Departamento de Matemáticas UAM

About a hundred years ago, an interpolation theorem was studied by Nevanlinna (in 1919) and Pick (in 1918). It asks, given points in the open unit disc and points in for some , when is there a holomorphic function of sup norm no more than on the unit disc, mapping the points to ? It was completely solved at that time by them, independently, using complex analysis. A landmark paper by Sarason in 1966, related this interpolation problem with functional analysis. Thus was born the celebrated Commutant Lifting Theorem. Since then, Hilbert space operator theorists have been greatly intrigued. The Nevanlinna-Pick interpolation problem has been discussed in relation to reproducing kernel Hilbert spaces in different domains. One of the ways in which it is now proved is via the so-called Realization Formula for a function in with sup norm no more than . We shall see how this was ingeniously generalized to the unit bidisc by Agler and then to a very general setting by Dritschel and McCullough. The interpolation problem then takes an altogether new shape. This talk will outline this journey from the time of Navanlinna and Pick to present day state of research.

El profesor Tirthankar Bhattacharyya es un catedrático en el Instituto Indio de Ciencias (IISc), que es una de instituciones científicas de India de primer nivel. Ha hecho su tesis en Análisis Funcional en Instituto Estadístico de India y después de hacer postdoc en Canadá, en 2000 ingresó en IISc. Trabaja en problemas contiguos de la Teoría de Operadores y Análisis Complejo. Tiene colaboradores en muchas partes del mundo. Muchos de sus alumnos de doctorado se han convertido en investigadores de primer nivel. Tiene premios por su labor docente. Como reconocimiento de su trayectoria investigadora, fue elegido miembro de La Academia de Ciencias de India.

After a short introduction to some of the basic puzzles surrounding Quantum Mechanics I will study the effective quantum dynamics of systems interacting with a long chain of independent probes, which, afterwards, are subject to a projective measurement and are then lost. This leads us to develop a theory of indirect measurements of time-independent quantities (non-demolition measurements). Subsequently, the theory of indirect weak measurements of time-dependent quantities is outlined, and a new family of diffusion processes, quantum jump processes, is introduced. To conclude, some open problems are described.

 Many physical phenomema lead to tracking moving fronts whose speed depends on the curvature. The "level set method" has been tremendously succesful for this, but the solutions are typically only continuous. We will discuss results that show that the level set flow has twice differentiable solutions. This is optimal.

These analytical questions crucially rely on understanding the underlying geometry. The proofs draws inspiration from real algebraic geometry and the theory of analytical functions.

The talk will be accesible to a general audience.

Tobias Colding's research studies problems in differential geometry, geometric analysis, PDEs and low-dimensional topology. His works covers several different fields, with strong implications on all of them. Early in his career he introduced new ideas coming from analysis to study the geometry of manifolds with bounds on Ricci curvature. This started a collaboration with Jeff Cheeger, that gave a new perspective on the study of the Gromov-Hausdorff closure of such manifolds. Soon after that, he worked with William Minicozzi, first on harmonic functions of polynomial growth over manifolds (solving a conjecture of S.T.Yau on the dimension of the space of such functions), and later in groundbreaking work on the structure of minimal surfaces. As a result of this work, they were awarded the 2010 AMS Oswald Veblen prize in Geometry; the prize statement for that year states:

"The 2010 Veblen Prize in Geometry is awarded to Tobias H. Colding and William P. Minicozzi II for their profound work on minimal surfaces. In a series of papers they have developed a structure theory for minimal surfaces with bounded genus in 3-manifolds, which yields a remarkable global picture for an arbitrary minimal surface of bounded genus. This contribution led to the resolution of long-standing conjectures and initiated a wave of new results.

Nowadays he and Minicozzi continue with their work on mean curvature flow of hypersurfaces in Euclidean space.

He received his Ph.D. in mathematics in 1992 at the University of Pennsylvania under the direction of Chris Croke. He was on the faculty at the Courant Institute of New York University in various positions from 1992 to 2008. He has also been a visiting professor at MIT (2000–01) and at Princeton University (2001–02) and a postdoctoral fellow at MSRI (1993–94).

