Wodzicki-Chern Classes

So I took a look today at this new paper on the arXiv, by Steve Rosenberg and collaborators, that considers bundles (typically over infinite-dimensional manifolds) whose structural group is a subgroup of the invertible pseudodifferential operators of order $\le 0$ (on some other manifold).

A natural context for such a situation to arise is where your infinite-dimensional manifold is some (Sobolev-type completion $\mathcal M$ of) Maps(N,M), where N and M are ordinary finite dimensional closed manifolds - loop spaces being the canonical example of this kind of thing. Choosing metrics on N and M allows one to give the tangent bundle $T{\mathcal M}$, etc, the structure described above.

Now suppose we have a bundle with structure group $G\subseteq \Psi_{\le 0}$. We can try to use Chern-Weil theory to generate characteristic classes. In order to do this we need a trace on $\Psi_{\le 0}$ itself (just as standard Chern-Weil theory uses the trace on $M_n$ which is the Lie algebra of $GL(n)$).

The most natural example of such a trace is the famous Wodzicki residue which is in fact a trace on the algebra of all pseudodifferential operators. Inputting this trace into Chern-Weil theory yields characteristic classes called the Wodzicki-Chern classes.

The authors conjecture, and prove in some special cases, that these Wodzicki-Chern classes always vanish. (Of course this would mean that it is open season on secondary invariants related to these classes. The authors reference an earlier paper, "Riemannian geometry of loop spaces", where they discuss some nontrivial examples of Wodzicki-Chern-Simons classes.

In view of the importance of the Dixmier trace/Wodzicki residue in Connes' noncommutative geometry scheme, it would be interersting to know how this paper fits with the Connes picture.

A rigid framework

Well, I am back from Yosemite, but not in quite the way I had hoped. I was climbing the Prow on Washington Column with Aaron McMillan (a grad student from Berkeley, student of Weinstein's) and on our second day I took a fall resulting in a broken ankle and the end of our climbing vacation. If you are interested in the long version you can find the story here.

Anyhow, I am sitting at home now with my leg encased in a rigid boot which will have to stay on for the next six weeks or so while the bones rejoin themselves. It got me thinking about the idea of such 'rigid frames' in teaching - actually in teaching analysis, since I'm thinking about my course for next fall. Bear with me for a moment while I try to explain what I mean.

Suppose that you're asked to give a proof of something like "the limit of a uniformly convergent sequence of continuous functions is continuous". As a professional mathematician you might just say "$3\epsilon$ argument", or you might write out a more detailed proof. But whatever you did - or, at least, whatever I would do - probably doesn't explicitly express every one of the many quantifiers that are involved in this statement, or explicitly delimit the scope of every one of the free variables that may be introduced during the proof. To be required to do so would be excessively rigid and constraining, like my ankle boot. As mathematicians we've developed the bones and muscles that allow us to work correctly with a less than wholly formal style of argumentation; and that's a vital skill. But, I'm wondering, do we give our students enough "rigid support" so that their mathematical "bones" can develop? Or do we overload them by presuming on strength that isn't there yet? If I just took off my boot now and tried to walk, the results would be disastrous; my bones aren't ready for that yet.

Specifically, one of the things that I try to emphasize in teaching analysis is taking apart an argument or definition involving multiple quantifiers into a hierarchy of more elementary units, which are nested within each other like subroutines in a computer program. And I then try to explain that to each of these elementary units corresponds a "proof skeleton", so that for instance to the elementary unit $\forall x\in A, P(x)$ ($P$ being some possibly complex proposition) corresponds the proof skeleton:

Let $x$ (or some other symbol not yet used) be an arbitrary member of $A$. Then (argument), leading to the conclusion $P(x)$. We have shown that $P(x)$ is truw for an arbitrary member $x\in A$, so we have proved $\forall x\in A, P(x)$. (end of scope of symbol $x$)

Nesting these proof skeleta in a way corresponding to the multiply-quantified statement to be proved gives a quite rigid framework - a "cast" - for the proof. Of course it is still necessary to supply the actual argument! In my experience though students sometimes need more guidance with the structure of the proof than the individual computations comprising it; and this system supplies it.

