# Various topics in discrete and comparison geometry

*Algebra/Geometry/Topology Seminar*

*by Chris Goddard*

*Institution:*University of Melbourne

*Date: Mon 2nd March 2009*

*Time: 2:15 PM*

*Location: David Caro Building, Room 211*

*Abstract*: In this talk I will discuss some investigations of mine within the general area of comparison geometry. The structure will be as follows:

(i) Basic definitions from differential geometry

(ii) Quick survey of the core concepts within Morse theory

(iii) Some emphasis on the connection between Morse theory and comparison geometry

(iv) Mention of the extension of Morse theory to stratified spaces

(v) Brief mention/motivation of the first research project "discrete comparison geometry"

I will conclude in the last 15-20 minutes with a relatively condensed summary of my proof of an interesting new structure result for the cut locus of a Riemannian manifold, which states that any sufficiently "nice" spine of a manifold (that is not a 2-sphere) can be deformed into a cut locus for the same space. This will be motivated by a quick synopsis of a paper due to Alan Weinstein which was the basis of his PhD thesis.