# Differential Geometry and Geometric Analysis

## Decision problems, curvature and topology

I shall discuss a range of problems in which groups mediate between topological/geometric constructions and algorithmic problems elsewhere in mathematics, with impact in both directions. I shall begin with a discussion of sphere recognition in different dimensions. I'll explain why there is no algorithm that can determine if a compact homology sphere of dimension 5 or more has a non-trivial finite-sheeted covering. I'll sketch how ideas coming from the study of CAT(0) cube complexes were used by Henry Wilton and me to settle isomorphism problems for profinite groups, and to settle a conjecture in combinatorics concerning the extension problem for sets of partial permutations.

## Adam Clay Lecture 2 of 2

This lecture is part of a course organized by Dale Rolfsen.

- Read more about Adam Clay Lecture 2 of 2
- 2217 reads

## Adam Clay Lecture 1 of 2

This lecture is part of a course organized by Dale Rolfsen.

- Read more about Adam Clay Lecture 1 of 2
- 2681 reads

## The Work of Misha Gromov, a Truly Original Thinker

The work of Misha Gromov has revolutionized geometry in many respects, but at the same time introduced a geometric point of view in many questions. His impact is very broad and one can say without exaggeration that many fields are not the same after the introduction of Gromov's ideas.I will try and explain several avenues that Gromov has been pursuing, stressing the changes in points of view that he brought in non-technical terms.Here is a list of topics that the lecture will touch:

- The h-Principle
- Distance and Riemannian Geometry
- Group Theory and Negative Curvature
- Symplectic Geometry
- A wealth of Geometric Invariants
- Interface with other Sciences
- Conceptualizing Concept Creation

## A glimpse into the differential geometry and topology of optimal transportation

The Monge-Kantorovich optimal transportation problem is to pair producers with consumers so as to minimize a given transportation cost. When the producers and consumers are modeled by probability densities on two given manifolds or subdomains, it is interesting to try to understand the structure of the optimal pairing as a subset of the product manifold. This subset may or may not be the graph of a map.

The talk will expose the differential topology and geometry underlying many basic phenomena in optimal transportation. It surveys questions concerning Monge maps and Kantorovich measures: existence and regularity of the former, uniqueness of the latter, and estimates for the dimension of its support, as well as the associated linear programming duality. It shows the answers to these questions concern the differential geometry and topology of the chosen transportation cost. It establishes new connections --- some heuristic and others rigorous ---based on the properties of the cross-difference of this cost, and its Taylor expansion at the diagonal.

See preprint at www.math.toronto.edu/mccann/publications

## Gauge Theory and Khovanov Homology

After reviewing ordinary finite-dimensional Morse theory, I will explain how Morse generalized Morse theory to loop spaces, and how Floer generalized it to gauge theory on a three-manifold. Then I will describe an analog of Floer cohomology with the gauge group taken to be a complex Lie group (rather than a compact group as assumed by Floer), and how this is expected to be related to the Jones polynomial of knots and Khovanov homology.

## Embedding questions in symplectic geometry

As has been known since the time of Gromov's Nonsqueezing Theorem, symplectic embedding questions lie at the heart of symplectic geometry.

In the past few years we have gained significant new insight into the question of when there is a symplectic embedding of one basic geometric shape (such as a ball or ellipsoid)into another (such as an ellipsoid or torus). After a brief introduction to symplectic geometry, this talk will describe some of this progress, with particular emphasis on results in dimension four.

## Introduction to Marsden & Symmetry

Alan Weinstein is a Professor of the Graduate School in the Department of Mathematics at the University of California, Berkeley. He was a colleague of Jerry Marsden throughout Jerry’s career at Berkeley, and their joint papers on “Reduction of symplectic manifolds with symmetry” and “The Hamiltonian structure of the Maxwell-Vlasov equations” were fundamental contributions to geometric mechanics.

## New geometric and functional analytic ideas arising from problems in symplectic geometry

The study of moduli spaces of holomorphic curves in symplectic geometry is the key ingredient for the construction of symplectic invariants. These moduli spaces are suitable compactifications of solution spaces of a first order nonlinear Cauchy-Riemann type operator. The solution spaces are usually not compact due to bubbling-off phenomena and other analytical difficulties.