Related papers: A short course on Witten Helffer-Sj\"ostrand theor…

We survey the different versions of Floer homology that can be associated to three-manifolds. We also discuss their applications, particularly to questions about surgery, homology cobordism, and four-manifolds with boundary. We then…

The purpose of this paper is twofold: 1. we prove the triangulability of smooth orbifolds with corners, generalizing the same statement for orbifolds. 2. based on 1, we propose a new homology theory. We call it geometric homology theory…

We prove a 20-year-old conjecture concerning two quantum invariants of three manifolds that are constructed from finite dimensional Hopf algebras, namely, the Kuperberg invariant and the Hennings-Kauffman-Radford invariant. The two…

For a closed Riemannian manifold we extend the definition of analytic and Reidemeister torsion associated to an orthogonal representation of fundamental group on a Hilbert module of finite type over a finite von Neumann algebra. If the…

We construct a scattering theory for harmonic one-forms on Riemann surfaces, obtained from boundary value problems involving systems of curves and the jump problem. We obtain an explicit expression for the scattering matrix in terms of…

Many fundamental structures of Riemannian geometry have found discrete counterparts for graphs or combinatorial ones for simplicial complexes. These include those discussed in this survey, Hodge theory, Morse theory, the spectral theory of…

The relation between the number of solutions of a nonlinear equation on a Riemannian manifold and the topology of the manifold itself is studied. The technique is based on Ljusternik-Schnirelmann category and Morse theory.

Riemann surfaces with nodes can be described by introducing simple composite operators in matrix models. In the case of the Kontsevich model, it is sufficient to add the quadratic, but ``non-propagating'', term (tr[X])^2 to the Lagrangian.…

This is the first of two papers devoted to showing how the rich algebraic formalism of Eliashberg-Givental-Hofer's symplectic field theory (SFT) can be used to define higher algebraic structures on the symplectic cohomology of open…

We give a simple proof of the cobordism invariance of the index of an elliptic operator. The proof is based on a study of a Witten-type deformation of an extension of the operator to a complete Riemannian manifold. One of the advantages of…

The S-expansion method is a generalization of the In\"{o}n\"{u}-Wigner (IW) contraction that allows to study new non-trivial relations between different Lie algebras. Basically, this method combines a Lie algebra $\mathcal{G}$ with a finite…

We consider quotients of string and M-theory by discrete subgroups of the U-duality group. This results in what we call O-folds, which are generalisations of orbifolds and orientifolds, and generically involve non-geometric identifications…

We construct equivariant and Bott-type Seiberg-Witten Floer homology and cohomology for 3-manifolds, in particular rational homology spheres, and prove their diffeomorphism invariance. We present several versions of the equivariant theory:…

Monge-Kantorovich optimal mass transport (OMT) provides a blueprint for geometries in the space of positive densities -- it quantifies the cost of transporting a mass distribution into another. In particular, it provides natural options for…

Using the Witten deformation and localization algebra techniques, we compute the $G$-equivariant $K$-homology class of the de Rham operator on a proper cocompact $G$-spin manifold, where $G$ is an almost connected Lie group. By applying a…

We revisit a paper of Li-Gabor-Yau on quantum tunneling on graphs using the approach of Helffer-Sjoestrand for the study of tunneling for Schroedinger operators on manifolds.

We extend the well-known Denjoy-Ahlfors theorem on the number of different asymptotic tracts of holomorphic functions to subharmonic functions on arbitrary Riemannian manifolds. We obtain some new versions of the Liouville theorem for…

Consider a Hamiltonian action of a compact Lie group on a symplectic manifold which has the strong Lefschetz property. We establish an equivariant version of the Merkulov-Guillemin $d\delta$-lemma and an improved version of the…

We develop the theory of twisted L^2-cohomology and twisted spectral invariants for flat Hilbertian bundles over compact manifolds. They can be viewed as functions on the first de Rham cohomology of M and they generalize the standard…

As a generalization of holomorphic submersions, anti-invariant submersions and slant submersions, we introduce slant Riemannian maps from almost Hermitian manifolds to Riemannian manifolds. We give examples, obtain the existence conditions…