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May 26, 2015

A 2-Categorical Approach to the Pi Calculus

Posted by John Baez

guest post by Mike Stay

Greg Meredith and I have a short paper that’s been accepted for Higher-Dimensional Rewriting and Applications (HDRA) 2015 on modeling the asynchronous polyadic pi calculus with 2-categories. We avoid domain theory entirely and model the operational semantics directly; full abstraction is almost trivial. As a nice side-effect, we get a new tool for reasoning about consumption of resources during a computation.

It’s a small piece of a much larger project, which I’d like to describe here in a series of posts. This post will talk about lambda calculus for a few reasons. First, lambda calculus is simpler, but complex enough to illustrate one of our fundamental insights. Lambda calculus is to serial computation what pi calculus is to concurrent computation; lambda calculus talks about a single machine doing a computation, while pi calculus talks about a network of machines communicating over a network with potentially random delays. There is at most one possible outcome for a computation in the lambda calculus, while there are many possible outcomes in a computation in the pi calculus. Both the lazy lambda calculus and the pi calculus, however, have as an integral part of their semantics the notion of waiting for a sub-computation to complete before moving onto another one. Second, the denotational semantics of lambda calculus in Set is well understood, as is its generalization to cartesian closed categories; this semantics is far simpler than the denotational semantics of pi calculus and serves as a good introduction. The operational semantics of lambda calculus is also simpler than that of pi calculus and there is previous work on modeling it using higher categories.

Posted at 9:28 PM UTC | Permalink | Followups (22)

SoTFoM III and The Hyperuniverse Programme

Posted by David Corfield

Following SoTFom II, which managed to feature three talks on Homotopy Type Theory, there is now a call for papers announced for SoTFoM III and The Hyperuniverse Programme, to be held in Vienna, September 21-23, 2015.

Here are the details:

The Hyperuniverse Programme, launched in 2012, and currently pursued within a Templeton-funded research project at the Kurt Gödel Research Center in Vienna, aims to identify and philosophically motivate the adoption of new set-theoretic axioms.

The programme intersects several topics in the philosophy of set theory and of mathematics, such as the nature of mathematical (set-theoretic) truth, the universe/multiverse dichotomy, the alternative conceptions of the set-theoretic multiverse, the conceptual and epistemological status of new axioms and their alternative justificatory frameworks.

The aim of SotFoM III+The Hyperuniverse Programme Joint Conference is to bring together scholars who, over the last years, have contributed mathematically and philosophically to the ongoing work and debate on the foundations and the philosophy of set theory, in particular, to the understanding and the elucidation of the aforementioned topics. The three-day conference, taking place September 21-23 at the KGRC in Vienna, will feature invited and contributed speakers.

I wonder if anyone will bring some category theory along to the meeting. Perhaps they can answer my question here.

Further details:

Posted at 2:50 PM UTC | Permalink | Followups (3)

May 22, 2015

PROPs for Linear Systems

Posted by John Baez

PROPs were developed in topology, along with operads, to describe spaces with lots of operations on them. But now some of us are using them to think about ‘signal-flow diagrams’ in control theory—an important branch of engineering. I talked about that here on the n-Café a while ago, but it’s time for an update.

Posted at 8:43 PM UTC | Permalink | Followups (3)

How to Acknowledge Your Funder

Posted by Tom Leinster

A comment today by Stefan Forcey points out ways in which US citizens can try to place legal limits on the surveillance powers of the National Security Agency, which we were discussing in the context of its links with the American Mathematical Society. If you want to act, time is of the essence!

But Stefan also tells us how he resolved a dilemma. Back here, he asked Café patrons what he should do about the fact that the NSA was offering him a grant (for non-classified work). Take their money and contribute to the normalization of the NSA’s presence within the math community, or refuse it and cause less mathematics to be created?

What he decided was to accept the funding and — in this paper at least — include a kind of protesting acknowledgement, citing his previous article for the Notices of the AMS.

