Related papers: Relative functoriality and functional equations vi…

We present topics in the Langlands Program to graduate students and a wider mathematically mature audience. We study both global and local aspects in characteristic zero as well as characteristic $p$. We look at modern approaches to the…

We study Translation functors and Wall-Crossing functors on infinite dimensional representations of a complex semisimple Lie algebra using D-modules. This functorial machinery is then used to prove the Endomorphism-theorem and the…

The Fock transform recently introduced by the authors in a previous paper is applied to investigate convergence of generalized functional sequences of a discrete-time normal martingale $M$. A necessary and sufficient condition in terms of…

We define the local trace function for subspaces of $\ltworn$ which are invariant under integer translation. Our trace function contains the dimension function and the spectral function defined by Bownik and Rzeszotnik and completely…

Genestier--Lafforgue and Fargues--Scholze have constructed a semisimple local Langlands paramterization for reductive groups over equicharacteristic local fields. Assuming a version of the stable twisted trace formula for function fields,…

We introduce a general approach to traces that we consider as linear continuous functionals on some function space where we focus on some special choices for that space. This leads to an integral calculus for the computation of the precise…

After explaining the concepts of Langlands dual and miniscule representations, we define an analog of the Gauss sum for any compact, simple Lie group with a simply laced Lie algebra. We then show a reciprocity property when a Lie group is…

This article gives a proof of the Langlands-Shelstad fundamental lemma for the spherical Hecke algebra for every unramified p-adic reductive group G in large positive characteristic. The proof is based on the transfer principle for…

This is an informal note that explains that the classical Langlands theory over function fields can be obtained from the geometric one by taking the trace of Frobenius. The operation of taking the trace of Frobenius takes place at the…

By solving a differential-functional equation inposed by the MaxEnt principle we obtain a class of two-parameter deformed logarithms and construct the corresponding two-parameter generalized trace-form entropies. Generalized distributions…

For any block of a finite group over an algebraically closed field of characteristic $2$ which has dihedral, semidihedral, or generalized quaternion defect groups, we determine explicitly the decomposition of the associated diagonal…

We prove that the Ramanujan conjecture is true under the assumption that the expected analytic properties of triple product $L$-functions hold. Further, we explain how these analytic properties imply certain reduction steps in the…

We prove that the endoscopic transfer on real Lie algebras commutes with the Fourier transform, using methods that are purely local.

We construct a functor valued invariant of oriented tangles on certain singular blocks of category O. Parabolic subcategories of these blocks categorify tensor products of various fundamental sl(k) representations. Projective functors…

Transformers flexibly operate over sets of real-valued vectors representing task-specific entities and their attributes, where each vector might encode one word-piece token and its position in a sequence, or some piece of information that…

We introduce a new class of symmetric space orbital integrals important for applications in certain relative trace formulas appearing in the theory of automorphic representations. We verify a fundamental lemma for $U_2\times…

Linear algebra's main concerns are sets of vectors, linear functions, subspaces, linear systems, matrices and concepts about those, such as whether the solution of linear system exists or is unique; a set of vectors is linearly independent…

Relational parametricity was first introduced by Reynolds for System F. Although System F provides a strong model for the type systems at the core of modern functional programming languages, it lacks features of daily programming practice…

A transfunction is a function which maps between sets of finite measures on measurable spaces. Push-forward operators form one important class of examples of transfunctions and are identified with their respective measurable functions. In…

We give another proof of the existence of the endoscopic transfer for unitary Lie algebras and its compatibility with Fourier transforms. By the work of Kazhdan and Vashavsky, this implies the corresponding endoscopic fundamental lemma…