2011

24-01-2011, 16:00
Sala Naranja ICMAT 		Kurusch Ebrahimi-Fard (ICMAT)
Exponential renormalization
In this talk we report on recent joint work with F. Patras (CNRS, Univ. of Nice) on a new perturbative renormalization method, developed in the context of Connes-Kreimer's Hopf algebra of Feynman graphs. Due to its form we dubbed it "exponential" method. Using Dyson's identity for Green's functions as well as the link between the Faa di Bruno Hopf algebra and Hopf algebras of Feynman graphs, its relation to the composition of formal power series is analyzed. This leads to the introduction of the notion of counterfactors and order n bare coupling constants. Eventually we analyze the role of the Rota-Baxter property for renormalization scheme maps. (KEF, F. Patras, "Exponential renormalization", Ann. Henri Poincaré 11 (2010), 943-971)
31-01-2011, 16:00
Sala Naranja ICMAT 		Francisco Presas (CSIC Madrid)
Geometric Quantization: cohomological approach
We explain the usual Geometric Quantization scheme for a Hamiltonian System. Once discussed we introduce an approach based in computations of the cohomology of certain sheaves and we discuss the main limitations of it as the possible solutions to these problems.
7-02-2011, 16:00
Sala Naranja ICMAT 		Dishant Pancholi (ICTP Trieste)
On Lutz twists in dimension bigger than three
Lutz twists play an important role in answering various questions about contact structures on a three manifold. After reviewing Lutz twists in dimension three we would give a possible generalization in dimension bigger than three.
21-02-2011, 16:00
Sala Naranja ICMAT 		Andres Pedroza (Univ. Colima, Mexico)
On the Bounded Isometry Conjecture
F. Lalonde and L. Polterovich study the isometries of the group of Hamiltonian diffeomorphisms with respect to the Hofer metric. They defined a symplectic diffeomorphism ? to be bounded, if the Hofer norm of [?, h] remains bounded as h varies on Ham(M, ?). The set of bounded symplectic diffeomorphisms, BI0 (M), of (M, ?) is a group that contains all Hamiltonian diffeomorphisms. They conjectured that these two groups are equal, Ham(M, ?) = BI0 (M, ?) for every closed symplectic manifold. They prove this conjecture in the case when the symplectic manifold is a product of closed surfaces of positive genus. In this talk we give an outline of a new class of manifolds for which bounded isometry conjecture holds.
7-03-2011, 16:00
Sala Naranja ICMAT 		Marco Zambon (UAM-ICMAT)
How graded manifolds turn useful in classical geometry.
Several classical geometric structures (Lie algebras, Lie algebroids, Courant algebroids...) can be conveniently described in terms of graded manifolds. The latter are a refined version of super-manifolds (spaces where some of the coordinates anti-commute). I will present some examples and discuss instances in which the graded point of view tells us something new about classical geometric structures.
14-03-2011, 16:00
Sala Naranja ICMAT 		Luis Álvarez-Cónsul (ICMAT)
Coupled equations for Kähler metrics and Yang-Mills connections
We study equations on a principal bundle over a compact complex manifold coupling a connection on the bundle with a Kähler structure on the base. These equations generalize the conditions of constant scalar curvature for a Kähler metric and Hermite-Yang-Mills for a connection. We provide a moment map interpretation of the equations and study obstructions for the existence of solutions, generalizing the Futaki invariant, the Mabuchi K-energy and geodesic stability. We finish by giving some examples of solutions. This is joint work with Mario García Fernández and Oscar García Prada (arXiv:1102.0991 [math.DG]).
21-03-2011, 16:00
Sala Naranja ICMAT 		Alberto Enciso (ICMAT)
Generic spectral properties of the Laplacian on forms on 3-manifolds
From a qualitative point of view, one the most attractive results in spectral geometry is K. Uhlenbeck's proof of the fact that, for a "generic" set of C^r metrics, the eigenvalues of the Laplacian on a closed manifold are simple and its eigenfunctions are Morse (1972). The extension of this result to the case of differential forms is problematic; indeed, topological obstructions to its validity were found shortly thereafter by Millman (1980). In this talk we will show how Uhlenbeck's theorem can be extended to the case of differential forms on 3-manifolds using an auxiliary Dirac-type operator. This talk will be based on joint work with Daniel Peralta-Salas.
