Four-dimensional manifold theory is remarkable for a variety of reasons. It has the only outstanding generalized smooth Poincare conjecture. It is the only dimension where vector spaces have more than one smooth structure. The only dimension with an unresolved generalized Shoenflies problem. The list goes on. One issue that is perhaps not discussed enough is the paucity of theorems about smooth isotopy. In dimensions 2 and 3, the Schoenflies and Alexander theorems are the backbone of all theorems about isotopy, allowing one to work from the ground-up.

## January 1, 2020

## October 2, 2016

### A gorgeous but incomplete proof of “The Smale Conjecture”

In 1959 Stephen Smale gave a proof that the group of diffeomorphisms of the 2-sphere has the homotopy-type of the subgroup of linear diffeomorphisms, i.e. the Lie Group O_3. His proof went in two steps: (more…)

## January 5, 2015

### Topology of musical data

A few years ago a musician friend asked me “there’s this new tool topologists have called Persistent Homology. I’d like to see what it can do when you apply it to data from music. Want to help?”

That friend is also an electrical engineer and knows some things about signal processing. This was important to me — we had some external criterion (from outside of mathematics) for determining whether or not the insights from Persistent Homology were interesting or not.

So I said “okay!” Not really knowing what I was getting myself into.

## May 11, 2011

### MO-problems: codimension zero embeddings

Greetings,

Jesse recently recruited me as a *special correspondent* for the goings-on at Math Overflow. Perhaps he’ll eventually let me blog about other things! To begin I’d like to point out a lovely and easy-to-state but not-so-little problem that appeared on MO.

The above problem is perhaps a representative problem in a family of problems that have received little attention by the geometric topology community, which is the issue of low co-dimension embeddings. They are not well understood. This is because these can be rather difficult problems. More than that, there isn’t an edifice — there’s no standard machine to play with.

## May 3, 2010

### What is L-theory and why should I care: Part I

In the middle of February, Mark Powell was in Kyoto, where he taught me about algebraic surgery and other interesting topics. These are tools which were developed by high-dimensional algebraic topologists to tackle the kind of problems which **they** are interested in, and moreover they have a rather fierce reputation. So why should the low-dimensional topologist care about L-theory and suchlike? In this short series of posts, I’ll summarize the basics of what Mark taught me in February, and I’ll tell you why I care about L-theory and why I think you should care about it too (for those who can’t wait: because of Cochran-Orr-Teichner and because it provides a natural language to think about Blanchfield pairings).

(more…)