My mathematical genealogy

 June 11, 2009 general

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According to the Mathematics Genealogy Project, my mathematical genealogy is:

There are some branches to choose among, but I think the branch starting with Pacioli is the most appropriate.

Growth series.

 November 30, 2006 general

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In seminar today, Okun pointed out the following interesting observation; for any finitely generated group G , you can define its growth series G(t) = \sum_{g \in G} t^{\ell(g)} , where \ell(g) is the length of the shortest word for g . The first observation is that G(t) is often a rational function, in which case G(1) makes sense. The second observation is that G(1) is “often” equal to \chi(G) . This is an example of weighted L^2 cohomology.

Grigorchuk’s group (and generally any group with intermediate (i.e., subexponential but not polynomial) growth) does not have a rational growth function; the coefficients in a power series for a rational function grow either polynomially or exponentially. This observation appears in [1]. More significantly, this paper constructs groups which, being nilpotent, have polynomial growth, but nonetheless have generating sets for which that the corresponding growth series is not rational.

[1] M. Stoll, Rational and transcendental growth series for the higher Heisenberg groups, Invent. Math. 126 (1996) 85–109.

Research Blog

 March 5, 2006 general

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I’ve been thinking for a while that I ought to start a research blog–something just to keep myself organized about the things I am thinking about, my thoughts on the papers I’ve read, my ideas, my questions. I figure I might as well make it public, though I seriously doubt anyone is going to read this.

Anyway, hence this blog. We’ll see how it works.