The n-Category Café

In his article – The Invisible Link Between Mathematics and Theology, in Perspectives on Science and Christian Faith, Vol. 56, pp. 111-116 – Ladislav Kvasz argues for the thesis that

…monotheistic theology with its idea of the omniscient and omnipotent God, who created the world, influenced in an indirect way the process of this mathematicization. In separating ontology from epistemology, monotheistic theology opened the possibility to explain all the ambiguity connected to these phenomena as a result of human finitude and so to understand the phenomena themselves as unambiguous, and therefore accessible to mathematical description.

If pigs could fly…

… what would the newspapers say?

The $n$-Café is buzzing with activity. It’s frustrating! My pal Minhyong is busy explaining how to visualize the algebraic completion of the rational numbers, and my student Alex is busy introducing us to integral transforms in derived algebraic geometry. But I’m too busy to read any of that stuff. I’m supposed to write a history of the Earth for the British magazine PhysicsWorld by May 1st. And it’s supposed to be 3000 words or less! So each word needs to describe 1.5 million years — a heavy responsibility.

• John Baez, The Earth — For Physicists.

In our Journal Club on geometric $\infty$-function theory this first official week starts with Alex Hoffnung talking about section 1 of “Integral Transforms”.

This is to get us going and hopefully also reduce the intimidation level. If it looks interesting, have a look at our schedule. We are still looking for volunteers who would like to have a look at section 4,5, and 6 of “Integral Transforms” and write some kind of report for us all, to start further discussion.

Big news! It seems Mike Hopkins, Doug Ravenel and Mike Hill have cracked the Kervaire Invariant One problem.

Hopkins announced this in a maximally dramatic fashion, as explained below…

Fishing about at the Café for material to extract for $n$Lab, I was reminded of a question I had never got around to asking. I was also led to the third of the talks listed here (note caveat there concerning change of sentiments, and here concerning the second half), where Minhyong Kim writes:

For years, I’ve felt the need to deny the popular conception of mathematics that equates it with the study of numbers. It is only recently that I’m returning to a suspicion that mathematics is perhaps about numbers after all. It is said that in ancient Greece the comparison of large quantities was regarded as a very difficult problem. So it was debated by the best thinkers of the era whether there were more grains of sand on the beach or more leaves on the trees of the forest. Equipped now with systematic notation and fluency in the arithmetic of large integers, it is a straightforward (albeit tedious) matter for even a schoolchild to give an intelligent answer to such a question. At present, our understanding of the complex numbers is about as primitive as the understanding of large integers was in ancient Greece.

For some time now we’ve been comparing different approaches to ‘smooth spaces’ — generalizations of manifolds that have proved handy in math and physics. Here’s a thesis on the subject:

• Martin Laubinger, Differential Geometry in Cartesian Closed Categories of Smooth Spaces, Ph.D. thesis, Louisiana State University, February 2008.

I thank Eugene Lerman for pointing this out. Alex Hoffnung has contacted Laubinger and let him know that there’s a community of people out here studying this subject.

Starting in fall 2009, Hamburg University will offer a research oriented master program leading to the degree of a Master in Mathematical Physics.

The program is taught by researchers of the Mathematics and Physics department of Hamburg University and the theory group of DESY. It aims at students holding a bachelor degree in mathematics or physics and prepares in particular for PhD programs in mathematics and physics.

The program is comprised of a one year course including lectures on:

- Differential Geometry

- Topology

- Lie Theory

- Homological Algebra

- Complex Algebraic Geometry

- Quantum Field Theory

- String Theory

- Conformal Field Theory

- Supersymmetry and Unification

- General Relativity

and a second year of individually supervised research. The courses offered are under annual review, please consult the webpage for up-to-date information.

For more information, including deadlines, see here.

A pdf-sheet with more information is here.

This announcement itself in pdf-form is here.

I just read Mike Shulman’s report on talks by people including Tom Leinster and Aaron Lauda at the 88th PSSL in Cambridge — and it’s a funny feeling, because I’m in Glasgow and having fun talking to them here! They sure get around. Together with about 50 other mathematicians, we’re at this workshop:

• Categorification and Geometrisation from Representation Theory, April 13–18, 2009, Department of Mathematics, University of Glasgow, organized by Ken Brown, Iain Gordon, Catharina Stroppel, Nicolai Reshetikhin and Raphael Rouquier.

Every talk at this workshop is worth blogging about, but I won’t have energy for that. Let me talk about the very first one and see what happens from there.

guest post by Mike Shulman

Hi everyone! Urs invited me to update everyone on the highlights of the PSSL 88 in Cambridge, U.K.

One of the highlights for me was meeting people who I’d previously only known virtually, including some nStuff patrons like Bruce Bartlett, Finn Lawler, and Ronnie Brown. I would also be remiss if I didn’t give honor to Peter Johnstone and Martin Hyland, in recognition of whose 60th birthdays the conference was being held; I met them both when I was doing Part III in Cambridge and was very inspired by them. There were also lots of great talks, which I’ll try to summarize; also, I think that some slides will be posted eventually.

