Tilting perverse sheaves arise in geometric representation theory as natural bases for many categories. (See [1] or Section 3 below for definitions and some background results.) For example, for category O of a complex reductive group G, tilting modules correspond to tilting sheaves on the Schubert stratification S of the flag variety B. In this setting, there are many characterizations of tilt...

In this paper we consider the φF category associated to the family of circle valued Morse functions defined on a closed manifold M . This number will be called the Morse-Smale characteristic of manifold M for circle-valued Morse functions and and it will be denoted by γs1(M). We present some basic notions and results concerning circle-valued Morse functions and we prove some properties for γs1(...

The tame flows are “nice” flows on “nice” spaces. The nice (tame) sets are the pfaffian sets introduced by Khovanski, and a flow Φ : R×X → X on pfaffian set X is tame if the graph of Φ is a pfaffian subset of R×X×X . Any compact tame set admits plenty tame flows. We prove that the flow determined by the gradient of a generic real analytic function with respect to a generic real analytic metric ...

The tame flows are “nice” flows on “nice” spaces. The nice (tame) sets are the pfaffian sets introduced by Khovanski, and a flow Φ : R×X → X on pfaffian set X is tame if the graph of Φ is a pfaffian subset of R × X × X. Any compact tame set admits plenty tame flows. We prove that the flow determined by the gradient of a generic real analytic function with respect to a generic real analytic metr...

In [10] we began a study of the 3-dimensional Seiberg-Witten equations on Seifert manifolds with two goals in mind: to compute the Seiberg-Witten-Floer (SWF) homology of these manifolds and ultimately, to produce gluing formulæ for the 4dimensional Seiberg-Witten invariants. A first difficulty one must overcome has to do with the less than obvious nature of the solutions of the 3-dimensional Se...

We extend an approach to Morse theory due to Harvey and Lawson to non compact manifolds. This provides a modified and arguably more natural version of Morse-Novikov theory, where the Novikov Ring is replaced by a new " Forward Laurent Ring ". A detailed development of standard Morse-Novikov theory is clarified in several ways. Geometrically: The Novikov complex is described as a subcomplex of t...

The effect of anharmonicity in the intramolecular modes of a model system for exothermic intramolecular nonadiabatic electron transfer is probed by examining the dependence of the transition probability on the exoergicity. The Franck-Condon factor for the Morse potential is written in terms of the Gauss hypergeometric function both for a ground initial state and for the general case, and compar...

Morse decompositions have been proposed to compute and represent the topological structure of steady vector fields. Compared to the conventional differential topology, Morse decomposition and the resulting Morse Connection Graph (MCG) is numerically stable. However, the granularity of the original Morse decomposition is constrained by the resolution of the underlying spatial discretization, whi...

For orientation-reversing diffeomorphisms on the torus necessary and sufficient conditions are given for an isotopy class to admit a Morse-Smale diffeomorphism with a specified periodic behavior. A diffeomorphism is Morse-Smale provided it is structurally stable and has a finite nonwandering set [P-S]. Several recent papers have explored the relationship between the topology of these maps and t...

We present a short proof of the existence of the normal cycle of a compact subanalytic set. The approach is inspired by some old ides of Joseph Fu, uses Morse theoretic techniques and o-minimal topology.