Mathematical Physics Seminar 

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.

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 $d$-dimensional Minkowski-space times a circle $M^d\times S^1$. 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.

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.

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.

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).

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.

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.

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.

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.

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.

This talk will be the starting point for a series of working seminars on the subject.

Let $M$ be an oriented $n$-dimensional manifold. We study the causal relations between the wave fronts $W_1$ and $W_2$ that originated at some points of $M$. We introduce a numerical topological invariant $\mathrm{CR}(W_1,W_2)$ (the so- called causality relation invariant) that, in particular, gives the algebraic number of times the wave front $W_1$ passed through the point that was the source of $W_2$ before the front $W_2$ originated. This invariant can be easily calculated from the current picture of wave fronts on $M$ without the knowledge of the propagation law for the wave fronts. Moreover, in fact we even do not need to know the topology of $M$ outside of a part $\bar{M}$ of $M$ such that $W_1$ and $W_2$ are null-homotopic in $\bar{M}$. We also construct the affine winding number invariant $\mathrm{win}$ which is the generalization of the winding number to the case of nonzero-homologous shapes and manifolds other than $R^2$. The $\mathrm{win}$ 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 $\mathrm{al}$ of nonzero homologous submanifolds $N_1$ and $N_2$ in $M$ introduced by us. To construct $\mathrm{al}$ we introduce a new pairing on the bordism groups of space of mappings of $N_1$ and $N_2$ into $M$. For the case $N_1=N_2=S^1$ 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.

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.

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.

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 ${ℝ}^3$ 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.

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 ${\mathrm{ℂ\mathbb{P}}}^N$.

O ingrediente fundamental desta abordagem é a noção de diástase em variedades Kähler, introduzida por Calabi em 1952.

First, within the algebro-geometric approach, we will read from the fan F the orbit decomposition of the algebraic toric variety $X_F$ 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.

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.

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.