I admire Stefan for openly discussing his dilemma, and I think there’s a lot to be said for how he’s handled it.

Posted at 2:28 PM UTC | Permalink | Followups (2)

May 21, 2015

The Origin of the Word “Quandle”

Posted by John Baez

A quandle is a set equipped with a binary operation with number of properties, the most important being that it distributes over itself:

a(bc)=(ab)(ac) a \triangleright (b \triangleright c) = (a \triangleright b)\triangleright (a \triangleright c)

They show up in knot theory, where they capture the essence of how the strands of a knot cross over each other… yet they manage to give an invariant of a knot, independent of the way you draw it. Even better, the quandle is a complete invariant of knots: if two knots have isomorphic quandles, there’s a diffeomorphism of 3\mathbb{R}^3 mapping one knot to the other.

I’ve always wondered where the name ‘quandle’ came from. So I decided to ask their inventor, David Joyce—who also proved the theorem I just mentioned.

Posted at 7:41 AM UTC | Permalink | Followups (10)

May 18, 2015

The Revolution Will Not Be Formalized

Posted by Mike Shulman

After a discussion with Michael Harris over at the blog about his book Mathematics without apologies, I realized that there is a lot of confusion surrounding the relationship between homotopy type theory and computer formalization — and that moreover, this confusion may be causing people to react negatively to one or the other due to incorrect associations. There are good reasons to be confused, because the relationship is complicated, and various statements by prominent members of both communities about a “revolution” haven’t helped matters. This post and its sequel(s) are my attempt to clear things up.

Posted at 8:46 PM UTC | Permalink | Followups (7)

May 13, 2015

Categorifying the Magnitude of a Graph

Posted by Simon Willerton

Tom Leinster introduced the idea of the magnitude of graphs (first at the Café and then in a paper). I’ve been working with my mathematical brother Richard Hepworth on categorifying this and our paper has just appeared on the arXiv.

Categorifying the magnitude of a graph, Richard Hepworth and Simon Willerton.

The magnitude of a graph can be thought of as an integer power series. For example, consider the Petersen graph.

Petersen graph

Its magnitude starts in the following way. #P =1030q+30q 2+90q 3450q 4 +810q 5+270q 65670q 7+. \begin{aligned} \#P&=10-30q+30q^{2}+90q ^{3}-450q^{4}\\ &\quad\quad+810q^{5} + 270 q^{6} - 5670 q^{7} +\dots. \end{aligned}

Richard observed that associated to each graph GG there is a bigraded group MH *,*(G)\mathrm{MH}_{\ast ,\ast }(G), the graph magnitude homology of GG, that has the graph magnitude #G# G as its graded Euler characteristic. #G = k,l0(1) krank(MH k,l(G))q l = l0χ(MH *,l(G))q l. \begin{aligned} #G &= \sum _{k,l\geqslant 0} (-1)^{k}\cdot \mathrm{rank}\bigl (\mathrm{MH}_{k,l}(G)\bigr )\cdot q^{l}\\ &= \sum _{l\geqslant 0} \chi \bigl (\mathrm{MH}_{\ast ,l}(G)\bigr )\cdot q^{l}. \end{aligned} So graph magnitude homology categorifies graph magnitude in the same sense that Khovanov homology categorifies the Jones polynomial.

For instance, for the Petersen graph, the ranks of MH k,l(P)\mathrm{MH}_{k,l}(P) are given in the following table. You can check that the alternating sum of each row gives a coefficient in the above power series.

k 0 1 2 3 4 5 6 7 0 10 1 30 2 30 3 120 30 l 4 480 30 5 840 30 6 1440 1200 30 7 7200 1560 30 \begin{array}{rrrrrrrrrr} &&&&&&k\\ &&0&1&2&3&4&5&6&7 \\ &0 & 10\\ & 1 & & 30 \\ &2 & && 30 \\ &3 &&& 120 & 30 \\ l &4 &&&& 480 & 30 \\ &5 &&&&& 840 & 30 \\ &6 &&&&& 1440 & 1200 & 30 \\ &7 &&&&&& 7200 & 1560 & 30 \\ \\ \end{array}

Many of the properties that Tom proved for the magnitude are shadows of properties of magnitude homology and I’ll describe them here.

