PLENARY SESSIONS

José Luis Balcázar (Universitat Politècnica de Catalunya y Universidad de Cantabria)

Towards a Logic of Association Rules: Deduction, Optimum Axiomatizations, and Objective Novelty

An association rule is a form of partial implication between two terms (sets of propositional variables, understood conjunctively). In the case of standard implications, we are just back in Horn logic; but, in association rules, the notion of implication is redefined to allow exceptions or different populations. Association rules are among the most widely employed data analysismethods in the field of Data Mining.

Naive uses of association miners end up often providing far too large amounts of mined associations to result actually useful in practice. Many proposals exist for selecting appropriate association rules, trying to measure their interest in various ways; most of these approaches are statistical in nature, or share their main traits with statistical notions. In the most common approach, association rules are parameterized by a lower bound on their confidence, which is the empirical conditional probability of their consequent given the antecedent, and/or by some other parameter bounds such as ``support'' or deviation from independence.

Alternatively, some existing notions of redundancy among association rules allow for a logical-style characterization and lead to irredundant bases (axiomatizations) of absolutely minimum size. We will discuss notions of redundancy, that is, of logicalentailment, among association rules, and how to complement the association rule mining process by filtering also the obtained rules according to their novelty, measured in a relative way with respect to the confidences of related rules.

Recent papers describing these advances are available from the author's webpage. Additionally, we can actually offer a preliminary version of a rule-mining proof-of-concept system implementing our contributions.

Roberto Emparan (ICREA y Universitat de Barcelona) 

Black holes in higher dimensions

Over the last decade we have realized that black holes in more than four dimensions are much more complex objects than four-dimensional ones. Their horizons admit a much wider class of geometries and topologies, and much richer dynamics. In five dimensions we may be close to having a complete classification of them, but in dimensions six or higher their study requires the development of novel techniques. In this talk I will review the motivations to study these higher-dimensional black holes, and the recent progress in understanding them.

Joaquim Júdice (Universidade de Coimbra)

The Eingenvalue Complementarity Problem: Theory and Algorithms

Given a real matrix A and a real (symmetric or asymmetric) Positive Definite matrix B, the Eigenvalue Complementarity Problem (EiCP) is an extension of the well-known Generalized Eigenvalue Problem GEiP(A,B) where some of the variables of the problem are required to be nonnegative and to satisfy a complementarity constraint. This problem finds interesting applications in contact problems. A few nonlinear programming formulations are introduced for the symmetric EiCP, such that stationary points of the corresponding objective functions on appropriate convex sets lead to solutions of the problem. When at least one of the matrices A or B is asymmetric, the EiCP reduces to a Finite-Dimensional Variational Inequality and to a Global Optimization Problem. Projected gradient methods and an enumerative algorithm are introduced for finding a solution to the EiCP. The computation of several complementary eigenvalues and of the maximum and minimum of these eigenvalues is also discussed. Computational experience is reported to illustrate the efficiency of the algorithms to deal with the EiCP.

Juan Ignacio Montijano (Universidad de Zaragoza)

Some recent research on Runge-Kutta methods

The study of Runge-Kutta methods has been a very active field of research since the work of  C. Runge (1895), K. Heun (1900) and W. Kutta (1901). Nowadays, this class of methods has become a fundamental tool in the practical solution of differential equations and it is present in the most used packages of software. One can think that there is no much else to study in relation with RK formulas but however, the development of new hardware and software technologies that provide more powerful capabilities of computation, and the numerical simulation of every time more complex real life problems lead to new requirements to the numerical integrators that demand further research.  As an example we will consider some current problems, such as those arising in computational aeroacoustic, and we will see how the research on RK methods can contribute to improve their numerical simulation.

José Mourão (Instituto Superior Técnico)

Large complex structure limits in geometric quantization

One important issue in geometric quantization consists in studying the dependence of quantization on the choice of polarization, an additional technical ingredient of which the outcome of quantization should be independent.

We will review this general problem and will study it in some examples corresponding to the degeneration of Kaehler polarizations to real polarizations (large complex structure limits).

