Thursday, June 20, 2013


Penn State is moving our departmental web pages to a new Wordpress site.  As part of that process, I have a new blog at

All the content from this blog has been imported there and future posts will be there, not here.  (Okay, I know there haven't been any posts here for years anyhow, but if someone out there is still listening, you might want to know.)

Thursday, October 07, 2010

Existence of collisional trajectories of Mercury, Mars and Venus with the Earth

Vadim Kaloshin was telling me about this article by Laskar and Gastineau.  In 2500 numerical simulations of solar system evolution, one trajectory had other planets hitting the Earth - in a few billion years.

A Hollywood movie is confidently expected.

Saturday, October 02, 2010

The Peter Principle Revisited

Not really "coarse mathematics" in the sense I initially intended, but feels rather relevant to the work of a department head.   This paper uses a simulation to demonstrate that, under certain hypotheses, a "hierarchical" organization will function more efficiently by promoting people at random than by always promoting those who are most competent in their current position.

Wednesday, September 29, 2010

"Mass endomorphism, surgery and perturbations"

That is the title of an interesting paper just posted on the arXiv. I had never heard of the "mass endomorphism" so this was new to me... Take a compact Riemannian spin manifold and suppose that the metric is flat in the neighborhood of a point p.  If there are no harmonic spinors (so that the Dirac operator is invertible) then the Dirac Green's function, i.e. the inverse of the Dirac operator, has an asymptotic expansion near p in which the zero term is an endomorphism of the spinor bundle called the mass operator.  It is known that if the mass operator is non-zero then a solution exists to the classical Yamabe problem.  In this paper it is shown that the mass operator is "generically" non zero - using a lot of the machinery from positive-scalar-curvature land: psc surgery, results of Stolz, etc...

Thursday, September 16, 2010

Coarse Math at MSRI!

Forwarded from Vincent Lafforgue

From August 15, 2011 to December 16, 2011, MSRI (Berkeley) hosts a program on Quantitative Geometry. It is organized by Keith Ball, Emmanuel Breuillard, Jeff Cheeger, Marianna Csornyei, Mikhail Gromov, Bruce Kleiner, Vincent Lafforgue, Manor Mendel, Assaf Naor (main organizer), Yuval Peres, and Terence Tao. This is a big program with many available positions.

Examples of areas that will be covered by the program are: geometric group theory, the theory of Lipschitz functions (e.g., Lipschitz extension problems and structural aspects such as quantitative differentiation), large scale and coarse geometry, embeddings of metric spaces and their applications to algorithm design, geometric aspects of harmonic analysis and probability, quantitative aspects of linear and non-linear Banach space theory, quantitative aspects of geometric measure theory and isoperimetry, and metric invariants arising from embedding theory and Riemannian geometry. Go to for more details.

The deadline for applications is
-October 1, 2010 for Research Professors
-December 1, 2010 for Research Members and Postdoctoral Fellows.
Look at
for more details.

Permanence properties in coarse geometry

I hate to think how long it has been since I last posted here. My apologies - it has been a difficult summer for various non-mathematical reasons. Anyhow, trying to get back on track let me mention a survey article that Erik Guentner sent me called "Permanence properties in coarse geometry". What Erik means by "permanence properties" is statements like "the property of having finite asymptotic dimension is closed under group extensions". Many statements of this kind, for a variety of coarse properties (asymptotic dimension, embeddability in Hilbert space, property A/exactness, etc) have by now been proved and this is a very nice survey bringing together general techniques for obtaining such results with specific applications.


Thursday, July 15, 2010

Metric spaces with dilations, and metric trees

Here is a gentle introduction to the theory of metric spaces with dilations ("rescaling maps", so that one can define an appropriate notion of tangent space.) This appears on the arXiv today.

Also a couple of elegant-looking papers on metric trees and their embeddings into Banach spaces

Lots of interesting stuff on the arXiv today. I probably won't be posting much for a few weeks as I have some personal business to take care of.

