22/04/2004, 15:00 — 16:00 — Sala P3.10, Pavilhão de Matemática
José Mourão, Instituto Superior Técnico
Krichever-Phong universal configuration spaces and symplectic forms
One problem in Hamiltonian systems admiting a Lax pair formulation (with spectral parameter) consists in identifying the action variables corresponding to the angle variables on the Jacobian of the (genus g) spectral curve C. Krichever and Phong identified the action variables on g dimensional leaves of moduli spaces of genus g curves with one (non singular) puncture and two meromorphic differentials with poles of given order at the puncture. I will describe their construction and will consider a few examples.
22/01/2004, 15:00 — 16:00 — Sala P3.10, Pavilhão de Matemática
Troels Harmark, Niels Bohr Institute
Phases of Kaluza-Klein Black Holes
In this talk I will go through the latest progress
on understanding the phase structure of Kaluza-Klein black holes,
i.e. gravity solutions with an event horizon that asymptotes
to a
-dimensional Minkowski-space times a circle
.
Among the phases are uniform and non-uniform black strings wrapped
on the circle, black holes that are localized on the circle
and furthermore solutions that are combinations of black objects
and static Kaluza-Klein bubbles.
17/12/2003, 15:00 — 16:00 — Sala P3.10, Pavilhão de Matemática
João Pimentel Nunes, Instituto Superior Técnico
An introduction to Lax equations
We will describe the notion of a Lax pair, its relation to Riemann
surfaces, and discuss a few examples.
This will be the seventh of a series of seminars devoted to
completely integrable systems.
10/12/2003, 15:00 — 16:00 — Sala P3.10, Pavilhão de Matemática
Carlos Florentino, Instituto Superior Técnico
Riemann surfaces, jacobians and moduli spaces of flat connections
We present a short introduction to the theory of Riemann surfaces,
algebraic curves and their jacobians, describing some of the main
constructions: sheaves, divisors and line bundles, and stating the
theorems of Riemann-Roch and Abel-Jacobi. We also describe some
natural integrable systems associated with these objects, such as
the jacobian itself, and more generally, the moduli space of flat
connections on a Riemann surface.
This will be the sixth of a series of seminars devoted to
completely integrable systems.
20/11/2003, 14:00 — 15:00 — Sala P3.10, Pavilhão de Matemática
Patrícia Engrácia, Instituto Superior Técnico
Apresentação de TFC: Universo de Gödel
Descreveremos resumidamente como as equações de Einstein para um espaçotempo estacionário podem ser reduzidas a um sistema de equações não lineares numa variedade Riemanniana de dimensão 3, e usaremos esta redução para obter a solução de Gödel e descrever a sua geometria (que está intimamente relacionada com a do plano hiperbólico).
19/11/2003, 15:00 — 16:00 — Sala P3.10, Pavilhão de Matemática
Rui Loja Fernandes, Instituto Superior Técnico
Integrability and Galois groups II
In this talk we will discuss obstructions to the existence of first
integrals. First, we start with a geometric result, due to
Poincaré. Then, we discuss Ziglin's method and, more
generally, we will explain how one can use differential Galois
theory to obtain obstructions to the existence of first integrals.
This talk corresponds to Chapter III of M. Audin's book.
This will be the fifth of a series of seminars devoted to
completely integrable systems.
05/11/2003, 15:00 — 16:00 — Sala P3.10, Pavilhão de Matemática
Rui Loja Fernandes, Instituto Superior Técnico
Integrability and Galois groups
In this talk we will discuss obstructions to the existence of first
integrals. First, we start with a geometric result, due to
Poincaré. Then, we discuss Ziglin's method and, more
generally, we will explain how one can use differential Galois
theory to obtain obstructions to the existence of first integrals.
This talk corresponds to Chapter III of M. Audin's book.
This will be the fourth of a series of seminars devoted to
completely integrable systems.
22/10/2003, 15:00 — 16:00 — Sala P3.10, Pavilhão de Matemática
Pedro Ferreira dos Santos, Instituto Superior Técnico
Differential Galois Theory
We will discuss differential fields, Picard-Vessiot estensions and
the Galois group Hamiltonian systems. As an example, we will show
that Airy's equation is not solvable by elementary functions.
This will be the third of a series of seminars devoted to
completely integrable systems.
08/10/2003, 16:00 — 17:00 — Sala P3.10, Pavilhão de Matemática
Leonor Godinho, Instituto Superior Técnico
Action-Angle Variables
We will discuss Hamiltonian actions of tori, the Arnold-Liouville
theorem for completely integrable systems and the construction of
action-angle variables. Several examples will be given.
