The Milnor fibration of a complex, projective hypersurface produces a smooth manifold as a regular, cyclic cover of the hypersurface complement. When the hypersurface is a union of complex hyperplanes, the Milnor fibre is part of the study of hyperplane arrangements. In this case, the hypersurface complement is well known and studied. In particular, it is a Stein manifold, a rationally formal space, and it admits a perfect Morse function.

The cohomology and the monodromy of the Milnor fibre can be understood in terms of the cohomology jump loci of the hypersurface complement. For generic hyperplane arrangements, this cohomology and monodromy representation are known and fairly straightforward, although current technique still falls short of being able to describe even the betti numbers in the case of reflection arrangements. Some combinatorial techniques can be used to construct arrangements with Milnor fibres with interesting properties that constrast with the well-behaved nature of the arrangement complements. These include integer homology torsion, non-formality, and non-trivial monodromy representations in all cohomological degrees.

This talk is based on joint work with Alex Suciu.

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.

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.

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?

Automatic sequences are sequences produced by automata, which can be seen as directed graphs with extra decoration. Most sequences arising in combinatorics are automatic when reduced modulo a prime power. Cayley graphs, on the other hand, are directed graphs obtained from finite groups with distinguished generators.

Following an observation by Rowland, we study those sequences which can be produced by an automaton which is a Cayley graph (with extra information). For 2-automatic sequences (for which the n-th term is a computed from the digits of n in base 2, essentially) the result is particularly satisfying: a given sequence comes from a Cayley graph if and only if it enjoys a certain symmetry, which we call self-similarity.

We give an application to the computation of certain rational fractions associated to automatic sequences.

Pro objects, such as the absolute Galois group of a field, are pervasive in algebra. They formed the basis for the original applications of homotopy theory in geometry and number theory, via étale homotopy theory. For some time, local homotopy theory and étale homotopy theory were almost orthogonal as theories, but the relationship between the two is much better understood now, and there is a theory which engulfs both.

Perhaps there is an even more general theory that is not based on pro objects, with potential geometric applications which are not bound to the étale topology. I will describe a candidate in this talk, after some teaching moments.