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.

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.

Chern-Simons –form

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.

A unified quantum $SO(3)$ invariant for rational homology –spheres

On the unification of quantum –manifold invariants

Let’s first recall the background to this paper.

The Kontsevich invariant is the universal quantum invariant for links. Quantum link invariants lift to Witten–Reshetikhin–Turaev (WRT) invariants of –manifolds, while the Kontsevich integral lifts to the LMO invariant of –manifolds. It turns out to be a deep and difficult question whether the LMO invariant dominates the WRT invariants (surprisingly so- for links the corresponding fact is easy). For a semisimple Lie algebra, for a prime root of unity , and for a rational homology –sphere (), T. Ohtsuki famously showed that this is indeed the case by showing, roughly, that the perturbative expansion of the WRT invariants (the Ohtsuki series) coincides with the LMO invariant composed with the weight system, which is determined modulo by the WRT invariant at a th root of unity. As I mentioned in the introductory paragraph, and as shown by L. Rozansky, the Ohtsuki series is the contribution of the flat connection to Witten’s invariant, so this is the part of the story which mathematicians can be said to partially understand in any serious sense (forget that we can’t really calculate it outside the simplest examples- even the Kontsevich invariant for links can’t be calculated for any but the very simplest links). Anything beyond the Ohtsuki series would be new.

Ohtsuki’s techniques relied heavily on being a prime root of unity. What happens when is a non-prime root of unity? For integral homology –spheres () K. Habiro constructed an unified WRT invariant whose evaluation at any root of unity coincides with the value of the WRT invariant at that root, and, roughly, whose Taylor series is the Ohtsuki series. Habiro’s invariant is valued in the Habiro ring , which is a ring with all kinds of nice properties, which is related to all kinds of sexy ideas like the field with one element. The bottom line is that belonging to the Habiro ring implies that the WRT invariants are seen to be not just a random collection of algebraic integers, but rather as coming together to form “an analytic function on roots of unity”. This is a conceptual breakthough, and it solves our problem. One key property of analytic functions is that they are uniquely determined by their values at countably many points; so it is for , so it turns out that knowing that the LMO invariant dominates WRT invariants for prime roots of unity implies the corresponding statement for all roots of unity.

The Beliakova-Bühler-Le paper extends Habiro’s idea to s. Namely, with some effort, the authors construct a unified WRT invariant for a , with , containing a link coloured by odd numbers. This unified invariant dominates the WRT invariants at all roots of unity, and is valued in a modified Habiro ring . The case recovers Habiro’s .

Now here’s the point: this paper gives us new and interesting Ohtsuki series. This strongly points to the existence of a refined LMO invariant for s which captures more information from the Chern–Simons theory than just the contribution of the trivial connection, one mathematical step closer to the full preternatural power of the Chern–Simons quantum field theory invariant, as seen by physicists. For s there are non-flat connections which should be contributing, so this actually looks really interesting.

Now to counterbalance the hype, I’m going to start griping… this isn’t really a criticism of this paper at all, but just a personal opinion about this whole branch of mathematics:

1. Nothing is calculated explicitly. Maybe nothing can be calculated explicitly except in completely degenerate cases. So (cynically speaking) what good is it?
2. There is a lot of sophisticated algebra and number theory in the paper, but almost no topology. Quantum topology of –manifolds pays homage to topology via the Kirby Theorem, which translates everything into a statement about links (or tangles) modulo combinatorial moves, which in turn are viewed as morphisms in some category of representations, which makes everything into algebra, which you then fire super-heavy howitzers of representation theory and of number theory at. Philosophically, I think that quantum –manifold topology ought to be more about… well… topology!

The second paper is Instantons beyond topological theory I by E. Frenkel, A. Losev, and N. Nekrasov. This paper was discussed five years ago (when it was still a preprint) in a post in Not Even Wrong by Peter Woit, who knows immeasurably more about the subject than I do, and commented on by other experts; it was also the topic of an erotic film which won a Grand Festival Award at the Berkeley Film Festival.

This paper is actually the first of a trilogy, which propose a new perspective on the non-perturbative regime of quantum field theory (QFT). It only discusses quantum mechanical aspects (QFT is to be discussed later), which (not knowing any physics at all, really) I interpret to mean “-manifold invariants are to be constructed later”.