Posted at 4:18 PM UTC | Permalink | Followups (19)

April 30, 2015

Breakfast at the n-Category Café

Posted by Emily Riehl

Michael Harris recently joined us for breakfast at the n-category café. Perhaps some readers here would be interested in following the postprandial discussion that is underway on his blog: Mathematics Without Apologies.

Posted at 6:30 PM UTC | Permalink | Followups (2)

April 28, 2015

Categories in Control

Posted by John Baez

To understand ecosystems, ultimately will be to understand networks. - B. C. Patten and M. Witkamp

A while back I decided one way to apply my math skills to help save the planet was to start pushing toward green mathematics: a kind of mathematics that can interact with biology and ecology just as fruitfully as traditional mathematics interacts with physics. As usual with math, the payoffs will come slowly, but they may be large. It’s not a substitute for doing other, more urgent things—but if mathematicians don’t do this, who will?

As a first step in this direction, I decided to study networks.

This May, a small group of mathematicians is meeting in Turin for a workshop on the categorical foundations of network theory, organized by Jacob Biamonte. I’m trying to get us mentally prepared for this. We all have different ideas, yet they should fit together somehow.

Tobias Fritz, Eugene Lerman and David Spivak have all written articles here about their work, though I suspect Eugene will have a lot of completely new things to say, too. Now I want to say a bit about what I’ve been doing with Jason Erbele.

Posted at 10:42 PM UTC | Permalink | Followups (17)

April 24, 2015

A Synthetic Approach to Higher Equalities

Posted by Mike Shulman

At last, I have a complete draft of my chapter for Elaine Landry’s book Categories for the working philosopher. It’s currently titled

  • Homotopy Type Theory: A synthetic approach to higher equalities. pdf

As you can see (if you read it), not much is left of the one fragment of draft that I posted earlier; I decided to spend the available space on HoTT itself rather than detour into synthetic mathematics more generally. Although the conversations arising from that draft were still helpful, and my other recent ramblings did make it in.

Comments, questions, and suggestions would be very much appreciated! It’s due this Sunday (I got an extension from the original deadline), so there’s a very short window of time to make changes before I have to submit it. I expect I’ll be able to revise it again later in the process, though.

Posted at 4:06 AM UTC | Permalink | Followups (82)

April 12, 2015

The Structure of A

Posted by David Corfield

I attended a workshop last week down in Bristol organised by James Ladyman and Stuart Presnell, as part of their Homotopy Type Theory project.

Urs was there, showing everyone his magical conjuring trick where the world emerges out of the opposition between \emptyset and *\ast\; in Modern Physics formalized in Modal Homotopy Type Theory.

Jamie Vicary spoke on the Categorified Heisenberg Algebra. (See also John’s page.) After the talk, interesting connections were discussed with dependent linear type theory and tangent (infinity, 1)-toposes. It seems that André Joyal and colleagues are working on the latter. This should link up with Urs’s Quantization via Linear homotopy types at some stage.

As for me, I was speaking on the subject of my chapter for the book that Mike’s Introduction to Synthetic Mathematics and John’s Concepts of Sameness will appear in. It’s on reviving the philosophy of geometry through the (synthetic) approach of cohesion.

In the talk I mentioned the outcome of some further thinking about how to treat the phrase ‘the structure of AA’ for a mathematical entity. It occurred to me to combine what I wrote in that discussion we once had on The covariance of coloured balls with the analysis of ‘the’ from The King of France thread. After the event I thought I’d write out a note explaining this point of view, and it can be found here. Thanks to Mike and Urs for suggestions and comments.

The long and the short of it is that there’s no great need for the word ‘structure’ when using homotopy type theory. If anyone has any thoughts, I’d like to hear them.

