two papers that describe a compactness theorem for sequences of solutions of certain SL(2;C) analogs of the anti-self dual equations on oriented, 4-dimensional Riemannian manifolds. This paper proves theorems that…

For a finitely-generated vertex operator algebra of central charge c, a locally convex topological completion is constructed. We construct on the completion a structure of an algebra over the operad of the c/2-th power of the determinant…

The Schur-Weyl duality, which started as the study of the commuting actions of the symmetric group $S_d$ and $\mathrm{GL}(n,\mathbb{C})$ on $V^{\otimes d}$ where $V=\mathbb{C}^n$, was extended by Drinfeld and Jimbo to the context of the…