The most compelling aspect of Quantum Topology for me is its connection to analytic number theory. Today I’d like to draw your attention to recent work of Minhyong Kim on Arithmetic Chern-Simons Theory (see his paper for more details). I was fortunate to hear him give a talk about this last Wednesday and a colloquium talk on related subjects the day before. People have been talking about such “quantum topological number theory” for a long time- e.g. this 2010 MO question – but we haven’t seen much of an uptake so far. This isn’t an easy direction to pursue because one needs to know both quantum topology and analytic number theory, but I was left with the strong feeling that “There’s gold in them thar hills”, both for topologists and for number theorists. (more…)
April 9, 2016
March 6, 2016
An interesting piece has come out in the New York Times about sexual harassment in an academic setting:
A. Hope Jahren, She Wanted to Do Her Research. He Wanted to Talk ‘Feelings.’, New York Times, March 4, 2016.
What makes this piece especially interesting for me is that it’s written so that one understands the harasser, and is made to realize that “it could be me”. The pattern she describes sounds more common than one might like to admit- and the person writing the e-mail would almost certainly not be cogniscent of it being harassment. A male TA, professor, or supervisor, using the excuse of an altered state of mind (haven’t slept, drank too much) e-mails a love confession to a female student or colleague in a way that blames her, is a total power play, and is creepy and maybe a bit threatening (although of course he doesn’t see it that way). A wrong response to this first e-mail might mean that the victim gets harassed for a long time.
The author says that this first e-mail must be answered by firmly telling him (not asking him) to stop. But, Jahren laments, it never, never stops. While surely Jahren’s suggestion is sensible, a firm, “Dude, I have zero romantic interest in you. In addition you might want to read this piece by Jahren,” might, I think, be even more effective.
What do you all think? How prevalent is this type of sexual harassment in mathematics, and what can be done to effectively nip such harassment patterns in the bud?
February 28, 2016
It doesn’t have much to do with topology, but I’d like to share with you something Avishy Carmi and I have been thinking about quite a bit lately, that is the EPR paradox and the meaning of (non)locality. Avishy and I have a preprint about this:
A.Y. Carmi and D.M., Statistics Limits Nonlocality, arXiv:1507.07514.
It offers a statistical explanation for a Physics inequality called Tsirelson’s bound (perhaps to be compared to a known explanation called Information Causality). Behind the fold I will sketch how it works. (more…)
December 20, 2015
This is just a short post to draw attention to a new preprint by Friedl and Powell The presentation of the Blanchfield pairing of a knot via a Seifert matrix.
The Blanchfield pairing on the Alexander module occurs in various places in knot theory, including in quantum topology. Levine’s 1977 argument for its expression in terms of the Seifert matrix doesn’t make easy reading (the authors suggest it’s incomplete- I can’t judge), and it is notoriously difficult to prove that the Blanchfield pairing is Hermitian. The authors deal deftly with both problems using a more modern but clearly sensible toolbox. Time to rewrite the textbooks.
I wish there were more papers like this. Some aspects of low dimensional topology could use a careful, sensible, modern reboot such as that of this paper.
November 30, 2015
Drew Zemke, who is a grad student of Jason Manning, posted a proof of the Simple Loop Conjecture for 3-manifolds modeled on Sol last week.
The Simple Loop Conjecture fits into that family of statements such as Dehn’s Lemma and the Sphere Theorem which translate statements about fundamental groups into statements about 3-manifolds. Such theorems allow us to trade 3-manifolds for their fundamental groups (which are much simpler mathematical objects). (more…)
November 22, 2015
A few days ago, two people named Joshua (one Howie and one Greene) independently posted to arXiv a similar solution to an old question of Ralph Fox:
Question: What is an alternating knot?
The preprints are:
This post will briefly introduce the problem; I look forward to reading the solutions themselves! (more…)
September 29, 2015
Back in January, I wrote a post about my experience with the differences and trade-offs between academic careers and private sector careers. In this post, I want to present some practical advice for anyone with an academic math background who might be seeking a non-academic job. This advice is based on both my own experience and the advice I found while trying to make the transition. Much of it is very similar to the advice you’ll find (in more detail) in a book called What Are You Going To Do With That? which I read early on and found very helpful. (The book is about finding a non-academic job with any type of PhD.)
September 27, 2015
A few days ago, my co-blogger Nathan Dunfield posted a counterexample to the Strong Neuwirth Conjecture.
N. Dunfield, A knot without a nonorientable essential spanning surface, arXiv:1509.06653
The Neuwirth Conjecture, posed by Neuwirth in 1963, asks roughly whether all knots can be embedded in surfaces in a way analogous to how a torus knot can be embedded in an unknotted torus. A weaker version, the “Weak Neuwirth Conjecture”, asks whether the knot group of any non-trivial knot in the 3-sphere can be presented as a product of free groups amalgameted along some subgroup. This was proven by Culler and Shalen in 1984. But nothing is proven about the ranks of these groups. The Neuwirth Conjecture would give the ranks as the genus of the surface. Thus, the Neuwirth Conjecture is an important conjecture for the structure theory of knot groups.
The Neuwirth Conjecture has been proven for many classes of knots, all via basically the same construction using a nonorientable essential spanning surface. The “Strong Neuwirth Conjecture” of Ozawa and Rubinstein asserts that this construction is always applicable because such a surface always exists.
Dunfield’s counterexample, verified by Snappea, indicates that we will need a different technique to prove the Neuwirth conjecture. Neuwirth’s Conjecture has just become even more alluring and interesting!
August 25, 2015
This interesting-looking preprint has just appeared on ArXiv:
Theo Johnson-Freyd, Heisenberg-picture quantum field theory, arXiv:1508.05908
It argues for a different category-theoretical formalism for TQFT than the `Schroedinger-picture‘ Atiyah-Segal-type axiomatization that we are used to. The `Heisenberg-picture‘ functor it proposes has as its target a category whose top level is pointed vector spaces instead of numbers, and whose second to top level is associative algebras instead of vector spaces. The preprint argues that this formalism is better physically motivated, and one might dream that it is better-suited to analyze “semiclassical limit” conjectures such as the AJ conjecture and its variants.
I’m very happy to see this sort of playing-around with the foundations of TQFT, which I am happy to believe are too rigid. I expect there should be a useful Dirac picture also, and that there are other alternative axiomatizations also. Let’s see where this all leads!
August 19, 2015
I’m sad to announce the untimely passing of Dongseok Kim, a specialist on Kuperberg spiders and their generalizations. He was also a really nice guy whose conference talks were always well-worth listening to.
Although he’s better known for his quantum and stuff, the part of Kim’s work which was most intriguing for me personally was his work on spiders for Lie superalgebras. His paper on the topic doesn’t seem to be cited, despite superalgebra-related quantum invariants being a hot topic (work of Geer, Patureau-Mirand, Costantino, Turaev…)- has anyone noticed it? (more…)