This is based on joint work with T.Baier, W.Kirwin and J.P.Nunes.

Amílcar Sernadas (IT y Instituto Superior Técnico)

The practical significance of the problem of combining logics is widely recognized, namely in knowledge representation (within artificial intelligence) and in formal specification and verification of algorithms and protocols (within software engineering and information security). In these fields, the need for working with several calculi at the same time is the rule rather than the exception. The topic is also of interest on purely theoretical grounds. For instance, one might be tempted to look at predicate temporal logic as resulting from the combination of first-order logic and propositional temporal logic. However, the approach will be significant only if general preservation results are available about the combination mechanism at hand, namely preservation of completeness. For these reasons, different forms of combining logics have been studied and several such transference results have been reported in the literature. To name just a few, fusion (of modal logics), temporalization and fibring are now well understood, although some interesting open problems remain, namely concerning transference results. Fibring [1] is the most general form of combination and its recent graphic-theoretic account makes it applicable to a wide class of logics, including substructural and non truth-functional logics. Capitalizing on these latest developments in the theory of fibring [2] and inspired by parallel composition of processes, a novel form of combination of logics, subsuming fibring as a special case, is proposed together with conservativeness results [3,4]. Special attention is given to the parallel composition of calculi via a generalization of the notion of 2-category.

[1] D. Gabbay. Fibred semantics and the weaving of logics: part 1. Journal of Symbolic Logic, 61(4):1057–1120, 1996.

[2] A. Sernadas, C. Sernadas, J. Rasga, and M. Coniglio. On graph-theoretic fibring of logics. Journal of Logic and Computation, 19:1321--1357, 2009.

[3] A. Sernadas, C. Sernadas and J. Rasga. Parallel composition of logics - semantics. Submitted for publication.

[4] A. Sernadas, C. Sernadas and J. Rasga. Parallel composition of logic calculi. Submitted for publication.

THEMATIC SESSIONS

Mathematical logic, foundations and computing

Carlos D´Andrea (Universitat de Barcelona)

Computing singularities of parametric plane curves

Given a birational parameterization of a curve C, a lot of information about its singularities can be extracted from simple matrices naturally arising from elimination theory. 

In this talk we will review different methods for computing the singularities of the curves, and focus on the particular case of plane curves for which the geometry is very rich.

Luís Antunes (Universidade do Porto)

Sophistication Revisited

Kolmogorov complexity measures the amount of information in a string as the size of the shortest program that computes the string. The Kolmogorov structure function divides the smallest program producing a string in two parts: the useful information present in the string, called sophistication if based on total functions, and the remaining accidental information. We formalize a connection between sophistication (due to Koppel) and a variation of computational depth (intuitively the useful or nonrandom information in a string), prove the existence of strings with maximum sophistication and show that they are the deepest of all strings.

Carlos Beltrán (Universidad de Cantabria)

Certified Numerical Solutions of Systems of Polynomial Equations

Homotopy or path-following methods are among the most popular numerical solvers for systems of polynomial equations. The general idea of these methods is as follows: for a given input system f, we generate another system g (usually with the same number of unknowns and degrees) which has a known solution. Then, we define some path (in the vector space of systems) whose extremes are g and f, and some path-following strategy is used to follow the "solution path", in such a way that the known zero of g is continued to approximate some zero of f. There has been much progress in the understanding of these methods during the last two decades. For example, we know how to describe this processin total detail, in such a way that, assuming exact computations, the method is certified (i.e. the output is an "approximate zero" of f), and the average complexity is polynomial in the number of monomials (dense encoding). I will summarize some of the last results in this framework. Many of these results are due to M.Shub & S. Smale, or joint work with L.M. Pardo, M. Shub, A. Leykin, J.P. Dedieu and G. Malajovich. I will also discuss the non-deterministic components of the present algorithm, and a recent advance by P. Burguisser and F. Cucker which shows that the algorithm can be made deterministic if quasi-polynomial running time is accepted.

