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

Friday, August 9, 2019
2:10 p.m.

PhD Candidate:  Justin Martel
Supervisor:   Robert McCann
Thesis title:  Applications of Optimal Transport to Algebraic Topology: A Method for Constructing Spines from Singularity


Our thesis describes new applications of optimal transport to algebraic topology. We use a variational definition of singularity based on semicouplings and Kantorovich duality, and develop a method for building Spines (Souls) of manifolds from singularities. For example, given a complete finite-volume manifold X we identify subvarieties Z of X and construct continuous homotopy-reductions from X onto Z using the above variational definition of singularities.

The main goal of the thesis is constructing compact Z with maximal codimension in X. The subvarieties Z are assembled from a contravariant functor arising from Kantorovich duality and solutions to a semicoupling program.

The program seeks semicoupling measures from a source (X,σ) to target (Y, τ) which minimize total transport with respect to a cost c. Best results are obtained with a class of anti-quadratic costs we call “repulsion costs”. We apply the above homotopy-reductions to the problem of constructing explicit small-dimensional EΓ classifying space models, where Γ is a finite-dimensional Bieri-Eckmann duality group.

A copy of the thesis can be found here: ut-thesis


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