He is a foreign member of the Royal Danish Academy of Science and Letters, and received an honorary professorship at the University of Copenhagen in 2006. He was elected Fellow of the American Academy of Arts & Sciences in 2008. He was selected by the MIT Mathematics Department as the holder of the Norman Levinson Professorship, 2009-2014. He was appointed Senior Scholar of the Clay Mathematics Institute, 2011-2012. In 2013, he was recognized for his commitment, service and scholarship when appointed the Cecil and Ida B. Green Distinguished Professorship of Mathematics. In 2016, Colding received the Carlsberg Foundation Research Prize for ground-breaking research in differential geometry and geometric analysis. For 2015-16, he received a second Senior Scholar appointment by the Clay Mathematics Institute. In 2017, he received the Simons Fellowship in Mathematics.

 We present an introduction to L²-Betti numbers including their basic properties, main immediate applications and a status report. We discuss open prominent problems. If time allows, we will also treat L²-torsion, a much more sophisticated invariant. The talk is for non-experts and shall shed light on the forthcoming thematic program about L²-invariants.

Wolfgang Lück es un experto internacional en topología algebraica. Destacan sus trabajos en la algebraización de los invariantes L^2 (como los números L^2 de Betti o la cohomología L^2), que fueron definidos originalmente por Atiyah en términos de álgebras de operadores, y sus aplicaciones a la geometría y a la teoría K. Wofgang Lück es un experto en conjeturas de isomorfismo (Baum-Connes, Farrell-Jones, Borel). Ha esclarecido la conexión entre distintas generalizaciones de estas conjeturas, y ha probado varias de ellas para importantes familias de grupos. Wolfgang Lück ha sido conferenciante invitado en el 5 congreso de la EMS y en el ICM 2010. Entre otros honores, ha sido director del HIM (Hausdorff Research Institute for Mathematics), presidente de la sociedad matemática alemana, es fellow de la American Mathematical Society, y ganador de los premios Max-Planck-Forschungspreis y Gottfried Wilhelm Leibniz Prize. Actualmente es el investigador principal de una ERC Advanced grant.

To any knot in R^3 one can associate a 2-variable polynomial, related to representations of the fundamental group of the complement into SL_2(C). I will motivate this construction and consider the Mahler measure of this polynomial (the logarithmic average on the torus) which happens to be computable only in this case, as discovered by number theorists.

One of the fundamental discoveries in higher order Fourier analysis is that functions on large finite abelian groups can be regarded as approximations of functions on inherently non-commutative objects such as nilmanifolds. This discovery led to the development of the theory of certain fascinating and exotic objects called nilspaces. Nilspaces are common generalisations of abelian groups and nilmanifolds. In this talk we give an overview on the subject.

