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29/05/2012, Terça

16:30, Geometria em Lisboa

Richard Wentworth, University of Maryland
Gluing formulas for determinants of Dolbeault laplacians on Riemann surfaces.

We present gluing formulas for zeta regularized determinants of Dolbeault laplacians on Riemann surfaces. These are expressed in terms of determinants of associated operators on surfaces with boundary satisfying local elliptic boundary conditions. The conditions are defined using the additional structure of a framing, or trivialization of the bundle near the boundary. An application to the computation of bosonization constants follows directly from these formulas.

28/05/2012, Segunda

16:30, Teoria de Cordas

Michele Cirafici, CAMGSD

21/05/2012, Segunda

16:30, Teoria de Cordas

Tadashi Takayanagi
Holographic Entanglement Entropy and Its Applications.

Video talk by Tadashi Takayanagi @ PI 2011 (1h 13min)

Vídeo: http://pirsa.org/11040059/

14/05/2012, Segunda

16:30, Teoria de Cordas

Gaetan Borot, University of Geneva
Matrix models, non-perturbative topological recursion, and knot invariants.

At least heuristically, the asymptotics of convergent matrix integrals are described by a non-perturbative version of the topological recursion, which is not in general an expansion in powers of the coupling constant $g_s$. However, if a Boutroux and a quantification condition hold (these notions will be explained), one recovers a perturbative expansion from the non-perturbative answer provided $g_s$ is quantified, and the non-perturbative effects just result in renormalizations at all orders by derivatives of theta functions. In the second part of the talk, I will explain how this framework can be used in knot theory. I will describe a Chern-Simons matrix model computation for torus knots invariants inspired by a recent work of Brini, Eynard and Marino. By generalizing to the case of hyperbolic knots (even though no matrix model is known in this case), this leads us to a conjecture (completing a former one of Dijkgraaf, Fuji and Manabe) for the all-order asymptotic expansion of the Jones polynomial of hyperbolic knots.

10/05/2012, Quinta

16:30, Geometria em Lisboa

Ethan Cotterill, Universidade de Coimbra
Some Brill-Noether theory on curves and metric graphs.

Brill-Noether theory seeks to answer the basic question of when an abstract complex curve $C$ of genus $g$ comes equipped with a nondegenerate degree-$d$ morphism to $P^r$.

When $C$ is general in the moduli space $M_g$, an answer is provided by a celebrated theorem of Griffiths and Harris, which establishes that $C$'s behavior is as "expected" whenever $\rho =g-(r+1)(g-d+r)$ is positive, and that there are no morphisms when $\rho$ is negative.

Payne, et al. recently gave a proof of the nonexistence statement using a combinatorial analysis of metric graphs of a particular combinatorial type. These belong to a moduli space $M_g^{\mathrm{trop}}$ for stable metric graphs, which admits a stratification that is dual to that of the Deligne-Mumford space of stable curves.

We will discuss work in progress with Melo, Neves, and Viviani aimed at understanding the Brill-Noether-type behavior of metric graphs of other combinatorial types, focusing on a remarkable infinite family of these.

09/05/2012, Quarta

16:15, Teoria Quântica do Campo Topológica

Anne-Laure Thiel, Instituto Superior Técnico
Diagrammatic categorification of extended Hecke algebra and quantum Schur algebra of affine type A.

Joint work with Marco Mackaay.

14:00, Teoria Quântica do Campo Topológica

Marco Mackaay, Univ. Algarve
sl3 web algebras.

This is joint work with Weiwei Pan and Daniel Tubbenhauer from Gottingen University, Germany.

08/05/2012, Terça

16:30, Geometria em Lisboa

Leonor Godinho, IST
Polygons in Minkowski 3-space: A quest for a long-lost family.

We consider moduli spaces of polygons in Minkowski 3-space with a fixed number of sides lying in the future time-like cone and the others in the past (where each side has a fixed Minkowski length) and study their relation with the fixed-point set of a natural involution on the space of rank-2 parabolic Higgs bundles over ${\mathrm{CP}}^1$.

This is joint work with I. Biswas, C. Florentino and A. Mandini.

07/05/2012, Segunda

16:30, Teoria de Cordas

Luca Mazzucato
The Konishi multiplet at strong coupling.

Video talk by Luca Mazzucato @ PI 2011 (1h 13min)

Vídeo: http://pirsa.org/11030078/

03/05/2012, Quinta

16:30, Geometria em Lisboa

Nuno M Romão, Max-Planck-Institut für Mathematik, Bonn
Vortices and Jacobian varieties.

