Related papers: Local and global Langlands conjecture(s) over func…

We introduce local iterated function systems and present some of their basic properties. A new class of local attractors of local iterated function systems, namely local fractal functions, is constructed. We derive formulas so that these…

This paper discusses the simplest examples of spectral zeta functions, especially those associated with graphs, a subject which has not been much studied. The analogy and the similar structure of these functions, such as their parallel…

We construct examples of p-adic L-functions over universal deformation spaces for GL(2). We formulate a conjecture predicting that the natural parameter spaces for p-adic L-functions are not the usual eigenvarieties (parametrising…

All kinds of global correspondences of Langlands are evaluated from the functional representation spaces of the algebraic bilinear semigroups GL2(.x.) with entries in products,right by left,of sets of archimedean increasing completions.…

These are lectures given at the 2022 Arizona Winter School. It gives an introduction to the rigidity method for constructing automorphic forms for semisimple groups over function fields. The rigidity method leads to explicit constructions…

This paper performs the following steps toward the proof of GLC in the de Rham setting: (i) We deduce GLC for G=GL_n; (ii) We prove that the Langlands functor L_G constructed in [GLC1], when restricted to the cuspidal category, is…

The usual way to interpret language models (LMs) is to test their performance on different benchmarks and subsequently infer their internal processes. In this paper, we present an alternative approach, concentrating on the quality of LM…

This work is concerned with the convex analysis of functions defined on (not necessarily finite-dimensional) Hilbert spaces whose values depend solely on a certain ``spectrum'' of the arguments, a class we term ``spectral functions.'' We…

A consistent functional calculus approach to the spectral theorem for strongly commuting normal operators on Hilbert spaces is presented. In contrast to the common approaches using projection-valued measures or multiplication operators,…

In our previous work, we proposed that engrams in the brain could be biologically implemented as autoencoders over recurrent neural networks. These autoencoders would comprise basic excitatory/inhibitory motifs, with credit assignment…

In this article, new anisotropic grand Lorentz spaces are defined and their properties are studied. These spaces are a new structure that provides a unified parameter for the study of various functional spaces. The consideration of grand…

This is an attempt at a practical and essentially self-contained theory of automorphic representations in the framework $$\hbox{$L^2(\varGamma\backslash\r{G})$ with $\r{G}=\r{PSL}(2,\B{R})$ and $\varGamma=\r{PSL}(2,\B{Z})$.}$$

Neural models learn data representations that lie on low-dimensional manifolds, yet modeling the relation between these representational spaces is an ongoing challenge. By integrating spectral geometry principles into neural modeling, we…

We prove the compatibility of the local and global Langlands correspondences at places dividing l for the l-adic Galois representations associated to regular algebraic conjugate self-dual cuspidal automorphic representations of GL_n over an…

We introduce a local multifractal formalism adapted to functions, measures or distributions which display multifractal characteristics that can change with time, or location. We develop this formalism in a general framework and we work out…

I derive analytically the spectrum of the CMB fluctuations. The final result for C_l is presented in terms of elementary functions with an explicit dependence on the basic cosmological parameters. This result is in a rather good agreement…

This work is intended as an introduction to the statement and the construction of the local Langlands correspondence for GL(n) over p-adic fields. The emphasis lies on the statement and the explanation of the correspondence.

In this paper, we prove that the arithmetic automorphic periods for $GL_{n}$ over a CM field factorize through the infinite places. This generalizes a conjecture of Shimura in 1983, and is predicted by the Langlands correspondence between…

These notes give a statement of the "fundamental lemma," which is a conjectural identity between p-adic integrals that arises as part of the Langlands program.

In [J14], a conjecture was proposed on a relation between the global Arthur parameters and the structure of Fourier coefficients of the automorphic representations in the corresponding global Arthur packets. In this paper, we discuss the…