Related papers: Geometric Langlands for hypergeometric sheaves
In this paper we establish some foundations regarding sheaves of vector spaces on graphs and their invariants, such as homology groups and their limits. We then use these ideas to prove the Hanna Neumann Conjecture of the 1950's; in fact,…
We introduce a hypergoemetirc series with two complex variables, which generalizes Appell's, Lauricella's and Kemp\'e de F\'eriet's hypergeometric series, and study the system of differential equations that it satisfies. We determine the…
The identifiability problem arises naturally in a number of contexts in mathematics and computer science. Specific instances include local or global rigidity of graphs and unique completability of partially-filled tensors subject to rank…
We study skew-product dynamics for a large class of finitely-generated semi--hyperbolic semigroups of rational maps acting on the Riemann sphere, which generalizes both the theory of iteration of a single rational map of a single complex…
We show how a novel construction of the sheaf of Cherednik algebras on a quotient orbifold Y=X/G by virtue of formal geometry in author's prior work leads to results for the sheaf of Cherednik algebra which until recently were viewed as…
In this paper we use the quantization of fields based on Geometric Langlands Correspondence \cite{diep1} to realize the automorphic representations of some concrete series of groups: for the affine Heisenberg (loop) groups it is reduced to…
Huang's Lemma is an important tool in CR geometry to study rigidity problems. This paper introduces a generalization of Huang's Lemma based on the rigidity properties of holomorphic mappings preserving certain orthogonality on projective…
Let G be a quasi-split reductive group over a non-archimedean local field. We establish a local Langlands correspondence for all irreducible smooth complex G-representations in the principal series. The parametrization map is injective, and…
Capacity control, the bias/variance dilemma, and learning unknown functions from data, are all concerned with identifying effective and consistent fits of unknown geometric loci to random data points. A geometric locus is a curve or surface…
In this paper we consider aspects of geometric observability for hypergraphs, extending our earlier work from the uniform to the nonuniform case. Hypergraphs, a generalization of graphs, allow hyperedges to connect multiple nodes and…
We construct yet another ${\mathcal N}=(4,4)$ gauged linear sigma model for the $A_N$-type ALE space. In our construction the toric data of the ALE space are manifest. Due to the $SU(2)_R$ symmetry, the F-term is automatically determined.…
A rich class of network models associate each node with a low-dimensional latent coordinate that controls the propensity for connections to form. Models of this type are well established in the network analysis literature, where it is…
We develop a discrete gauge-theoretic framework for superposition in large language models (LLMs) that replaces the single-global-dictionary premise with a sheaf-theoretic atlas of local semantic charts. Contexts are clustered into a…
In this article, we determine the trace of some Hecke operators on the spaces of level one automorphic forms on the special orthogonal groups of the euclidean lattices $\mathrm{E}_7$, $\mathrm{E}_8$ and $\mathrm{E}_8\oplus \mathrm{A}_1$,…
A geometric characterization of the structure of the group of automorphisms of an arbitrary Birkhoff-Grothendieck bundle splitting $\bigoplus_{i=1}^{r} \mathcal(m_{i})$ over $\mathbb{C}\mathbb{P}^{1}$ is provided, in terms of its action on…
We study automorphic categories of nilpotent sheaves under degenerations of smooth curves to nodal Deligne-Mumford curves. Our constructions realize affine Hecke operators as the result of bubbling projective lines from marked points. We…
A brief survey is given of the classical Langlands correspondence between n-dimensional representations of Galois groups of local and global fields of dimension 1 and irreducible representations of the groups GL(n). A generalization of the…
The purpose of this paper is to introduce and study certain irreducible perverse l-adic sheaves on a reductive group G over a finite field (we call them gamma-sheaves). One can construct such a sheaf starting with (almost) every…
Let X be a smooth projective curve over an algebraically closed field of characteristic >2. Consider the dual pair H=GSO_{2m}, G=GSp_{2n} over X, where H splits over an etale two-sheeted covering of X. Write Bun_G and Bun_H for the stacks…
We construct a sheaf-theoretic representation of quantum observables algebras over a base category equipped with a Grothendieck topology, consisting of epimorphic families of commutative observables algebras, playing the role of local…