Gauged vortices are configurations of fields for certain gauge theories on fibre bundles over a surface S. Their moduli spaces support natural L^2-metrics, which are Kaehler, and whose geodesic flow approximates vortex dynamics at low speed. My talk will focus on vortices in line bundles, for which the moduli spaces are modelled on the spaces of effective divisors on S with a fixed degree k; I shall describe the behaviour of the underlying L^2-metrics in a "dissolving limit" where the L^2-geometry simplifies and can be related to the geometry of the Jacobian variety of the surface. Some intuition about multivortex dynamics in this limit will be provided by analysing the simplest nontrivial example (two dissolving vortices moving on a hyperelliptic curve of genus three). This is joint work with N. Manton.

30/04/2012, Segunda

16:30, Teoria de Cordas

João Penedones, Universidade do Porto
Conformal Regge Theory.

In this talk, we will review the basic properties of the Mellin space representation of conformal correlation functions and use it to study high energy scattering in the dual string theory on Anti-de Sitter space. We shall see that this regime is dominated by the exchange of the leading Regge trajectory (i.e. leading twist fields), whose resumed contribution can be described by pomeron exchange. In the process, we will obtain new predictions for 3pt-functions involving leading twist operators.

26/04/2012, Quinta

16:30, Geometria em Lisboa

Enrique Arrondo, Univ. Complutense de Madrid
Subvarieties of small codimension.

We intend to give an elementary talk to explain why (smooth) subvarieties of small codimension are expected to be quite special. We will concentrate on a theorem by Barth stating that a subvariety of dimension $n$ in a projective space of dimension $N$ inherits much of the topology of the projective space, namely the integral cohomology up to order $2n-N$ must be the same. We will give a new geometrical approach to this theorem, which will allow us to extend Barth's theorem to other ambient spaces different from the projective space. We will put all this in relation with the famous Hartshorne's conjecture about subvarieties of small codimension in the projective space

24/04/2012, Terça

15:00, Análise, Geometria e Sistemas Dinâmicos

Henrique Oliveira, Instituto Superior Técnico
New Results on the Collatz Problem.

23/04/2012, Segunda

16:30, Teoria de Cordas

Yolanda Lozano, Universidad de Oviedo
Non-singlet baryons in gauge / gravity duality.

The AdS/CFT correspondence predicts the existence of non-singlet baryons, i.e. baryons with a number of quarks less than the rank of the gauge group, at strong 't Hooft coupling. Using gauge/gravity duality we will see that these configurations also exist in more realistic, less supersymmetric and/or confining, gauge theories. We will also explore their realization in the gravity side beyond the strong 't Hooft coupling regime.

18/04/2012, Quarta

15:00, Equações Diferenciais Parciais

Radoslaw Czaja, CAMGSD, IST
Introduction to Pullback Attractors.

We present the notion of the pullback attractor and variations of its definition. This concept is helpful to describe the long-time behavior of solutions of nonautonomous differential equations. We formulate the theorem on the existence of a pullback attractor for a closed process and show its application to nonautonomous reaction-diffusion equations.

17/04/2012, Terça

16:30, Geometria em Lisboa

Simone Marchesi , Univ. Complutense de Madrid
Steiner and Schwarzenberger bundles on Grassmannians.

In this talk we will present Steiner and Schwarzenberger bundles on the Grassmannians and show the theory that relates them. In the first part we will define the two families of bundles mentioned before and study their properties. We will introduce then the concept of jumping pair for a Steiner bundle and we will study the dimension of the jumping locus of the bundle. Finally, we will give a complete classification of Steiner bundles on the Grassmannian whose jumping locus has maximal dimension and will describe them as Schwarzenberger bundles.

16/04/2012, Segunda

16:30, Teoria de Cordas

Samuel Monnier, ENS Paris
Geometric quantization and the metric dependence of the self-dual field theory.

We will review Witten's ideas for the construction of the partition function of the self-dual field on a Riemannian manifold $M$. Then we will explain how geometric quantization on the intermediate Jacobian of $M$ can be used to find the metric dependence of the partition function. We will also show how the local gravitational anomaly of the theory is recovered in this formalism. Applying these results to the $(2,0)$ supermultiplet on a Calabi-Yau threefold, we will show that its one-loop determinant coincides with the one-loop determinant of the B-model.

11:00, Equações Diferenciais Parciais

Enrico Valdinoci, Università di Milano, Italia
Local and nonlocal phase transition interfaces.

We will discuss some symmetry results for the equation $(-\Delta {)}^{s}u+u-u^3$ and for the limit phase separation interface. Here $s\in (0,1)$ is a fractional parameter. Several open problems will be presented as well.

13/04/2012, Sexta

17:00, Equações Diferenciais Parciais

Serena Dipierro, SISSA, Trieste, Italia
Concentration of solutions for a singularly perturbed elliptic PDE problem in non-smooth domains.

