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Archive for March, 2012

In absolutely big news, the Norwegian Academy of Science and Letters has made a fantastic decision by awarding the 2012 Abel Prize to Prof. Endre Szemeredi, one of the greatest mathematicians of our time. We must remember that such decisions are made by committees, and hence I would congratulate the Abel Committee (comprising of Ragni Piene, Terence Tao, Dave Donoho, M. S. Raghunathan and Noga Alon) for such an excellent decision !

Some months ago I told one of my über-cool-dude supervisors (Gabor) that Endre would win the Abel prize this year (guessing was no rocket science!)! I usually don’t like making such statements as there are always many great mathematicians who could win at any given time and there are a lot of other factors too. But Gabor actually told this to Endre, who ofcourse didn’t think it was serious. But apparently he did win it this year! A very well deserved award!

It’s pointless to make an attempt to talk about (not that I am competent to do so anyway) some of Prof. Szemeredi’s deep results and the resulting fundamental contributions to mathematics. Timothy Gowers wrote a good article on the same for the non-mathematical audience. Especially see a mention of Machine Learning on page 7. However, other than the Regularity Lemma that I find absolutely beautiful, my other favorite result of Szemeredi is his Crossing Lemma. A brief discussion on the Regularity Lemma in an older blog post can be found here.

An Irregular Mind: Szemeredi is 70. Book from Szemeredi’s 70th birthday conference recently.

For a short background Prof. Szemeredi was born in Budapest and initially studied at Eötvös before getting his PhD from Moscow State University, where he was advised by the legendary Soviet mathematician Israel Gelfand. He presently holds a position both at the Alfréd Rényi Institute of Mathematics and Rutgers and has had held visiting positions at numerous other places. Recently on his 70th birthday The János Bolyai Mathematical Society organized a conference in his honour, the proceedings of which were published as an appropriately titled book – “An Irregular Mind” (obviously a play on his “Regularity Lemma” and related work and as stated in the book “Szemerédi has an ‘irregular mind’; his brain is wired differently than for most mathematicians. Many of us admire his unique way of thinking, his extraordinary vision.”).

Congratulations to Endre Szemeredi and the great, absolutely unique Hungarian way of doing mathematics.

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See Also: Short Course on Additive Combinatorics focused on the Regularity Lemma and Szemeredi’s Theorem, Princeton University. (h/t Ayan Acharya)

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I thought I understood Spectral Clustering well enough till I came across these two paragraphs:

Graph Laplacians are interesting linear operators attached to graphs. They are the discrete analogues of the Laplace-Beltrami operators that appear in the Hodge theory of Riemannian manifolds, whose null spaces provide particularly nice representative forms for de Rham cohomology. In particular, their eigenfunctions produce functions on the vertex set of the graph. They can be used, for example, to produce cluster decompositions of data sets when the graph is the 1-skeleton of a Vietoris-Rips complex. We find that these eigenfunctions (again applied to the 1-skeleton of the Vietoris-Rips complex of a point cloud) also can produce useful filters in the Mapper analysis of data sets

– From Prof. Gunnar Carlsson’s survey Topology and Data. (More on this survey as a manifesto for “Topology and Data” in a later post). That aside, I do like how the image on the wiki entry for Vietoris-Rips complex looks like:

A little less intimidating ( now this almost borders on “ofcourse that’s how it is”. I am interested in the same reaction for the paragraph above some months later):

A related application [of the graph laplacian] is “Spectral Clustering”, which is based on the observation that nodal domains of the first eigenvectors of the graph laplacian can be used as indicators for suitably size-balanced minimum cuts.

– From Laplacian Eigenvectors of Graphs linked in the previous post. While this isn’t really as compressed as the lines above, they made me think since I did not know about Courant’s Nodal domain theorem. Like I did in the previous blog post, I would highly recommend this (about 120 page) book. It soon covers the Nodal Domain theorem and things make sense (even in context of links between PCA and k-means and Non-Negative Matrix Factorization and Spectral Clustering, at least in an abstract sense).

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