The study of correlation functions of BPS (or topological) observables in supersymmetric models of QFT is a traditional bridge between quantum gravity and mathematics in four dimensions. As discussed in the introductory paragraphs, these correlation functions give rise to important topological invariants such as Gromov–Witten and Donaldson invariants. This paper studies models (not supersymmetric per se) which have a topological sector, but they are also interested in the correlation functions of the non-BPS (non-topological, dynamical, or `off-shell’) observables. At a certain limit of the coupling constant (), they can solve the model, and, in two and in four dimensions (I suppose that we only really care about four), the Hamiltonians turn out not always to be diagonalizable and the theories turns out to be logarithmic conformal field theories.

If you didn’t understand the paragraph above then don’t worry- I’m not sure I properly understand the above paragraph myself. The key point for me is just that “quantum topology” might be seeing beyond topology into dynamics. To define a QFT you need a manifold with a metric, and then in topological quantum field theory (TQFT) the partition function is independent of the metric and you get topological invariants of manifolds. BPS observables give rise to powerful topological invariants of the underlying manifold (which is equipped with a metric). What we’re now seeing is non-BPS observables which we can “get at” mathematically, and which might allow us to construct powerful dynamical invariants of .

Quantum topology beyond topology… not to mention Equation 5.6, the formula for love.

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.

This sounds like a most interesting experiment- adding a video and a seminar to a paper seems like a good way to get people to read and to understand it! I wish more papers came with videos and seminars.
The content itself also seems interesting to me. Quantum topology of tangles is notoriously hard, because it is based on the Kontsevich invariant (the universal finite-type invariant) which is an extremely sophisticated piece of mathematics which is difficult to understand and to make polynomial-time calculations with. W-knotted objects are a generalization of tangles, for which the universal finite-type invariant is much simpler, and for which polynomial-time calculations are possible. I’m looking forward to hearing all about them on wClips!

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.

Note that the hyperbolic condition is necessary because there are toroidal manifolds, particularly Seifert fibered, spaces with infinitely many minimal genus Heegaard splittings [1]. Minimal genus (rather than just irreducible) is also necessary since there are hyperbolic manifolds with infinitely many distinct irreducible splittings [2]. On the other hand, Lustig and Moriah have constructed manifolds with arbitrarily many minimal genus Heegaard splittings [3], but in order to increase the number of splittings, they have to increase the genus of the the Heegaard splittings. Note that Lustig and Moriah’s examples show that if exists, it grows at least exponentially with . I’ve been toying lately with constructions that produce many distinct irreducible splittings, but again I need to increase the genus in order to increase the number of minimal genus splittings.

My intuition is that for a relatively simple hyperbolic 3-manifold, there are relatively few ways to (efficiently) cut up its topology. In order to find a manifold that can be cut up in more ways, you need to make it more complicated, which increases its genus. However, “more complicated” in this case can’t mean simply higher volume because there are hyperbolic 3-manifolds of arbitrarily high volume with bounded Heegaard genus, though their minimal genus Heegaard splittings will, generally speaking, be unique.

Instead, the picture of what a 3-manifold with lots of minimal genus Heegaard splittings should look like seems to be related to the conjecture that Ian Biringer wrote about a few years ago here. The conjecture is, roughly, that all hyperbolic manifolds with bounded rank and injectivity radius are made up of a finitely many types of pieces glued together along their (incompressible) boundaries. More “complcated” gluing maps should correspond to higher volume manifolds, but once you choose the types of pieces and pair up their boundary components, there will be a bound on the number of minimal genus Heegaard splittings. To increase this number, you would need to either choose more blocks (which increases the genus) or choose more complicated blocks by allowing higher rank (and higher genus) or lower injectivity radius.

[2] Unpublished preprint by Andrew Casson and Cameron Gordon. See this paper for a generalization of the same construction.

[3] Lustig, Martin and Moriah, Yoav. On the complexity of the Heegaard structure of hyperbolic 3-manifolds. Math. Z. 226 (1997), no. 3, 349–358.