We consider the following singularly perturbed equation $$-{\epsilon }^2\Delta u+u=u^p\text{in}\Omega ,$$ where $\Omega \subset {ℝ}^n$ is a bounded domain whose boundary has an $(n-2)$-dimensional smooth singularity. We study the problem both with Neumann and with mixed Dirichlet and Neumann boundary conditions. Assuming $1<p<(n+2)/(n-2)$ we prove that, in both cases, concentration of solutions occurs at suitable points of the non smooth part of the boundary as the singular perturbation parameter tends to zero.

11/04/2012, Quarta

16:30, Geometria em Lisboa

André Neves
Willmore conjecture and min-max methods.

In 1965, Willmore conjectured that for every torus the integral of the square of the mean curvature is bigger or equal to $2{\pi }^2$. I will explain how to prove the conjecture using min-max methods. This is joint work with Fernando Marques (IMPA).

10/04/2012, Terça

10:30, Seminário de Trabalho em Geometria/Topologia Simpléctica/de Contacto

Daniele Sepe, CAMGSD/IST
Lecture VII - Lagrangian fibrations with elliptic singularities.

The Eliasson-Miranda-Zung linearisation theorem provides a symplectic model in a neighbourhood of a compact non-degenerate orbit of a completely integrable Hamiltonian system. A natural question to ask is whether such a model exists for singular fibres of Lagrangian fibrations (which, generally, consist of several disjoint orbits), the aim being a symplectic classification of Lagrangian fibrations with singularities. In this lecture the simplest case of purely elliptic singularities is studied in order to illustrate some of the difficulties that arise in developing such a classification theory.

03/04/2012, Terça

14:00, Análise, Geometria e Sistemas Dinâmicos

Saber Elaydi, Trinity University, San Antonio, USA
Application of singularity theory in planar discrete dynamical systems and applications to competition models.

10:30, Seminário de Trabalho em Geometria/Topologia Simpléctica/de Contacto

Daniele Sepe, CAMGSD/IST
Lecture VI - The Eliasson-Miranda-Zung linearisation theorem.

The definition of non-degenerate singular orbits of completely integrable Hamiltonian systems gives local linear models for the underlying Hamiltonian ${ℝ}^n$-action. A natural question is whether there exists a symplectomorphism from an open neighbourhood of a fixed singular orbit to the local linear model which preserves the respective Lagrangian fibrations. The Eliasson-Miranda-Zung theorem answers the above questions affirmatively when the orbit is compact. In this lecture this theorem is stated and a strategy for its proof is outlined, so as to explain the main underlying geometric ideas.

02/04/2012, Segunda

16:30, Teoria de Cordas

Shing-Tung Yau
The Shape of Inner Space.

Video talk by Shing-Tung Yau @ PI 2011 (1h 30min)

Vídeo: http://pirsa.org/11010091/

27/03/2012, Terça

14:00, Série de Palestras LARSyS em Engenharia e Matemática

Bernold Fiedler, Free University Berlin
Determining nodes for regulatory networks.

We consider systems of differential equations which model complex regulatory networks by a graph structure of dependencies. We show that the concepts of informative nodes (Mochizuki) and determining nodes (Foias, Temam) coincide with the notion of feedback vertex sets from graph theory. As a result we can determine the long-time dynamics of the entire network from observations on the feedback vertex set, only. We present biological examples of gene-regulatory and signal transduction networks where the required observation set is much smaller than the entire regulatory network. Since the mathematical scope of our approach is much broader, however, we actively seek discussion of regulatory and control aspects in other networks, for example in electrical engineering, multi-agent coupling, neural networks, and the like.

10:30, Seminário de Trabalho em Geometria/Topologia Simpléctica/de Contacto

Daniele Sepe, CAMGSD/IST
Lecture V - Non-degenerate singularities.

Having established several structural results for Lagrangian fibrations which are submersions in the first four sessions, this lecture aims to study a more general class of fibrations which allows for topologically well-behaved (i.e. Morse-Bott in some sense) singularities. These naturally arise in the theory of completely integrable Hamiltonian systems (e.g. if the phase space is compact, they must exist) and in mirror symmetry. The notion of non-degenerate singularities is introduced and illustrated with several examples. Time permitting, the Eliasson-Miranda linearization theorem for non-degenerate singularities is going to be studied in some detail (but probably without proof).

26/03/2012, Segunda

16:30, Teoria de Cordas

Igor Klebanov
Testing the $F$-theorem.

Video talk by Igor Klebanov @ KITP 2012 (1h 11min)

Vídeo: http://online.itp.ucsb.edu/online/joint98/klebanov2/

21/03/2012, Quarta

15:00, Equações Diferenciais Parciais

Vardan Voskanian, IST, LIsboa
Extended mean field games - existence.

