Stop Elsevier!

Mathematicians have been complaining for years about Elsevier‘s business practices. In 2006, topologists rose up against it, when the editorial board of Topology resigned and established the Journal of Topology.

A few days ago, Tim Gowers wrote a blog post HERE suggesting that it might now be a good time to act together to stop Elsevier by refusing to submit to their journals or to referee or do editorial work. A petition is HERE.

As a random thought, I wonder whether it would be possible for representatives of the scientific community to sue large publishers to gain open access to papers written before year X, where X is around 2005 or something.

Student Conferences

I just added two student oriented conferences to the LDTopology conference list, which I think deserve special attention. The first is the UnKnot conference, which is specifically intended for undergraduate students. Since I don’t think many undergraduates look at the conference page (or read this blog at all for that matter), you should encourage any promising students that you know to consider going. Since Colin Adams is one of the organizers, it promises to be fun.

The second conference is the Topology Student Workshop at Georgia Tech, which appears to be intended for graduate students. It looks like this one is going to have a lot of professional development, which is becoming more and more important these days.  A few years ago, there was an apparently unrelated “Topology Student Forum” at Tulane University. I think they had plans of making it an annual event, but I couldn’t find any references to a 2012 meeting. If anyone has information about this (or any other upcoming topology conference), please leave a comment on this page or on the conferences page.

Update: Right after I posted this, I found out about two other student conferences. The first is the Graduate Student Topology Conference at Indiana University, March 31-April 1st. The second is the Underrepresented Students in Topology and Algebra Research Symposium (USTARS) at University of Iowa, April 13-15.

Beyond the trivial connection

One of the foundational papers in Quantum Topology, and one of the main reasons that the subject is called Quantum Topology, is Edward Witten’s landmark paper Quantum field theory and the Jones polynomial. One of the things Witten did in that paper was to define a –manifold invariant as a partition function with action functional proportional to the Chern-Simons –form. A partition function is a path integral, so Witten’s invariant is a physical construction rather than a mathematical one. Quantum topology of –manifolds is, to a large extent, the field whose goal is to mathematically reconstruct, and to understand, Witten’s invariant. Meanwhile, for –manifolds with a metric, Witten defined a –manifold invariant as a partition function in another landmark paper Topological quantum field theory.

I should warn you that I don’t know any physics so some (all?) of what I say below might be rubbish. Still, pressing boldly ahead…

Up until recently, mathematicians only understood tiny corners of Witten’s invariants, or, more broadly, of invariants (topological or otherwise) of manifolds (with or without extra structure) which come from quantum field theory partition functions. But I’ve recently glanced through two papers which seem to finally be going further, seeing more. The tiny corners we have seen already give rise to mathematical invariants of preternatural power (surely that’s the best word to describe it!), such as Ohtsuki series of rational homology –spheres (), Donaldson invariants, and Seiberg–Witten invariants.

Bar-Natan – Dancso paper comes with seminar and video

Dror Bar-Natan makes the following announcement:

Dear Friends,

With help from my students, in the next semester I will be running the “wClips Seminar”, which will be a combination of a class, a seminar, and an experiment. We will meeting on Wednesdays at noon starting January 11, 2012 – follow us on http://www.math.toronto.edu/drorbn/papers/WKO/!

The “class” part of this affair is that we will slowly and systematically go over my in-progress joint paper with Zsuzsanna Dancso, “Finite Type Invariants of W-Knotted Objects: From Alexander to Kashiwara and Vergne” (short “WKO”, and again see http://www.math.toronto.edu/drorbn/papers/WKO/), section by section, lemma by lemma, and covering all necessary prerequisites as they arise.

The “seminar” component is the usual. Occasionally people other than me will be telling the story.

The “experiment” part is that every lecture will be video taped and every blackboard will be photographed and everything will be immediately put on the WKO website, so that at the end we will have along with the paper a “video companion” – series of video clips explaining every bit of it. The paper will be mathematically self-contained, yet in addition every section thereof will include a link/reference to the corresponding clip in its video companion. And every video clip will have its written counterpart in one of the sections of the paper.

Feel free to follow almost in real time! Also, please let me know if you want to be added to the wClips mailing list.

Best,

Dror.

The minimal genus Heegaard splitting conjecture

Today, I will continue on my quest to find the most interesting conjectures about Heegaard splittings. (Most of these conjectures, including this one, fail criteria one and two in Daniel’s recent post, but strive to satisfy criteria three.) Here’s the latest:

The minimal genus Heegaard splitting conjecture: For every positive integer , there is a constant such that if is a hyperbolic 3-manifold with Heegaard genus then has at most isotopy classes of (minimal) genus Heegaard splittings.

