Abstract: In the first part we continue with:

--- Heat kernel and its asymptotic expansion, Gilkey's formula, Mackean-Singer formula, ---The index problem for elliptic PDE's, characteristic classes via Chern-Weil theory.

In the second part, Baris Ugurcan (Western) will talk about IFS (iterated function systems) and boundaries.

Abstract: Let $M$ be a real analytic Riemannian manifold. An adapted complex structure on $TM$ is a complex sturcture on a neighborhood of the zero section such that the leaves of the Riemann foliations are complex submanifolds. This complex structure is called entire if it may be extended to the whole of $TM$. In this complex structure, the energy function $E = g(x,v)$ is strictly plurisubharmonic and the length function $\sqrt{E}$ satisfies the homogeneous complex Monge-Amp\`ere equation. Thus $TM$ is a Stein manifold. If the leaves of the Riemann foliations are ``nice'' enough, in our case, $M$ being a Zoll sphere, we prove that $(M,g)$ must be the round sphere. The technique in the proof can be used in a more general setting to prove an ``algebraization'' result.

Abstract: Many of us are familiar with the Jordan canonical form for a linear operator over C, and also the rational canonical form for a linear operator over an arbitrary field F.

In this talk we consider linear operators over the field C((t)) of formal power series, and we identify a canonical form (which we call standard canonical form) for such operators based on the theorem of Newton-Puiseux. To do this we introduce the standard matrix of an irreducible polynomial over C((t)). These results provide a departure from the companion matrix approach to producing canonical forms over C((t)).

The interesting open problem here is to identify other fields F for which there is a notion of "standard canonical form". Does this depend on some kind of generalized Newton-Puiseux Theorem for F? Or is it enough to start with any field F that comes equipped with a discrete valuation R?