I am wondering about writing some software which will generate these "skeleta" semi-automatically and will force students to write proofs into them. Not for ever of course - just until the "bones" grow strong. Of course the worry is that then the teaching suddenly becomes about software and not about mathematics. Still, I think it could be a helpful tool. Maybe something like this exists already. Does anyone know?

Brief pause

I won't be posting for a couple of weeks as I will be away climbing in Yosemite. I hope to get back to coarse remarks when I return :-)

Around soficity

Andreas Thom just posted the article [1005.0823] About the metric approximation of Higman's group on the arXiv today. It is quite short with a specific result about Higman's group, but the introduction was most helpful to me in learning a bit about the ideas related to "soficity" of groups. It refers to another interesting paper: Elek, Gábor, and Endre Szabó. “Hyperlinearity, essentially free actions and L2-invariants. The sofic property.” Mathematische Annalen 332, no. 2 (4, 2005): 421-441.

It seems that these authors use some words like "hyperlinear" and "amenable action" in a sense different to that which is common to us in Baum-Connes land. for instance, for Elek-Szabo, the trivial action of a group on a point is *always* amenable.

Packing Tetrahedra

This is some way from what this blog is supposed to be about, but like many packing problems it is fascinating and difficult. The question: How densely can regular tetrahedra be packed in 3-dimensional Euclidean space? Nobody knows, but here are some very interesting packings...


[1005.0011] Exact Constructions of a Family of Dense Periodic Packings of Tetrahedra

More about characterizations of exactness

Following up an earlier post with some notes on the three papers below:

• Invariant expectations and vanishing of bounded cohomology for exact groups by Douglas and Nowak [DN]
• Amenable actions, invariant means and bounded cohomology by Brodzki, Niblo, Nowak and Wright [BNNW]
• A note on topological amenability by Monod. [M]

All of these papers focus on the question of characterizing in "homological" terms what it is for a discrete group $G$ to be exact (or, more generally, to act amenably on some compact space --- it is known that exactness is equivalent to the amenability of the action of $G$ on its Stone-Cech compactification $\beta G$).

A necessary step along the way (again, in all the papers) is to relate the notion of exactness to some kind of "invariant mean". This follows a path explained by Johnson for ordinary amenability (Johnson, Barry Edward. 1972. Cohomology in Banach algebras. Providence, R.I.: American Mathematical Society.)

The paper [M] gives the greatest number of equivalent conditions (it is a legal requirement that all papers on amenability show that many conditions are equivalent). In particular let us consider the appropriate notion of "invariant mean". This is an element $\phi$ of the bidual $A^{**}$, where $A$ is the algebra of continuous functions on $\beta G$ with values in $\ell^1 G$ (also equivalent to the algebra of unconditionally convergent formal series $\sum_g f_g [g]$, with $f_g \in \ell^\infty(G)$); $\phi$ must be $G$-invariant and must sum to the constant function $1 \in \ell^\infty(G)$.

Aside: Monod also gives a couple of interesting alternative characterizations of $A$:

• The space of compact operators on $\ell^1(G)$
• The space of weak-$*$ - to - weak continuous operators on $\ell^\infty(G)$

The characterizations in [BNNW] seem quite close to that of [M]. Their invariant means are elements of a double dual $W_0^{**}$, where $W_0$ is defined a little differently but appears to be the subspace of $A$ consisting of elements that sum to a multiple of 1.

In [DN] the double dual of $W_0$ is taken in a more ambitious sense, as $hom(hom(W_0,C),C)$, where $C$ is the Banach space $\ell^\infty(G)$. But then one looks inside this double dual at the weak-$*$ closure, in an appropriate sense, of the members of $W_0$ itself. Now, let $R$ be some suitable ring of endomorphisms of the Banach space $C$ (e.g., the translation algebra). Then both $C$ and $hom(W_0,C)$ are $R$-modules and the elements of $hom(hom(W_0,C),C)$ coming from $W_0$ are $R$-module maps. Thus, it seems to me, one might as well restrict to the subspace of $R$-module maps from the start, and then much of the extra "size" of the double dual goes away. I think this may make a connection between the [DN] approach and the other two.