Domingo Gómez (Universidad de Cantabria)

Dynamical Systems and Pseudorandom Number Generation

There are many problems that are known to be dificult or impossible to solve using deterministic algorithms. One example is the following: Given two computer programs, prove that they are equivalent, i.e for same inputs, the outputs are equals. The normal approach to solve this problems are the Monte Carlo methods. A Monte Carlo method, in simple and general terms, is any algorithm that uses random numbers. Random number generation is dificult by definition. Although there are ways to generate random numbers using a computer, it is ineficient by definition, a computer is a deterministic machine. A way around would be the following, to generate sequence of numbers that simulate "randomness". Any algorithm that uses pseudorandom numbers sequences is known as a Quasi-Monte Carlo method. The main purpose of this talk is to introduce a collection of good pseudorandom sequences generated by dynamical systems from a purely theoretical point of view as well as in view of diferent applications. We analyze the pseudorandom sequences using mainly number theoretic methods as for example exponential sum techniques.

Mário Jorge Edmundo (Universidade Aberta)

The universal covering map in o-minimal structures

We prove the existence of locally definable universal covering maps for (locally) definable manifolds in o-minimal expansions of groups as well as invariance results for o-minimal fundamental groups.

This is joint work with P. Eleftheriuo and L. Prelli

José Espírito Santo (Universidade do Minho)

Towards a canonical classical natural deduction system

This talk is about a new classical natural deduction system, presented as a typed lambda-calculus. It is designed to be isomorphic to Curien-Herbelin's calculus, both at the level of proofs and reduction, and the isomorphism is based on the correct correspondence between cut (resp. left-introduction) in sequent calculus, and substitution (resp. elimination) in natural deduction. It is a combination of Parigot's calculus with the idea of "coercion calculus" due to Cervesato-Pfenning, accommodating let-expressions in a surprising way: they expand Parigot's syntactic class of named terms.

This calculus aims to be the simultaneous answer to three problems. The first problem is the lack of a canonical natural deduction system for classical logic. The proposed calculus is not yet another classical calculus, but rather a canonical reflection in natural deduction of the impeccable treatment of classical logic by sequent calculus. The second problem is the lack of a formalization of the usual semantics of Curien-Herbelin's calculus, that explains co-terms and cuts as, respectively, contexts and hole-filling instructions. The mentioned isomorphism is the required formalization, based on the precise notions of context and hole-expression offered by the proposed calculus. The third problem is the lack of a robust process of  "read-back'' into natural deduction syntax of calculi in the sequent calculus format, that affects mainly the recent proof-theoretic efforts of derivation of  lambda-calculi for call-by-value. An isomorphic counterpart to the Q-subsystem of Curien-Herbelin's calculus is derived, obtaining a new lambda-calculus for call-by-value, combining control and let-expressions.

Ignacio García-Marco (Universidad de la Laguna)

An algorithm for checking whether a simplicial toric ideal is a complete intersection

Let k be an arbitrary field and k[x] = k[x_1, … ,x_n] and k[t] = k[t₁, …, t_m] two polynomial rings over k. Let A = {a1, …, a_n} be a set of nonzero vectors in N^m; each vector a_i = (a_i1, ... , a_im) corresponds to a monomial t^ai = t^ai1  …t^aim m in k[t]. The kernel of the homomorphism of k-algebras f: k [x] ®k[t]; x_i ® t^ai is called a toric ideal and will be denoted by IA. It is an A-homogeneous binomial ideal, i.e., if one sets the A-degree of a monomial  x^a Îk[x] as a₁a₁ + …+a_na_n ÎN^m, and says that a polynomial fÎk [x] is A-homogeneous if its monomials have the same A-degree, then I_A is generated by A-homogeneous binomials.

The ideal IA is a complete intersection if there exists a system of A-homogeneous binomials g_1,…, g_s such that I_A = (g₁,…, g_s) , where s = n – rk (ZA).

If n > m and A = {d₁e_1,…, d_me_m; a_m+_1,…, a_n}, where {e₁, …, e_m} is the canonical basis of Z^m, the toric ideal I_A is said to be a simplicial toric ideal.