Who cares about the Volume Conjecture?

Yesterday, I attended a very interesting informal talk by Roland van der Veen in which, among other things, he told me a little bit about why he cares about the Volume Conjecture. The Volume Conjecture is considered somehow to be the `big open problem’ in quantum topology. I had never understood why though (I had even asked an MO question but hadn’t really been convinced by any of the very good answers). Why, after all, should people care about any mathematical conjecture?

Videos for NSF-CBMS Cubulationathon

I’ve mentioned here several times the work of Wise on residual properties of certain word-hyperbolic groups, specifically those of Haken hyperbolic 3-manifolds. You can now view all 10 of Dani’s talks at the NSF-CBMS conference (as well as all the other talks) at the conference webpage. The picture and audio quality is quite reasonable considering the setup that was used, and they are certainly watchable.

I really wish more conferences did this. While it’s certainly true that the benefits of attending a conference go far beyond the content of the talks themselves, I think it’s still quite valuable to have this online for those who weren’t able to addend.

The reducible automorphism conjecture

Recall that the mapping class group of a Heegaard splitting is the group of automorphisms of the ambient 3-manifold that take the Heegaard surface onto itself, modulo isotopies of that keep on itself. The isotopy subgroup is the group of such maps that are isotopy trivial on , when you ignore the Heegaard surface. Hyam Rubinstein and I constructed a number of examples of Heegaard splittings with non-trivial isotopy subgroups [1], but all of these groups were generated by reducible automorphisms. Before our paper, Darren Long had constructed a strongly irreducible Heegaard splitting with a pseudo-Anosov element in its mapping class group [2] but the ambient manifold is a Seifert fibered space over a sphere with five singular fibers and Heegaard splittings of Seifert fibered spaces have large mapping class groups, usually including lots of reducible maps. This raises the question: Is there a Heegaard splitting with a non-trivial isotopy subgroup consisting entirely of pseudo-Anosov maps (other than the identity)?  I will go a step further and, in the spirit of my last two posts, suggest a reckless conjecture:

The reducible automorphism conjecture: The isotopy subgroup of every Heegaard splitting is generated by reducible automorphisms.

Dehn filling and genus dropping

A common problem in low-dimensional topology is to ask how the topology and geometry of a manifold changes if you glue a solid torus into one of its torus boundary components (also known as Dehn filling) or more generally, if you glue a handlebody into a higher genus boundary component.  One topological version of this problem is to ask how the isotopy classes of Heegaard surfaces change. Every Heegaard surface  for the unfilled manifold becomes a Heegaard surface for the filled manifold, but there may be other properly embedded non-Heegaard surfaces that also become Heegaard surfaces if you cap them off after the filling. In particular these new Heegaard surfaces may have lower genus, so the Heegaard genus of the manifold could drop after filling. The quintessential example of this is a knot complement in the 3-sphere: There are knot complements with arbitrarily high Heegaard genus, but if you Dehn fill to produce the 3-sphere, then the genus drops to zero.

Of course, for such a manifold there is exactly one filling that produces the 3-sphere and one can ask how much the genus can drop for the other fillings. There are examples where Heegaard genus drops by one for a line of slopes, and the resulting Heegaard surfaces are often called horizontal.  However, Moriah-Rubinstein [1] (and later Rieck-Sedgwick [2]) showed that there are only finitely many slopes for which the genus can drop by more than one (and only finitely many lines of slopes where it drops by one.) As far as I know there are no examples where there are two slopes for which the genus drops by more than one. So one can ask:

Question: Is there a 3-manifold with Heegaard genus , a torus boundary component and two slopes on such that Dehn filling along each slope produces a 3-manifold with Heegaard genus less than or equal to ?

The generalized Scharlemann-Tomova conjecture

Soon after John Hempel introduced the notion of (curve complex) distance for Heegaard splittings, Kevin Hartshorn showed that the existence of an incompressible surface implies a bound on the distance for any Heegaard splitting of the same 3-manifold. Scharlemann and Tomova noted that a strongly irreducible Heegaard surface behaves much like an incompressible surface, and generalized Hartshorn’s Theorem as follows: If admits a distance Heegaard surface then every other genus Heegaard surface with is a stabilization of . This is a great theorem and has had huge consequences for the field, but there is one thing that has always bothered me about it: It leaves open the possibility that, for example, there may a 3-manifold with a genus three, distance 100 Heegaard surface and a second Heegaard surface of genus 201 that is unrelated to the first one. This has always seemed very unlikely to me, so I propose the following conjecture:

The generalized Scharlemann-Tomova conjecture: For every genus , there is a constant such that if is a genus , distance Heegaard surface then every Heegaard surface for is a stabilization of .