An update on random groups

rgupdates.pdf application/pdf Object

technological toys

Inspired by Nigel (who is often on the bleeding edge of technology) I ordered a Fujitsu ScanSnap - see below - which arrived a day ago. It's about the size of a loaf of bread, has one control button ("Scan"), and scans 20 double-sided pages per minute to PDF. I hope to use it to organize the piles of preprints, handwritten notes and manuscripts that I have accumulated in nearly thirty years as a mathematician.

That then begs the question - what software should I use to keep track of the resulting huge pile of PDFs? Right how I am working with Zotero which will organize pdfs, archive them on a WebDAV server, and also integrate with Penn State's library and other sources of bibliographic info. And it will seamlessly import the bibtex bibliography that I have maintained since I started using TeX. But there may well be other useful software packages out there that will do the same or better - any suggestions?

Various characterizations of exactness

This post is a place to list a number of papers that have recently appeared on the arXiv which reformulate the notion of exactness for groups (or property A for spaces or amenable actions) in different ways:

• A cohomological characterisation of Yu's Property A for metric spaces by Brodzki, Niblo and Wright.
• Invariant expectations and vanishing of bounded cohomology for exact groups by Douglas and Nowak.
• Amenable actions, invariant means and bounded cohomology by Brodzki, Niblo, Nowak and Wright.
• A note on topological amenability by Monod.

I'll post more later about the relations between these.

Invariant translation approximation

At the very end of my book Lectures on Coarse Geometry I asked the following question: suppose you take a discrete group $\Gamma$, consider it as a metric space and form the uniform translation algebra $UC^*(|\Gamma|)$. This algebra has a natural $\Gamma$-action and the $\Gamma$-fixed subalgebra, $UC^*(|\Gamma|)^\Gamma$, clearly contains the reduced $C^*$-algebra of the group $\Gamma$. Are these objects equal? In the book I showed that they are equal for amenable groups and outlined an argument, invented by Nigel Higson, which shows that they are also equal for free groups - this uses Haagerup's results about rapid decay.

It is clear that some kind of approximation property is involved here and in the book I called it the "invariant translation approximation property". (In our earlier discussions Nigel, Jerry and I were so irritated by this question that we called it the "completely stupid approximation property" but fortunately we were not completely stupid enough to use this term in print. Ooops...) Which groups possess this property?

While talking with Nowak in Texas I learned about a paper by Joachim Zacharias, On the invariant translation approximation property for discrete groups , which makes significant progress on this question. Zacharias' paper works as follows. First consider a strengthening of the ITAP by allowing coefficients: one looks at $UC^*(|\Gamma|;S)$ where $S$ is an auxiliary $C^*$-algebra (or operator space) and asks whether the $\Gamma$-invariant part of that is equal to $C^*_r(\Gamma)\otimes S$ (minimal tensor product). (N.B. There is no $\Gamma$-action on $S$ - no 'twisting'.) Zacharias proves that for exact groups this strengthened ITAP is equivalent to the Haagerup-Kraus approximation property ( Approximation properties for group $C^*$-algebras and group von Neumann algebras , Transactions of the American Mathematical Society, Vol. 344, No. 2 (Aug., 1994), pp. 667-699, which says that there is a net in the Fourier algebra $A(\Gamma)$ converging to 1 in a certain weak topology on the completely bounded multipliers on $C^*_r(\Gamma)$. Unfortunately no example of an exact discrete group without this property is known, but it has been conjectured that $SL(3,Z)$ is such a group.

On the way the author proves another characterization of exact groups, namely that $\Gamma$ is exact iff the map $S \mapsto UC^*(|\Gamma|;S)$ is an exact functor.

Geometry and complexity theory

I'm at TAMU today, at the invitation of Piotr Nowak and Ron Douglas. Along with a number of others, they have made significant progress with understanding exactness of groups/property A in terms of appropriate notions of "invariant means" and "vanishing of bounded cohomology". I will probably write about this later.