Mean field games is a new class of problems recently introduced by Lions and Lasry, and independently by Caines and his co-workers which attempts to understand the limiting behavior of systems involving large numbers of rational agents which play dynamic games under partial information and symmetry assumptions. In this talk we present a reformulation of this problem as coupled system of an ordinary differential equations in an $L^2$ space, together with a Hamilton-Jacobi equation. This allows us to consider an extension of the original mean-field problem where the interaction a player and the mean field also takes into account the collective behavior of the players but not only its state. Finally we establish an existence result for this extended mean-field games.

20/03/2012, Terça

10:30, Seminário de Trabalho em Geometria/Topologia Simpléctica/de Contacto

Daniele Sepe, CAMGSD/IST
Lecture IV - (Integral) affine manifolds and Lagrangian fibrations II.

This lecture continues with the study of (integral) affine manifolds. First, some important invariants associated to these manifolds (the affine and linear holonomies and the radiance obstruction) are constructed. Particular attention is devoted to the radiance obstruction, a cohomology class constructed by Goldman and Hirsch which contains important information about the given (integral) affine structure. Secondly, it is proved that a manifold is the base of any Lagrangian fibre bundle with compact and connected fibres if and only if it is an integral affine manifold, which allows to study the problem of constructing all Lagrangian fibre bundles with compact and connected fibres over a fixed integral affine manifold. As usual, the theory developed is illustrated by means of examples.

19/03/2012, Segunda

16:30, Teoria de Cordas

José Mourão, CAMGSD/DMIST
Geometric quantization and simple non-equivalent quantizations of the harmonic oscilator.

I will recall the basics of geometric quantization and what is usually meant by quantizing a mechanical (a hamiltonian) system.

Will then consider two one-parameter families of Kahler polarizations on the plane intersecting at one point and degenerating to two important real polarizations: the vertical or Schrodinger one and the (singular) harmonic-oscilator-energy one.

Eventhough the quantizations associated with the above real degenerations are equivalent, on the coherent-space-transform driven quantum path from one to the other [more specifically all along the half way from the crossing point to the energy representation] I will argue that the Kahler quantizations are inequivalent to the two real ones above.

Will then comment briefly on the relevance of this analysis to loop quantum gravity.

Based on joint work with William Kirwin and João P. Nunes.

14/03/2012, Quarta

15:00, Equações Diferenciais Parciais

Levon Nurbekian, IST, Lisboa / UT|Austin, Texas
Lagrangian Dynamics and a Weak KAM theorem on the $d$-infinite dimensional torus.

The space $L^2[0,1]$ has a natural Riemannian structure on the basis of which in their recent work W. Gangbo and A. Tudorascu introduced an infinite dimensional torus $𝕋$. For a certain class of Hamiltonians they prove an existence of a viscosity solution to the cell problem on $𝕋$. Furthermore they exploit the solution to prove the existence of the solution to the so called nonlinear $1$-dimensional Vlasov system and obtain asymptotics for the solution. In the current work we try to generalize results obtained by Gangbo and Tudorascu to the so called higher dimensional case, where the ambient space is the $L^2([0,1{]}^d;{ℝ}^d)$ instead of the $L^2[0,1]$. More precisely, for a certain class of Hamiltonians $H$, defined on the cotangent bundle of the infinite dimensional Hilbert space $L^2([0,1{]}^d;{ℝ}^d)$, and for any $c\in {ℝ}^d$ we prove the existence of the periodic continuous viscosity solution $U$ to the cell problem $H(M,Du+c)=\lambda ,$ where $\lambda \in ℝ$ is a constant depending on $c$. Once we prove the existence of the solution to that problem we are able to prove the existence of the solution to the nonlinear $d$-dimensional Vlasov system and obtain an asymptotics for the solution.

13/03/2012, Terça

16:30, Geometria em Lisboa

Ciro Ciliberto, Università di Roma II
Gonality of nodal curves on general $K3$ surfaces.

This is a report on joint work in collaboration with A. Knutsen (University of Bergen, Norway).

The pairs $(S,H)$ with $S$ a complex K3 surface (i.e. $S$ is simply connected and ${\Omega }_S^2$ is trivial), and $H$ is an ample line bundle of genus $p$ (i.e. $H^2=2p-2$), have an irreducible moduli space ${𝒦}_p$ of dimension 19. Let $(S,H)\in {𝒦}_p$ be general. The linear system $\mid H\mid$ has dimension $p$ and its general element is a smooth curve of genus $p$. I consider $V_{\mid H\mid ,\delta }$, the Severi variety of $\delta$-nodal curves in $\mid H\mid$, which is locally closed of codimension $\delta$ in $\mid H\mid$ and its general element is an irreducible curve with exactly $\delta$ nodes and no other singularities, so that its geometric genus (i.e. the genus of its desingularization) is $g=p-\delta$. The image of $V_{\mid H\mid ,\delta }$ in ${ℳ}_g$, the moduli space of curves of genus $g$, has dimension $g$ and its importance in moduli problems is well known. For any integer $k\ge 2$, consider the subscheme $V_{\mid H\mid ,\delta }^k\subset V_{\mid H\mid ,\delta }$ given by $$\{C\in V_{\mid H\mid ,\delta }:\text{the normalization of}C\text{is a}k\text{-tuple cover of}{\mathbb{P}}^1\}.$$

