Not Even Wrong

My colleague Johan de Jong for the last few years has been working on an amazing mathematical endeavor he calls the “Stacks Project”. As boring 20th century technology, this is a work-in-progress document (now nearly 4000 pages), available here. But from the beginning Johan (known to his friends as “the Linus Torvalds of algebraic geometry”) has conceptualized this as an open-source project using 21st century technology, including a blog and a github repository.

As of last night, the Stacks Project has many new features, courtesy of impressive work by Johan’s collaborator on this, Pieter Belmans. I was going to write something here describing the new features and how cool they are, but a much better job of this has been done by Cathy O’Neil, aka Mathbabe (by the way, if you’re not reading Cathy’s blog, you should be…). With her permission, I’m cross-posting her new blog entry about this, so, what follows is from Cathy:

The Stacks Project gets ever awesomer with new viz

Here’s a completely biased interview I did with my husband A. Johan de Jong, who has been working with Pieter Belmans on a very cool online math project using d3js. I even made up some of his answers (with his approval).

Q: What is the Stacks Project?

A: It’s an open source textbook and reference for my field, which is algebraic geometry. It builds foundations starting from elementary college algebra and going up to algebraic stacks. It’s a self-contained exposition of all the material there, which makes it different from a research textbook or the experience you’d have reading a bunch of papers.

We were quite neurotic setting it up – everything has a proof, other results are referenced explicitly, and it’s strictly linear, which is to say there’s a strict ordering of the text so that all references are always to earlier results.

Of course the field itself has different directions, some of which are represented in the stacks project, but we had to choose a way of presenting it which allowed for this idea of linearity (of course, any mathematician thinks we can do that for all of mathematics).

Q: How has the Stacks Project website changed?

A: It started out as just a place you could download the pdf and tex files, but then Pieter Belmans came on board and he added features such as full text search, tag look-up, and a commenting system. In this latest version, we’ve added a whole bunch of features, but the most interesting one is the dynamic generation of dependency graphs.

We’ve had some crude visualizations for a while, and we made t-shirts from those pictures. I even had this deal where, if people found mathematical mistakes in the Stacks Project, they’d get a free t-shirt, and I’m happy to report that I just last week gave away my last t-shirt. Here’s an old picture of me with my adorable son (who’s now huge).

Q: Talk a little bit about the new viz.

A: First a word about the tags, which we need to understand the viz.

Every mathematical result in the Stacks Project has a “tag”, which is a four letter code, and which is a permanent reference for that result, even as other results are added before or after that one (by the way, Cathy O’Neil figured this system out).

The graphs show the logical dependencies between these tags, represented by arrows between nodes. You can see this structure in the above picture already.

So for example, if tag ABCD refers to Zariski’s Main Theorem, and tag ADFG refers to Nakayama’s Lemma, then since Zariski depends on Nakayama, there’s a logical dependency, which means the node labeled ABCD points to the node labeled ADFG in the entire graph.

Of course, we don’t really look at the entire graph, we look at the subgraph of results which a given result depends on. And we don’t draw all the arrows either, we only draw the arrows corresponding to direct references in the proofs. Which is to say, in the subgraph for Zariski, there will be a path from node ABCD to node ADFG, but not necessarily a direct link.

Q: Can we see an example?

Let’s move to an example for result 01WC, which refers to the proof that “a locally projective morphism is proper”.

First, there are two kinds of heat maps. Here’s one that defines distance as the maximum (directed) distance from the root node. In other words, how far down in the proof is this result needed? In this case the main result 01WC is bright red with a black dotted border, and any result that 01WC depends on is represented as a node. The edges are directed, although the arrows aren’t drawn, but you can figure out the direction by how the color changes. The dark blue colors are the leaf nodes that are farthest away from the root.

Another way of saying this is that the redder results are the results that are closer to it in meaning and sophistication level.

Note if we had defined the distance as the minimum distance from the root node (to come soon hopefully), then we’d have a slightly different and also meaningful way of thinking about “redness” as “relevance” to the root node.

This is a screenshot but feel free to play with it directly here. For all of the graphs, hovering over a result will cause the statement of the result to appear, which is awesome.

Next, let’s look at another kind of heat map where the color is defined as maximum distance from some leaf note in the overall graph. So dark blue nodes are basic results in algebra, sheaves, sites, cohomology, simplicial methods, and other chapters. The link is the same, you can just toggle between the different metric.