1. Because it helps you to compute something you are interested in.
   The conjecture, if true, provides us with a practical way (an algorithm?) to calculate something we care about. The Baum-Connes Conjecture, in my limited understanding, is an example of such a conjecture. The Volume Conjecture is not. SnapPea can calculate hyperbolic volumes quite efficiently, thank you very much, without any help from any NP-hard Jones polynomial calculations.
2. Because it gives you a qualitative understanding of a class of objects which you care about.
   The conjecture, if true, tells you that all possible objects in some class are nice and simple, and nothing “pathological” occurs. The Hodge conjecture is such a conjecture. The Poincaré Conjecture was another one. But the Volume Conjecture is nothing like that. As far as I can make out, the Volume Conjecture alone doesn’t seem to give much of a qualitative understanding of anything (certainly not of hyperbolic geometry!), except maybe for some far off corner of the Jones polynomial which nobody would otherwise care about.
3. Because the conjecture sharply highlights a gap in your understanding, and suggests a mathematical journey on which you should embark.
   For such conjectures, it isn’t the statement of the conjecture itself that we care deeply about per se, but the mathematical understanding which would (hopefully) have to emerge as part of any conceivable proof. An example of such a conjecture would be Fermat’s Last Theorem, in that we never really cared about integer solutions to I don’t think, but rather about gaining a qualitatively better understanding of certain classes of Diophantine equations. The Littlewood Conjecture would be a second example, and the Collatz Conjecture would be a third. It is into this class of interesting conjectures that the Volume Conjecture falls.
   The main problem in quantum topology has always been “what do quantum invariants mean”? In particular, “what does the Jones polynomial mean”? The Jones polynomial was introduced through the representation theory of braid groups into Temperley-Leib algebras, which, on the face of it, has nothing at all to do with topology and everything to do with algebra. But we know empirically that the Jones polynomial in fact has everything to do with topology, because it is a super-strong invariant of knots and links. We can draw suggestive 2D diagrams to interpret the Jones polynomial by means of a skein relation, but mathematically we don’t really have any sense of a good explanation for why the Jones polynomial should have anything at all to do with dimension 3 (physically there’s Witten’s TQFT for the Jones polynomial, but that’s a different story).
   The Volume Conjecture is interesting, perhaps, because it gives us a simple precise concrete statement whose proof would necessarily entail understanding (at least on some level) what exactly it is that makes the Jones polynomial an invariant of 3-dimensional objects. What, after all, could possibly be more 3-dimensional than volume?
   I like Roland’s answer a lot (although I probably misrepresented it quite badly). The Volume Conjecture should be thought of not as a useful formula, nor as a classification of nice objects of some sort, but rather as an easily visible flag to run to and to grab, where the real mathematical value is going to be in the journey rather than in its eventual destination. And, if the flag is well placed, this is exactly what an interesting mathematical conjecture should be.

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

I picked the isotopy subgroup rather than the whole mapping class group because the mapping class group may have finite order elements coming from automorphisms of the ambient 3-manifold. On the other hand, Hyam and I show that if is hyperbolic then the isotopy subgroup is torsion free, so we don’t need to worry about finite order elements.

Constructing irreducible Heegaard splittings with reducible automorphisms turns out not to be too difficult once you’ve had some practice. For example, the union of any two pages of an open book decomposition forms a Heegaard surface, and we can “spin” this Heegaard surface around the monodromy of this open book. The fixed set of this automorphism is the binding of the open book, which is separating in the Heegaard surface, and the restriction to each of the complementary components is equal to the monodromy map for the open book. These automorphisms form a (usually infinite) cyclic subgroup, and I recently showed that for many open book decompositions, this is the entire isotopy subgroup of the induced Heegaard surface [3].

A second, more general, construction is to find a handle on one side of the Heegaard surface that has some flexibility. For example if the Heegaard surface has a pair of disks on opposite sides that intersect in exactly two points, then a regular neighborhood of the two disks is a solid torus. The Heegaard surface intersects the boundary of this solid torus in two or four points depending on the signs of the two intersections between the disks. You can take the handle dual to one of these disks and pull it around the longitude of the solid torus, inducing Dehn twists along the two or four loops.  Pairs of disks like this are quite common and the set of automorphisms defined in this way generate the isotopy subgroup for the (unique) genus three Heegaard splitting of the 3-torus [4].

