We initiate the study of harmonic Cheeger-Simons characters, with applications to smooth versions of the Geometric Langlands program in the abelian case.

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 exchanged with its Langlands dual. We also explore the relation with theta functions and modular transformations. In the non-simply laced case, we construct a...

3 Relative cycles. 15 3.1 Relative cycles. . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.2 Cycles associated with flat subschemes. . . . . . . . . . . . . . 20 3.3 Chow presheaves. . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.4 Relative cycles over geometrically unibranch schemes. . . . . . 32 3.5 Multiplicities of components of inverse images of equidimensional cycles over...

The arguments of this article support the view that in TGD Universe number theoretic and geometric Langlands conjectures could be understood very naturally. The basic notions are following. 1. Zero energy ontology (ZEO) and the related notion of causal diamond CD (CD is short hand for the cartesian product of causal diamond of M and of CP2). ZEO leads to the notion of partonic 2-surfaces at the...

Let N be a closed spin manifold with positive scalar curvature and DN the Dirac operator on N. M1 M2 two Galois covers of such that is quotient M1. Then map from to naturally induces maps between geometric C⁎-algebras associated manifolds. We prove, by finite-propagation argument, maximal higher rho invariants lifts behave functorially respect above map. This can applied computation invariants,...

Abstract Let $f$ f be a cuspidal Hecke eigenform without complex multiplication. We prove the automorphy of symmetric power lifting $\operatorname{Sym}^{n} f$ Sym n for every $n \geq 1$ ≥ 1 .

Notwithstanding known obstructions to this idea, we formulate an attempt to turn quantization into a functorial procedure. We define a category Poisson of Poisson manifolds, whose objects are integrable Poisson manifolds and whose arrows are isomorphism classes of regular Weinstein dual pairs; it follows that identity arrows are symplectic groupoids, and that two objects are isomorphic in Poiss...

We propose a generalization of Gysin maps for DM-type morphisms of stacks F → G that admit a perfect relative obstruction theory E F/G. We prove functoriality properties of the generalized Gysin maps. As applications, we analyze Gromov-Witten invariants of blow-ups and we give a short proof of Costello’s push-forward formula.