The Typographical Equivalent of a Knife Fight

I've been thinking recently about typefaces -- the four to eight readers of this blog may have noticed that the font used in the body text of these posts has changed.  I've also been thinking about best practices for mathematical typesetting, for my next paper.  I might write a more serious post on this topic another time.

While researching the topic, I ran into a few interesting articles:

  • Adam Townsend has written a nice article about choosing a font for mathematics writing at Chalkdust Magazine.
  • Dan Rhatigan wrote an interesting master's thesis about mathematical typesetting -- one of the pleasures of reading these sorts of documents is that they are almost invariably beautifully typeset -- Dan's thesis is no exception.
  • In sadder news, I just found out that the venerable type foundry Hoefler & Frere-Jones (now Hoefler & Co.) has split up in what this article refers to as "the legal equivalent of a knife fight in the street."  My CV is typeset in Hoefler Text; the Rhatigan thesis above is typeset in Whitney, also created by the firm.  Frere-Jones alleged that Hoefler promised him a 50-50 partnership and then delayed giving him equity for 13 years, even after Frere-Jones transferred ownership of valuable typefaces to the firm for a nominal sum of $10.  You can see Hoefler and Frere-Jones, enjoying happier times, in the clip below (from the hit documentary Helvetica).

Hoefler and Frere-Jones in the film Helvetica. Credit: Helvetica (documentary) Directed by Gary Hustwit.

Man After Man

One of my favorite books growing up was Dougal Dixon's Man after man: an anthropology of the future, which imagines the development and speciation of humanity in the far future -- under the influence of both genetic engineering and apocalyptic disaster.

Somehow the genetically engineered humans of the far future have mullets.

Somehow the genetically engineered humans of the far future have mullets.

I looked back on the book recently and was struck by how imaginative Dixon is, but also how his imagination is limited in some ways -- the future he imagines is visibly an '80s future (see: the haircuts of the "hivers" he imagines in the picture above).  You can find a semi-legal copy of the book online here.

Morita Theory, Tannaka Duality, and Approximate Tannaka Duality

Let \(R\) be a ring -- it is well-known that the category \(R\text{-mod}\) of (left) \(R\)-modules does not determine \(R\).  For example, the functor $$-\otimes R^n: R\text{-mod}\to \text{Mat}_{n\times n}(R)\text{-mod}$$ is an equivalence of categories.

That said, there is one additional piece of data that lets us recover the ring \(R\) from the category \(R\text{-mod}\). 

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Integral House

I'm teaching Calculus III at Columbia this semester, and am kind of amazed at the exploitative prices charged for Stewart's Calculus, 8th Edition, "Early Transcendentals".  (Not that anyone knows what "Early Transcendentals" are).  On the other hand, I now know how Stewart was able to build the famous "integral house":

Who says that math doesn't pay?

The man loved calculus...

Stewart's concert hall, in which he could, for a brief moment, forget the cries of the millions of calculus students whose textbook purchases paid for his mansion habit

More pictures at Sotheby's and HuffPost.


Still at UGA, at the very enjoyable but weirdly named conference TAAAG.  The conference featured a very interesting talk by Padmavathi Srinivasan, as well as mini-courses by Michel Raibaut, Ben Williams, and Arnav Tripathy (whose website I can't find).

Arnav's great talk on the integral Hodge and Tate conjecture, with lots of gestures

There were also many short talks, by many of the other participants.  

Padma gave a great talk about her work comparing

  1. The discriminant of a hyperelliptic curve, with
  2. The conductor of the curve.

As I understand it, the conductor only depends on the family of curves (it is more or less the difference in Euler characteristic between the general fiber and the special fiber of a minimal regular model), whereas the discriminant only depends on the covering of the base given by the family of Weierstrass points.  In the case of a minimal regular model, these numbers are just the Euler characteristics of the vanishing cycles sheaf on the family of curves (resp. the family of Weierstrass points).  It would be nice to enhance her inequality to a map of sheaves...


My Hero

This is a picture of Alexander Grothendieck, taken near the end of his life:

You shall not pass!

I took this picture of a page in a book that was just lying around in Oberwolfach a couple of weeks ago (I think the book was published by the Fields Institute).  Unfortunately, I failed to note down an exact attribution -- if anyone knows the name of this book/the photographer, please let me know.

EDIT:  Commenter DH writes that the source of this photo is Masters of Abstraction, by photographer Peter Badge.

Krashen the party

I'm at UGA for the week, in between SWAG and TAAAG.  Today Danny Krashen gave a great talk on this paper of Auel, First, and Williams.  The paper is one of the latest in a long tradition of papers which construct counterexamples by making a topological computation and then approximating the relevant topological spaces by algebraic varieties.  (To my knowledge, this technique began with Totaro, but it has been exploited to great effect by Antieau, Williams, and most recently, fellow Ravi student Arnav Tripathy.) ...

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Families of Curves Wanted

An interesting problem

Let \(n\) be a large positive integer.  Recently I've been looking for a family of curves \(f_n: \mathscr{C}_n\to \mathbb{P}^1\) with the following properties:

  • \(f_n\) is flat and proper of relative dimension \(1\),
  • the general fiber of \(f_n\) is smooth, and the family is not isotrivial
  • every singularity that appears in a fiber of \(f_n\) is etale-locally of the form $$xy=t^n$$ where \(t\) is a parameter on \(\mathbb{P}^1\)...
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I'm currently at SWAG, the awesomely named Summer Workshop in Algebraic Geometry at Georgia.  

Jonathan Wise gave a really amazing talk today about logarithmic geometry, which got me very excited about the possibility of using log geometry to compute various structures on fundamental groups of degenerating curves (e.g. Hodge structures, Galois actions, etc).  It was also by far the best introduction to the subject I've seen.  Yu-jong Tzeng gave a nice introduction to algebraic cobordism, and sketched a proof of Goettsche's formula for the Euler characteristic of the Hilbert scheme of points of a surface, which was quite elegant.  Renzo Cavalieri gave a beautiful overview of Pixton's work on the double ramification cycle(s); I'm really interested in understanding where the products over graphs in Pixton's formulas "come from" geometrically, but so far I've been unable to figure out a rigorous statement.  Finally, Dmitry Zakharov spoke about his recent work with Clader, Janda, and Zvonkine on applying Pixton's work to prove various classical tautological relations.  Overall a really great conference so far.

Here's a question Dmitry asked at dinner, which I think is pretty interesting.  Given that we expect the motion of the 10+ objects in the solar system to be chaotic, can we explain the fact that it has been relatively stable over the past 4 or so billion years?  How long can we expect this stability to continue.  Dimitry reassures me that numerical evidence indicates we have at least a few million years, but I think the Bayesian argument that we have at least a billion is kind of weak.  I'd be very interested to hear an argument from physical (rather than probabilistic) principles analyzing the long-term stability of the solar system.

Starting a blog?

I'm thinking of starting a blog.  Let's see how this goes.