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Student Topology SeminarWednesday March 23, 2016LCB323 — 3:15 - 4:05 Self Quasi-isometries of Hyperbolic Space Abstract: We will show that a quasi-isometry from n-dimensional hyperbolic space to itself induces a homeomorphism on the sphere at infinity. Along the way, we will prove the Morse-Mostow lemma. Wednesday March 9, 2016LCB323 — 3:15 - 4:05 Soap Bubbles and Harmonic Maps Abstract: When soap bubbles form, their shape is determined by a harmonic function. A classical theorem tells us that functions on Euclidean spaces are harmonic precisely when they minimize an "energy" functional. We can construct a similar energy functional for maps between hyperbolic surfaces, and define harmonic maps to be minima of this functional. Harmonic maps have strong geometric properties. In particular, we will discuss how harmonic maps lead to quadratic differentials and a proof of Teichmuller's theorem. Wednesday February 24, 2016LCB323 — 3:15 - 4:05 Finite Presentation of the Mapping Class Group Abstract: We will give a brief introduction to the mapping class group Mod(S) and prove it's finitely presented. The idea of the proof is to construct a K(Mod(S),1) complex with finite 2-skeleton. In the process, we will prove contractibility of the arc complex and use some standard constructions from the cohomology of groups. Wednesday February 17, 2016LCB323 — 3:15 - 4:05 Quasi-Conformal Maps Abstract: In this talk, we will give two definitions of quasi-conformal maps: a geometric one and an analytic one. The goal of the talk will be to prove the equivalence of these two definitions. Wednesday February 10, 2016LCB323 — 3:15 - 4:05 Green's Functions on Riemann Surfaces Abstract: We will define Green's Functions subsurfaces with boundary of Riemann surfaces and prove their existence. Tuesday February 2, 2016LCB323 — 2:00 - 3:00 Extremal Lengths Abstract: We will discuss finding the extremal length for a set of arcs in the complex plane. We will explicitly compute the extremal length for a rectangle and an annulus. Lastly, we will discuss extremal metrics. This talk is based on material from Conformal Invariants by Ahlfors. Wednesday January 20 & 27, 2016LCB323 — 3:15 - 4:05 Thurston's Earthquake Maps Abstract: We will discuss earthquake maps on the hyperbolic plane and outline Thurston's proof of the earthquake theorem. We will also discuss the role that Thurston's theorem plays in hyperbolic geometry and Teichmuller theory. Past Semesters |
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