We will revisit two important operations on differential forms, namely, contraction and Lie derivative. In particular, if M is a $G$-manifold, then contraction and Lie derivative (both with respect to the fundamental vector fields), make the de Rham complex $\Omega(M)$ a $\mathfrak{g}$-DGA. Next, we will return to the Weil model for circle actions and show that it is isomorphic to the polynomial ring with circle-invariant forms on $M$ as the coefficients.

Meeting ID: 997 4840 9440 Passcode: 911104

The main idea of D-modules (that is, of algebraic analysis) is to re-interpret systems of linear PDEs as modules over a ring of differential operators. You then prove things about these modules using the tools of algebra and algebraic geometry, and then translate the results back into PDE language. An archetypal example of such a result is the Cauchy-Kovalevskaya-Kashiwara theorem, which is a vast generalization of the Cauchy-Kovalevskaya theorem from the classical study of PDEs. I will give a sketch of the ideas that are needed to digest this theorem. Along the way, you'll be introduced to D-modules (of course!) and their characteristic varieties.

Many systems across the sciences evolve through the interaction of multiplicative growth and diffusive transport. In the presence of disorder, these opposing forces can generate localized structures and bursty dynamics, a phenomenon known as "intermittency" in non-equilibrium physics and as "punctuated equilibrium" in evolutionary theory. This behaviour is difficult to forecast; in particular there is no general principle to locate the regions where the system will settle, how long it will stay there, or where it will jump next. In this talk I will introduce a Markovian representation of growth-transport dynamics that closes these gaps. This retrospective view of evolution unifies the concepts of linear intermittency and metastability, and provides a generally applicable method to reduce, and predict, the dynamics of disordered linear systems. Applications range from Zeld'dovich's parabolic Anderson model to Eigen's quasispecies model of molecular evolution.