Events for 03/02/2015 from all calendars

Free Probability Seminar

Time: 11:30AM - 12:20PM

Location: BLOC 628

Speaker: Igor Klep, University of Auckland

Title: Dilations, inclusion of free spectrahedra and beta distributions

Abstract: Given a tuple A=(A₁,...,A_g) of symmetric matrices of the same size, consider the affine linear matrix polynomial L(x):=I - ∑ A_j x_j. The solution set S_L of the corresponding linear matrix inequality, consisting of those x in R^g for which L(x) is positive semidefinite (PsD), is called a spectrahedron. The set D_L of tuples X=(X₁,...,X_g) of symmetric matrices (of the same size) for which L(X):=I - ∑ A_j ⊗ X_j is PsD, is a free spectrahedron. We explain that any tuple X of symmetric matrices in a bounded free spectrahedron D_L dilates, up to a scale factor, to a tuple T of commuting self-adjoint operators with joint spectrum in the corresponding spectrahedron S_L. From another viewpoint, this scale factor measures the extent that a positive map can fail to be completely positive. In the case when S_L is the hypercube [-1,1]^g, we derive an analytical formula for the scale factor, which as a by-product gives new probabilistic results for the binomial and beta distributions.

Frontiers in Mathematics Lecture Series

Time: 4:00PM - 5:00PM

Location: BLOCKER 220

Speaker: Alain Goriely, University of Oxford

Title: Magnetic chains: from self-buckling to self-assembly (Graduate Lecture)

Abstract: Spherical neodymium-iron-boron magnets are marketed as toys as they can be assembled into different shapes due to their high magnetic strength. In this talk, I will consider two simple structures, chains and cylinders of magnets. By manipulating these structures, it quickly appears that they exhibit an elastic response to small deformations. Indeed, chains buckle on their own weight, rings oscillate, and cylinders resist bending but recover their shape after poking. Akin to the fundamental problem of materials science that consists in relating microscopic properties to macroscopic response, a natural question is to understand these macroscopic behaviours based on the individual physical properties of the magnets. I will show through illustrative experiments and simple model calculations that the idea of an effective magnetic bending stiffness is, in fact, an excellent macroscopic characterisation for the mechanical response of magnetic chains. I will then propose a more formal approach of the problem by considering discrete-to-continuum asymptotic analysis to derive a continuum model for the energy of a deformed chain of magnets based on the magnetostatic interactions between individual spheres.