# MATPUR

## MAPTUR. PURE INTERTHEMATIC MATHEMATICS.

The aim of this programm is to promote the study and research of important problems in Pure Mathematics. The adjective interthematic is therefore superfluous, addressing a characteristic of all significant Mathematics: it touches on differents topics. Indeed the directors of this programm come from three different areas: Algebra, Analysis and Geometry & Topology. This has a central role in chosing specific problems to work on. Anyway all selected problems should have a big variety of objectives and involve researchers coming from a wide spectrum of subjects.

In the Mathematical Department of the UCM work many different mathematicians who have reached a considerable success in their respective fields, as any quality index shows. This is a good starting point. It seems the moment has come to take adventage of the critical mass of researchers and to improve the communication between them; an aspect sometimes forgotten in the past. The research background and experience of all these scientists make us confident about the possibilities of progessing a step forward in the way research has been carried out for many years. The philosophy of our programm may be explained by saying that we will focus efforts on teaching and learning, in one single word, on studying. A sure road to sucessful discoveries.

Up to now there are three research directions:

1. The study of the Topology of Complex Algebraic Varieties has not only reached deep results but also posed some fundamental questions. The most relevant case is the Barth-Larsen Theorem from the 70s: The lower codimension a projective variety has, the more topological propierties it shares with the projective space. More or less at that time, Hartshorne stated the most important open problem in the subject: Are all projective varieties of codimension equal to or lower than the dimension of the variety minus 2 complete intersections? From the Barth-Larsen theorem it follows that a variety with codimension equal to or lower than its dimension minus 2 shares the Picard group with the projective space. And therefore a hypersuface of the variety is a complete intersection in the variety. Recent results due to the UCM Research group Geometry of Projective Algebraic Varieties show that the same holds for other ambient spaces and suggest a more general Barth-Larsen Theorem.

The first aim is to dealt with other ambient spaces like Grassmannians and products of projective spaces. This requires to manage the many different techniques used in Larsen"s proof: homotopy and homology groups, Morse theory, geodesics, Levi forms, filtrations and spectral sequences, sheaves cohomology... This will bring together several specialists from Algebra, Analysis, Geometry and Topology.

Key words: Barth-Larsen Theorem, Morse Theory, Picard group, Homotopy group, Spectral sequence, small codimension.

2. The second research direction is Subdifferential Calculus and Hamilton-Jacobi equations on Riemannian manifolds. The subdifferential of a convex function is a classical tool in Convex Analysis, Optimization and Control Theory. The subdifferential of a non necessarily convex function on a Banach space was introduced by Crandall and Lions in order to study Hamilton-Jacobi equations. This notion allowed them to define the concept of viscosity solution and to prove its existance and uniqueness, particularly in many cases where the classical solution does not exist. More recently, the basics of first-order subdifferential calculus in Riemannian manifolds has been settled and applied to Hamilton-Jacobi equations for uniformly continuous Hamiltonians. In the same vein, it is interesting to develop a second-order subdifferential calculus on Riemannian manifolds and apply it to second-order Hamilton-Jacobi equations. In this context it is also useful to study the regularization of Lipschitz functions on the manifold. A key question related to the topic is the classical Myers-Nakai theorem. This result states that the Riemannian structure of a finite-dimensional manifold is determined by the structure of the Banach algebra associated to the space of bounded C^1-functions with bounded differential on the manifold. The development of the above techniques will allow the study of the validity of a Myers-Nakai theorem for infinite dimensions.

Key words: Hamilton-Jacobi equations, Subdifferential calculus, infinite-dimensional Riemannian structures, Algebras of differential functions on manifolds.

3. In order to present the research line Geometry of Large Scale Metric Spaces, K-theory of C*-algebras and Complements of Z-setslet us consider the shape of the earth. Nowadays it is easy to claim it is round because we were able to travel to outer space and to look at it from there. The key question is: Had we been able to get out of the earth"s surface without knowing it is round? We propose to study from outside the compact metric spaces embedded in certain special way (called Z-embeddings) in the universe (Hilbert cube). To compare different structures we study functors between categories. Mathematics is full of such beautiful examples. Algebraic topology relates topological spaces and groups, rings and modules... A special situation are total equivalences between categories, like in the case of Topological Algebra. Gelfand and Naimark showed that the study of locally compact Haussdorf spaces is equivalent to the study of commutative C*-algebras. In the same direction, it was T.C. Chapman in 1972 who built an equivalence between the Shape Theory of Z-subsets of the Hilbert cube and the Proper and Weak Homotopy Theory of their complements and, in some cases, their topological types. Later M. Gromov introduced the Coarse Geometries developed by himself and many others like N. Higson, J. Roe, Guoliang Yu, S. Ferry, S. Weinberger, N. Wright, A. Dranishnikov... Coarse Geometries have been mainly used in Geometric Theory of Groups to give partial answers to the conjecture of Novikov, to assign Atiyah-Singer indexes to operators in non compact Riemannian manifolds and to the study of the non-commutative geometry of A. Connes. It is also very important the study of the so called C_0 coarse geometry of metric spaces, which was introduced by N. Wright. It is clear the wide spectrum of methods and problems involved in this line and its broad interest for mathematicians from very differents areas.

Keywords: Hilbert cube, Z-embeddings, Coarse geometries, Functions vanishing at infinity, C*-algebras, K-theory, Homotopy, Shape.