However, while here I also had a chance to talk with Joseph Landsberg about his preprint P versus NP and geometry. Who could resist such a title? Here is my summary of what he told me. Suppose that you have to compute some huge polynomial in many variables. Then (obviously) in general it will take you a long time. As an example, consider the determinant of an $n\times n$ matrix, which is a polynomial of degree $n$ in $n^2$ variables. A brute force, term-by term evaluation takes a very long time. But in this case there is a trick - Gaussian elimination - which allows one to compute the polynomial much more quickly (with roughly $O(n^4)$ arithmetic operations I think). This is a hidden symmetry leading to speedy computation. Other polynomials, e.g. most famously the permanent (which is the same as the determinant but with all signs $+$), cannot (so far as is known) be computed in this easy way.

This leads to the notion of determinantal complexity (Valiant). You can envisage computing some polynomial such as the permanent of an $n\times n$ matrix by computing instead the determinant of some larger matrix built out of the original one in some way. (If the "larger matrix" is allowed to be sufficiently much larger one can always do this.) Define the determinantal complexity of the (sequence of) given polynomials to be the function that tells you how much you must increase the size of an $n\times n$ matrix to build a larger matrix that computes your polynomial via a determinat. Valiant conjectured that for the permanent, the determinantal compelxity grows faster than any polynomial. If I understand correctly, the falsity of this conjecture (i.e. a polynomial bound for $dc$ of the permanent) would imply $P=NP$.

To address this, the program of "geometric complexity theory" transfers the problem to one in algebraic or differential geometry. One looks at the variety defined by the determinant (in projective space) and allows the general (or special) linear group of the $n^2$ coordinates to act. The permanent (in some smaller degree $d$) defines a point in this space and the question becomes whether this point (or the Zariski closure of its orbit) lies in the Zariski closure of the orbit of the determinant. This question is then addressed either by representation theory (the ring of regular functions on a $G$-orbit can be completely described in terms of representation theory) or by local differential geometry.

I can write TeX!

Seems as though I figured out how to include some TeX: $x^2+y^2+z^2=r^2$, $\int_{-\infty}^\infty e^{-x^2}\,dx = \sqrt{\pi}$. I got this from http://sumidiot.blogspot.com/2008/01/latex-in-blogger.html if you want to try it. You need to allow a javascript file to run from Nottingham University (UK).

The Atiyah conjecture is false (sort of)

Diarmuid Crowley visited for a couple of days last week to talk about the Manifold Atlas Project which is a plan to produce a sort of online encyclopedia/journal of information about all sorts of manifolds. Of course we talked about other things also, and I learned from Diarmuid about a paper by Tim Austin of UCLA which gives a counterexample to a version of the "Atiyah Conjecture".

Atiyah would want me to point out that the conjecture is misnamed. In the paper in Asterisque where he introduces the L2 Betti numbers, Atiyah asked as a problem: "Give examples where these invariants are not integers or perhaps even irrational". The name "Atiyah conjecture" got attached to the claim that there are no such examples! And the question about the integrality of the invariants (for torsion free groups) is still open. But for groups with torsion, Zuk gave a counterexample some years ago to the claim that the denominators of the L2 Betti numbers must be generated by the torsion orders in the group; and now Austin shows that there are groups with irrational (even transcendental) L2 Betti numbers.

The argument is a non-constructive one: Austin builds an uncountable family of groups, and in the group ring of each group a particular element, such that the von Neumann dimensions of the kernels of these elements are all different. (The groups are certain "lamplighter-type" groups built on the free group.) Thus there are uncountably many different real numbers which are L2 Betti numbers, and some of them must not be rational (or algebraic). But the process doesn't identify a particular group for which this is true.

If I could figure out how to get TeX into this thing I might post more...

(Added later: See Thomas' comments below for a follow-up paper by Lukasz Grabowski, which begins "The main point of this article is to show some connections between Turing machines and von Neumann algebras".

Lin Shan's index theory

OK, I am going to try to revive this blog in the hope that it will encourage me to read and keep up with the mathematical literature. We shall see...

Anyhow, I just wrote a review for Mathematical Reviews of the paper "Equivariant higher index theory and nonpositively curved manifolds" by Lin Shan (JFA 255(2008), 1480-1496. This paper defines and studies an analytic assembly map that includes both the coarse assembly map and the Baum-Connes assembly map as special cases, and it proves a Novikov conjecture type statement.

Infinite Expanders

In a note published at

http://www.wisdom.weizmann.ac.il/~itai/infexp.ps

it is asked (by Binjamini I think), "Is there an infinite expander?".

