Dublin Area Mathematics Colloquium

G. Tomassini (Scuola Normale Superiore, Pisa)
Cohomology and extension problem
Friday 11 May 2007, 4 pm, Salmon Lecture Theatre, Hamilton Building, TCD.

Abstract. We are interested in cohomology of semi $1$-coronae in the complex space $C^n$. By definition, a semi $1$-corona is a bounded domain $D\subset C^n$ whose boundary is formed by a $1$-pseudoconvex part, a $1$-pseudoconcave part and a Levi flat part. We prove first a ''bump lemma''. Next, applying the Andreotti-Grauert method, we show a finiteness theorem for coherent sheaves $F$ on $D$ satisfying $depth F_x\ge 3$ for every $x\in D$. As a consequence, an extension theorem for analytic subset is obtained.

Ch. Athanasiadis (University of Athens)
Face numbers of simplicial polytopes and spheres
Friday 9 February 2007, 5 pm, Salmon Lecture Theatre, Hamilton Building, TCD.

Abstract. Convex polytopes (convex hulls of finite point sets in d-dimensional Euclidean space) have been studied since antiquity. The f-vector of a convex polytope P is the sequence of integers which records the number of faces of P in each dimension. It is an important problem in polyhedral combinatorics to determine which sequences of integers arise in this way. In the first part of the talk we survey results, techniques and open questions related to this problem, focusing on the simplicial case (meaning that every proper face of P is a simplex). In the second part (joint work with Thomas Brady, Jon McCammond and Colum Watt) we discuss generalized associahedra, a special class of simplicial polytopes arising in the theory of cluster algebras, and some of the interesting combinatorial problems related to their face numbers.

G. Villari (University of Florence)
Limit cycles of dynamical systems in the plane. A generalization of the Massera uniqueness theorem.
Friday 26 January 2007, 4.30 pm, East End Lecture Theatre 1, Hamilton Building, TCD.

S. Bechtluft-Sachs (NUIM)
Natural Functionals in Homotopy Theory
Friday 8 December 2006, 4.00 pm, East End Lecture Theatre 1, Hamilton Building, TCD.

Abstract. A natural functional is defined on smooth functions $f:M^n -> X^k$ of compact Riemannian manifolds $M$, $X$ by an integral $$E_p(f)=\int_M p(df) dvol_M .$$ Herein $p:M(n \times k)\to \bf R$ is a function defined on matrices and invariant under the action of $O(n) \times O(k)$. Thus the above integral becomes well defined. Let $inf E_p(f)$ denote the infimum of $E_p$ on the homotopy class of $f$. We are interested in the homotopy information that can be seen from the $inf E_p(f)$. It turns out that a crucial property which can almost be detected is that of factorization in homotopy of $f$ through a subskeleton of a triangulation of $X$. In most of this the conformal Jacobians play a central role.

W. Kaup (University of Tübingen)
Homogeneous Levi degenerate CR-manifolds
Friday 1 December 2006, 4.00 pm, East End Lecture Theatre 1, Hamilton Building, TCD.

Abstract. Usually in the study of CR-manifolds Levi-nondegeneracy is assumed. In contrast we are interested in CR-manifolds that are Levi-degenerate at every point, but are nondegenerate in a higher order sense. For locally homogeneous manifolds of this type we give simple (and computable) criteria for k-nondegeneracy as well as construction principles for interesting examples. In dimension 5 all locally homogeneous 2-nondegenerate CR-manifolds can be classified.

T. Kappeler (University of Zürich)
Fermi-Pasta-Ulam lattices: normal form and KAM theorem
Friday 24 November 2006, 4.00 pm, East End Lecture Theatre 1, Hamilton Building, TCD.

Abstract. In the fifties, Fermi, Pasta, and Ulam discovered numerically recurrence phenomena for solutions of 1-dimensional lattices with nearest neighbor interaction. This work had a profound impact on the theory of integrable systems and led to the discovery of integrable Hamiltonian PDE's. In this talk I will report on recent joint work with A. Henrici leading to an explanation of the recurrence phenomena observed numerically at least for small energy solution.