The first question is: when is $V_k\subset V_{\mid H\mid ,\delta }$ not empty? Using a (by now classical) vector bundle technique due to Lazarsfeld, we show that a necessary condition is $\delta \ge \alpha (g-(k-1)(\alpha +1))$ where $\alpha :=\lfloor \frac{g}{2(k-1)}\rfloor$. A dimension count shows that, if $V_{\mid H\mid ,\delta }^k$ is not empty, its expected dimension should be $2k-2$. As a matter of fact, we prove that for all even $p$ and all $\delta ,k$ verifying the above relation, $V_{\mid H\mid ,\delta }^k$ is actually not empty, of dimension $2k-2$. For $p$ odd we have a slightly weaker result. Finally, I will talk about some interesting relations of these results with a conjecture by Hassett and Tschinkel on the Mori cone of the hyperkhähler manifold ${\text{Hilb}}^k(S)$ (which parametrizes subschemes of $S$ of finite lenght $k$).

10:30, Seminário de Trabalho em Geometria/Topologia Simpléctica/de Contacto

Daniele Sepe, CAMGSD/IST
Lecture III - (Integral) affine manifolds and Lagrangian fibrations I.

A theorem due to Weinstein states that the leaves of any Lagrangian foliation admit a flat, torsion-free connection. A manifold admitting such a connection is called affine; these have been extensively studied since the '50s as a generalisation of flat Riemannian manifolds. This lecture first proves the above result directly for Lagrangian fibrations and then introduces more formally (integral) affine manifolds, illustrating the theory with examples.

12/03/2012, Segunda

16:30, Teoria de Cordas

João Esteves, CAMGSD
Geometric Quantization: a beginner's perspective.

Geometric Quantization is an attempt to give a rigorous mathematical formulation for the process of quantization of some physical systems, like the ones studied in Quantum Mechanics. In this first talk we review some well known facts and techniques in this framework with emphasis on the quantization of the harmonic oscillator as an explicit example. We will present some classical results on this subject that follow a geometrical perspective, in contrast with the traditional approach of solving explicitly the Schrodinger equation.

07/03/2012, Quarta

15:00, Equações Diferenciais Parciais

Marco Morandotti, Carnegie Mellon University, Pittsburgh, USA
Self-propulsion in viscous fluids through shape deformation.

I will present a model for micro-swimmers in viscous fluids, both plain and particulate. Given the Reynolds number is very low, Stokes' and Brinkman's equations can be used to govern the velocity and the pressure of the surrounding, infinite fluid. Imposing a no-slip boundary condition allows to relate the deformation of the swimmer to the fluid velocity field, while self-propulsion is the constraint through which we can reduce, via an integral representation of the viscous forces and momenta, the equations of motion for the swimmer to a system of six ODEs. Under mild regularity assumptions, an existence and uniqueness theorem for the motion is proved. Eventually, I will focus on the case of a flagellum swimming in a viscous fluid. In this case, the equations of motion are derived from an approximate theory, and optimality results are discussed. This is partially joint work with Gianni Dal Maso and Antonio DeSimone (SISSA, Trieste, Italy).

06/03/2012, Terça

15:00, Geometria em Lisboa

Éveline Legendre, Université Paul Sabatier, Toulouse
Commonness of Kähler-Einstein metrics in toric geometry.

I will explain why each integral convex compact polytope is the moment polytope of a Kähler-Einstein orbifold (unique up to a dilatation). More generally, I will explain how to prove that each convex compact polytope admits a unique ray of symplectic potentials solving the toric Kähler-Einstein equation. Then, I'll bring out some geometric applications.

10:30, Seminário de Trabalho em Geometria/Topologia Simpléctica/de Contacto

Daniele Sepe, CAMGSD/IST
Lecture II - Topological and symplectic classification.

A theorem due to Liouville, Mineur and Arnol'd states that if a Lagrangian fibration admits a compact and connected fibre $F$, then the fibre is diffeomorphic to a torus, nearby fibres are also tori and there exists a symplectomorphism between a neighbourhood of $F$ and the zero section of the cotangent bundle to the torus which preserves the fibrations. In this lecture, a generalisation of this theorem for complete Lagrangian fibrations is proved, using the natural fibrewise action of the cotangent bundle to the base on the total space of the fibration. This construction allows to develop a topological (in fact, smooth) and symplectic classification theory for such fibrations, which yields two topological invariants, the period net and Chern class, and one symplectic characteristic class, the Lagrangian Chern class.