8-04-2011, 15:00
Sala Naranja ICMAT 		Margherita Sandon (Universite de Nantes)
On existence of translated points for contactomorphisms
A point p in a contact manifold is called a translated point for a contactomorphism \phi with respect to a fixed contact form if p and \phi(p) belong to the same Reeb orbit and if the contact form is preserved at p. In my talk I will discuss the problem of existence of translated points, and its relation with the Arnold conjecture, the chord conjecture and the problem of leafwise coisotropic intersections. If I have the time I will also explain how to use generating functions techniques to study this problem for contactomorphisms of the euclidean space, the sphere and the projective space.
11-04-2011, 16:00
Sala Naranja ICMAT 		Daniel Peralta Salas (ICMAT)
Knots and links in fluid mechanics
In 1965 V.I. Arnold classified the steady solutions of the Euler equation, implying in particular that the types of knots and links that the stream (or vortex) lines of a fluid can exhibit are quite restricted except for the so called Beltrami fields. Arnold's work gave rise to the topological hydrodynamics conjecture that any knot and link can be realized as a stream (or vortex) line of a steady solution of the Euler equation (typically of Beltrami type). The importance of this conjecture is that it tests the topological complexity of fluid flows and hence it is directly related to phenomena like turbulence and hydrodynamics instability. The goal of this talk is to review the strategy which has recently led to the proof of this conjecture (with A. Enciso), as well as some interesting applications as the solution to the Etnyre-Ghrist problem: there exists a steady solution of the Euler equation containing all knot and link types.
9-05-2011, 16:00
Sala Naranja ICMAT 		Vicente Muñoz (UCM)
Hodge polynomials of SL(2,C)-character varieties
The character variety of a surface of genus g is the space parametrizing representations of the fundamental group of the surface into GL(r,C). Twisted character varieties appear when taking representations of the fundamental group of a once punctured surface and fixing the holonomy around the puncture. We shall study the character variety for the case of rank r=2, and small genus g=1,2,3. We propose a new technique to compute the Hodge-Deligne polynomials of these varieties. (joint work with M. Logares & P. Newstead)
23-05-2011 16:00
Sala Naranja ICMAT 		Jaime Arboleda (CSIC)
Aproximaciones no conmutativas de espacios topológicos
Toda variedad topológica puede ponerse como límite proyectivo de espacios topológicos finitos. Revisaremos esta construcción y enunciaremos algunos teoremas asociados (cálculo del grupo fundamental y los grupos de cohomología). A continuación, mostraremos que todo espacio finito es a su vez el espectro de una C*-álgebra no conmutativa, que se puede construir de manera natural. Esto permite dualizar la construcción primera, mediante la cual obtenemos un límite directo de C*-álgebras cuyo centro son las funciones continuas sobre la variedad original. Este trabajo es una explicación geométrica de varios artículos de Giovanni Landi, con idea a futuras aplicaciones al cálculo numérico.
5-09-2011, 12:00
Sala Naranja ICMAT 		Sergiu I. Vacaru (University Alexandru Ioan Cuza, Iasi, Romania)
Geometric Flows of Lagrange and Hamilton Geometries and Deformation Quantization
We study geometric evolution on nonholonomic manifolds modelling regular Lagrange systems. There are defined analogs of Perelman?s functionals, associated entropy and thermodynamical values and derived Hamilton type nonholonomic evolution equations. We provide a deformation quantization formalism for Ricci flow evolution in geometric mechanics using almost Kaehler-Lagrange variables. Further perspectives and developments to fractional/ noncommutative Lagrangians and Hamiltonians and Clifford-algebroid system are discussed.
8-09-2011, 16:00
Sala Naranja ICMAT 		Florent Schaffhauser (Universidad de Los Andes, Bogotá)
Topology of moduli spaces of vector bundles on a real algebraic curve
Moduli spaces of real and quaternionic vector bundles on a curve can be expressed as Lagrangian quotients and embedded into the symplectic quotient corresponding to the moduli variety of holomorphic vector bundles of fixed rank and degree on a smooth complex projective curve. From the algebraic point of view, these Lagrangian quotients are irreducible sets of real points inside a complex moduli variety endowed with an anti-holomorphic invo- lution. This presentation as a quotient enables us to generalise the equivariant methods of Atiyah and Bott to a setting with involutions, and compute the mod 2 Poincaré series of these real algebraic varieties. This is joint work with Chiu-Chu Melissa Liu from Columbia University.
12-09-2011, 11:00
Sala Naranja ICMAT 		Noah Kieserman (IMPA)
What is... an infinitesimally multiplicative structure?