Next we delved further into how results depend on those different topics. Here, again for the same result, we can see the extent to which that result depends on the different on results from the various chapters. If you scroll over the nodes you can see more details. This is just a screenshot but you can play with it yourself here and you can collapse it in various ways corresponding to the internal hierarchy of the project.

Finally, we have a way of looking at the logical dependency graph directly, where result node is labeled with a tag and colored by “type”: whether it’s a lemma, proposition, theorem, or something else, and it also annotates the results which have separate names. Again a screenshot but play with it here, it rotates!

Check out the whole project here, and feel free to leave comments using the comment feature!

• I confess to mostly finding “philosophy of physics” arguments not very helpful for understanding anything, but for those who feel differently, some new things to look at are a Scientific American article Physicists Debate Whether the World is Made of Particles or Fields or Something Else Entirely, an interview with Jonathan Bain, an interview with Tim Maudlin, a debate between John Ellis, Lawrence Krauss and theologian Don Cupitt about Why is there something rather than nothing?, and the talks at a UCSC Philosophy of Cosmology Summer School. Since the last of these was funded by the Templeton Foundation, it ended with several talks on “Implications of cosmology for the philosophy of religion”. These included a detailed argument that the explanation for the laws of nature is “there is a perfect being”, contrasting this to another argument favored at the Summer School “the multiverse did it”.
• This week the Perimeter Institute will host Loops 13, devoted to loop quantum gravity and other quantum gravity approaches. While it’s also funded by Templeton, the organizers seem to have managed to keep God out of this one.
• At CERN, Amplitudes, Strings and Branes is on-going. Philip Gibbs has an amusing argument that this and Loops 13 are The Same Bloody Thing.
• One thing the LQG and Amplitudes people do share is that some of their most important ideas come from the same person: Roger Penrose (who, but the way, would be a good candidate for the Fundamental Physics Prize, although his distaste for string theory might be a disqualifier). There’s a long interview with him at The Ideas Roadshow, mainly about his “Cyclic Universe” ideas.
• The Simons Foundation has been publishing some excellent science reporting, and now has an online publication they’re calling Quanta Magazine. The latest story there is a very good piece on the search for dark matter from Jennifer Ouellette. The Simons Center at Stony Brook now has a newsletter about their activities.
• Another on-going conference is one of the big yearly HEP conferences, EPS HEP 2013 in Stockholm. CMS and LHCb have impressive new results about rare B decays, timed for this conference. For the details, see Tommaso Dorigo. There are also CMS and CERN press releases.

  Last year similar but less accurate results were advertised as putting SUSY “in the hospital”, which some people objected to, on the grounds that it was already in trouble and this kind of result doesn’t make things much worse. Resonaances had the details, summarizing this a “another handful of earth upon the coffin”. The CERN press office tries to put the best SUSY spin on this that it can:

  One popular theory is known as supersymmetry, SUSY for short. It postulates the existence of a new type of particle for every Standard Model particle we know, and some of these particles would have just the right properties to make up a large part of the dark universe. There are many SUSY models in circulation, and SUSY is just one of many theoretical routes to physics beyond the Standard Model. Today’s measurements allow physicists to sort between them. Many are incompatible with the new measurements, and so must be discarded, allowing the theory community to work on those that are still in the running.

• Finally, for those with mathematical interests who have waded through the above, Terry Tao has a remarkable long expository piece about the Riemann hypothesis, ranging from analytic number theory aspects through the function field case and l-adic cohomology.

Update: For more from Penrose, see this recent talk in Warsaw.

Update: Studies in History and Philosophy of Modern Physics is planning a special issue on the significance of the Higgs discovery, the call for papers is here.

Back now from vacation, and while I was away several people sent me links to point out that string theory promoters definitely aren’t taking a vacation. Links here with a few quick comments, followed by something about the issue of making fun of string theorists.

• Lenny Susskind has a quite good new book out about classical mechanics (see here), but the Economist doesn’t want to talk to him about this, instead it’s the usual string theory promotional effort:
  

  These extra dimensions can be arranged and put together in many different patterns, in a variety of different ways. Not billions, trillions or quintillions of ways, but many more than that. The ways these dimensions are put together into these tiny little spaces determine how particles will behave, what particles will exist, what the constants of nature are—quantities like dark energy or the electric charge of an electron. In string theory all those things are features of the ways that these tiny dimensions are put together. The tiny dimensions are like the DNA of the universe.

