Abstract Subdifferential Calculus and Semi-Convex Functions

∗ The work is partially supported by NSFR Grant No MM 409/94.

Conference on Geometry and Mathematical Physics, Bulgaria, Zlatograd, 28.08-01.09.2005

List of Participants

Highest Weight Modules of W1+∞, Darboux Transformations and the Bispectral Problem

This paper is a survey of our recent results on the bispectral problem. We describe a new method for constructing bispectral algebras of any rank and illustrate the method by a series of new examples as well as by all previously known ones. Next we exhibit a close connection of the bispectral problem to the representation theory of W1+∞–algerba. This connection allows us to explain and generalise to any rank the result of Magri and Zubelli on the symmetries of the manifold of the bispectral operators of rank and order two.

Toward Clemens' Conjecture in Degrees between 10 and 24

1 Supported in part by the Norwegian Research Council for Science and the Humanities. It is a pleasure for this author to thank the Department of Mathematics of the University of Sofia for organizing the remarkable conference in Zlatograd during the period August 28-September 2, 1995. It is also a pleasure to thank the M.I.T. Department of Mathematics for its hospitality from January 1 to July 31, 1993, when this work was started. 2Supported in part by NSF grant 9400918-DMS.

Existence and Uniqueness of Solutions for Partial Differential-Functional Equations of the First Order with Deviating Argument of the Derivative of Unknown Function

We consider the existence and uniqueness problem for partial differential-functional equations of the first order with the initial condition for which the right-hand side depends on the derivative of unknown function with deviating argument.

Involutivity and Symple Waves in R^2

A strictly hyperbolic quasi-linear 2×2 system in two independent variables with C2 coefficients is considered. The existence of a simple wave solution in the sense that the solution is a 2-dimensional vector-valued function of the so called Riemann invariant is discussed. It is shown, through a purely geometrical approach, that there always exists simple wave solution for the general system when the coefficients are arbitrary C^2 functions depending on both, dependent and independent variables.

Functionally Countable Spaces and Baire Functions

The concept of the distinguished sets is applied to the investigation of the functionally countable spaces. It is proved that every Baire function on a functionally countable space has a countable image. This is a positive answer to a question of R. Levy and W. D. Rice.

Commuting Nonselfadjoint Operators and their Characteristic Operator-Functions

* Partially supported by Grant MM-428/94 of MESC.

Decomposition of Banach Space into a Direct Sum of Separable and Reflexive Subspaces and Borel Maps

* This paper was supported in part by the Bulgarian Ministry of Education, Science and Technologies under contract MM-506/95.

Uniform Eberlein Compacta and Uniformly Gâteaux Smooth Norms

* Supported by grants: AV ĈR 101-95-02, GAĈR 201-94-0069 (Czech Republic) and NSERC 7926 (Canada).

Coincidence of Vietoris and Wijsman Topologies: A New Proof

Let (X, d) be a metric space and CL(X) the family of all nonempty closed subsets of X. We provide a new proof of the fact that the coincidence of the Vietoris and Wijsman topologies induced by the metric d forces X to be a compact space. In the literature only a more involved and indirect proof using the proximal topology is known. Here we do not need this intermediate step. Moreover we prove that (X, d) is boundedly compact if and only if the bounded Vietoris and Wijsman topologies on CL(X) coincide.

Representable Banach Spaces and Uniformly Gateaux-Smooth Norms

It is proved that a representable non-separable Banach space does not admit uniformly Gâteaux-smooth norms. This is true in particular for C(K) spaces where K is a separable non-metrizable Rosenthal compact space.

A Note on Coercivity of Lower Semicontinuous Functions and Nonsmooth Critical Point Theory

The first motivation for this note is to obtain a general version of the following result: let E be a Banach space and f : E → R be a differentiable function, bounded below and satisfying the Palais-Smale condition; then, f is coercive, i.e., f(x) goes to infinity as ||x|| goes to infinity. In recent years, many variants and extensions of this result appeared, see [3], [5], [6], [9], [14], [18], [19] and the references therein. A general result of this type was given in [3, Theorem 5.1] for a lower semicontinuous function defined on a Banach space, through an approach based on an abstract notion of subdifferential operator, and taking into account the “smoothness” of the Banach space. Here, we give (Theorem 1) an extension in a metric setting, based on the notion of slope from [11] and coercivity is considered in a generalized sense, inspired by [9]; our result allows to recover, for example, the coercivity result of [19], where a weakened version of the Palais-Smale condition is used. Our main tool (Proposition 1) is a consequence of Ekeland’s variational principle extending [12, Corollary 3.4], and deals with a function f which is, in some sense, the “uniform” Γ-limit of a sequence of functions.

Stability of Supporting and Exposing Elements of Convex Sets in Banach Spaces

* This work was supported by the CNR while the author was visiting the University of Milan.

Subdifferentials of Performance Functions and Calculus of Coderivatives of Set-Valued Mappings

The paper contains calculus rules for coderivatives of compositions, sums and intersections of set-valued mappings. The types of coderivatives considered correspond to Dini-Hadamard and limiting Dini-Hadamard subdifferentials in Gˆateaux differentiable spaces, Fréchet and limiting Fréchet subdifferentials in Asplund spaces and approximate subdifferentials in arbitrary Banach spaces. The key element of the unified approach to obtaining various calculus rules for various types of derivatives presented in the paper are simple formulas for subdifferentials of marginal, or performance functions.