A Character variety X_FG is a space of representations of a finitely generated group F into a Lie group G. The most interesting cases are when F is the fundamental group of a Kähler manifold M, and G is a reductive group, since then X_FG is homeomorphic to a space of so-called G-Higgs bundles over M. Typically, X_FG are singular algebraic varieties, defined over the integers, and many of its topological and arithmetic properties can be encoded in a polynomial generalization of the Euler-Poincaré characteristic: the E-polynomial.

# Seminars

We give some geometric conditions which are necessary and sufficient for the existence of Sinai-Ruelle-Bowen (SRB) measures for

C^{1+\alpha} surface diffeomorphisms. As part of our argument we give an original method for constructing first return Young Towers,

which demonstrates that every hyperbolic measure, and in particular every SRB measure, can be lifted to such a tower. This method relies

on a new result in non-ununiform hyperbolicity theory which is independent of interest. Joint work with V. Climenhaga and Y. Pesin.

The concept of chaos is widely used in the field of Dynamical Systems, and several approaches which aim to establish the presence of chaotic dynamics have been developed in the literature. At this juncture, a prototypical example comes from the geometric structure associated with the Smale’s horseshoe, cf. [4]. In recent years, several different approaches have been proposed to extend this classical geometry in a topological direction. This way, the so-called concept of “topological horseshoes” was introduced in [2].

J Ize conjectured that for any absolutely irreducible representation of a compact Lie group G on a finite dimensional real vectorspace there exists an isotropy subgroup which has an odd dimensional fixed point space. If it were true it had immediate consequences in equivariant bifurcation. Lauterbach & Matthews showed that this is not the case. Their findings of three infinite families of finite groups were supplemented by extensive computer analysis showing a very difficult zoo of groups acting on R4. In this talk we will give a complete list of counter examples in R4.

Hyper-bent Boolean functions were introduced in 2001 by Youssef and Gong (and initially proposed by Golomb and Gong in 1999 as a component of S-boxes) to ensure the security of symmetric cryptosystems but no cryptographic attack has been identified till 2016.

Hyper-bent functions have properties still stronger than the well-known bent functions which were already studied by Dillon and

A right R-module M is called a Utumi Module (U-module) if, whenever A and B are isomorphic submodules of M with A ∩ B = 0, there exist two summands K and T of M such that A is an essential submodule of K, B is an essential submodule of T and K ⊕ T is a direct summand of M . The class of U -modules is a simultaneous and strict generalization of three fundamental classes of modules; namely the quasi-continuous, the square-free and the automorphism-invariant modules.

Dynamical systems with an underlying network structure are a subject of great interest as they arise frequently in applications and exhibit many staggering phenomena some of which resemble those in equivariant dynamics. We introduce a theory developed by Rink and Sanders that connects a class of network dynamical systems - namely homogeneous coupled cell systems - to equivariant dynamical systems. The symmetries, however, are generalized in the sense that they do not necessarily form a group but more general structures such as monoids or semigroups.

This talk is about describing dynamics on primitive network objects and finding conditions that allow a good "reductive" description of network dynamics. We will give a number of examples when feedback is added. The examples range from surprising synchrony, an example of the "bullwhip" effect and a remarkable layered network mixing synchrony and chaotic dynamics. Some of this work is part of a joint project with Ana Dias and Manuela Aguiar.

We examine the relation between a heteroclinic network as a flow-invariant set and directed graphs of possible connections between nodes. In particular, we show that there are robust realizations of a large class of transitive directed graphs that are not complete (i.e. not all unstable manifolds of nodes are included) but almost complete (i.e. complete up to a set of zero measure in the unstable manifold) and equable (i.e. all sets of connections from a node have the same dimension).

Pseudovarieties of ordered semigroups have been introduced as a refined algebraic classifying tool for regular languages in the sense of Eilenberg's Correspondence Theorem. Forgetting the order, each such pseudovariety generates a pseudovariety of semigroups. A natural question is which pseudovarieties of semigroups arise in this way from non-selfdual pseudovarieties of ordered semigroups which, for the purpose of the talk, we call orderable.