05/03/2012, Segunda

16:30, Teoria de Cordas

Miguel Costa, Universidade do Porto
Deeply Virtual Compton Scattering from Gauge/Gravity Duality.

We use gauge/gravity duality to study deeply virtual Compton scattering (DVCS) in the limit of high center of mass energy at fixed momentum transfer, corresponding to the limit of low Bjorken $x$, where the process is dominated by the exchange of the pomeron. Using conformal Regge theory we review the form of the amplitude for pomeron exchange, both at strong and weak $\prime t$ Hooft coupling. At strong coupling, the pomeron is described as the graviton Regge trajectory in AdS space, with a hard wall to mimic confinement effects. This model agrees with HERA data in a large kinematical range. The behavior of the DVCS cross section for very high energies, inside saturation, can be explained by a simple AdS black disk model. In a restricted kinematical window, this model agrees with HERA data as well.

29/02/2012, Quarta

14:00, Equações Diferenciais Parciais

Carlos Rocha, IST, Lisboa
Connection Graphs for Sturm Attractors of $S^1$-equivariant Parabolic Equations.

We consider a semilinear parabolic equation of the form $u_t=u_{xx}+f(u,u_x)$ defined on the circle $x\in S^1=ℝ/2\pi \mathbb{Z}$. For a dissipative nonlinearity $f$ this equation generates a dissipative semiflow in the appropriate function space, and the corresponding global attractor $A_f$ is called a Sturm attractor. We use the Sturm permutation ${\sigma }_f$ introduced for the characterization of Neumann flows to obtain a purely combinatorial characterization of the Sturm attractors $A_f$ on the circle. With this Sturm permutation ${\sigma }_f$ we show how to construct a connection graph $G_f$ representing the Sturm attractor $A_f$.

28/02/2012, Terça

10:30, Seminário de Trabalho em Geometria/Topologia Simpléctica/de Contacto

Daniele Sepe, CAMGSD/IST
Lecture I - An introduction to Lagrangian fibrations.

This lecture presents an overview on the whole series, introducing some fundamental concepts in the study of Lagrangian fibrations by means of some examples. Special attention is dedicated to completely integrable Hamiltonian systems, which are a type of mechanical systems that are intimately connected to Lagrangian fibrations. The structure of the cotangent bundle as a Lagrangian fibration (in fact a fibre bundle) is studied, thus illustrating the ideas that lead to the topological and symplectic classification of Lagrangian fibrations.

16/02/2012, Quinta

15:00, Análise, Geometria e Sistemas Dinâmicos

José Ferreira Alves, Universidade do Porto
Gibbs-Markov structures vs statistical properties in dynamical systems.

In a classical approach to dynamical systems one frequently uses certain geometric structures of the system to deduce statistical properties, such as invariant measures with stochastic-like behaviour, large deviations or decay of correlations. Such geometric structures are generally highly non-trivial and thus a natural question is the extent to which this approach can be applied. We show that in many cases stochastic-like behaviour itself implies that the system has certain geometric properties, which are therefore necessary as well as sufficient conditions for the occurrence of the statistical properties under consideration.

15/02/2012, Quarta

14:00, Equações Diferenciais Parciais

Jorge Drumond Silva, Instituto Superior Técnico
Dispersive PDE techniques and local well-posedness results for generalized KP-II type equations on cylinders.

We present recent results for the KP-II equations with generalized dispersion terms, in two and three spacial dimensions, periodic only in the $x$ variable. We will start by reviewing some modern methods for dispersive partial differential equations, and then show how the solutions to the linearized KP-II equations satisfy bilinear Strichartz-type estimates, which are independent of the intensity of the dispersion. We then use these estimates to establish local well-posedness for the Cauchy problem associated to the full nonlinear equations for low regularity data, in the framework of Bourgain spaces. For certain ranges of dispersion, these local results are optimal.

This is a joint work with Axel Grünrock and Mahendra Panthee.

14/02/2012, Terça

10:30, Seminário de Trabalho em Geometria/Topologia Simpléctica/de Contacto

Rosa Sena Dias, CAMGSD/IST
Hyperkähler manifolds - II.

Hypertoric manifolds are $4n$ dimensional hyperkähler manifolds admiting a tri-hamiltonian ${𝕋}^n$ action. We describe a construction of such manifolds as quotients of ${ℂ}^{2d}$ by subtori of ${𝕋}^d$. This construction mimics Delzant's construction of toric Kahler manifolds as quotients of ${ℂ}^d$ by subtori of ${𝕋}^d$. We will discuss relations between the two constructions and explain how to use the hyperkahler version to give a description of the hyperkähler metric in hyper action-angle coordinates.