• Last month Cumrun Vafa was in Bangalore, explaining (see here, slides here) among other things about the significance of the web of string dualities that makes up M-theory:
  

  String dualities [are] in my opinion perhaps the most fundamental discovery that physicists have made in a century.

  Note that the past century includes General Relativity (barely), quantum mechanics, gauge theory, the Standard Model, as well as quite a few discoveries in other areas of physics.

• The Strings 2013 conference included the usual public talks promoting string theory. Witten’s was String Theory and the Universe, which was pretty much unchanged (minus optimism about SUSY at the LHC) from similar talks he has been giving for nearly 20 years (since 1995 and the M-theory proposal). Linde’s was Universe or Multiverse?, about the “new scientific paradigm” of Multiverse Mania. He argues that the virtue of this is that it is “impossible to disprove”.
• David Gross’s public talk on The Frontiers of Particle Physics had nothing to do with string theory, focusing on explaining the standard model and some of the questions it leaves unanswered. Much more interesting was his outlook talk at the conference which included the usual exhortations about string theory being alive and healthy, flourishing with many new and brilliant string theorists, but also included some material unusual in such a venue and much more challenging for his audience. His reference to connections between string theory and condensed matter physics described this as having been “overhyped by our community”. About AdS/CFT, he noted that it “does not provide a satisfactory non-perturbative quantum gravity”.

  He commented on the lack of connection between the talks and HEP physics, saying that it was “important that string theorists not retreat into quantum gravity”. About SUSY, he characterized it as “still alive, but not kicking”, and he argued that the LHC results of the past year have made more likely the “Extremely pessimistic scenario” of an SM Higgs, no SUSY, no dark matter, no indication of the next energy threshold. Since “HEP is where string theory connects to reality”, he made the point to the audience that “if this scenario materializes we are all going to suffer.”

  I’m not sure why he picked this date, but he encouraged those with post-1999 Ph.D.s to realize that it was now quite possible that those who came before them had “somehow got it wrong”. This was the first time I’ve seen an influential member of the string theory community raise this possibility and call for people to consider it seriously.

• Sean Carroll deals here with arguments from the “Popperazi” that the string theory anthropic multiverse is pseudo-science by ignoring the serious arguments being made. He has his own definition of what science is, which looks to me to open far more questions than it answers. About string theory unification itself, the question has never been whether it’s science, but whether it’s an idea worth pursuing given the ways in which it has so far failed. The best argument for continued pursuit of the idea is of course that there aren’t obviously better ones around, but this raises its own issues.

Sabine Hossenfelder has a posting about an introduction to a Lawrence Krauss talk where the joke was made that “String theorists have to sit in the back”. The context for this was a controversy about the place of women at a discussion involving Krauss hosted by an Islamic group. Like Sabine, I don’t want to discuss here that controversy, just agree there’s a good case to be made that it’s no joking matter. I think she makes a mistake by interpreting the joke as an attack on string theorists (it’s a joke, after all, open to many interpretations), but I was struck by her perception of string theorists as an embattled minority under unfair attack, as well as the claim that it’s not all right to in any way make fun of them.

The situation these days is clearly very different than it was back in 2004 when I started this blog, partly because of the past decade of failure of string theory unification to get anywhere, partly because of the negative LHC results, partly because of the multiverse, and partly because of the public behavior of some in the string theory community in reaction to criticism. Given the high profile ongoing promotional campaign exemplified above, I don’t think we’re yet at the point where criticism of string theorists is “kicking them when they’re down”, and humor is sometimes the best way to make a point concisely. Probably the most incisive criticism of string theory ever made was made in a cartoon, and personally I’ve never understood how it is even possible to take arguments like Linde’s seriously (and am not even sure he does…), so I see a role for humor in charting the continuing story of the collapse of the heavily influential string theory unification paradigm.

Update: If you noticed more than the usual sloppiness, incompleteness and incoherence, maybe it was because this got published early, while I was in the middle of writing it.

Update: The latest on the philosophy of science from Linde (see here), who is at a workshop in Bad Honnef this week.

The multiverse is the only known explanation so in a sense it has already been tested

Is it all right to make fun of this, or should one seriously discuss the scientific merits here?

I’ll be heading North tomorrow, ultimately ending up in backwoods Maine, hiking the Appalachian Trail, then back home after a week or so. The comment section here will likely be closed for the duration.

Progress is being made on my notes on quantum mechanics and representation theory, based on the course I taught this past year. About 3/4 done with the writing at this point, the latest version is available here.