Related papers: Local and global Langlands conjecture(s) over func…
The global Jacquet--Langlands correspondence is an instance of Langlands functoriality, namely the expected lifting of the irreducible automorphic representations of an inner form of the general linear group to the split form via the…
This paper explores relations between two separate worlds. They are the algebraic geometry of Alexander Grothendieck and the automorphic representation theory of Robert Langlands. The relation between them would be a very broad example of…
This is a report on the global aspects of the Langlands-Shahidi method which in conjunction with converse theorems of Cogdell and Piatetski-Shapiro has recently been instrumental in establishing a significant number of new and surprising…
We give a proof of the existence of Asai, exterior square, and symmetric square local $L$-functions, $\gamma$-factors and root numbers in characteristic $p$, including the case of $p = 2$. Our study is made possible by developing the…
We establish the Langlands-Shahidi method over a global field of characteristic p. We then focus on the unitary groups and prove global and local Langlands functoriality to general linear groups for generic representations. Main…
Computing volumetric correspondences between 3D shapes is a prominent tool for medical and industrial applications. In this work, we pave the way for spectral volume mapping, extending for the first time the surface-based functional maps…
The aim of this article is to explore in all remaining aspects the spectral theory of locally normal operators. In a previous article we proved the spectral theorem in terms of locally spectral measures. Here we prove the spectral theorem…
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…
The analytic properties of automorphic L-functions have historically been obtained either through integral representations (the "Rankin-Selberg method"), or properties of the Fourier expansions of Eisenstein series (the "Langlands-Shahidi…
We consider the universal part of entanglement entropy across a plane in flat space for a QFT, giving a non-perturbative expression in terms of a spectral function. We study the change in entanglement entropy under a deformation by a…
Functionals (i.e. functions of functions) are widely used in quantum field theory and solid-state physics. In this paper, functionals are given a rigorous mathematical framework and their main properties are described. The choice of the…
Generalisations of the bent property of a boolean function are presented, by proposing spectral analysis with respect to a well-chosen set of local unitary transforms. Quadratic boolean functions are related to simple graphs and it is shown…
We prove a theorem which provides a method for constructing points on varieties defined by certain smooth functions. We require that the functions are definable in a definably complete expansion of a real closed field and are locally…
In an earlier paper, we considered several restriction problems in the representation theory of classical groups over local and global fields. Assuming the Langlands-Vogan parameterization of irreducible representations, we formulated…
Langlands' beyond endoscopy proposal for establishing functoriality motivates interesting and concrete problems in the representation theory of algebraic groups. We study these problems in a setting related to the Langlands $L$-functions…
We establish an explicit global spectral decomposition of shifted convolution sums and the second moment of automorphic $L$-functions for Maass forms with explicit integral transforms as well as explicit inversion formulae over every local…
Sampling theory in spaces other than the space of band-limited functions has recently received considerable attention. This is in part because the band-limitedness assumption is not very realistic in many applications. In addition,…
The paper proposes a vector generalization of the basic concepts of the theory of complex variable: the concept of modulus and argument of complex number. The author introduces some generalizations of the notion of holomorphic functions and…
$L$-functions can be viewed axiomatically, such as in the formulation due to Selberg, or they can be seen as arising from cuspidal automorphic representations of $\textrm{GL}(n)$, as first described by Langlands. Conjecturally these two…
Let $\pi$ be a cuspidal automorphic representation of a general linear group over the rational numbers. We establish a subconvex bound for the standard $L$-function of $\pi$ in the $t$-aspect. More generally, we address the spectral aspect…