Tema 1. “Lenguajes” algebraicos en los siglos XVI y XVII

# Courses

Higgs bundles appear in several guises including (a) as solutions to gauge-theoretic equations for connections and sections of a bundle (b) as holomorphic realizations of fundamental group representations or, equivalently, local systems and (c) as special cases of principal bundles with extra structure (principal pairs). Each point of view leads to a construction of a moduli space, i.e.

16/04/2012. Introduction to Theory of Prime Ends

18/04/2012. Applications I: Index of iterates of planar homeomorphisms

20/04/2012. Applications II: Rotation number of planar attractors

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12-15 Setembro. Total de 8 horas.

There will be 3 lectures:

- The first is devoted to explaining the main concepts and we will take the opportunity to talk about other statistical properties of dynamical systems, such as Central Limit Theorems, Laws of large numbers or Birkhoff's ergodic theorem, Poincaré's recurrence theorem and Kac's theorem.

Lecture 1: Representations of surface groups and harmonic maps

In this series of two lectures I will review what we know about moduli space of instanton bundles on projective spaces. We will see that the situation drastically change when the dimension of the projective space increase. At the end of the two talks I will state some interesting open problems.

Graded manifolds are "manifolds where some coordinates anticommute".

Among other things, they provide a way to encode well-known

matematical structures in a geometric and concise fashion.

I plan to cover

- graded linear algebra, graded manifolds,

- how they encode $L_{\infty}$ algebras and Lie algebroids

15, 16 e 17 de Abril próximo

Em cada um destes dias haverá duas sessões de duas horas

Os slides do curso estão disponíveis na página:

The aim of the course is to introduce pseudodifferential calculus for Lie groupoids and study the index theory of elliptic such operators.

Lecture 1: 17 September (Wed): 14h30-15h30 (Room 1.02 )

Bundle Theory:

We shall review some results in the theory of Hopf algebras and their tensor categories of representations, putting special emphasis in the classification problem in the semisimple case. We shall discuss the construction of extensions and semisimple Hopf algebras arising from matched pairs of finite groups as well as the more recent construction of group theoretical Hopf algebras.

Lecture 1 (October 15: 14.30-16.00, Room 0.04) : Moduli and quotients

Sessão 1 - Segunda-feira, 4 de Junho, 14.30-16.00

Classical variational principle and Hausdorff dimension in conformal dynamics

Sessão 2 - Terça-feira, 5 de Junho, 14.30-16.00

Relativized variational principle and Hausdorff dimension in nonconformal dynamics

Mini-curso:

Mini curso:

Approximate Solutions of Second Kind Integral Equations

Sessão 1 - Quarta-feira, 7 de Junho, 11h, anf. 0:30

Sessão 2 - Quinta-feira, 8 de Junho, 14 h, anf. 0:30

A numerical semigroup is a subset of ${\mathbb N}$ closed under addition, containing the zero element, and with finite complement in ${\mathbb N}$. Even though the concept is easy to understand, there are many pro\-blems related to numerical semigroups that remain unsolved.

Os Expoentes de Lyapunov medem a taxa exponencial de aproximação, ou afastamento de pontos quando sujeitos a um sistema dinâmico. Neste curso iremos considerar sistemas dinâmicos conservativos como:

In order to find an approximate solution of an eigenvalue problem Tφ=λφ for a bounded operator T on a Banach space, one often considers a sequence (T_{n}) of bounded operators which converge to T in some sense and then one solves T_{n}φ_{n}=λ_{n}φ_{n}. When the convergence of (T_{n}) to T is slow, one needs to accelerate the convergence of (λ_{n}) to λ and of φ_{n} to φ.