You can find more interesting subgroups by generalizing this construction. For example, if is a one-sided surface, then a regular neighborhood of is a twisted interval bundle. If you attach a tube to along one of the vertical intervals, the resulting surface is a Heegaard surface. (This is left as an exercise.) Moreover, there is an automorphism of this Heegaard surface defined by draging this tube along any path in the one-sided surface and back to itself. Thus the isotopy subgroup of the Heegaard surface has a subgroup isomorphic to the fundamental group of the one-sided surface. (I’m currently finishing up a preprint showing that for many one-sided Heegaard surfaces, this is the entire isotopy subgroup of the induced two-sided Heegaard surface.)

There are a few other ways to generalize this tube dragging construction as well, which I won’t write about here. I haven’t figured out a way to generalize the construction to more than one handle at a time, but I think it would be very interesting to find examples of automorphisms defined by isotoping a larger portion of the surface, i.e. more than just a single handle, around inside of .

[2] Long, D. D., On pseudo-Anosov maps which extend over two handlebodies. Proc. Edinburgh Math. Soc. (2) 33 (1990), no. 2, 181–190.

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 ?

Note that this is closely related to the Berge conjecture, which asks which knots in the 3-sphere have a Dehn surgery that produces a 3-manifold with Heegaard genus one. If you could prove the answer to this question is “no” then you would reduce the Berge conjecture to a question of which tunnel-number-one knots (i.e. those with Heegaard genus two) have lens space surgeries.  However, I’m more inclined to bet on “yes” for this one. Perhaps it’s because I’ve been spending too much time trying to construct manifolds with multiple non-isotopic Heegaard splittings. But whichever answer turns out to be the case will be very interesting.

For gluing in higher genus handlebodies, things get much more complicated because there is no longer a single slope that determines the gluing. Tao Li [3] has generalized Moriah-Rubinstein’s result to higher genus, but rather than having a finite number of fillings in which the genus may drop, Li requires that the (infinitely many) meridian curves for the glued in handlebody have distance in the curve complex above some bound from a finite collection loops in the boundary. (The difficulty has to do with the fact that disjoint essential loops in the torus are parallel, but disjoint essential loops in high genus surfaces may not be.) It would be interesting to find examples where the Heegaard genus drops after gluing in handlebodies in two inequivalent ways, but I don’t know the best way to generalize the question above.

[1] Moriah, Yoav; Rubinstein, Hyam, Heegaard structures of negatively curved 3-manifolds. Comm. Anal. Geom. 5 (1997), no. 3, 375–412.

[2] Rieck, Yo’av;  Sedgwick, Eric, Persistence of Heegaard structures under Dehn filling. Topology Appl. 109 (2001), no. 1, 41–53.

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 .

I should mention that this conjecture is in the spirit of a comment Cameron Gordon made in his talk at Hyamfest last summer, along the lines of “If you’re going to ask a question, you may as well make a conjecture because people love to prove you wrong.” (As proof of this he noted, that someone had once found a counter example to a “question of Gordon” which he hadn’t even stated as a conjecture.) In this case, I have no particular reason to believe that the conjecture is true and no ideas for how to prove it. Moreover, in the last few weeks, I’ve been more inclined to look for counter examples.

The equivalent generalization of Hartshorn’s Theorem is not true, as Saul Schleimer pointed out to me a while ago: Take any 3-manifold with a genus Heegaard splitting and infinite first homology (such as a connect sum of copies of where the first homology has rank ). Choose a pseudo-Anosov map on the Heegaard surface that acts trivially on its first homology (a Torelli map) and whose stable and unstable laminations are not limits of disks in either handlebody. Hempel showed that the second condition implies that cutting the manifold along the Heegaard surface and regluing by composing with high powers of this map produces Heegaard splittings of arbitrarily high genus. The homology condition implies that the first homology group of each new manifold will be isomorphic to the first homology of the original manifold. Infinite first homology implies infinite second homology (by duality) so these manifolds with arbitrarily high distance Heegaard splitting are all Haken.

The generalized S-T conjecture doesn’t necessarily have weighty and far reaching consequences, but it seems to me like a good conjecture to help motivate progress in the field. Much of the recent work on Heegaard surfaces and bridge surfaces has been fueled by the ability to generalize the ideas in Scharlemann-Tomova’s proof to other situations. However, they all include the caveat that they only restrict the existence of Heegaard splittings below some genus bound. To keep the progress from stalling, we will need an injection of fundamentally new ideas to determine whether or not this caveat is necessary.