Flawed Groups
Sean Lawton (George Mason Univ.)
Abstract: A group is flawed if its moduli space of G-representations is homotopic to its moduli space of K-representations for all reductive affine algebraic groups G with maximal compact subgroup K. In this talk we discuss this definition, and associated examples, theorems, and conjectures. This work is in collaboration with Carlos Florentino.
Configuration spaces of points and their homotopy type
Ricardo Campos (CNRS/U. Montpellier)
Abstract: Given a topological space X, one can study the configuration space of n points on it: the subspace of X^n in which two points cannot share the same position. Despite their apparent simplicity such configuration spaces are remarkably complicated; the homology of these spaces is reasonably unknown, let alone their homotopy type. This classical problem in algebraic topology has much impact in more modern mathematics, namely in understanding how manifolds can embed in other manifolds, such as in knot theory. In this talk I will give a gentle introduction to this topic and explain how using ideas going back to Kontsevich we can obtain algebraic models for the rational homotopy type of configuration spaces of points.
Lie algebras and higher Teichmüller components
André Oliveira (Univ. Porto)
Abstract: Consider the moduli space M(G) of G-Higgs bundles on a compact Riemann surface X, for a real semisimple Lie group G. Hitchin components in the split real form case and maximal components in the Hermitian case were, for several years, the only known source of examples of higher Teichmüller components of M(G). These components (which are not fully distinguished by topological invariants) are important because the corresponding representations of the fundamental group of X have special properties, generalizing Teichmüller space, such as being discrete and faithful. Recently, the existence of new such higher Teichmüller components was proved for G = SO(p,q) which, in general, is not neither split nor Hermitian.
In this talk I will explain the new Lie theoretic notion of magical nilpotent, which gives rise to the classification of groups for which such components exist. It turns out that this classification agrees with the one of Guichard and Wienhard for groups admitting a positive structure. We provide a parametrization of higher Teichmüller components, generalizing the Hitchin section for split real forms and the Cayley correspondence for maximal components in the Hermitian (tube type) case. [joint work with S. Bradlow, B. Collier, O. García-Prada and P. Gothen]