Category: math
I'm a Numberphile!
The popular YouTube channel Numberphile has just released a video featuring yours truly — filming this with Brady Haran was a real pleasure! A couple of quick mathematical comments: in the video, I compute the Dehn invariant of the cube…
Guest Post: Ordinals and Hydras, by Brian Lawrence
A few days ago, the hydra game came up in discussion with Brian Lawrence; we'd both run into a somewhat mystifying analysis of the game via ordinals. Brian and I came up with a much more down-to-earth analysis (which experts will recogni…
The parity of zero, the primality of two, and other mysteries
From time to time, I try to speak or write about mathematics for general (non-mathematical) audiences. If you've done this, you know it's pretty hard -- in large part because it's hard to know what people know, despite my best attempts t…
Villani for Parliament!
The (relatively new) French party En Marche! (from which the winning presidential candidate, Macron, hails) has nominated spider-brooch wearing mathematician Cedric Villani as a parliamentary candidate! For the full list of candidates, s…
Uniformization over finite fields
I just asked this question on MathOverflow. It's basically idle curiosity, but I've now been idly curious about this for several years, so I figured I might as well ask it publicly. Please let me know if you have any thoughts! In short…
Constructive Criticism
Here is a classical example of a non-constructive proof. Thm 1. There exist irrational numbers \(x,y\) such that \(x^y\) is rational. Proof. If \(\sqrt{2}^{\sqrt{2}}\) is rational, then we may take \(x=y=\sqrt{2}\). Otherwise, we may tak…
Sawin on Severi's Conjecture
One of my favorite questions is: for which \(g, n, p\) is the moduli space of \(n\)-pointed genus \(g\) curves \(\mathscr{M}_{g,n, \mathbb{F}_p}\) unirational/uniruled? Will Sawin has just posted a beautiful paper on the ArXiv answering…
Mumford at the Met
I was at the Met yesterday, where by chance I ran across the Concinnitas Project . (Pictures below the fold.)
C.S. Lewis on Commutative Algebra
Read the whole essay here . Sorry for the pause in blogging -- I should start up again soon.
Graph Theory and \(\mathfrak{sl}_2\)
How many unlabeled graphs are there with \(n\) vertices and \(k\) edges? (More below the fold.)
Jie Liu on projective space
Exciting news! Jie Liu has proven a conjecture of mine -- thereby resolving an old conjecture of Sommese. I'll briefly explain his result in this post.
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 obsc…
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…
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 .)…
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 equivalenc…
Varieties with infinitely generated automorphism group
John Lesieutre has just sent me an exciting new preprint in which he constructs a smooth projective variety \(X\) such that \(\text{Aut}(X)\) is discrete, and \(\text{Aut}(X)\) is not finitely generated. Whether or not such varieties exi…
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…
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 webs…
\(SL_4/\mu_2\) and a mod \(8\) congruence
This is a continuation of a previous post . Recall that we wanted to prove the following claim: Claim. Let \(\rho\) be a representation of \(SL_4\) such that no irreducible subrepresentation of \(\rho\) descends to \(SL_4/\mu_2\). Then i…
More Tautological Classes?
Still at UGA -- I just saw a great talk by Jason van Zelm (a student of Nicola Pagani who apparently does not have a webpage), constructing non-tautological cycles on \(\overline{\mathscr{M}_g}\) for \(g\geq 12\)...
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…
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 \(1\)…