The purpose of this work is to provide and implement an algorithm for determining whether a simplicial toric ideal I_A is a complete intersection without having to compute a minimal system of A-homogeneous generators of I_A. This is a joint work with Isabel Bermejo. 

Fernando Ferreira (Universidade de Lisboa)

Brevemente…

Reinhard Kahle (Universidade Nova de Lisboa, Centria e DM)

Applicative Theories and Computational Complexity

Abstract: We give a short survey on applicative theories and how they can be used to characterize classes of computational complexity.

The presentation includes joint work with Isabel Oitavem (CMAF and DM, FCT-UNL).

Manuel António Martins (Universidade de Aveiro)

Abstract Algebraic Logic Tools in Program Development

Hidden k-logics, as a natural generalization of k-deductive systems, were introduced by Don Pigozzi and Manuel Martins in 2003 to specify object oriented software systems. The use of hidden k-logics unifies the study of several kinds of logics and provides a bridge between abstract algebraic logic (AAL) and specification theory.

In this talk, we discuss a new application of AAL to program development. We present an alternative approach to refinement of algebraic specifications in which signature morphisms are replaced by logic interpretations. Intuitively,an interpretation is a logic translation which preserves meaning. Originally used as a tool for studying equivalent algebraic semantics, this notion has proved to be an effective tool to capture a number of transformations difficult to deal with in classical terms, such as data encapsulation and the decomposition of operations into atomic transactions.

Ana Romero (Universidad de la Rioja)

Effective homology of groups and discrete Morse theory

In the talk we will present several algorithms related with the computation of the homology of groups, by means of different techniques of Algebraic Topology. More concretely, we have developed some algorithms which, making use of the effective homology method, construct the homology groups of Eilenberg-MacLane spaces K(G,1) for different groups G, allowing one in particular to determine the homology groups of G. These results can be applied to the computation of homology groups of central extensions and 2-types. Moreover, our initial algorithms have been improved by using discrete Morse theory (concretely, by constructing discrete vector fields which describe the homology of finite cyclic groups).

Numerical analysis and optimization

Paulo Amorim (CMAF e Universidade de Lisboa)

Numerical schemes for short wave long wave interaction equations

We prove convergence of some numerical schemes for short wave long wave interaction equations. These are systems of nonlinear PDEs consisting of a Schroedinger equation for the short waves and either a KdV equation or a nonlinear conservation law modeling the long waves. We present some computations and a numerical study of some open problems. This is joint work with M. Figueira of CMAF-UL.

Sílvia Barbeiro (Universidade de Coimbra) 

Coupling methods for poroelasticity

Poroelasticity modeling attracts  researchers from different areas and is of  great importance not only in soil mechanics but also in several applications in civil,  petroleum and even biomedical engineering. In this talk we consider the numerical solution of a coupled fluid flow and geomechanics in Biot's consolidation model for poroelasticity. The method combines mixed finite elements for Darcy flow and Galerkin finite elements for elasticity. The numerical discretization of this model gives rise to algorithmic challenges. Our focus is to discuss the development of efficient and accurate numerical solutions, and give some insight into the theoretical basis of the underlying methods. The fully coupled approach solves flow and elasticity equations simultaneously. Alternatively, operator splitting techniques can be used for solving the discrete system. The choice of the coupling scheme affects the stability and accuracy of the numerical solutions as well as the computational efficiency. In this talk we discuss a priori convergence estimates for fully coupled schemes  and for iteratively coupled schemes. We perform numerical experiments for verifying our theory and to compare different coupling techniques in engineering applications.

Ana Luísa Custódio (Universidade Nova de Lisboa)

Direct-Search Methods for Single and Multiobjective Derivative-Free Optimization: solving difficult problems in an efficient way

Abstract: Noisy functions, conflictual objectives, expensive function evaluation are some of the difficulties that we face when working in real applications. Since the 90’s the mathematical optimization community as renewed its interest in these classes of problems, in particular because convergence could be established for some of the algorithms which were commonly used in engineering. In this talk, we will survey the class of directional Direct-Search Methods (DSM), both for single and multiobjective derivative-free optimization problems. These algorithms are relatively easy to implement, only requiring comparisons among objective function values, and being suited for the optimization of nonsmooth functions. The major drawback of its use is related to efficiency. We will also describe several techniques which can be used to improve the performance of these methods, from the use of interpolation models to the natural algorithmic strategies of parallelization.

