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Monthly Archives: May 2017
Notes of Alina Vdovina’s Cambridge lecture 31-05-2017
Expanders, Beauville surfaces and buildings 1. Expanders For a graph, Cheeger’s constant enters in the isoperimetric inequality Example. For a -regular tree, . Definition 1 An expander is a sequence of finite bounded degree graphs whose Cheeger constants are bounded … Continue reading
Notes of Viktor Schroeder’s third informal Cambridge lecture 30-05-2017
Moebius structures on boundaries, III Today’s material is taken from Jonas Beyrer’s PhD. Given a space, Bourdon’s formula often takes value 0. Nevertheless, for higher rank symmetric spaces, or for products of spaces, the restriction to the Furstenberg boundary is … Continue reading
Notes of Richard Schwartz’ sixth Cambridge lecture 26-05-2017
PETs, pseudogroup actions, and renormalisation Started by group theorist B.H. Neumann in 1959. Outer billiard around a convex polygon composes 180 degrees rotation through vertices. Theorem 1 For a kite (a quadrilateral with one axial symmetry), there exist unbounded orbits … Continue reading
Notes of Richard Schwartz’ fifth Cambridge lecture 24-05-2017
The pentagram map and discrete integrable systems Joint work with Valentin Ovsienko and Serge Tabachnikov Start with a convex polygon. Draw diagonals between vertices at distance 2, they form a smaller polygon inside. Call this the pentagram map, although the … Continue reading
Notes of Viktor Schroeder’s second informal Cambridge lecture 23-05-2017
Moebius structures on boundaries, II 1. Ptolemaic Moebius structures Recall that a Moebius structure is Ptolemaic if cross-ratios satisfy the Ptolemaic inequality, This means that takes its values in the triangle with vertices at the extra points . Examples Boundaries … Continue reading
Notes of Nicolas Matte Bon’s Cambridge lecture 23-05-2017
Uniformly recurrent subgroups and rigidity of non-free minimal actions Joint work with A. Le Boudec and T. Tsankov. 1. The Chabauty space of a group Let be a locally compact group. The Chabauty space of is the set of subgroups … Continue reading
Notes of Erik Guentner’s Cambridge lecture 23-05-2017
Affine actions, cohomology and hyperbolicity When can a group act properly on a Hilbert space or an space? I start from scratch. 1. Affine actions a discrete group, a Banach space. We are interested in actions of on where each … Continue reading
Notes of David Kyed’s Cambridge lecture 18-05-2017
-Betti numbers of universal quantum groups Joint with Pichon, Arndt, Vaes,… I spoke on the same subject in the same room in 2006. I think my understading has improved. 1. Infinite discrete groups Let act on vectorspace . There are … Continue reading
Notes of Nina Friedrich’s Cambridge lecture 17-05-2017
Homological stability of moduli spaces of high-dimensional manifolds 1. Manifolds Say a sequence manifolds and maps satisfies homological stability if induced maps on homomology groups become eventually isomorphisms. How does one prove such a property? We are interested in the … Continue reading
Notes of Richard Schwartz’ third Cambridge lecture 17-05-2017
Iterated barycentric subdivisions and steerable semi-groups In two dimensions, there are many different affinely natural procedures on simplices: the barycentric subdivision (defined by coning and induction on dimension), yielding 6 triangles; the truncation of corners, yielding 4 triangles. The second … Continue reading
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