Category Archives: Course

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

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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

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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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Notes of Pierre-Emmanuel Caprace’s fifth Cambridge lecture 04-05-2017

Exotic lattices and simple locally compact groups, V Today, I conclude the discussion and explain how to get simple cocompact lattices in products of trees. Before, I need to complete the issue of irreducibility. 1. Recap We consider leafless trees … Continue reading

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Notes of Richard Schwartz’ first Cambridge lecture 03-05-2017

Pappus’s theorem and the modular group A series of lectures with a common theme: moduli spaces of geometric objects. The geometric objects change from lecture to lecture. 1. Context Deforming a representation into a larger space . Examples: , . … Continue reading

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Notes of Pierre-Emmanuel Caprace’s fourth Cambridge lecture 27-04-2017

Exotic lattices and simple locally compact groups, IV 1. Subgroup separability and residual finiteness Theorem 1 (Kropholler-Reid-Wesolek-Caprace) Let be a finitely generated group, let be a commensurated subgroup (every conjugate commensurates ). Let be the profinite closure of , i.e. … Continue reading

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Notes of Emmanuel Breuillard’s sixth Cambridge lecture 26-04-2017

Introduction to approximate groups, VI Recall the weak version of the structure theorem for approximate groups. Theorem 1 (Breuillard-Green-Tao) Let . Let be a finite group such that . Then there exists a virtually nilpotent subgroup of and such that … Continue reading

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