We consider the Yang-Mills problem for a quantum Heisenberg manifold, which is a C∗-algebra defined by the (strict) deformation quantization of the ordinary Heisenberg manifold, in the setting of non-commutative differential geometry following Connes and Rieffel [Co] [Co1]. 1. Preliminaries Classical Yang-Mills theory is concerned with the set of connections (i.e. gauge potentials) on a vector ...

Geometric Langlands duality can be understood from statements of mirror symmetry that can be formulated in purely topological terms for an oriented two-manifold C. But understanding these statements is extremely difficult without picking a complex structure on C and using Hitchin’s equations. We sketch the essential statements both for the “unramified” case that C is a compact oriented two-mani...

If a closed smooth n-manifold M admits a finite cover M̂ whose Z/2Z-cohomology has the maximal cup-length, then for any riemannian metric g on M , we show that the systole Sys(M, g) and the volume Vol(M, g) of the riemannian manifold (M, g) are related by the following isosystolic inequality: Sys(M, g) n ≤ n!Vol(M, g). The inequality can be regarded as a generalization of Burago and Hebda’s ineq...

1. Basic gradient estimate; different variations of the gradient estimates; 2. The theorem of Brascamp-Lieb, Barkey-Émery Riemannian geometry, relation of eigenvalue gap with respect to the first Neumann eigenvalue; the Friedlander-Solomayak theorem, 3. The definition of the Laplacian on L space, theorem of Sturm, 4. Theorem of Wang and its possible generalizations. 1 Gradient estimate of the f...

Cotangent bundle reduction theory is a basic and well developed subject in which one performs symplectic reduction on cotangent bundles. One starts with a (free and proper) action of a Lie group G on a configuration manifold Q, considers its natural cotangent lift to T ∗Q and then one seeks realizations of the corresponding symplectic or Poisson reduced space. We further develop this theory by ...

In this paper first it is proved that if ξ is a nontrivial closed conformal vector field on an n-dimensional compact Riemannian manifold (M, g) with constant scalar curvature S satisfying S ≤ λ1(n − 1), λ1 being first nonzero eigenvalue of the Laplacian operator ∆ on M and Ricci curvature in direction of a certain vector field is non-negative, then M is isometric to the n-sphere S(c), where S =...

We study the topology of the boundary manifold of a line arrangement in CP , with emphasis on the fundamental group G and associated invariants. We determine the Alexander polynomial ∆(G), and more generally, the twisted Alexander polynomial associated to the abelianization of G and an arbitrary complex representation. We give an explicit description of the unit ball in the Alexander norm, and ...

We find lower and upper bounds for the risk of estimating a man-ifold in Hausdorff distance under several models. We also show that there are close connections between manifold estimation and the problem of deconvolving a singular measure. 1. Introduction. Manifold learning is an area of intense research activity in machine learning and statistics. Yet a very basic question about manifold learn...