In this talk, we will cover some results on the nature of mixed Hodge structures on the cohomology of complex quasi-projective varieties. Specifically, we will be interested in varieties whose cohomology behaves as an exterior algebra. For those, we will manage to deduce the mixed Hodge structure on their symmetric products. These results will allows us to obtain a general formula for the mixed Hodge structure of free abelian character varieties for some reductive groups. We will study this formula in some detail for *G=Sp(n, C)*, a case handled by me and C.

# Seminars

The celebrated Birkhoff's ergodic theorem asserts that from a probabilistic viewpoint the times averages of "almost all" points converge to a space average. Motivated by the application of iterated function systems (IFS) to model central dynamics of partially hyperbolic diffeomorphisms, we will describe mild conditions that ensure that Birkhoff non-typical points form a Baire generic subset. If time permits we will provide some applicationsof this result in a partial hyperbolicity context. This is a ongoing joint work with my postdoctoral student G. Ferreira (UFMA).

I will present quantitative equidistribution results for the action of Abelian subgroups of the (2g + 1)-dimensional Heisenberg group on a compact homogeneous nilmanifold. The results are based on the study of the cohomology of the action of such groups on the algebra of smooth functions on the nilmanifold, on tame estimates of the associated cohomological equations with respect to a suitable Sobolev grading, and on renormalization in an appropriate moduli space (a method applied by Forni to surface flows and by Forni and Flaminio to other parabolic flows).

Let 0 < a < 1, 0 ≤ b < 1 and I = [0, 1). We call contracted rotation the interval map φ_{a,b}: x ∈ I → ax+b mod 1. Once the parameter a is fixed, we are interested in the family φ_{a,b}, where b runs on the interval I. We use the fact that, as in the case of circle homeomorphisms, any contracted rotation φa,b has a rotation number which depends only on the parameters a et b. We will discuss the dynamical and diophantine aspects of the subject.

We study the dynamics of planar diffeomorphisms having a unique fixed point that is a hyperbolic local saddle. We obtain sufficient conditions under which the fixed point is a global saddle. We also address the special case of D2-symmetric maps, for which we obtain a similar result for C1 homeomorphisms. Some applications to differential equations are also given.

This is joint work with B. Alarc ́on (UFF — Brasil) and S.B.S.D. Castro (CMUP) and is part of a project of using symmetries to obtain global stability results.

We consider point processes of rare events that keep record of the number of extreme occurrences on a certain time frame and also of the magnitude of the exceedances observed in every such occasion. We study the convergence of such point processes both in the presence and absence of clustering. As a result we prove the convergence of record times and record values point processes.

See the attached file.

A formal power series h(x) = sum a_n x^n is called algebraic if it satisfies a polynomial equation with coefficients in K[x]. Typical examples are rational series like (1+x)^(-1) or roots of polynomials like sqrt(1+x). In characteristic 0, it is an open problem to characterize the coefficient sequence (a_n) of an algebraic series, whereas in positive characteristic, the theory of automata provides a complete answer (which is, though, not entirely satisfactory).

We will describe a method for constructing geodesics on the space of Kähler metrics on a compact Kähler manifold. Applications to geometric quantization will also be discussed.

A numerical semigroup is a submonoid of the non-negative integers, under addition, whose complement in IN is finite. The cardinality of this complement is said to be the genus of the numerical semigroup. In 2008 Bras-Amorós conjectured that the sequence $(n_g)_g$, where $n_g$ is the number of numerical semigroups of genus $g$, behaves like the Fibonacci sequence.