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Thermodynamics

ThermoI’ve been trying to learn a bit about classical thermodynamics, using Fermi’s lecture notes which are available as a low-cost Dover reprint.

That’s partly just because the subject has always been a bit mysterious to me and I would like to understand it better, but also because the Second Law of thermodynamics often gets invoked in environmental discussions – and I wonder whether it is being used accurately.  (See this blog post for extended discussion about that.)

As a mathematician, I expected the discussion of thermodynamics to be statistical, heavily engaged with probability theory.  But the main text of Fermi’s book is not about statistical mechanics at all.  Instead, it is about classical thermodynamics; the nineteenth century theory that attempted to quantify the properties of that mysterious fluid, “heat”, and its transmission from one body to another. Continue reading

Dydak’s axiomatization of Euclidean geometry

Jerzy Dydak sent me an interesting paper a couple of months ago, which you can find on his website here.   Entitled “A Topological Approach to the Axiomatization of Geometry”, it proposes a new way of building Euclidean plane geometry from he ground up.

Readers will know that although Euclid was the first to propose an axiomatic foundation for geometry, his axiom system is not precise enough by modern standards.  In the 19th century Hilbert gave a complete system of axioms for geometry, and other mathematicians followed in his steps.  In 1932 G.D.Birkhoff published a paper called A system of axioms for plane geometry based on scale and protractor.    This appeared in the Annals of Mathematics!  Birkhoff’s innovation was to assume the real numbers as given: his axioms stated that certain geometric quantities could be “measured” by real numbers.   This was of course quite different from Euclid’s approach, where the “theory of proportion”, equivalent to what we would call today the theory of the real numbers, was developed as part of geometry (Elements, book 5).   Birkhoff’s approach has been followed by many later writers of textbooks (such as my own Elementary Geometry for example).

Dydak wants to get back to the Euclidean order of business where the real numbers are developed concurrently with the foundations of geometry.  The fundamental undefined notion in his theory is the ternary relation of betweenness (as in “C is between A and B”).   This allows one to define line segments, rays, and so on.  What would conventionally be called “the completeness of the real numbers” is expressed in terms of connectedness and compactness properties of line segments (themselves defined in terms of betweenness, of course); and the “arithmetic” of real numbers appears as the result of adjoining and subdividing line segments, very much in the Euclidean manner.

It would be interesting to try teaching geometry this way.  You spend a lot of time thinking about one-dimensional geometry in this development; but, as a bonus, you understand the foundations of the real number system in an intuitively appealing way.  Dydak also suggests that the concept of lines as defined by an abstract “betweenness” might help make models of other geometries, such as the Klein model or Poincare model of hyperbolic geometry, less mysterious: no more “we are going to call this a ‘line’ even though it doesn’t look like one.”

 

“Finite part of operator K-theory” I

First of all, I apologize for the hiatus in posting over the past couple of weeks,  Organizing a (non-mathematical) conference has absorbed a big chunk of my time, and then getting back up to speed with routine tasks has absorbed another big chunk.   However…

So I started looking at the recent paper of Shmuel Weinberger and Guoliang Yu,  They are interested in looking at the part of the \(K\)-theory of the maximal C*-algebra of a group \(\Gamma\) which is generated by the projections

\[ p_H = \frac{1}{|H|} \sum_{h\in H} h\quad \in {\mathbb C}[G] \]

in the complex group algebra of \(G\), where \(H\) is a finite cyclic subgroup.   (Question: Why do they restrict attention to finite cyclic subgroups? Wouldn’t any finite subgroup work just as well.)

The claim is that these generate a “large” subgroup of \(K_0(C^*_{max}(G)) \) which is not in the image of the maximal assembly map from \(K_0(BG)\).  “Large” is expressed in terms of a lower bound for the rank of this abelian group.

The basic strategy, so far as I understand it, can be thought of in terms of a familiar argument for property T groups.  Let \(G\) be any group.  The maximal group C*-algebra has a homomorphism \(\alpha\) to \(\mathbb C\), which just is the regular representation (as a linear map on \({\mathbb C}[G]\) it sends every group element to 1.   On the other hand, the reduced (and therefore also the maximal) group \(C^*\) algebras have a different trace \(\tau\) which sends the identity element to 1 and every other element of \(G\) to 0 – this is the tracial vector state associated to the unit vector \(\xi_e\) in the regular representation \(\ell^2(G)\).  At the level of K-theory we get a diagram

\[\begin{array}{ccc} K_0(C^*_{max}(G))&\to^\alpha &{\mathbb Z}\\ \downarrow&&\downarrow\\ K_0(C^*_r(G)&\to^\tau & {\mathbb R}\end{array} \]

This diagram need not commute.  In fact, if \(G\) has property T and we consider at the top left corner the K-theory class of the Kazhdan projection – the projection (whose existence is guaranteed by property T) which maps, under any representation, to the projection onto the G-invariant subspace of that representation – then this class maps to 1 by traversing the diagram via the top right corner and to 0 traversing via the lower left corner.   However, it must commute for any element in the image of the (maximal) assembly map, as follows essentially from Atiyah’s \(L^2\) index theorem.  Thus, as is well known, we infer that the class of the Kazhdan projection is not in the image of the maximal assembly map.

Weinberger and Yu point out that a similar argument can be applied to the projection \(p_H\) associated to a finite cyclic subgroup \(H\) of \(G\). In fact, the homomorphism \(\alpha\) takes \([p_H]\) to 1, whereas the trace \(\tau\) takes it to \(|H|^{-1}\).  This is independent of any property T considerations.  Motivated by this, they conjecture that the rank of the subgroup of \(K_0(C^*_{max}(G)))\) generated by the \([p_H]\) (they call this the “finite part” of this group) is at least equal to the number of distinct orders of cyclic subgroups of \(G\), and that no non-identity element in the finite part lies in the image of the assembly map.

Next time I hope to talk about their approach to proving this in  certain cases.

 

 

Weinberger, Shmuel, and Guoliang Yu. Finite Part of Operator K-theory for Groups Finitely Embeddable into Hilbert Space and the Degree of Non-rigidity of Manifolds. ArXiv e-print, August 21, 2013. http://arxiv.org/abs/1308.4744.