# Talking More, Flying Less: Coarse Index Theory Lectures at Freiburg

At the beginning of this year (which now seems a very long time ago) I accepted an invitation from Thomas Schick to speak in a week-long summer school at Freiburg on the subject of “Coarse methods in index theory”.   This was before the upheavals began in my life this year, one of which has been a severe illness making it impossible for me to travel.

From one point of view this is big disappointment, but from another it’s an opportunity.  I’ve long been worried by the inconsistency between the “green” values embraced by many academics, including me, and the ease with which we seem to justify jetting round the world to talk to one another about “Coarse methods in index theory”, or whatever it may be.  I totally agree that there are things we can learn face-to-face which are much harder to learn through alternative media.  Some conferences have been life and career changing for me.  But it’s hard to argue that this justifies every conference, in the face of the existential threat posed by climate change. Such is the argument made by the group of academics at flyingless.org:

Flying is an elite activity. The vast majority of the world’s population has never flown. Academics–particularly those from the world’s most prosperous countries–fly more frequently than most people do. University communities typically embrace sustainable practices in other areas of daily life. It would be inconsistent to ignore sustainability just in the case of flying.

University-based faculty, staff, and students can make large reductions in their total greenhouse gas emissions with moderate sacrifice in terms of institutional goals, professional advancement, and quality of life. However, they require mechanisms that are institutionally sensitive to differences in status, power, and position, as well as the right structural supports.

So I have an opportunity to try to implement a “flyingless” policy by delivering my lectures using remote streaming technology.  There will be two lectures which I’ll give to an audience at Penn State and which will also be livestreamed to Freiburg (assuming we can all figure out the technology in time).  The Freiburg audience will have the opportunity to interact with the speaker (me) as well as with the Penn State audience.    By not sending me physically to Germany, we save $$2\frac12$$ tons of carbon dioxide emissions (a conservative estimate) and this compares with the 4 tons per person per year which is the current global average. There may be some disadvantage to the conferees in not having me physically present but I would guess it’s small. We shall see!

Here’s a schedule of the talks for those who are interested.  It is possible that we may be able to make the stream public – in which case I’ll post the information here so anyone can watch!

Lecture 1: Title: Coarse geometry and index theory

Abstract: I will try to explain why there is a close connection between the underlying idea of coarse geometry (that geometric information is encoded in the “large scale structure” of metric spaces) and the underlying idea of index theory (that topological information is encoded in the “low energy structure” of elliptic operators).  This lecture will be livestreamed to the Coarse Index Theory conference in Freiburg, Germany (and the audience there will participate by livestream too).

Lecture 2:  Title: Coarse geometry and structure invariants

Abstract:  If an elliptic operator has index zero, then it is “stably invertible”.   The reasons for such stable invertibility can themselves be analyzed and classified; they are called analytic structures associated to the operator in question.  In this talk I’ll give an introduction to the theory and application of analytic structures.   This lecture will be livestreamed to the Coarse Index Theory conference in Freiburg, Germany (and the audience there will participate by livestream too).

# Topology, Moore or Less – Concluded

We’ve now finished the Moore method topology class that I wrote about in Topology: Moore or Less It’s been an intense experience for everyone, I think.  Many students have surprised themselves by what they have achieved.  At the end of the course we printed off copies of the co-written course textbook (complete with frontispiece photo of the authors!) and everyone received one in time for the “open book” final exam.  I hope that many students will hold on to these as a reminder of our common achievement.  Here’s the cover.

The first part of the course was descended from Bing’s notes  (and thus, indirectly, from Moore himself) as they are reproduced in the Journal of Inquiry-Based Learning in Mathematics.  After mid-semester we digressed into product and quotient topologies and then, briefly, into function spaces and the compact-open topology – the last proof that the students were guided through was a special case of the exponential law for function spaces,

$X^{Y\times Z} \cong (X^Y)^Z.$

That seemed like a good place to stop.