This will be the second of a series of seminars devoted to
completely integrable systems.
01/10/2003, 16:00 — 17:00 — Sala P3.10, Pavilhão de Matemática
José Natário, Instituto Superior Técnico
Introduction to Integrable Systems
In this introductory seminar we will review the notions of
symplectic manifold, Hamiltonian vector field, Poisson bracket and
completely integrable system, of which several examples will be
given.
This will be the first of a series of seminars devoted to
completely integrable systems.
17/09/2003, 15:00 — 16:00 — Sala P3.10, Pavilhão de Matemática
Pol Vanhaecke, Université de Poitiers
An introduction to algebraically completely integrable systems
This talk will be the starting point for a series of working
seminars on the subject.
17/07/2003, 11:00 — 12:00 — Sala P3.10, Pavilhão de Matemática
Vladimir Chernov, Dartmouth College
Affine linking and winding numbers and the study of front
propagation
Let
be an oriented
-dimensional manifold. We study the causal relations between the wave fronts
and
that originated at some points of
. We introduce a numerical topological invariant
(the so- called causality relation invariant) that, in particular, gives the algebraic number of times the wave front
passed through the point that was the source of
before the front
originated. This invariant can be easily calculated from the current picture of wave fronts on
without the knowledge of the propagation law for the wave fronts. Moreover, in fact we even do not need to know the topology of
outside of a part
of
such that
and
are null-homotopic in
. We also construct the affine winding number invariant
which is the generalization of the winding number to the case of nonzero-homologous shapes and manifolds other than
. The
invariant gives the algebraic number of times the wave front has passed through a given point between two different time moments without the knowledge of the wave front propagation law. The invariants described above are particular cases of the general affine linking invariant
of nonzero homologous submanifolds
and
in
introduced by us. To construct
we introduce a new pairing on the bordism groups of space of mappings of
and
into
. For the case
this pairing can be regarded as an analog of the string-homology pairing constructed by Chas and Sullivan, and it is a generalization of the Goldman Lie bracket.
27/11/2002, 14:30 — 15:30 — Sala P3.10, Pavilhão de Matemática
Diogo Gomes, Instituto Superior Técnico
Perturbation theory for Hamilton-Jacobi equations and stability of Mather sets (II)
We study the stability of integrable Hamiltonian systems under small perturbations using viscosity solutions of Hamilton-Jacobi equations.
We obtain an asymptotic expansion for viscosity solutions of Hamilton-Jacobi equations. The main advantage of our approach is that only a finite number of terms in this asymptotic expansion are needed in order to obtain uniform control. Therefore there are no convergence issues involved.
An application of these results is to show that Diophantine invariant tori and Aubry-Mather sets are stable under small perturbations.
20/11/2002, 14:30 — 15:30 — Sala P3.10, Pavilhão de Matemática
Diogo Gomes, Instituto Superior Técnico
Perturbation theory for Hamilton-Jacobi equations and stability of Mather sets
We study the stability of integrable Hamiltonian systems under small perturbations using viscosity solutions of Hamilton-Jacobi equations.
We obtain an asymptotic expansion for viscosity solutions of Hamilton-Jacobi equations. The main advantage of our approach is that only a finite number of terms in this asymptotic expansion are needed in order to obtain uniform control. Therefore there are no convergence issues involved.
An application of these results is to show that Diophantine invariant tori and Aubry-Mather sets are stable under small perturbations.
10/10/2002, 15:00 — 16:00 — Sala P3.10, Pavilhão de Matemática
Jacek Tafel, Institute of Theoretical Physics, University of Warsaw
Solutions of the Einstein Equations with 2 Killing vectors
The generalized Kaluza-Klein reduction of the Einstein equations and properties of the Ernst equation are recalled. A relation between solutions of the latter equation and surfaces in is discussed. It is shown how the theory of such surfaces can be used to obtain solutions of the Einstein equations with matter fields (cylindrical waves in aperfect fluid). Some examples are presented.
31/07/2002, 15:00 — 16:00 — Sala P3.10, Pavilhão de Matemática
Marcos Mariño, Harvard University
Counting Strings with Knots and Links
18/07/2002, 15:00 — 16:00 — Sala P3.10, Pavilhão de Matemática
Tiago Requeijo, Instituto Superior Técnico
Diástase de Calabi em Variedades Tóricas
Neste trabalho faz-se uma demonstração alternativa de um resultado de Guillemin de 1994 que caracteriza as métricas Kähler em variedades tóricas que admitem um mergulho isométrico em .
