Introduction Lecture 1. Algebraic properties of correlators in 2D topological field theory. Moduli of a 2D TFT and WDVV equations of associativity. Lecture 2. Equations of associativity and Frobenius manifolds. Deformed flat connection and its monodromy at the origin. Lecture 3. Semisimplicity and canonical coordinates. Lecture 4. Classification of semisimple Frobenius manifolds. Lecture 5. Mon...

By dimensional reduction, Einstein-Hermitian equations of n + 1 dimensional closed Kähler manifolds lead to vortex equations of n dimensional closed Kähler manifolds. A Yang-Mills-Higgs functional to unitary bundles over closed Kähler manifolds has topological invariance by adding the additional terms which have ghost fields. Henceforth we achieve the matter (Higgs field) coupled topological fi...

We prove an analog of the classical Hartogs extension theorem for certain (possibly unbounded) domains on coverings of strongly pseudoconvex manifolds. Our result is related to a problem posed in the paper of Gromov, Henkin and Shubin [GHS] on holomorphic L2 functions on coverings of pseudoconvex manifolds. Previously in [Br3] similar results were established for some domains on coverings of St...

By studying modular invariance properties of some characteristic forms, which are related to elliptic genera, we obtain twisted cancellation formulas for characteristic forms. We apply these twisted cancellation formulas to study divisibilities on spin manifolds and congruences on spinc manifolds. In particular, we obtain twisted Rokhlin congruences for 8k + 4 dimensional spinc manifolds.

For certain 3-manifolds M with finite fundamental group, we construct smooth, negative definite 4-manifolds, with boundary containingM and some orthogonal spherical spaces forms. This allows a translation of the existence problem for finite fundamental groups of 3-manifolds into a problem in equivariant gauge theory.

Building on the theory of elliptic operators, we give a unified treatment of the following topics: • the problem of homotopy invariance of Novikov’s higher signatures on closed manifolds; • the problem of cut-and-paste invariance of Novikov’s higher signatures on closed manifolds; • the problem of defining higher signatures on manifolds with boundary and proving their homotopy invariance.

In this paper, we introduce the concept of principal bundles on statistical manifolds. After necessary preliminaries on information geometry and principal bundles on manifolds, we study the α-structure of frame bundles over statistical manifolds with respect to α-connections, by giving geometric structures. The manifold of one-dimensional normal distributions appears in the end as an applicatio...

We extend Witten’s spinor proof of the positive mass theorem to large classes of complete asymptotically flat non-spin manifolds, including all manifolds of dimension less than or equal to 11 and all manifolds of dimension n less than 26 which admit codimension 3 immersions into Euclidean space.

We calculate the cohomology rings of a collection of seven dimensional manifolds supporting an S × S-action with one dimensional orbit space. These manifolds are of interest to differential geometers studying non-negative and positive sectional curvature. From this collection, we identify several families of manifolds for which there exist well-known topological invariants distinguishing homeom...

Building on the theory of elliptic operators, we give a unified treatment of the following topics: • the problem of homotopy invariance of Novikov's higher signatures on closed manifolds; • the problem of cut-and-paste invariance of Novikov's higher signatures on closed manifolds; • the problem of defining higher signatures on manifolds with boundary and proving their homotopy invariance.