Speaker: Masato Wakayama, Institute of Mathematics for Industry
Kyushu University, Fukuoka, Japan
[Seminar for undergraduates]
An introduction and application of Lie theory - a theme from spherical
harmonics
Spherical harmonics are orthogonal functions and span rotation invariant
spaces on the two dimensional sphere S＾2. They are a basis of the space
of the square integrable functions on S^2, as the name would suggest.
Spherical harmonics are used extensively in various fields. They have
been used to solve problems in physics, such as in heat equations, the
gravitational and electric fields. They have also been used in quantum
chemistry and physics to model the electron configuration in atoms, and
recently in computer graphics. In this talk, I will present fundamental
elements of a (Lie) group theoretical background of spherical harmonics
(representation theory of the rotation group). If time permits, I will
try to touch its application to the spherical harmonic lighting in
computer graphics.
Speaker: Masato Wakayama (IMI Kyushu U.)
Title: Quantum Rabi’s model and non-commutative harmonic oscillators – between
physics and number theory
Abstract: The quantum Rabi model is known to be the simplest model used in quantum optics to describe interaction of light and matter beyond the harmonic
oscillator. Although this model has had an impressive impact on many
fields of physics, only recently in 2011 could this model be declared
solved by D. Braak. Introduced over 70 years ago, its applications range
from quantum optics, magnetic resonance to solid state and molecular
physics. The non-commutative harmonics oscillator (NcHO) is a self-
adjoint, parity-preserving ordinary differential operator of order two
with non-commutative coefficient. The NcHO was introduced purely in
mathematics context in 1999. Although its spectrum is not still very
clear, the deep number theoretic properties of the spectral zeta
function shows the family of NcHOs has rich mathematical structure and
that’s why finding out an inter connection via representation theory
between NcHOs and Rabi -kind models would be interesting in both
mathematics and physics. In this talk, a non-trivial relation between
the quantum Rabi model and the NcHO discovered recently from the
representation theoretic viewpoint in terms of their Heun ODE pictures
will be given. Further, some number theoretical results, which may
relate to the Rabi model, will be also discussed.
Title: Stability of solitary waves of nonlinear Schrodinger equation
Abstract:
In this talk, we consider Nonlinear Schrodinger equation (NLS) which appears in many regions of mathematical physics such as Bose-Einstein condensation, plasma physics and the motion of filament vortex. One of the typical long time dynamics of NLS is the solitary wave which is a spatially localized solution which moves in a constant speed. We consider orbital and asymptotic stability of solitary waves of NLS and related nonlinear dispersive equations.
Title: K-theory, representation theory, and the Baum-Connes conjecture.
Speaker: Rufus Willet (U. Hawai`i)
Abstract: I’ll describe what K-theory is, and why it’s relevant to representations of groups via group algebras. If the group is infinite, I’ll discuss how analytic data enters the picture via various completions of the group algebras involved, and some examples. I’ll finish by describing the Baum-Connes conjecture, which posits a topological formula for the relevant K-groups, and some recent attempts to fix counterexamples.
Some of this is joint work with Paul Baum and Erik Guentner, and some with Alcides Buss and Siegfried Echterhoff.