Tuesday, July 06, 2010

Smith theory made coarse

"Smith theory" is the name given to the investigation (starting with the work of P.A. Smith in the late 1930s) of the homological properties of the fixed-sets for finite groups (especially $p$-groups) acting on spheres. See Dwyer, William G., and Clarence W. Wilkerson. “Smith Theory Revisited.” The Annals of Mathematics 127, no. 1. Second Series (January 1988): 191-198.

Ian Hambleton and his student Lucian Savin have just posted an article about a coarse-geometric counterpart of Smith Theory, Hambleton, Ian, and Lucian Savin. “Coarse Geometry and P. A. Smith Theory.” 1007.0495 (July 3, 2010).

The notion of fixed set is replaced by a "coarse fixed set", which is the coarse structure (if it stabilizes) of the sequence of "approximate fixed sets" $\{x: d(x,f(x))\le n\}$ for $n=1,2,\ldots$.

This notion is well behaved for finite group actions.

Friday, July 02, 2010

"Holomorphic Functional Calculus"

Writing up the Connes-Renault notes, which I mentioned in a previous post, leads to a number of interesting digressions. For instance, the notion of "holomorphic closure" is discussed at some length in these early notes. But what exactly is the relationship between "holomorphic closure", "inverse closure", "complete holomorphic closure" (= holomorphic closure when tensored with any matrix algebra), and so on? I was aware that there had been some progress in this area but had not really sorted it out in my mind. Here's a summary (all these results are pretty old, so perhaps everyone knows this but me...)

Theorem 1. Let G be a group which has a complete metric topology for which multiplication is jointly continuous. Then inversion is continuous (i.e., G is a topological group).

There is a short, slick proof in Pfister, "Continuity of the inverse", Proc AMS 95(1985) 312-314, though the result is older.

For the rest of this note I follow Schweitzer, Larry B. “A Short Proof that $M_{n}(A)$ is local if $A$ is local and Fr\'echet.” International Journal of Mathematics 3 (1992): 581-589. Suppose that B is a C* or Banach algebra and that A is a dense subalgebra which is a Fr\'echet algebra under some topology stronger than the topology of B.

Theorem 2. A is holomorphically closed in B iff it is inverse closed.

For the proof, one notes that inverse closure implies that the invertibles are open in A, so their topology (in A) can be given by a complete metric. By Theorem 1, inversion is continuous. This means that the Cauchy integral formula for the holomorphic calculus converges in A.

Theorem 3. A is inverse closed in B iff every irreducible A module is a submodule of some B module.

This involves some topological-algebraic manipulation with maximal left ideals. An irreducible A module is of the form $A/m$, where $m$ is some maximal left ideal. Let M be the closure of m in B. Using density and inverse closure one sees that $m=M\cap A$. But then
$A/m = A/A\cap M = (A+M)/M \subseteq B/M $ is contained in the B-module $B/M$.

Corollary. $A$ is inverse (or holomorphically) closed in $B$ iff every matrix algebra $M_n(A)$ is inverse closed in $M_n(B)$.

This follows since an irreducible $M_n(A)$-module is just the direct sum of $n$ copies of some irreducible $A$-module.

Tuesday, June 29, 2010

Connes Embeddings and von Neumann Regular Closures of Group Algebras

This is an interesting paper of Gabor Elek's which touches on some things I've posted about recently - especially (i) the Atiyah conjecture and (ii) the idea (which shows up in the work of Ara et al) that one can use some kind of "asymptotic rank" instead of "asymptotic trace" in some contexts where you want to build "continuous dimension" type invariants.

Friday, June 25, 2010

Exchange rings and translation algebras

Recall that a ring $R$ is von Neumann regular if given any $x\in R$ there is $y\in R$ such that $xyx=x$. (Examples: fields; matrix algebras; various rings of unbounded operators, where $y$ is "the inverse of $x$ away from the kernel".) A ring $R$ is called an exchange ring if, for every $x\in R$, there is an idempotent $e\in R$ such that $e\in xR$ and $(1-e)\in (1-x)R$. Von Neumann regular rings are examples of exchange rings. There is a developed algebraic theory of these rings, with links to real rank 0 C*-algebras among other things.

