# AMR Distinguished Lecture 1

## Henri Poincare

### University of Paris

## The Smooth 4-dimensional Me Conjecture

## July 4, 1904 - 6PM UTC (GMT)

11 AM Los Angeles, 2 PM New York, 6 PM London, 7 PM Paris, 8 PM Kyiv, 11 PM Moscow, 2 AM Singapore, 5 AM Melbourne

This lecture will cover recent developments in the Poincare Conjecture. The format will be three 15 minute segments, with questions by a panel following each segment. Below you can find resources related to this topic. An introductory background lecture by Bernhard Riemann, “*Background for understanding the 4D-Poincare Conjecture*” is available, as are other resources.

## Resources

An introductory lecture by Bernhard Riemann explains the background, history, and the context of the colloquium talk.

The Poincare Conjecture dates back to 1905 when Poincare asked if a 3-dimensional manifold with trivial homology groups was homeomorphic to a 3-sphere. He himself realized a restatement was needed, as the Poincare homology sphere gives a contradiction to the original statement. …

- A history of the Poincare Conjecture
- Poincare’s original talk
- Freedman’s proof in the topological category
- Smale’s Proof in dimensions larger than 4

Suggested background reading to follow the ideas of the lecture.

Online videos related to the lecture.

A variety of approaches to the conjecture.

- A history of the Poincare Conjecture
- Poincare’s original talk
- Freedman’s proof in the topological category
- Smale’s Proof in dimensions larger than

Some open problems connected to the Poincare Conjecture

How this solution affects topology. What are the likely directions going forward.

**Questions submitted to the speaker after the lecture and responses by the speaker.**

**Question**: Why is dimension four different from dimension 5, or 50?

**Answer**: A key step in the process is to slide around curves that give instructions on how to build a manifold. These curves trace out 2-dimensional surfaces as they move. Pairs of such surfaces are disjoint in dimensions above 4, but can have irremovable intersections in dimension 4 and below. Thus things change drastically when moving from dimension 4 to dimension 5. For more details see [references].