El profesor Balázs Szegedy ha sido invitado por Pablo Candela, profesor de nuestro departamento, a él le debemos las siguientes lineas sobre el próximo coloquio:
Balázs Szegedy is a researcher at the Alfréd Rényi Institute of Mathematics in Budapest, where he leads several research projects supported by an ERC consolidator grant and by the Hungarian Academy of Sciences. After obtaining his PhD from Eötvös Loránd University in Budapest, he held positions at Microsoft Research, the Institute for Advanced Study, and the University of Toronto Scarborough, before taking his present position at the Rényi Institute in 2013.
Szegedy's main research areas are combinatorics and group theory. In recent years he has been working on various topics related to the general field of limits of discrete structures. This field is connected to combinatorics, ergodic theory and probability theory, and builds on decades of deep research in these areas, surrounding in particular Szemeredi's regularity theory, which gives very general and powerful descriptions of large networks (graphs and hypergraphs), and also the ergodic theoretic approach to Szemerédi's theorem pioneered by Furstenberg, which provides general descriptions of measure-preserving systems. The main idea of this new field of limit theories is to regard very large structures in combinatorics and algebra as approximations of infinite analytic objects. This viewpoint brings new tools from analysis and topology into these subjects. The success of this branch of mathematics has already been demonstrated through numerous applications in computer science, extremal combinatorics, probability theory and group theory.
Related to the broad subject of limit theories is a vibrant new area known as higher order Fourier analysis, an area in which Szegedy is one of the world leading experts. This area, which will be the main object of his colloquium, originates in a seminal paper by Gowers which gave a new and quantitatively effective proof of Szemerédi’s theorem. This theorem states that for any fixed positive integer k and positive constant c, if A is a set of density c in a sufficiently long interval of integers, then A contains an arithmetic progression of length k. Gowers norms were also used in the celebrated result by Green and Tao proving the existence of arithmetic progressions of arbitrary finite length in the prime numbers. Within the framework of limit theories, one arrives at the subject of higher order Fourier analysis by studying limits (or ultralimits) for sequences of functions on increasingly larger abelian groups. Szegedy's work in this direction has led in particular to important conceptual progress concerning the relation between Gowers norms and nilmanifolds.
Szegedy has been awarded several honours for his research. In particular he was one of two winners of the 2009 European Prize in Combinatorics, he was awarded the 2012 Fulkerson Prize jointly with László Lovász for their work on graph limits, and he was the 2013 winner of the Coxeter-James Prize of the Canadian Mathematical Society. He will also be an invited speaker at the ICM in Rio de Janeiro in 2018.

The motivation of the problem comes from a problem of visual perception. Indeed the visual cortex has been modelled as a Lie group, with a totally degenerate metric, also called subriemannian. The associated PDE can be expressed as a sum of squares of vector fields, and the study of these type of equations under suitable geometric condition, has been started by Hormander and Rothschild – Stein. However regularity at the boundary, well known in the Euclidean setting, was still open even for the subriemannian Laplacian. We will face the problem with a suitable extension of a result of Caffarelli and Silvestre to this degenerate setting.

La profesora Giovanna Citti ha sido invitada por Davide Barbieri, profesor de nuestro departamento, a él le debemos las siguientes lineas sobre el próximo coloquio:
Giovanna Citti is full professor of mathematical analysis at the University of Bologna, where she coordinates the PhD program in mathematics. She has given important contributions to the study of linear and nonlinear PDE in different settings, from complex analysis, with crucial results on the Levi equation, to Hormander-type PDE. A PDE on a manifold that is defined by the sum of squares of vector fields, or even in more general forms, can have many of the properties of an elliptic equation (or of a parabolic one, if the associated diffusion is considered) if the involved vector fields together with their commutators span the entire tangent space at any point of the manifold. In this case it is called a Hormander-type PDE. Such vector fields can be recast, up to an approximating limiting procedure, into elements of a nilpotent Lie algebra with a notion of dilations, whose corresponding Lie groups are called Carnot groups, and their left invariant metrics are called subriemannian. Giovanna is one of the main world expert in the analysis of linear and nonlinear PDE on such structures. Her results have appeared in most of the more relevant journals of the area, such as Acta Math., J. Math. Pures Appl., Adv. Math., Calc. Var. Partial Differential Equations, J. Funct. Anal., Crelle's J., Trans. Amer. Math. Soc.
She is also, thanks to her long term collaboration with A. Sarti (CNRS, Paris), one of the founders and leading exponents of the so-called neurogeometry. Neurogeometry is a mathematical modelling approach to visual perception which relies on both the phenomenology of vision as a cognitive experience and the quantitative analysis of the physiological properties of families of cells in the brain's cortical area dedicated to the processing of visual stimuli. One of their most important works consists of a neural model for the generation of visual illusions showing the appearance of illusory contours by means of an algorithm of Hormander-type diffusion and concentration to a regular surface. More recently, they have introduced neural models based on spectral analysis and gauge theory in subriemannian structures that reproduce perceptual image segmentations.
Giovanna is also very active in the organization of initiatives for the stimulus of mathematical research. In the recent years she has coordinated an FP6 Adventure STREP project, an FP7 ITN project, a EU-US Atlantis project and recently a H2020 RISE project, the last two with the participation of the department of mathematics at UAM.
In this colloquium she will address the solution of one of the oldest open problems concerning PDE in Carnot groups. The problem of Hoelder regularity of solutions at the boundary of smooth domains is indeed a central issue in the theory of classical solutions to elliptic PDE. However, it was still open for Carnot groups. Only when the vector fields are the generators of the Lie algebra of the Heisenberg group this could be solved, more than 35 years ago, by D. Jerison, but with techniques that can not be extended to other settings because they rely on the special relationship between the Heisenberg group and the Fourier transform. G. Citti, together with A. Baldi and G. Cupini at the University of Bologna, have recently developed a new set of techniques that allow to control the restrictions of heat kernels to smooth manifolds for all Carnot groups. This turned out to be a powerful instrument to attack several problems, and in particular to obtain boundary regularity.