Main reference: R. Bielawski and A. Dancer, The geometry and topology of toric hyperkähler manifolds

08/02/2012, Quarta

14:00, Equações Diferenciais Parciais

Fabio Camilli, Università di Roma, La Sapienza
Hamilton-Jacobi equations on ramified spaces.

In this talk I will present an extension of the theory of viscosity solutions for Hamilton-Jacobi equations on a class of ramified spaces, the LEP spaces. Ramified spaces were introduced by Lumer and they are a generalization of topological graphs to higher dimensional spaces. I will give the definition of ramified manifolds and, as special cases, the one of locally elementary polygonal ramified spaces (LEP spaces). On LEP spaces I will define a notion of viscosity solutions for Hamilton-Jacobi equations, which provides existence and uniqueness results.

07/02/2012, Terça

16:30, Geometria em Lisboa

Sheila Sandon, CNRS and Institute for Advanced Study
On translated points of contactomorphisms.

The Arnold conjecture in Symplectic Topology states existence of many fixed points for Hamiltonian symplectomorphisms of a compact symplectic manifold. In my talk I will discuss an analogue of this conjecture in Contact Topology, based on the notion of translated points.

10:30, Seminário de Trabalho em Geometria/Topologia Simpléctica/de Contacto

José Mourão, CAMGSD/IST
Hyperkahler manifolds - I.

Motivational introduction

Q1

Why would a differential/symplectic/algebraic geometer care about hyperkähler manifolds?

Q2

Why would a quantum geometer care?

Technical introduction

Definition and basic properties of hyperkähler manifolds. Relation with the holomorphic symplectic point of view. Hyperkähler quotient/holomorphic symplectic quotient as a tool to construct interesting examples.

01/02/2012, Quarta

14:00, Equações Diferenciais Parciais

Joana Mohr, Universidade Federal do Rio Grande do Sul, Brazil
On the general One-Dimensional $\mathrm{XY}$ Model: positive and zero temperature, selection and non-selection.

In the Bernoulli space $ℬ=({𝕊}^1{)}^{\mathbb{N}}$ we consider the dynamical system given by the shift, $\sigma :ℬ\to ℬ$, and a potential $A:ℬ\to ℝ$ that describes the interaction between sites in the one-dimensional lattice $({𝕊}^1{)}^{\mathbb{N}}$. For each value $\beta =1/T$, where $T$ is the temperature, we shall consider a $\sigma$-invariant measure that we call Gibbs state ${\mu }_{\beta A}$, and study the limit of the family ${\mu }_{\beta A}$, when $\beta \to \mathrm{\infty }$ (or $T\to 0)$. An interesting question is the selections of probability measures when the temperature goes to zero. By selection of a measure ${\mu }_{\mathrm{\infty }}$, when $T\to 0$, we mean the existence of the limit (in the weak${}^{*}$ sense) ${\displaystyle {\mu }_{\mathrm{\infty }}:=\underset{\beta \to \mathrm{\infty }}{\lim }{\mu }_{\beta A}.}$

31/01/2012, Terça

16:30, Geometria em Lisboa

Matias del Hoyo, CAMGSD/IST
Vector bundles over Lie groupoids and algebroids.

In a real vector space addition is settled by scalar multiplication. This lead us to the following description of smooth vector bundles: they are manifolds endowed with a nice action of the multiplicative monoid of the real numbers. In this talk I will deal with vector bundles over Lie groupoids and algebroids. Lie groupoids are presentations for singular smooth spaces, constitute a concrete unified framework to deal with manifolds, Lie groups, actions, foliations and others. Lie algebroids are their infinitesimal counterpart, together they play a rich theory in actual development. Vector bundles over groupoids and algebroids are generalized representations and help us to understand the geometry of our singular spaces. I will present definitions and examples, provide a characterization by using the multiplicative monoid, and discuss the integration of these structures. This is part of a joint work with H. Bursztyn and A. Cabrera.

25/01/2012, Quarta

14:00, Equações Diferenciais Parciais

Rafael Rigão Souza, Universidade Federal do Rio Grande do Sul
Discrete state space mean field games.

I will present a continuous time, discrete state space model for mean field games. Such kind of models can describe situations of competition between a large number of rational players, where all we control is the statistical distribution of players and its strategies. After presenting our model, I will derive forward and backwards equations, prove the existence of solutions and discuss uniqueness, and if time allows, show that, for short times, the mean field model can be approximated by a $N$-player game, where the number $N$ of players is large but finite.

18/01/2012, Quarta

14:00, Equações Diferenciais Parciais

Wladimir Neves, Instituto de Matemática, Universidade Federal do Rio de Janeiro
The multidimensional Muskat Problem.

