- Programmes

**Brief description:**

This program is in the interaction between analysis and geometry, focusing on certain aspects and geometric nonlinear functional analysis and classical analysis. The program is structured around the following three lines: 1) Global analysis and geometric analysis; 2) Differentiability, convexity and geometry of Banach spaces; 3) Multilinear Analysis, hiperciclicity and linearizability.

The overall objective of the program is a series of activities to promote and develop research in the lines mentioned.

**Aims of the program:**

- Fine approximation and smooth extension of convex functions in Euclidean space and on Riemannian manifolds. Smooth surgery of convex bodies in Euclidean space. Development of different tools of non-regular analysis, such as the Clarke generalized jacobian or the second order subdifferential in Riemannian and finslerians manifolds, with applications to parcial differential equations of second order. Global inversion theorems in Riemannian and finslerians manifolds and their relationship with non-regular analysis. Differentiability and Poincaré inequalities in metric measure spaces. Bornologies and real compactifications in metric spaces.
- Differentiable surjections between Banach spaces. Approximation of continuous functions on Banach spaces by real-analytic functions without critical points. Approximation of differentiable functions in Banach spaces, for the fine topology. Extension of differentiable functions between Banach spaces. Fine and uniform approximation of convex functions and smooth surgery of convex bodies in Banach spaces. Convex geometry in Banach spaces and study of diametrically maximal sets. Factorization of holomorphic or differentiable functions between Banach spaces. Reproductive kernels of Banach spaces and learning machines.
- Polynomial inequalities, estimation of polarization linear constants in Banach spaces, and constants of type Bohnenblust-Hille. Linearity, spaciability and hiperciclicity. Control of quantum states and Wigner transform.

**Website: **http://www.ams.org/mathscinet/search/publications.html?pg1=IID&s1=93775

http://www.ams.org/mathscinet/search/publications.html?pg1=IID&s1=621076

**Researchers:** **Jesús Á. Jaramillo Aguado (Full Professor, Fac. Math. UCM)**

Daniel Azagra (Associate Professor, Fac. Math. UCM)

José Luis Gámez Merino (Associate Professor, Fac. Math. UCM)

Mª Isabel Garrido (Associate Professor, Fac. Math. UCM)

Javier Gómez Gil (Associate Professor, Fac. Math. UCM)

José Luis González Llavona (Full Professor, Fac. Math. UCM)

Mar Jiménez Sevilla (Associate Professor, Fac. Math. UCM)

Ángeles Prieto (Associate Professor, Fac. Math. UCM)

Gustavo A. Muñoz (Associate Professor, Fac. Math. UCM)

Juan B. Seoane (Associate Professor, Fac. Math. UCM)

Óscar Madiedo (Ph. D. Student, Fac. Math. UCM)

**External collaborators:** L. Ambrosio, Scuola Normale Superiore di Pisa, Italia.

R. Aron, Kent State University, USA.

Z. Balogh, Universität Bern, Switzerland.

G. Beer, California State University, USA

J. Björn, Linköping University, Sweden.

D. Carando y S. Lassalle, Universidad de Buenos Aires.

F. Clarke, Université de Lyon.

R. Deville, Université de Bordeaux I, France.

V. Dimant, Universidad de San Andrés, Buenos Aires.

G. Godefroy, Université Paris 6.

M. Fernández, CIMAT, Guanajuato, México.

P. Hájek, Academia de Ciencias, Praga, R. Checa.

J. Kinunen, Aalto University, Finland.

S. Levi, Università degli studi di Milano Bicocca.

D. Pellegrino, Universidade de Paraiba, Brasil.

Y. Sarantopoulos, National Technical University, Atenas, Greece.

A. Seeger, Université d’Avignon.

N. Shanmugalingam, University of Cincinnatti, USA.

J. Tyson, University of Illinois, USA.

- RedIUMIMI is a member of RedIUM, Red de Institutos Universitarios de Matemáticas. See List of members. See official website.
- Ph.D. programsIMI researchers are currently involved in the following Ph.D. programs in Mathematics: