## Departmental PhD Thesis Exam – Gregory Chambers

Everyone is welcome to attend. Refreshments will be served in the Math Lounge before the exam.

Thursday, April 10, 2014
2:00 p.m.
BA6183, 40 St George St.

PhD Candidate: Gregory Chambers

Supervisors: Alex Nabutovsky/Rina Rotman

Thesis title: Optimal homotopies of curves on surfaces

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Abstract:

We prove several results in quantitative topology. First, we prove that if two simple closed curves on a 2-dimensional Riemannian manifold are homotopic through curves of length at most $L$, then for any $\varepsilon > 0$, they are homotopic through simple closed curves of length at most $L + \varepsilon$. This can be seen as an e ffective version of a theorem by Baer and Epstein.

Second, we prove that if a simple closed curve is contractible on a 2-dimensional Riemannian manifold through curves of length at most $L$, then for any $\varepsilon > 0$, the curve is contractible through based loops of length at most 3$L+$2$d+ \varepsilon$, where $d$ is the diameter of the manifold. If the manifold is a 2-disc and the initial curve is the disc’s boundary, then this bound is improved to $L+$2$d+ \varepsilon$.

The thesis can be found here: Chambers_Thesis