O ingrediente fundamental desta abordagem é a noção de diástase em variedades Kähler, introduzida por Calabi em 1952.
07/06/2002, 14:30 — 15:30 — Sala P5, Pavilhão de Matemática
José Mourão, Instituto Superior Técnico
Mirror symmetry on toric hypersurfaces
First, within the algebro-geometric approach,
we will read from the fan F the orbit decomposition
of the algebraic toric variety
under the action of the
algebraic torus. This will associate to every one dimensional cone
in the fan a divisor and will provide us with a natural
description of equivariant line bundles and of
their sections.
To an ample divisor there
corresponds an integral polytope which
coincides with the image of a moment map in the
symplectic approach.
We then describe briefly the Batyrev mirror
construction for Calabi-Yau hypersurfaces in
certain toric varieties.
10/05/2002, 14:30 — 15:30 — Sala P5, Pavilhão de Matemática
Leonor Godinho, Instituto Superior Técnico
Topology of Toric Manifolds
This will be the fifth talk of a series in a reading seminar devoted to Toric Manifolds. This lecture is about how we can use a moment map polytope to learn about some topological properties of the corresponding symplectic toric manifold. In particular, we will show how we can use Morse theory to compute Betti numbers, and we will discuss the Duistermaat-Heckman theorem on the variation in cohomology of reduced symplectic forms.
03/05/2002, 10:00 — 11:00 — Sala P3.10, Pavilhão de Matemática
José Mourão, Instituto Superior Técnico
Path Integral and Mathematics: Love or Hate?
In the interface between contemporary fundamental physics and mathematics there is a long lasting (55 years (!)) mystery. This mistery is known by the name of Feynman Path Integral (PI) and its role in contemporary mathematics is becoming more and more prominent. In this introductory seminar I will comment only on a very small part of the aspects mentioned in the following introduction.
Historical Introduction
In the last 15 years we have witnessed to what is perhaps an unprecedented moment in the history of contributions to leading areas of mathematics originated in "physical ideas" (see Atiyah, Michael Mathematics in the 20th century. Bull. London Math. Soc. 34 (2002), no. 1, 1–15).
The main actor behind the scene is (I believe) the PI (quite often under the "disguise" of string theory).
So which is the mystery? In my view it consists in the following facts:
- F1
-
In most of the cases the PI (still) does not have a rigorous mathematical definition. In few cases it does though: either through rigorous measures studied in stochastic analysis or via axiomatic approaches proposed by Atiyah and Segal.
- F2
-
Physicists have a well defined set of rules on how to manipulate the PI in practically all situations. The correct application of the rules involves quite sophisticated mathematics and is a very well guarded secret by a few leading mathematicians like Kontsevich, Borcherds (if you read the web pages of 1998 fields medals you will see the role of PI emphasized). Still I do not think that it is reasonable to argue that these rules can be used as mathematical definition of the PI.
- F3
-
By applying the above rules (F2) to different geometrical problems it has been possible to formulate very precise mathematical conjectures in different geometric situations (examples follow below).
- F4
-
A large number of the above conjectures have then been proved by standard mathematical techniques.
A few examples are:
- aspects of mirror symmetry (reducing very difficult problems in enumerative geometry to the calculation of simple integrals)
- knot invariants
- Seiberg-Witten invariants of four manifolds
- The Moonshine conjecture/theorem
- Kontsevich quantization of Poisson manifolds
- Intersection theory on moduli spaces
- (...) the list continues
My point: I believe that for a mathematician with experience in geometry to become a (PI) secret holder it is certainly not trivial but it is much simpler than usually believed: Maybe 2 or 3 months of dedicated investment (with a lot of patience needed due to the fact that the path through F2 is quite sinuous ...) Note: I am not an expert in most aspects (F2) of PI. I wish I were ...
A few references to the historical introduction:
- Richard Feynman, Space-time approach to non-relativistic quantum mechanics, Reviews of Modern Physics, 20 (1948) 367.
- Atiyah, Michael Mathematics in the 20th century. Bull. London Math. Soc. 34 (2002), no. 1, 1–15.
- R. E. Borcherds, A. Barnard, math-ph/0204014 Lectures on Quantum Field Theory.
- Jose M. F. Labastida, hep-th/0107079 Math and Physics.