Javier M. Moguerza (Universidad Rey Juan Carlos)

Building Improved Directions of Negative Curvature for Constrained Optimization

In this work, we provide an approach to the computation of improved directions of negative curvature for nonlinearly constrained optimization problems. In particular, we focus on the use of low cost procedures to improve directions of negative curvature obtained from a direct factorization of the Hessian matrix of the objective function. The key feature is that the directions are computed within the null subspace of the Jacobian matrix of the constraints. In addition, we show how to include the above procedure within an interior-point algorithm for constrained optimization.Finally, some numerical experiments showing the successful performance of our proposal are presented.

This is a joint work with Javier Cano and Francisco J. Prieto.

Julia Novo (Universidad Autónoma de Madrid) 

Stabilization and adaptivity of transient convection-dominated convection diffusion problems

In this work we study a procedure to stabilize Galerkin finite element approximations to linear evolutionary convection-reaction-diffusion equations in the convection dominated regime. It is well known that standard finite element approximations to this kind of equations develop spurious oscillations when convection dominates diffusion. We propose a postprocessing that is able to eliminate spurious oscillations at a fixed time. The idea is the following. One first compute the standard Galerkin approximation at a given time and then solve a steady convection-reaction-diffusion problem with data based on the previously computed Galerkin approximation over the same finite element space but using the SUPG stabilized method. In the second part of the talk we propose an adaptive algorithm based on postprocessing that is able to compute an oscillation-free Galerkin approximation over an automatically adapted mesh that locates the smaller elements at the boundary layers.

This is a joint work with Javier de Frutos and Bosco García-Archilla

Maria do Rosário Pinho (Universidade do Porto) 

Constrained free time optimal control problems

In this talk we will concentrate on necessary conditions of optimality for free time optimal control problems involving control constraints. We show how recent developed maximum principles for optimal control problems involving mixed state-control constraints can be successfully applied to derive new and interesting results for such problems. Other  applications of interest will also be treated.

Pilar Salgado (Universidad de Santiago de Compostela)

Numerical solution of a transient eddy current problem with current intensities as boundary data

The objective of this work is to analyze a time-dependent eddy current problem defined in a 3D bounded domain including conducting and dielectric materials and giving the current source in terms of current intensities. The input current intensities will be introduced in the model as non-local boundary conditions (see, for instance, [1]). Thus, by following [2], we will analyze a formulation based on the magnetic field in the conductor regions and a multivalued scalar magnetic potential in the dielectric part.

From a mathematical point of view, we will obtain a parabolic problem and prove its well posedness as well as some regularity results. We will propose a finite element combined with an implicit Euler time discretization to numerically solve the problem. Concerning the space discretization, the magnetic field is approximated by the lowest N´ed´elec edge finite elements and the magnetic potential by standard piecewise linear continuous elements. The current intensities are imposed as jumps of the multivalued magnetic potential on some prescribed cut surfaces. We will obtain convergence results for the main physical quantities, namely, the magnetic field and the current density. Finally, we will present some numerical results corresponding to the simulation of an application of electromagnetic forming.

This is a joint work with A. Bermúdez, B. López-Rodríguez, R. Rodríguez, P. Salgado.

References

[1] A. Bossavit, Most general non-local boundary conditions for the Maxwell equation in a bounded region, COMPEL (2000), 19, pp. 239–245.

[2] A. Berm´udez, R. Rodr´ıguez and P. Salgado, Numerical solution of eddy current problems in bounded domains using realistic boundary conditions, Comput. Methods Appl. Mech. Engrg.

