Francis Brown : Periods, Galois theory and particle physics: Applications
- Gergen Lectures ( 286 Views )In the final lecture, I will propose how the Galois theory of periods should lead to a classification of periods by types. When applied to the set of Feynman integrals occurring in particle physics, experiments suggest the emergence of a `cosmic? Galois group of symmetries acting on the constants of high-energy physics.
Jordan S. Ellenberg : Stability and Representations
- Gergen Lectures ( 278 Views )The notion of stability --speaking loosely, "sometimes an infinite sequence of vector spaces eventually starts being constant" -- appears in many branches of mathematics, perhaps most notably topology, where Harer's theorem about the stability of the homology of mapping class groups has driven decades of work. Some natural sequences of vector spaces are evidently NOT eventually constant: for instance, the space Q_n of quadratic polynomials in n variables has dimension (1/2)n(n-1), so gets larger and larger as n goes to infinity. On the other hand, Q_n carries an action of the symmetric group S_n by permutation of coordinates. We will discuss a new framework which allows us to speak meaningfully about what it means for a sequence of representations of S_n to be stable. It turns out that the structures we define are ubiquitous, appearing in topology (e.g. homology groups of configuration spaces and of moduli spaces of curves) algebraic combinatorics (e.g. the graded pieces of diagonal coinvariant algebras) and algebraic geometry (e.g. spaces of polynomials on discriminant and rank varieties.) We prove, for instance, that all these sequences of vector spaces have dimension which is eventually a polynomial in n.
Alice Guionnet : The spectrum of non-normal matrices, II: the Brown measure.
- Gergen Lectures ( 268 Views )In this talk, which is a continuation of Wednesday's lecture, we shall describe the natural candidate for the limit of the empirical measure of the eigenvalues of non-normal matrices, the so-called Brown measure. We will give some details about how to prove convergence towards such a limit, but also discuss the instability of such convergence.
Leo P. Kadanoff : Drips and Jets: Singularities, Topology Changes, and Scaling for Fluid Interfaces
- Gergen Lectures ( 38 Views )We investigate the behavior of the interface between two fluids. We are interested in the singularities which develop when the bridge connecting two pieces of fluid goes to zero thickness. One physical situation is Hele-Shaw flow: two fluids are trapped between parallel glass plates and feel frictional forces. Another situation is a cylindrically symmetrical stream, for example, in a dripping faucet. In a third case, a fluid in an electric or magnetic field which pulls off a piece of fluid ending in a sharp point. The last case has an interface between a light and a heavier fluid being sucked up as in a drinking straw. At a critical value of the sucking, a very thin bridge of the heavy fluid is formed. Analytical, numerical, and experimental methods are used to describe what happens right around the pinchoff of these bridges. Much of the resulting behavior can be understood via scaling arguments.