Fermat's Like Equation
Updated: 2011-12-22 01:52:30
For which positive integers k does the equation X^(2k-1) + y^(2k) = z^(2k+1) have a solution in positive integers x, y, and z- For which positive integers k does the equation X^(2k-1) + y^(2k) = z^(2k+1) have a solution in positive integers x, y, and z
A nilpotent Freiman dimension lemma
Updated: 2011-12-22 00:23:53Emmanuel Breuillard, Ben Green and I have just uploaded to the arXiv the short paper “A nilpotent Freiman dimension lemma“, submitted to the special volume of the European Journal of Combinatorics in honour of Yahya Ould Hamidoune. This paper is a nonabelian (or more precisely, nilpotent) variant of the following additive combinatorics lemma of Freiman: [...]The spectral theorem and its converses for unbounded symmetric operators
Updated: 2011-12-20 20:41:10Let be a self-adjoint operator on a finite-dimensional Hilbert space . The behaviour of this operator can be completely described by the spectral theorem for finite-dimensional self-adjoint operators (i.e. Hermitian matrices, when viewed in coordinates), which provides a sequence of eigenvalues and an orthonormal basis of eigenfunctions such that for all . In particular, given [...]254B, Notes 2: Cayley graphs and Kazhdan’s property (T)
Updated: 2011-12-06 18:39:01In the previous set of notes we introduced the notion of expansion in arbitrary -regular graphs. For the rest of the course, we will now focus attention primarily to a special type of -regular graph, namely a Cayley graph. Definition 1 (Cayley graph) Let be a group, and let be a finite subset of . [...]