Ana Leonor Silvestre (Instituto Superior Técnico) 

On the detection of immersed obstacles by boundary measurements

This talk is devoted to the inverse problem of determining the shape and location of a body immersed in a fluid, based on measured quantities on the exterior boundary of the fluid domain. For certain fluid models, we suggest an inverse algorithm that combines Newton method and the method of fundamental solutions for the associated direct problems. Some numerical simulations are presented to illustrate the feasibility of this approach in the two-dimensional case.

This is joint work with Nuno Martins (Faculdade de Ciências e Tecnologia, Lisbon).

Tatiana Tchemisova (Universidade de Aveiro) 

On a new approach to optimality conditions for Convex Semi-Infinite Programming

We consider convex problem of Semi-Infinite Programming (SIP) with multi - dimensional index set. In study of these problems we apply a new approach based on the notions of immobile indices and their immobility orders. We formulate the first order optimality conditions for conves SIP that are explicit and have the form of criterion. We compare this criterion  with other known optimality conditions for SIP and show its efficiency in the convex case.

Thomas Baier (Instituto Superior Técnico)

Metric degeneration and quantization

We discuss further examples of geometric quantization in KŠhler and (singular) real polarizations. This is based on joint work with W.Kirwin, J.Mouro and J.P.Nunes.

José Cariñena (Universidad de Zaragoza)

Motion on constant curvature surfaces and a quantization of some position dependent mass systems

Classical motion on constant curvature surfaces will be analysed and in particular the harmonic oscillator-like system in two dimensions will be proved to be superintegrable. A quantization of these systems will be carried out in several alternative ways. This is a report of results of several papers coworked with M.F. Rañada and M. Santander.

References

[1] P.M. Mathews and M. Lakshmanan, "On a unique nonlinear oscillator", Quart. Appl. Math. 32, 215{218 (1974)

[2] J.F. Cariñena, M.F. Rañada, M. Santander and M. Senthilvelan, "A non-linear Oscillator with quasi-Harmonic behaviour: two-and n-dimensional Oscillators", Nonlinearity 17, 1941{63 (2004)

[3] J.F. Cariñena, M.F. Rañada and M. Santander, "One-dimensional model of a Quantum non-linear Harmonic Oscillator", Rep. Math. Phys. 54, 375{83 (2004)

[4] J.F. Cariñena, M.F. Rañada and M. Santander, "A quantum exactly solvable nonlinear oscillator with quasi-harmonic behaviour", Ann. Phys. 322, 434{459 (2007)

[5] J.F. Cariñena, M.F. Rañada and M. Santander, "The quantum harmonic oscillator on the sphere and the hyperbolic plane", Ann. Phys. 322, 2249{2278 (2007)

[6] J.F. Cariñena, M.F. Rañada and M. Santander, "The quantum harmonic oscillator on the sphere and the hyperbolic plane: k- dependent formalism, polar coordinates and hypergeometric functions", J. Math. Phys. 48, 102106 (2007)

Michele Cirafici (Instituto Superior Técnico)

Quivers and Donaldson-Thomas theory on local threefolds 

Donaldson-Thomas invariants of Calabi-Yau manifolds represent stable BPS states in string theory. They can be studied as generalized instanton of a topological gauge theory defined on the Calabi-Yau. I will present an ADHM-like formalism to study these instantons and its extension to orbifolds.

Carlos Herdeiro (Universidade do Porto)

Black hole collisions in D dimensions and phenomenology for the LHC

String theory suggests scenarios with large extra dimensions in which the fundamental Planck scale could be as low as the TeV scale. It is expected that particle collision with "trans-Planckian" centre of mass energies will produce black holes. This could therefore occur at the Large Hadron Collider (which will reach energies of 14 TeV). Dedicated Monte Carlo event generators have been designed to model the formation and evaporation of such black holes, and are being used to filter experimental data at the LHC. These need as input some observables, such as cross sections and energy lost into gravitational radiation, which must be provided by theory.

In this talk I shall decribe the mathematical framework and the first physical results of a current research programme to obtain such observables. More concretely I shall describe how we study black hole collisions in D dimensions, by solving numerically the full non-linear Einstein equations.