Lie algebroids arise naturally across geometry and mechanics when describing infinitesimal structures. Often they come with extra data - for instance, the Lie algebroid associated to a Poisson manifold has skew-symmetric anchor, and its bracket preserves closed 1-forms. These extra data are explained by the discovery that several important infinitesimal structures (Poisson, Dirac, generalized complex, Lie bialgebroid) integrate to structures on Lie groupoids, distinguished by the fact that they are multiplicative. I will discuss various perspectives on the outcome of differentiating multiplicative structures.
16-09-2011, 12:45
Sala Naranja ICMAT 		José Ignacio Iglesias Curto (U Salamanca)
Códigos convolucionales algebro-geométricos
Los códigos correctores de errores permiten evitar la pérdida de información durante su transmisión o almacenamiento. Tras un breve repaso a los conceptos básicos y los problemas fundamentales que este área pretende resolver, nos detendremos en la construcción de códigos de bloques mediante objetos geométricos, y las ventajas que acarrea dicha construcción. Junto con los códigos de bloques, la otra gran familia de códigos son los códigos convolucionales. En la parte final se expondrá cómo los métodos algebro-geométricos son también aplicables a esta familia y las consecuencias del uso de dichos métodos en los aspectos que son propios de este tipo de códigos.
3-10-2011, 16:00
Sala Naranja ICMAT 		Javier Fernández de Bobadilla (ICMAT)
The Nash conjecture for surfaces
The space of arcs through the singular set of an algebraic variety has a infinite dimensional scheme structure. In the late sixties Nash proved that it has finitely many irreducible components. He defined a natural mapping from this set of irreducible components to the set of essential divisors of a resolution of singularities. Roughly speaking the set of essential divisors is the set of irreducible components of the exceptional divisor of a resolution whose birational transform is an irreducible component of the exceptional divisor of any other resolution. Nash proved that this mapping is injective and proposed to study its bijectivity. In 2003 S. Ishii and J. Kollar gave a counterexample to the surjectivity in dimension at least 4. Recently, in a joint work with M. Pe Pereira, the speaker has settled the bijectivity for surfaces. In this talk I will explain the proof.
17-10-2011, 16:00
Sala Naranja ICMAT 		James Lewis (U Alberta, Canada)
An Archimedean height pairing on the equivalence relation defining Bloch's higher Chow groups
The existence of a height pairing on the equivalence relation defining Bloch's higher Chow groups is a surprising consequence of some recent joint work by myself and Xi Chen on a nontrivial K_1-class on a self-product of a general K3 surface. I will explain how this pairing comes about.
24-10-2011, 16:00
Sala Naranja ICMAT 		Marco Zambon (UAM y ICMAT)
Singular foliations and holonomy groupoids
I will recall the notion of holonomy for a regular foliation on a smooth manifold, and the related notion of holonomy groupoid. I will then introduce singular foliations (the simplest example is given by a vector field, possibly with singularities) and sketch a construction of an associated holonomy groupoid by Skandalis-Androulidakis. The latter is in general not smooth. I will show that, however, under suitable conditions the restriction to leaves is smooth. The content of this talk is the first part of a project to find a normal form theorem for singular foliations, and is joint work with Androulidakis.
14-11-2011, 16:00
Sala Naranja ICMAT 		Domingo Toledo (University of Utah)
Energy on Teichmüller Space and Siu-Sampson Rigidity Theory
Let M be a closed surface of genus at least two, N a manifold of non-positive Hermitian curvature (the Siu-Sampson condition) and fix a homotopy class of maps from M to N. For each complex structure J on M there is a harmonic map f:M->N, and, if this map is unique, it depends smoothly on J and its energy E defines a smooth function on the Teichmüller space of M. We prove that this function is plurisubharmonic, study conditions when it is strictly plurisubharmonic, and give some results on the zeros of its complex Hessian.

This result was suggested by Gromov as an alternative way of developing and strengthening the Siu-Sampson rigidity theory. We will sketch one stregthening developed by Gromov using this result, namely new sufficient conditions for a harmonic map to be holomorphic.

5-12-2011, 16:00
Sala Naranja ICMAT 		Alfonso Zamora (ICMAT)
A GIT interpretation of the Harder-Narasimhan filtration
An unstable torsion free sheaf on a smooth projective variety gives a GIT-unstable point in certain Quot scheme. To a GIT-unstable point, Kempf associates a "maximally destabilizing" 1-parameter subgroup, and this induces a filtration of the torsion free sheaf. We show that this coincides with the Harder-Narasimhan filtration (joint work with T. Gomez and I. Sols).
Last updated 3 November 2011