Abstract: In this talk, we will define a special class of nilpotent elements of complex semisimple Lie algebra and describe how a Slodowy slice construction defines connected components of moduli spaces of G-Higgs bundles on a compact Riemann surface for specific real groups G. Through the nonabelian Hodge correspondence, these components define connected components of the G-character variety with many nice features. We will then describe how these nice features, together with recent work of Guichard-Labourie-Wienhard on positive representations imply that such components are higher Teichmuller spaces, that is, they consist entirely of discrete and faithful representations. In the remaining time, we will describe what remains to be proven to finish the classification of higher Teichmuller spaces.
Abstract: A fundamental result in 3-manifold topology due to Lickorish and Wallace says that every closed oriented connected 3-manifold can be realized as surgery on a link in the 3-sphere. One may therefore ask: which 3-manifolds can be obtained by surgery on a link with a single component, i.e. a knot, in the 3-sphere? More specifically, one can ask: which 3-manifolds are obtained by zero surgery on a knot in the 3-sphere? In this talk, we give a brief outline of some known results to this question in the context of small Seifert fibered spaces. We then sketch a new method, using involutive Heegaard Floer homology, to show that certain 3-manifolds cannot be obtained by zero surgery on a knot in the three sphere. In particular, we produce a new infinite family of weight 1 irreducible small Seifert fibered spaces with first homology Z which cannot be obtained by zero surgery on a knot in the 3-sphere, extending a result of Hedden, Kim, Mark and Park.
Abstract: In the 60s and 70s, there was a flurry of activity concerning the question of whether or not various subgroups of homeomorphism groups of manifolds are simple, with beautiful contributions by Fathi, Kirby, Mather, Thurston, and many others. A funnily stubborn case that remained open was the case of area-preserving homeomorphisms of surfaces. For example, for balls of dimension at least 3, the relevant group was shown to be simple by work of Fathi from the 1970s, but the answer in the two-dimensional case was not known. I will explain recent joint work proving that the group of compactly supported area preserving homeomorphisms of the two-disc is in fact not a simple group, which answers the ``Simplicity Conjecture” in the affirmative. Our proof uses a new tool for studying area-preserving surface homeomorphisms, called periodic Floer homology (PFH) spectral invariants; these recover the classical Calabi invariant of monotone twists. Time permitting, I will also briefly mention a generalization of our result to compact surfaces of any genus.
Abstract: This is joint work with Matei Coiculescu. Sol is probably the weirdest of the 8 Thurston geometries. In this talk I will give an exact characterization of when a geodesic segment in Sol is distance minimizing and then explain why the result implies that the metric spheres in Sol are topological spheres, smooth away from at most 4 arcs. One form of the characterization is a homogenous function on R^3 involving the arithmetic-geometric mean of Gauss. I'll illustrate the talk with some computer demos.
Abstract: Given a closed, orientable 3-manifold M, it is a subtle and often difficult problem to determine whether M may be smoothly embedded in R^4. Even among integer homology spheres, and restricting to special classes such as Seifert manifolds, the problem is open in general, with positive answers for some such manifolds and negative answers in other cases. However, recent work shows that if suitable geometric conditions are imposed then there is a uniform answer for an important class of 3-manifolds called Brieskorn homology spheres: no such 3-manifold admits an embedding as a hypersurface "of contact type" in R^4, which is to say as the boundary of a region that is convex from the point of view of symplectic geometry. I'll describe further context and background for this result, which is joint work with Bülent Tosun, give some highlights of the proof, and indicate connections with complex geometry and further potential directions.
Abstract: We will present the theory behind and new results on the cohomology of super-reflexive Banach G-modules X, where G is a countable discrete group. In particular, we shall show how the cohomology is controlled by the FC-centre of G, that is, the subgroup of elements having finite conjugacy classes. For example, using purely cohomological tools, we show that when X is an isometric super-reflexive Banach G-module so that X has no almost invariant unit vectors under the action of the FC-centre, then the associated cochain complex is split exact. Further connections to the work of Bader-Furman-Gelander-Monod, Nowak, and Bader-Rosendal-Sauer will be presented.
Abstract: In this talk, we will relate homological filling functions and the existence of coarse embeddings. In particular, we will demonstrate that a coarse embedding of a group into a group of geometric dimension 2 induces an inequality on homological Dehn functions in dimension 2. As an application of this, we are able to show that if a finitely presented group coarsely embeds into a hyperbolic group of geometric dimension 2, then it is hyperbolic. Another application is a characterization of subgroups of groups with quadratic Dehn function. If there is enough time, we will talk about various higher dimensional generalizations of our main result.
Abstract: The Lorentzian Lichnerowicz Conjecture is a Lorentzian analogue of the Ferrand-Obata Theorem on conformal transformation groups of Riemannian manifolds. I will discuss my verification of the conjecture in dimension three, for real-analytic metrics, in recent joint work with C. Frances.
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