# Topology

## WCATSS Problem Set 2

This is the second problem of the West Coast Algebraic Topology Summer School (WCATSS)

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## WCATSS Problem Set 1

This is the first problem of the West Coast Algebraic Topology Summer School (WCATSS)

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## Algebraic Topology- Methods, Computation and Science 6

## A short elementary survey of Symplectic Topology

Symplectic topology can be thought as the mathematical versant of String theory: they were born independently at the same time, the second one as a fantastic enterprise to unify large-scale and low-scale physics, and the first one to solve classical dynamical problems on periodic orbits of physical problems, the famous Arnold conjectures. In the 80's, Gromov's revolutionary work opened a new perspective by presenting symplectic topology as an almost Kähler geometry (a concept that he defined), and constructing the corresponding theory which is entirely covariant (whereas algebraic geometry is entirely contravariant). A few years later, Floer and Hofer established the bridge between the two interpretations of Symplectic topology, the one as a dynamical theory and the one as a Kähler theory. This bridge was confirmed for the first time by Lalonde-McDuff who related explicitly the first theory to the second by showing that Gromov's Non-Squeezing Theorem is equivalent to Hofer's energy-capacity inequality.

Nowadays, Symplectic Topology is a very vibrant subject, and there is perhaps no other subject that produces new and deep moduli spaces at such a pace ! More recent results will also be presented.

## Mathematics of Doodling

The Richard & Louise Guy Lecture Series is an annual lecture event bringing world-renowned mathematicians to Calgary to share the joy of mathematics with a public audience. The lecture series celebrates the joy of discovery and wonder in mathematics for everyone. Indeed, the lecture series was a 90th birthday present from Louise to Richard in recognition of his love of mathematics and his desire to share that love with the world.

This year's lecture is "The Mathematics of Doodling" delivered by Dr. Ravi Vakil (Stanford University):

"Doodling has many mathematical aspects: patterns, shapes, numbers,and more. Not surprisingly, there is often some sophisticated and fun mathematics buried inside common doodles. I’ll begin by doodling, and see where it takes us. It looks like play, but it reflects what mathematics is really about: finding patterns in nature, explaining them, and extending them. By the end, we’ll have seen important notions in geometry, topology, physics, and elsewhere; some fundamental ideas guiding the development of mathematics over the course of the last century; and ongoing work continuing today."

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## 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

## Asymptotic dimension

Most of the talk will be an introduction to (second) bounded cohomology of a discrete group. I will explain classical constructions of bounded cocycles and recent results (joint with Bromberg and Fujiwara) regarding mapping class groups and a construction of bounded cocycles with coefficients in an arbitrary unitary representation.

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## Second Bounded Cohomology

Most of the talk will be an introduction to (second) bounded cohomology of a discrete group. I will explain classical constructions of bounded cocycles and recent results (joint with Bromberg and Fujiwara) regarding mapping class groups and a construction of bounded cocycles with coefficients in an arbitrary unitary representation.

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## 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.