Cohomology and Morse theory for strongly indefinite functionals

Supported in part by the Swedish Natural Science Research Council     

On topological Morse theory

Starting from the concept of Morse critical point, introduced in [A. Ioffe and E. Schwartzman, J. Math. Pures Appl. (9), 75 (1996), 125?C153], we propose a purely topological approach to Morse theory.     

Morse lemma for functionals of variational calculus

Institute for Control Problems, Academy of the Sciences of the USSR. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 25, No. 3, pp. 1–11, July–September, 1991.     

Morse theory for some lower-C 2 functions in finite dimension

Using inf-regularization methods, we prove that Morse inequalities hold for some lower-C ² functions. For this purpose, we first recall some properties of the class of lower-C ² functions and of their Moreau-Yosida approximations. Then, we establish, under some qualification conditions on the critical points, that it is possible to define a Morse index for a lower-C ² functionf. This index is preserved by the Moreau-Yosida approximation process. We prove in particular that the Moreau-Yosida approximations are twice continuolusly differentiable around such a critical point which is shown to be a strict local minimum of the restriction off and of its approximations to some affine space. In a last step, Morse inequalities are written for Moreau-Yosida approximations and with the aid of deformation retractions we prove that these inequalities also hold for some lower-C ² functions.     

Morse theory for minimal surfaces in manifolds

A Morse theory of a given function gives information of the numbers of critical points of some topological type. A minimal surface, bounded by a given curve in a manifold, is characterized as a harmonic extension of a critical point of the functional \({\mathcal E}\) which corresponds to the Dirichlet integral. We want to obtain Morse theories for minimal surfaces in Riemannian manifolds. We first investigate the higher differentiabilities of \({\mathcal E}\). We then develop a Morse inequality for minimal surfaces of annulus type in a Riemannian manifold. Furthermore, we also construct body handle theories for minimal surfaces of annulus type as well as of disc type. Here we give a setting where the functional \({\mathcal E}\) is non-degenerated.     

Morse theory for indefinite nonlinear elliptic problems

Using the heat flow as a deformation, a Morse theory for the solutions of the nonlinear elliptic equation:

−Δu−λu=a₊(x)|u|^q−1u−a₋(x)|u|^p−1u+h(x,u)$-\mathrm{\Delta }u-\lambda u=a_{+}(x){|u|}^{q-1}u-a_{-}(x){|u|}^{p-1}u+h(x,u)$

Morse theory for analytic functions on surfaces

In this paper we deal with analytic functions defined on a compact two dimensional Riemannian surface S whose critical points are semi degenerated (critical points having a non identically vanishing Hessian). To any element p of the set of semi degenerated critical points Q we assign an unique index which can take the values −1, 0 or 1, and prove that Q is made up of finitely many (critical) points with non zero index and embedded circles. Further, we generalize the famous Morse result by showing that the sum of the indexes of the critical points of f equals χ (S), the Euler characteristic of S. As an intermediate result we locally describe the level set of f near a point p ∈Q. We show that the level set f ⁻¹(f (p)) is either a) the set {p}, or b) the graph of a smooth curve passing through p, or c) the graphs of two smooth curves tangent at p or d) the graphs of two smooth curves building at p a cusp shape.     

On uncomplemented subspaces ofL p , 1 <p <2

It is shown tat, for 1 <p < 2, there is an uncomplemented subspace ofL _p [0,1] that is isomorphic to Hilbert space.     

Some results for impulsive problems via Morse theory

We use Morse theory to study impulsive problems. First we consider asymptotically piecewise linear problems with superlinear impulses, and prove a new existence result for this class of problems using the saddle point theorem. Next we compute the critical groups at zero when the impulses are asymptotically linear near zero, in particular, we identify an important resonance set for this problem. As an application, we finally obtain a nontrivial solution for asymptotically piecewise linear problems with impulses that are asymptotically linear at zero and superlinear at infinity. Our results here are based on the simple observation that the underlying Sobolev space naturally splits into a certain finite dimensional subspace where all the impulses take place and its orthogonal complement that is free of impulsive effects.     

Discrete Morse theory for totally non-negative flag varieties

In a seminal 1994 paper Lusztig (1994) [26], Lusztig extended the theory of total positivity by introducing the totally non-negative part (G/P)_?0 of an arbitrary (generalized, partial) flag variety G/P. He referred to this space as a “remarkable polyhedral subspace”, and conjectured a decomposition into cells, which was subsequently proven by the first author Rietsch (1998) [33]. In Williams (2007) [40] the second author made the concrete conjecture that this cell decomposed space is the next best thing to a polyhedron, by conjecturing it to be a regular CW complex that is homeomorphic to a closed ball. In this article we use discrete Morse theory to prove this conjecture up to homotopy-equivalence. Explicitly, we prove that the boundaries of the cells are homotopic to spheres, and the closures of cells are contractible. The latter part generalizes a result of Lusztig's (1998) [28], that (G/P)_?0 - the closure of the top-dimensional cell - is contractible. Concerning our result on the boundaries of cells, even the special case that the boundary of the top-dimensional cell (G/P)_>0 is homotopic to a sphere, is new for all G/P other than projective space.     

Asymptotic distribution theory for general statistical functionals

Summary In this paper an asymptotic theory is developed for a general statistical functional which includesL-estimators,M-estimators, andR-estimators as special cases. It is shown that a proof of the asymptotic normality ofL-estimators,M-estimators, andR-estimators can be based on one important fact, namely, that the inverse c.d.f. is compactly differentiable.     

Morse 2-jet space and <Emphasis Type="Italic">h</Emphasis>-principle

A section in the 2-jet space of Morse functions is not always homotopic to a holonomic section. We give a necessary condition for being the case and we discuss the sufficiency.