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.
Thursday, May 27, 2010
Wednesday, May 26, 2010
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:
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?
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?
Friday, May 07, 2010
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 :-)
Thursday, May 06, 2010
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.
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.
Tuesday, May 04, 2010
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
[1005.0011] Exact Constructions of a Family of Dense Periodic Packings of Tetrahedra
Monday, May 03, 2010
More about characterizations of exactness
Following up an earlier post with some notes on the three papers below:
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 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.
- 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.
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