By definition an infinite expander is an infinite connected bounded geometry graph with the following property: there exists a positive constant, call it c, such that for any set S of vertices (whether finite or not) and any ball B, less than half of whose points are in S, the ratio

(size of boundary S intersect B)/(size of S intersect B)

is greater than c.


The conjecture is that no such "infinite expander" exists.

QUESTIONS:

(a) What would it take for the graph of a group to be an infinite expander?
(b) Relate to the coarse property T problem.

[math/0606120] Maximal rank maps between Riemannian manifolds with bounded geometry

[math/0606120] Maximal rank maps between Riemannian manifolds with bounded geometry

This paper by Abreu-Suzuki gives a condition under which a coarse submersion between Riemannian manifolds is coarsely a product.

Interesting not only in itself but for its references - which are to another author (Kumeu) who came up with the basic coarse ideas, in the context of bg Riemannian manifolds, in the middle 1980s. I had not been aware of this before.

GT Monographs: Volume 9

GT Monographs: Volume 9

This is the proceedings of an Oberwolfach conference on "exotic" homology manifolds. (Roughly speaking, these are manifolds for which the "zero'th Pontrjagin class" is not equal to 1.)

[math/0603675] The lower central series and pseudo-Anosov dilatations

[math/0603675] The lower central series and pseudo-Anosov dilatations

The lower central series and pseudo-Anosov dilatations

Authors:
Benson Farb,
Christopher J. Leininger,
Dan Margalit

Comments: 26 pages, 6 figures

Subj-class: Geometric Topology; Dynamical Systems


MSC-class: 37E30 (Primary) 57M60, 37B40 (Secondary)

The theme of this paper is that algebraic complexity implies dynamical
complexity for pseudo-Anosov homeomorphisms of a closed surface S_g of genus g.
Penner proved that the logarithm of the minimal dilatation for a pseudo-Anosov
homeomorphism of S_g tends to zero at the rate 1/g. We consider here the
smallest dilatation of any pseudo-Anosov homeomorphism of S_g acting trivially
on Gamma/Gamma_k, the quotient of Gamma = pi_1(S_g) by the k-th term of its
lower central series, k > 0. In contrast to Penner's asymptotics, we prove that
this minimal dilatation is bounded above and below, independently of g, with
bounds tending to infinity with k. For example, in the case of the Torelli
group I(S_g), we prove that L(I(S_g)), the logarithm of the minimal dilatation
in I(S_g), satisfies .197 < L(I(S_g))< 4.127. In contrast, we find
pseudo-Anosov mapping classes acting trivially on Gamma/Gamma_k whose
asymptotic translation lengths on the complex of curves tend to 0 as g tends
toward infinity.

Full-text: PostScript, PDF, or Other formats

[math/0603669] All generating sets of all property T von Neumann algebras have free entropy dimension $\leq 1$

[math/0603669] All generating sets of all property T von Neumann algebras have free entropy dimension $\leq 1$

All generating sets of all property T von Neumann algebras have free
entropy dimension $\leq 1$

Authors:
Kenley Jung,
Dimitri Shlyakhtenko

Comments: 6 pages

Subj-class: Operator Algebras


MSC-class: 46L54; 52C17

Suppose $N$ is a diffuse, property T von Neumann algebra and X is an
arbitrary finite generating set of selfadjoint elements for N. By using
rigidity/deformation arguments applied to representations of N in full matrix
algebras, we deduce that the microstate spaces of X are asymptotically discrete
up to unitary conjugacy. We use this description to show that the free entropy
dimension of X, $\delta_0(X)$, is less than or equal to 1. It follows that when
N embeds into the ultraproduct of the hyperfinite $\mathrm{II}_1$-factor, then
$\delta_0(X)=1$ and otherwise, $\delta_0(X)=-\infinity$. This generalizes the
earlier results of Voiculescu, and Ge, Shen pertaining to $SL_n(\mathbb Z)$ as
well as the results of Connes, Shlyakhtenko pertaining to group generators of
arbitrary property T algebras.

Full-text: PostScript, PDF, or Other formats

Which authors of this paper are endorsers?