References:

http://arXiv.org/abs/1001.2302

http://arXiv.org/abs/1006.3081

Marco Mackaay (Universidade do Algarve)

Categorifications of quantum groups, Hecke algebras and q-Schur algebras

For any Cartan datum, Khovanov and Lauda defined a diagrammatic 2-category whose Grothendieck algebra they conjectured to be isomorphic to the associated quantum group. For A_n, they proved their conjecture. For any simple Lie algebra, Soergel defined a monoidal category of bimodules whose Grothendieck algebra he proved to be the associated Hecke algebra. Elias and Khovanov gave a diagrammatic version of Soergel's monoidal category for the A_n series.

In my talk I will sketch these results for the series A_n and show how they are related via a quotient of the Khovanov-Lauda 2-category, whose Grothendieck algebrais isomorphic to the q-Schur algebra. This is joint work with M. Stosic and P. Vaz.

Juan Carlos Marrero (Uiversidad de La Laguna, Tenerife)

Hamiltonian dynamics on Lie algebroids, unimodularity and preservation of volumes

The modular class of a Lie algebroid A was introduced by Evens, Lu and Weinstein (Quart. J. Math. Oxford 50, (1999), 417-436) as a cohomology class of order 1 in the Lie algebroid cohomology of A with trivial coefficients. A is said to be unimodular if its modular class is zero. In this talk, I will present some recent results about the relation between the unimodularity of A and the existence of invariant volume forms for a hamiltonian system on the dual bundle A_ to A. Applications of these results to several Poisson mechanical systems will be also given.

Marc Mars (Universidad de Salamanca)

Marginally outer trapped surfaces as quasi-local black holes

Marginally outer trapped surfaces (MOTS) are generally believed to be suitablequasi-local replacements for the concept of black hole. In this talk I willdiscuss a number or rigorous results obtained in the last few years that put thisexpectation on firmer grounds. Specifically, I intend to mention existence of MOTS in an initial data set, smooth local evolution of MOTS, jumps of outermost MOTS, the Penrose inequality conjecture and uniqueness theorems of static spacetimes containing a MOTS.

Filipe Mena (Universidade do Minho)

Spacetime junctions and black hole formation

We will overview the theory of spacetime junctions in General Relativity and some applications to the study of black hole formation. In particular, we shall present a recent work about the existence and stability of spacetimes involved in the gravitational collapse to topological black holes in higher dimensions. This is a joint work with  Natario, Tod, Ann. H. Poincaré.

Vicente Muñoz (CSIC, Madrid)

Moduli Spaces of Pairs and of Bundles

Pairs are objects formed by a holomorphic bundle over a compact Riemann surface, together with a holomorphic section. There is a concept of stability for pairs depending on a real parameter tau and we can consider moduli spaces of tau-stable pairs. We show how these moduli spaces are related to the moduli spaces of stable bundles, and how they do depend on tau. This gives a method to prove properties of the moduli spaces of bundles and of pairs, by induction on the rank.

We show many instances of this technique: irreducibility, birationality, Brauer class, stably rationality, Hodge structures, Torelli theorems, Hodge numbers, motives, Hodge conjecture, ...

Ricardo Schiappa (Instituto Superior Técnico)

Borel and Stokes Analysis of Instantons in Topological Strings

I will review some recent developments concerning the nonperturbative structure of topological string theory. In particular, I will describe how Borel analysis and Stokes phenomena play fundamental roles in this construction.

German Sierra (Universidad Autónoma de Madrid)

The Riemann zeros as spectral lines

One hundred years ago it was conjectured by Polya and Hilbert that the zeros of the Riemann zeta function could be the eigenvalues of a quantum mechanical Hamiltonian. There are strong indications that this may be the case, however the so called Riemann Hamiltonian has remained elusive so far. Building on previous works by Berry, Keating and Connes we shall show that the Hamiltonian of an electron moving in a plane and subject to the action of magnetic and electric fields could provide hints to this long standing problem.

Luis Ugarte (Universidad de Zaragoza)