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October 30, 2013

The HoTT Approach to Physics

Posted by David Corfield

Summer saw the foundations of mathematics rocked by the publication of The HoTT Book. Here we are a few months later and the same has happened to physics with the appearance on the ArXiv of Urs’s Differential cohomology in a cohesive infinity-topos.

Physics clearly needs more than the bare homotopy types of HoTT. Field configurations may be groupoids (1-types) under gauge equivalence, or indeed groupoids\infty-groupoids (homotopy types) under gauge-of-gauge-of-… equivalence, but they also possess differentiable structure. The question then is how to cater for all of those principal bundles, connections, curvature forms, and in more recent times 2-bundles, orbifolds, Lie \infty-algebroids,…, while building on HoTT, or, in terms of the environments in which HoTT (or Univalent Foundations) functions, while adding structure to (,1)(\infty, 1)-toposes.

Posted at 10:21 AM UTC | Permalink | Followups (159)

October 21, 2013

Jet Categories at the nForum

Posted by David Corfield

Some people I talk to who have noticed a slackening off at the Café in recent months, and who know that some of this is due to John’s energy passing to his Azimuth project, don’t seem aware that another chunk of the energy didn’t just vanish, but got transmitted to the nForum. This venue has the advantage of democratically allowing anyone to initiate a discussion, but the disadvantage that people don’t seem to want to wade through every announcement of any alteration to an nLab entry for the occasional interesting nugget. For whatever reason, we don’t get visited much nowadays by some of the prestigious visitors of yesteryear. Still, if you want to see the day to day movements of Urs sweeping up great clumps of mathematical physics into a glorious synthetic package, the nForum is the place to be. Or, if you prefer to read the finished product, see his site.

Something we occasionally suggest to each other is a periodic digest of what’s happening at the nnLab, but I believe we’ve only managed two to date (I, II). Let me try something less ambitious.

Posted at 10:23 AM UTC | Permalink | Followups (3)

October 16, 2013

Announcing the Kan Extension Seminar

Posted by Emily Riehl

Daniel Kan’s influence at MIT persists through something called the Kan seminar, a graduate reading course in algebraic topology. Over the course of a semester, each student is asked to give a few one-hour lectures summarizing classic papers in the field and to engage with each other paper by writing a reading response. The lectures are preceded by a practice talk of unbounded length that is conducted in private, i.e., in the absence of the lead instructor, before the reading responses are due. This format aims to teach students how to read papers quickly and at various levels of depth, as well as to work on presentation skills. At the semester’s conclusion, Kan traditionally hosted a party that took advantage of Boston’s high concentration of mathematicians, giving his students an opportunity to meet senior people in the field.

This (northern hemisphere) spring, from early January to late June 2014, I plan to run an online (“extension”) Kan seminar in category theory with the aim of reading the twelve papers listed below. I am seeking between 6 and 12 participants who will compose one or two blog posts to appear here on the nn-Category Café over the course of the six months, which will be published every other week. Everyone will be expected to write comments, engaging with all of the papers.

  • Lawvere, An elementary theory of the category of sets

  • Street, The formal theory of monads

  • Freyd-Kelly, Categories of continuous functors, I

  • Lawvere, Metric spaces, generalized logic and closed categories

  • Kelly-Street, Review of the elements of 2-categories

  • Street-Walters, Yoneda structures on 2-categories

  • Johnstone, On a topological topos

  • Kelly, Elementary observations on 2-categorical limits

  • Blackwell-Kelly-Power, Two-dimensional monad theory

  • Adámek-Borceux-Lack-Rosický, A classification of accessible categories

  • Lack, Codescent objects and coherence

  • Shulman, Enriched indexed categories

Posted at 4:34 PM UTC | Permalink | Followups (5)

October 6, 2013

Unexpected Connections

Posted by Tom Leinster

On Wednesday I’ll give a half-hour talk to all the new maths PhD students in Scotland, called Unexpected connections. What should I put in it?

When British students arrive to do a PhD, they have already chosen a supervisor, and they have a fairly good idea of what their PhD topic will be. So, they’re in the mood to specialize. On the other hand, they’re obliged to take some courses, and here in Scotland the emphasis is on broadening — balancing that specialization by learning a wide range of subjects.

Some students don’t like this. They’re not undergraduates any more, they’ve decided what they want to work on, and they resent being made to study other things. My job is to enthuse them about the wider, wilder possibilities — to tell them about some of the amazing advances that have been made by bringing together parts of mathematics that might appear to be completely unconnected.

What are some compelling stories I could tell? What are your favourite examples of apparently disparate mathematical topics that have been brought together to extraordinary effect?

The more disconnected the topics seem to be, the better. Best of all would be stories that connect pure mathematics with either applied mathematics or statistics.

Posted at 11:08 PM UTC | Permalink | Followups (40)

October 3, 2013

Witten Looking Anew at the Jones Polynomial

Posted by David Corfield

Guest post by Bruce Bartlett

Today at the Clay Research Conference, Edward Witten gave a talk on A new look at the Jones polynomial of a knot. It was an opportune moment, 25 years after his original original paper.

Let me give a quick report-back. Hopefully the video and slides will be available on the Clay website at some point, but that may take some years!

Posted at 8:47 AM UTC | Permalink | Followups (5)

October 2, 2013

Who Ordered That?

Posted by Tom Leinster

Prize for the most peculiar theorem of the year must surely go to my colleague Natalia Iyudu and her collaborator Stanislav Shkarin, who recently proved the following conjecture of Kontsevich.

Start with a 3×33 \times 3 matrix.

Take its transpose, then take the reciprocal of each entry, then take the inverse of the whole matrix.

Take the transpose of that, then take the reciprocal of each entry, then take the matrix inverse.

Take the transpose of that, then take the reciprocal of each entry, and then, finally, take the matrix inverse.

Theorem: Up to a bit of messing about, you’re back where you started.

What on earth does this mean? It’s not clear that anyone really knows.

Posted at 9:53 PM UTC | Permalink | Followups (11)