Posted at 2:01 PM UTC | Permalink | Followups (144)

April 7, 2015

Information and Entropy in Biological Systems

Posted by John Baez

I’m helping run a workshop on Information and Entropy in Biological Systems at NIMBioS, the National Institute of Mathematical and Biological Synthesis, which is in Knoxville Tennessee.

I think you’ll be able to watch live streaming video of this workshop while it’s taking place from Wednesday April 8th to Friday April 10th. Later, videos will be made available in a permanent location.

To watch the workshop live, go here. Go down to where it says

Investigative Workshop: Information and Entropy in Biological Systems

Then click where it says live link. There’s nothing there now, but I’m hoping there will be when the show starts!

Posted at 4:17 AM UTC | Permalink | Post a Comment

April 6, 2015

Five Quickies

Posted by Tom Leinster

I’m leaving tomorrow for an “investigative workshop” on Information and Entropy in Biological Systems, co-organized by our own John Baez, in Knoxville, Tennessee. I’m excited! And I’m hoping to learn a lot.

A quick linkdump before I go:

Posted at 3:09 PM UTC | Permalink | Followups (56)

March 14, 2015

Split Octonions and the Rolling Ball

Posted by John Baez

You may enjoy these webpages:

because they explain a nice example of the Erlangen Program more tersely — and I hope more simply — than before, with the help of some animations made by Geoffrey Dixon using WebGL. You can actually get a ball to roll in way that illustrates the incidence geometry associated to the exceptional Lie group G 2\mathrm{G}_2!

Abstract. Understanding exceptional Lie groups as the symmetry groups of more familiar objects is a fascinating challenge. The compact form of the smallest exceptional Lie group, G 2\mathrm{G}_2, is the symmetry group of an 8-dimensional nonassociative algebra called the octonions. However, another form of this group arises as symmetries of a simple problem in classical mechanics! The space of configurations of a ball rolling on another ball without slipping or twisting defines a manifold where the tangent space of each point is equipped with a 2-dimensional subspace describing the allowed infinitesimal motions. Under certain special conditions, the split real form of G 2\mathrm{G}_2 acts as symmetries. We can understand this using the quaternions together with an 8-dimensional algebra called the ‘split octonions’. The rolling ball picture makes the geometry associated to G 2\mathrm{G}_2 quite vivid. This is joint work with James Dolan and John Huerta, with animations created by Geoffrey Dixon.

Posted at 1:18 AM UTC | Permalink | Followups (12)

March 11, 2015

A Scale-Dependent Notion of Dimension for Metric Spaces (Part 1)

Posted by Simon Willerton

Consider the following shape that we are zooming into and out of. What dimension would you say it was?

zooming dots

At small scales (or when it is very far away) it appears as a point, so seems zero-dimensional. At bigger scales it appears to be a line, so seems one-dimensional. At even bigger scales it appears to have breadth as well as length, so seems two-dimensional. Then, finally, at very large scales it appears to be made from many widely separated points, so seems zero-dimensional again.

We arrive at an important observation:

The perceived dimension varies with the scale at which the shape is viewed.

Here is a graph of my attempt to capture this perceived notion of dimension mathematically.

A graph

Hopefully you can see that, moving from the left, you start off at zero, then move into a region where the function takes value around one, then briefly moves up to two then drops down to zero again.

The ‘shape’ that we are zooming into above is actually a grid of 3000×163000\times 16 points as that’s all my computer could handle easily in making the above picture. If I’d had a much bigger computer and used say 10 7×10 210^{7}\times 10^{2} points then I would have got something that more obviously had both a one-dimensional and two-dimensional regime.

In this post I want to explain how this idea of dimension comes about. It relies on a notion of ‘size’ of metric spaces, for instance, you could use Tom’s notion of magnitude, but the above picture uses my notion of 22-spread.

I have mentioned this idea before at the Café in the post on my spread paper, but I wanted to expand on it somewhat.

Posted at 3:25 PM UTC | Permalink | Followups (19)