I just became aware of the paper Ara, P., K. C O'Meara, and F. Perera. Gromov translation algebras over discrete trees are exchange rings. Transactions of the American Mathematical Society 356, no. 5 (2004): 2067–2079 (electronic). In this paper it is shown that the algebraic translation algebra associated to a tree, with coefficients in a von Neumann regular ring, must be an exchange ring. The tree hypothesis is used in various places, but the authors don't know, apparently, whether there are examples of metric spaces $X$ for which the translation algebra is not an exchange ring. (The plane might be a good example to start with.)

Wednesday, June 23, 2010

C*-algebras, foliations and K-theory

In 1980, Alain Connes gave a course entitled "C*-algebras, foliations and K-theory". Jean Renault was a student in the course at that time and took notes, and photocopies of his meticulously handwritten manuscript have been passed around generations of students. I must have acquired mine some time around 1988.

The notes describe projective modules, Morita equivalence, K-theory, non-unital algebras and multipliers, quasi-isomorphisms, smooth subalgebras and "holomorphic closure", Bott periodicity, crossed products, the Thom isomorphism for crossed products, and the beginnings of noncommutative geometry. Its fascinating to see how early some of these ideas were germinating, and what they looked like at taht early stage.

In our seminar last year we assigned graduate students to read and lecture on various parts of the manuscript. This led to a rough English translation, which I'm now in the process of tidying up. I hope to post a more polished version to the arXiv before too long. I'm grateful to Alain and Jean for encouraging this project.

You can find scans (not too legible) of the lecture notes here.

Wednesday, June 16, 2010

Bounded groups

Today's post arises from a paper by my Penn State colleague Dima Burago and his collaborators, Conjugation-invariant norms on groups of geometric origin. (October 7, 2007).

The basic definition is a very simple one: say that a group $G$ is bounded if it has finite diameter with respect to any bi-invariant metric.
(Of course it is enough to consider the distance from a group element $g$ to the identity element, and this function $|g|$ then becomes a \emph{conjugation invariant norm} on the group, hence the title of the paper. A group is then bounded if every conjugation-invariant norm is bounded from above. One can also ask whether every conjugation-invariant norm is bounded from below (away from the identity element). When a norm is bounded from above and below and can be said to be trivial: it is equivalent to the norm which equals 1 on all non-identity elements. If every conjugation-invariant norm is trivial the group is called meager.

The basic phenomenon studied in the paper is the surprising fact that conjugation-invariant norms on large groups of diffeomorphisms tend to be trivial. In fact the main result is that the identity component of the diffeomorphism group of $M$ is meager whenever $M$ is a sphere or a closed connected 3-manifold. However the paper begins by proving some of the basic facts about bounded groups, and this part was already very interesting to me.

The groups $SL(n,R)$, $n\ge 2$, and $SL(n,Z)$, $n\ge 3$, are bounded. In the real case this follows from "a suitable version of the Gauss elimination process" to quote the paper... you have to show that any element of $SL(n,R)$ can be written as the product of a bounded number of conjugates of a bounded number of generators (namely, the elementary matrices with a 1 off the diagonal). Takes a moment's thought. Over the integers one needs also the bounded generation of $SL(n,Z)$ for $n\ge 3$ (Carter, David, and Gordon Keller. “Bounded Elementary Generation of SLn(O).” American Journal of Mathematics 105, no. 3 (June 1983): 673-687.)

An abelian group is bounded if and only if it is finite. This seems obvious, but there is no finite generation restriction here, so some care is needed in applying structure theory.

If $G$ surjects onto an unbounded group $H$, then $G$ is unbounded The basic idea is simple enough - take a conjugation-invariant norm on $H$ and pull it back via the surjection - but unfortunately the result need not be a norm (it vanishes on the whole kernel not just on the identity), so one needs to develop a more flexible theory of unbounded "quasi-norms" and prove that the existence of unbounded norms and quasi-norms are equivalent.

Thursday, May 27, 2010

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.

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:
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.

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

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 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.

Thursday, April 29, 2010

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?

Monday, April 26, 2010

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:

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.

Friday, April 23, 2010

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.