Unlike their counterparts on bounded domains, semilinear heat equations on $R^N$ admit bounded solutions with very diverse large-time behavior. I will first present several examples of solutions with interesting and sometimes entertaining behavior in compact regions. Then I will discuss a few general results describing the behavior of more specific classes of solutions. Some ideas and techniques of more general interest, such as the Sturmian zero number and the method of spatial trajectories, will also be discussed. 

The triangulation conjecture stated that any n-dimensional
topological manifold is homeomorphic to a simplicial complex. It is
true in dimensions at most 3, but false in dimension 4 by the work of
Casson and Freedman. In this talk I will explain the proof that the
conjecture is also false in higher dimensions. This result is based
on previous work of Galewski-Stern and Matumoto, who reduced the
problem to a question in low dimensions (the existence of elements of
order 2 and Rokhlin invariant one in the 3-dimensional homology
cobordism group). The low-dimensional question can be answered in the
negative using a variant of Floer homology, Pin(2)-equivariant
Seiberg-Witten Floer homology.

A subset of the real line is called a Cantor set if it is compact,
perfect, and nowhere dense. Cantor sets arise in many areas; in this
talk we will discuss their relevance in the spectral theory of
Schr\"odinger operators. We discuss several results showing that the
spectrum of such an operator is a Cantor set, from the discovery of the
first example by Moser to a genericity result by Avila, Bochi, and
Damanik. A Cantor measure is a probability measure on the real line
whose topological support is a Cantor set. A primary example in the
spectral theory context is the density of states measure in situations
where the spectrum is a Cantor set. A conjecture of Simon claims a
strict inequality between the dimensions of the set and the measure for
the Fibonacci potential. If time permits, we will discuss a recent
result of Damanik, Gorodetski, and Yessen, which establishes this
conjecture in full generality.

In the last century, Geometry underwent several
substantial extensions and revisions based on
the fundamental revolutions that it lived through
in the XIXth century.

The purpose of the lecture is to discuss several
aspects of these transformations: the new
concepts that emerged from these new points
of view, the new perspectives that could be
drawn from bringing together the continuous and
discrete viewpoints, some classical problems that could
be solved, and the new interactions with other
disciplines that went along.

It includes the presentation of the views of the late Professor
Chern Shiing Shen on some of these issues.

I will make a survey of recent results on the spectrum of periodic and, to a smaller extent, almost-periodic operators. I will consider two types of results:

1. Bethe-Sommerfeld Conjecture. For a large class of multidimensional periodic operators the numbers of spectral gaps is finite.

2. Asymptotic behaviour of the integrated density of states of periodic and almost-periodic operators for large energies.