Einstein's general theory of relativity provides a geometrical description of gravity in terms of space-time curvature. The Einstein field equations pose fascinating challenges that have stimulated a great deal of research in geometry and partial differential equations. Important questions include the well-posedness of the initial value problem, the linear and non-linear stability of space-times, the formation of black holes, and the boundary value problems arising from the classical aspects of the AdS/CFT correspondence. I will give a survey of some significant advances and open problems pertaining to these questions. No background knowledge of General Relativity will be assumed.

El profesor Niky Kamran ha sido invitado por Alberto Enciso, investigador del ICMAT, a él le debemos las siguientes lineas sobre el próximo coloquio:
En su charla, el profesor Kamran tiene previsto presentar una visión general de problemas actuales sobre ecuaciones de ondas en variedades lorentzianas. Entre otros aspectos, se revisarán sus espectaculares resultados sobre la interacción entre ondas y geometría en los agujeros negros de Kerr. Estos resultados fueron obtenidos en colaboración con F. Finster, J. Smoller y el medallista Fields S.T. Yau en una serie de siete artículos que abarca de 2001 a 2009 y que se ve resumido en el artículo “Linear waves in the Kerr geometry: a mathematical voyage to black hole physics” (Bull. Amer. Math. Soc., 2009). El estudio de ecuaciones hiperbólicas en variedades lorentzianas es un área de importancia creciente en Análisis Geométrico, y que en la que se están realizando avances notables en los últimos años merced a los trabajos de los autores anteriormente citados y de otros matemáticos de primer nivel como Christodoulou, Dafermos, Klainerman, Rodnianski y Tataru.
Los resultados del profesor Kamran demuestran, en particular, el fenómenos de la superradiancia, que es una propiedad de las ondas en presencia de un agujero negro análoga a la extracción de energía mediante partículas que constituye el célebre proceso de Penrose. Esta propiedad había sido conjeturada hace décadas pero su demostración había eludido todos los intentos anteriores. En lo referente al impacto de estos trabajos, cabe destacar que estos resultados han abierto el estudio de agujeros negros más allá de Schwarzschild (en particular, el problema de estabilidad para agujeros de Kerr, que se considera central en Relatividad General matemática). También considerará la existencia de métricas de Einstein asintóticamente anti-de Sitter con geometría conforme prescrita.
El profesor Kamran ostenta una cátedra James McGill en la universidad de McGill (Montreal) y es miembro de Centre de Recherches Mathématiques de la misma ciudad. Se trata de un investigador de gran prestigio en los campos de geometría diferencial y física matemática. Es miembro de la Academia de Ciencias de la Royal Society of Canada desde 2002 y Fellow de la AMS. Ha recibido el premio André Aisensadt (1992, primer galardonado), el CRM-Fields-PIMS prize (2014) y una Killam Fellowship (2006). Ha sido conferenciante en más de una veintena de Coloquios en diversas, incluyendo MIT, Minnesota, Toronto y el Fields Institute.