We decided here that it might be fun to go together through the recent Ben-Zvi/Francis/Nadler work on

geometric $\infty$-function theory

in a kind of online “journal club”. The idea would be to go slowly but surely, step-by-step through the material, discuss it, ask questions about it, understand it, and, crucially, distill the joint insight into entries on the $n$Lab.

To start with, I have created in parallel to this blog entry here an $n$Lab entry. The idea is that this blog entry here is the base for the discussion part of our “Journal club”, while that $n$Lab entry is the base for the write-up part of the undertaking.

We want to sort out questions, strategy and other things that require discussion here on the blog, but want to be sure that all stable insights that we gain in the process will eventually be distilled into $n$Lab entries, lest all the effort will leave no useful traces.

Therefore, in a combined manner similar to what I tentatively started doing for the book Categories and Sheaves and the book Higher Topos Theory and for a possibly similar joint seminar at Journal Club – $(\infty,1)$-categories, the idea is that at the entry geometric $\infty$-function theory we have a bit of background information, literature, summary, overviews, and then crucially a section-by-section list of links to $n$Lab entries which concern themselves with discussing the relavent keywords.

All this will be very incomplete and preliminary in the beginning, but it may be fun to eventually and incrementrally fill in substance and thereby eventually create a useful web of linked entries that become a useful resource for reading, learning and thinking about this stuff.

Currently, there is some first bit of content at that entry, as much as I could come up with using a bit of virtual time that I didn’t really have. As with the rest of the $n$Lab, the stuff that is there is not meant to imply to be complete or even necessarily good. The maxime is: incrementally improve.

If you see anything that makes you raise an eyebrow – or if your eyebrows raise because you don’t see something – then you are in the right state for contributing: either add a query box as described at HowTo and drop the rest of us a note on why you are unhappy and what you think should be improved, or – much better – hit the “edit” button on the bottom of the page and implement an improvement. But then, please, if it is anything nontrivial, drop the rest of us a brief note indicating what you did, either in the comment section below or at latest changes.

Any fan of diagrammatic reasoning, category theory, or computer-aided reasoning should be intrigued by this:

• Workshop on Computer Algebra Methods and Commutativity of Algebraic Diagrams, IRIT, Toulouse, France, October 16–17, 2009, organized by P. Damphousse, Y. Lafont, R. Matthes and S. Soloviev.

For some mysterious reason this conference doesn’t have a webpage. Maybe that’s why they call it theoretical computer science.

If you could commission a computer demonstration of any categorical idea, what would you ask for? Could such demonstrations have helped you, or your students, learn tricky ideas? And, would you be willing to share the visualisations and metaphors that you have devised to explain these ideas to yourself or others?

On Friday I’m going to the workshop on Categorification and Geometrisation From Representation Theory in Glasgow. I plan to learn a lot of algebra and give this talk:

• John Baez, Categorification and Topology.

The weekend after that I’ll give a more leisurely version of this talk at the Graduate Student Topology Conference in Wisconsin.

I started writing an entry

Heuristic introduction to sheaves, cohomology and higher stacks

This is (supposed to be) a pedagogical motivation of the concepts sheaf, stack, $\infty$-stack and higher topos theory. It assumes only that the reader has a heuristic knowledge of topological spaces and aims to provide from that a heuristic but useful idea of the relevance of the circle of ideas of categories and sheaves, nonabelian cohomology, sheaf cohomology and a bit of higher topos theory.

t should be readable, but will eventually need more polishing. I’d be grateful if readers with very little or no knowledge about sheaves and cohomology could tell me how helpful or not they find this.

In the course of writing this exposition, I also created an entry

cohomology

about the very general notion in the context of higher $(\infty,1)$-categorical topos theory.

I was sifting through The Oxford Handbook of Philosophy of Mathematics and Logic to see whether it included any trace of my philosophical heroes. Lautman was always going to be an outside bet, but I reckoned on finding Lakatos there.

Conspicuously absent in the index, I resorted to Google Books to help me search, and there, along with some mentions of Lakatos as editor of a book that Kreisel had contributed to, I finally found in the introduction:

It’s good to see other people talking about things we chat about here. So I was interested to see today Tobias Fritz’s paper Convex Spaces I: Definition and Examples:

We propose an abstract definition of convex spaces as sets where one can take convex combinations in a consistent way. A priori, a convex space is an algebra over a finitary version of the Giry monad. We identify the corresponding Lawvere theory as the category from arXiv:0902.2554 and use the results obtained there to extract a concrete definition of convex space in terms of a family of binary operations satisfying certain compatibility conditions. After giving an extensive list of examples of convex sets as they appear throughout mathematics and theoretical physics, we find that there also exist convex spaces that cannot be embedded into a vector space: semilattices are a class of examples of purely combinatorial type. In an information-theoretic interpretation, convex subsets of vector spaces are probabilistic, while semilattices are possibilistic. Convex spaces unify these two concepts.