A "minimal" proof of the fundamental theorem of algebra

When I was in graduate school, I came up with what I think is a nice proof of the fundamental theorem of algebra.  At the time, I wrote it up here somewhat formally; I thought it might make a nice blog post, since the formal write-up obscures the very simple underlying ideas.  The goal was to use the minimal amount of technology possible -- in the end I use just a little algebra and some elementary point-set topology, as well as the implicit function theorem...

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-Lemmas

A dilemma is a difficult choice between two alternatives.  I recently learned that there is a word for a choice between three alternatives:  trilemma.  But what if I have a hard choice between four options?  

I was curious, so I did some googling.  It turns out that there's some disagreement as to what a choice between four options should be called -- is it a quadrilemma or a tetralemma?  (The Greek and Latin prefixes for "three" are both "tri-," so there's no conflict in that case.)  I'd argue for the Greek tetralemma, as the suffix -lemma comes from the Greek word for premise, and Google seems to agree:  there are 25,100 hits for tetralemma and only 6,940 for quadrilemma.  

Here's a dilemma: should I be spending my time on this or on something more important?

I was curious as to how this played out for more -lemmas; the Greek "tetralemma" and "pentalemma" dominate the Latin "quadrilemma" and "quintilemma," respectively.  However, the internet seems to have found the Latin-Greek mix "sexalemma" irresistible, for obvious reasons, and the Latin "septalemma" easily won out over the Greek "heptalemma."  There are a huge number of "nonalemmas," apparently, and almost no "ennealemmas." And apparently people with 100 choices prefer the Latin-Greek creole "centilemma" to the pure Greek "hectalemma."  I was unable to find either the Latin or the Greek prefix for 99, but I'm sure that for those linguists with 99 problems, this ain't one.

Some interesting multilemmas:

  • The Lewis Trilemma:  the argument that Jesus was either "Lunatic, Liar, or Lord."  The eponymous Lewis is C.S. Lewis of Narnia fame, who was also a famous Christian apologist.

  • The Charleston Mercury argued, reported the New York Times in 1861, that the Confederate States were caught on the horns of a quadrilemma:  their options were to negotiate, engage in privateering, fight on the sea, or use the economic power provided by the cotton trade to forward their interests.

  • On the other hand, the "non-classical logic of India" preferred the tetralemma, or catuskoti, which was the claim that a statement could be either true, false, neither, or both(!).  This seems to be an originally Buddhist idea which has made its way into other parts of Indian philosophy.

  • Perhaps following Lewis, contemporary Christian apologists have come up with quintilemmas and other myrialemmas.  The pentalemma seems to be largely of interest to online dictionaries.

  • Please don't google "sexalemma."  On the other hand, there are several interesting hexalemmas -- for example, this paper by Campbell Brown argues that it is better to exist than not, rebutting previous depressing work of Benatar (David, unfortunately, not Pat). Brown postulates the existence of a person, named Jemima, who exists in many worlds, and compares those worlds to one in which she does not exists.  He extracts six alternatives from this comparison -- if you are interested despite my description, feel free to read more.

  • The Buddhist tetralemma above has been extended to a septalemma by the Jains.  Other literature refers to it as a heptalemma.

  • Apparently the problem of time in quantum gravity leads to an octalemma.

  • And most of the hits for "decalemma" are the result of Google generously interpreting my search as an interest in "decal Emma." 

Are Shimura Varieties \(K(\pi, 1)\)'s?

Let \(\mathscr{A}_g\) be the moduli space of principally polarized Abelian varieties of dimension \(g\).  The complex-analytic space (stack) associated to \(\mathscr{A}_g\) is a \(K(\pi,1)\); that is, its only non-vanishing homotopy group is \(\pi_1\), which is \(Sp_{2g}(\mathbb{Z})\).  In particular, the cohomology of \(\mathscr{A}_g\) is the same as the cohomlogy of \(Sp_{2g}(\mathbb{Z})\).

Jesse Silliman, a graduate student at Stanford, has told me an argument that shows that this is in some sense maximally untrue when one considers the "etale homotopy type" of \(\mathscr{A}_g\). 

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Weapons of Math Destruction

I've just finished reading Cathy O'Neil's book Weapons of Math Destruction, which I highly recommend.  (One notable feature of the book is that the skull and cross-bones on the cover is the second known example of mathematical piracy.)  

Wea-puns of mass destruction?

The book is, as you might guess from the title, quite negative about the use of big data and mathematical models in government and the corporate world.  This is a point of view that I felt some knee-jerk disagreement with; that said, Cathy is quite clear that her intent is only to discuss the negative features of big data and the use of mathematics in social and business planning:

“Big data has plenty of evangelists, but I’m not one of them. This book will focus sharply in the other direction, on the damage inflicted by WMDs [weapons of math destruction] and the injustices they perpetuate.”

And I think one should read the book with that comment in mind; of course the models the book complains about have some redeeming features and effects.  But that complaint (which is prevalent in the Amazon reviews) misses the point -- to decide whether, on balance, they are a good thing, one has to have a careful accounting of their evils.  This book is that accounting -- it makes no pretense at even-handedness and does not try to weigh the good against the bad, except in the most minimal way.  I don't view that as a strike against it.  

I do have some mild quibbles with the book -- I think that in some cases, the book is uncharitable to the users of the algorithms it objects to.  For example, on page 110, Cathy discusses the use of personality tests in job applications.  Certain answers on these tests reveal that the test-taker has "high narcissism."  "Who wants a workforce peopled with narcissists?," the text asks.  This section is at best misleading -- as the author probably knows, the narcissism these tests discuss is not necessarily pathological.  Rather, narcissism in this setting is a technical term, which may in fact be healthy.  And throughout, the book offers the potential for abuse of algorithms as a strike against them (or offers anecdotal cases of abuse). For example, in the discussion of car insurance companies' use of opt-in technology that tracks one's driving habits, Cathy suggests that soon this technology will be opt-out, at a significant cost.  I'm not necessarily skeptical that this will happen, but I'd argue that we should wait for the abuse to occur before objecting to the technology.

In any case, I think this is an important (and excellent) book, and a necessary counterweight to the techno-utopianism to which I, and many in government (in particular in the current administration), business, and academia are often prone.  I doubt the book will cure me completely of my faith in technocracy.  But I think its real goal is likely to temper that faith with some skepticism. At bottom, the book advocates for rigor in modeling, and for internalizing negative externalities -- who can argue with that?

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.

TAAAG

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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SWAG

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.