Students prepared their in-class presentations in “teams” of 4 rather than individually – this was a modification that I made to the standard Moore method. The online platform Piazza was the main mode of team collaboration – I held a couple of in-class sessions too where teams worked together and I acted as a roving consultant, but I should probably have done more of that.

What did the students think?  Here are a few comments:

“One of the biggest takeaways from this class was seeing how mathematics is constructed firsthand. This semester we constructed a complicated and powerful machine that I am eager to build upon in my later mathematics courses, and now I have the tools necessary to do so.”

“Having to prepare for class with what theorem or example we had to prove for the class and writing our own book has given me an understanding of the material that I don’t think would have happened if the course was taught similar to a traditional course.”

“I felt very engaged in the course as a result of the unique “Moore Method” used to teach the course.”

“The class structure challenged me to think differently than I ever had, and I genuinely appreciated that.”

I love these quotes, but of course not everyone feels the same way. I paraphrase the next comment:

“The method used to teach this class, while I can see its benefits, was really not right for me… I wanted my struggles with the material to be private, and because the class was so collaborative there wasn’t an easy way to do this.”

I hear what this student is saying, and wish I could have helped him/her better. One thing I would have liked to be able to share is that we all struggle, in math as in life, and that I had wanted the class to be a place where we could struggle together, not feel we have to project a brittle confidence or else stay silent.  That will be something to work on for next time, if I do this again.

Thanks, students! I really enjoyed the class and I hope you did too.

# Michael Atiyah’s Birthday!

Heads up!  In  a couple of days (April 22nd) it is the 87th birthday of “Britain’s mathematical pope”, (not just Britain’s, either, IMO), otherwise known as my doctoral advisor, Professor Sir Michael Atiyah.   HAPPY BIRTHDAY MICHAEL!

To celebrate, his son David is assembling an online tribute – see http://www.atiyah.eu/mfa87/    Please consider sending a tribute message to david@atiyah.eu  Here’s what hes ays:

We are collecting messages of congratulations on the occasion of Michael Atiyah‘s 87th birthday Friday, April 22, 2016.

If you have the time, memory, and an inclination, please also include your favourite personal story about Britain’s Mathematical Pope*. I keep hearing every mathematician has one – it would be a shame not to collect and archive them for posterity.

Bonus points awarded for photographs, with prizes for the best MP4 video message we can share on the night.

Pls include:
– your current position, & location (if appropriate)
– when and where you first met Michael

We will keep it simple and hope to collate and publish submisssions in due course.

* = with thanks to Siobhan Roberts for the expression used in her recent biog of J H Conway – i have simply extended his Popedom from England to Britain.

If you haven’t seen it, here is a great article from Wired last week: Mathematical Matchmaker Atiyah Dreams of a Quantum Union.

# Higher index theory with change of fundamental group

I gave a talk in our seminar yesterday which arises from trying to understand the paper of Chang, Weinberger and Yu (Chang, Stanley, Shmuel Weinberger, and Guoliang Yu. “Positive Scalar Curvature and a New Index Theorem for Noncompact Manifolds,” 2013) where they use relative index theory in a non $$\pi-\pi$$ situation to produce examples of manifolds with strange positive-scalar-curvature properties (e.g., a  non-compact manifold which has an exhaustion by compact manifolds with boundary carrying nice positive-scalar-curvature metrics, but which itself carries no such metric).

I wanted to develop an approach to this kind of index theory that was more accessible (to me) and the talk was a report on my efforts in that direction.  Here are the slides from that talk.

# Tychonoff’s theorem II

In my earlier post on Tychonoff’s theorem, I talked about the original proof, based on the following characterization of compactness which is due to Kuratowski.

Definition  Let $$S$$ be a subset of a topological space $$X$$.  A point $$x\in X$$ is a point of perfection of $$S$$ if, for every neighborhood $$U$$ of $$x$$, the set $$U\cap S$$ has the same cardinality as $$S$$.

Lemma (Kuratowski) A topological space $$X$$ is compact if and only if every infinite subset has a point of perfection.

Now I will give the proof of this characterization (again following Wright).