In this talk, we study the multidimensional Muskat Problem. We show the solvability of the initial-boundary value problem to a proposed generalized Buckley-Leverett system. Moreover, we discuss some important questions concerning singular limits of the proposed model.

17/01/2012, Terça

16:30, Geometria em Lisboa

Xavier Roulleau, CAMGSD/IST
Quotients of Fano surfaces.

Fano surfaces parametrize the lines on smooth cubic threefolds. In this talk, we will explain the remarkable geometric properties of these surfaces and we will explain how to use these properties in order to compute the invariants of the surfaces obtained as quotients of Fano surfaces by an automorphism group.

12/01/2012, Quinta

11:30, Teoria Quântica do Campo Topológica

Jeffrey C. Morton, IST
Groupoidification and Khovanov's Categorification of the Heisenberg Algebra.

The aim of this talk is to describe the connection between two approaches to categorification of the Heisenberg algebra. The groupoidification program of Baez and Dolan has been used to give a representation of the quantum harmonic oscillator in the category Span(Gpd) where the Fock space is represented by the groupoid of finite sets and bijections. This naturally gives a combinatorial interpretation of the (one-variable) Heisenberg algebra in the endomorphisms of this groupoid. On the other hand, Khovanov has given a categorification in which the integral part of the (many variable) Heisenberg algebra is recovered as the Grothendieck ring of a certain monoidal category described in terms of a calculus of diagrams. I will describe how an extension of the groupoidification program to a 2-categorical form of Span(Gpd) recovers the relations used by Khovanov's construction, and how to interpret them combinatorially in terms of the groupoid of finite sets.

11/01/2012, Quarta

14:00, Série de Palestras LARSyS em Engenharia e Matemática

Olivier Guéant, Université Paris-Diderot
Games with infinitely many players: the mean field games approach.

The PDEs corresponding to mean field games are two strongly coupled PDEs: one (backward) Hamilton-Jacobi-Bellman equation and one (forward) transport equation. We will present these equations and prove a very general criterion for uniqueness. Existence will be discussed but only proved in the case of quadratic hamiltonians for which many results are known and will be proved (constructive schemes, comparison principle,...).

Vídeo: http://media-camgsd.math.ist.utl.pt/LARSyS-II/LS2-1-F.WMV http://media-camgsd.math.ist.utl.pt/LARSyS-II/LS2-1-F.WMV

10/01/2012, Terça

16:30, Geometria em Lisboa

Brian Hall, Notre Dame
Complex structures and magnetic fields.

If $M$ is a real-analytic Riemannian manifold, then there exists a canonical "adapted complex structure" on a neighborhood of the zero-section in the tangent bundle $TM$. These structures were introduced independently by Guillemin and Stenzel and by Lempert and Szoke. It is possible to understand the adapted complex structure in terms of the "imaginary time geodesic flow." I will describe a new family of complex structures in which one modifies the problem by adding a "magnetic field" on $M$, described by a closed 2-form. These new complex structures are described in terms of the Hamiltonian flow for a particle in a magnetic field, evaluated (in a suitably interpreted way) at an imaginary time. For a constant magnetic field on ${ℝ}^2$ or $S^2$, the magnetic complex structure can be computed explicitly and is defined on the entire tangent bundle.

06/01/2012, Sexta

16:00, Teoria Quântica do Campo Topológica

Nuno Freitas, Universitat de Barcelona
From Fermat's Last Theorem to some generalized Fermat equations.

The proof of Fermat's Last Theorem was initiated by Frey, Hellegouarch, Serre, further developed by Ribet and ended with Wiles' proof of the Shimura-Tanyama conjecture for semi-stable elliptic curves. Their strategy, now called the modular approach, makes a remarkable use of elliptic curves, Galois representations and modular forms to show that $a^p+b^p=c^p$ has no solutions, such that $(a,b,c)=1$ if $p\ge 3$. Over the last 17 years, the modular approach has been continually extended and allowed people to solve many other Diophantine equations that previously seemed intractable. In this talk we will use the equation $x^p+2^{\alpha }{y}^p=z^p$ as the motivation to introduce informally the original strategy ($\alpha =0$) and illustrate one of its first refinements (for $\alpha =1$). Then we will discuss some further generalizations that recently led to the solution of equations of the form $x^5+y^5={\mathrm{dz}}^p$.

03/01/2012, Terça

16:30, Geometria em Lisboa

Alejandro Uribe, Michigan
On Donaldson's complexification of the group of Hamiltonian automorphisms of a symplectic manifold.

I will review the notion in the title (which, is not a group) and show how to construct in certain cases an "exponential" in the complexification. The construction is motivated by quantum mechanics.