Peter Krautzberger · on the web

Idempotent Ultrafilters, an introduction (Michigan Logic Seminar Nov 09, 2011)

Because of a power outage at the department my talk announced for October 29th was postponed by a week.

Idempotent Ultrafilters: An Introduction (University of Michigan Logic Seminar 2011-11-08) from Peter Krautzberger on Vimeo.

Here are transcripts of my notes (as well as the originals at the end).

Hindman's Theorem

Hindman's Theorem If , then such that

Imagine you'd like to prove this with an ultrafilter:

What do we need? We will build inductively!

Pick -- we can't really choose better than that (except maybe by shrinking the set first).

If we're looking for our result, we need

In other words, we need to begin with, i.e., -- for any !

Galvin in 1970: is almost left-translation invariant iff .

Is this enough?

But to continue the process, we need more!

We need such that:

What does this mean? by associativity.

Ah! But we have seen this before!

We needed , so we needed !

But that's ok!! & !

How do we get to the end?

Question: Do "almost left-translation invariant" ultrafilters exist?

Glazer, ~1975: Yes of course! These are the idempotent ultrafilters! We know these exist since Ellis 1958.

What does this mean?

The first set is in , the second in .

Now remember: what did Galvin need?