The geometry of Weyl groups and their maximal tori
Motivated by a question from Theoretical Physics, we classify the conjugacy
classes of maximal order Abelian subgroups of Weyl groups and establish a
version of Cartan's theorem on the conjugacy of maximal tori in this
context. The classification and applications will encounter many old friends
and some new ones (Symmetric spaces, branching rules, Del Pezzo surfaces,
Fano planes and qubits).
A new invariant of \(G_2\) structures
I will report on recent work where we define an invariant of diffeomorphisms
and homotopies of a \(G_2\) structure on a closed 7-manifold \(M\). The
\(\nu\)-invariant takes values in \(\Bbb Z/48\) and is defined via the Euler
characteristic and signature of a \(\text{Spin}(7)\)-coboundary of the \(G_2\)
structure.
An important motivation for defining the \(\nu\)-invariant is to investigate
the connectivity of moduli space of \(G_2\)-metrics on \(M\) in the case
where such metrics exist.
I will discuss examples, calculations for the invariant to date and its
relationship to the mapping class groups of spin 7-manifolds. This work is
joint with Johannes Nordström.
Beating and heating planar domains
If you had perfect pitch and listened to a recording of the sounds a drum
made when struck, could you determine the shape of the drum? Phrased more
mathematically, do the eigenvalues of the Dirichlet Laplace operator acting
on smooth functions on a piecewise smooth planar domain \(D\) determine
\(D\) up to isometry? This question was answered in the negative in 1992 by
Carolyn Gordon, David Webb, and Scott Wolpert. Their answer tells us that
these eigenvalues are an incomplete set of geometric invariants, and it is
natural to look for ways to distinguish such non-isometric sound-alike
drums. After introducing some examples of drums that sound alike, we will
examine what we can learn from heating these drums and studying the amount
of heat in them over time. This is joint work with Michiel van den Berg and
Thomas Kappeler.
On Novikov cohomology of cochain complexes
Informally, Novikov cohomology of a cochain complex is the cohomology of a
"completion" of the cochain complex. I will discuss the case of a cochain
complex over a Laurent polynomial ring in finitely many indeterminates, in
which case Novikov cohomology detects certain finiteness properties of the
original cochain complex. Concrete results proved in collaboration with
David Quinn make heavy use of a toric re-interpretation of a known
one-variable result due to Ranicki. I will explain the relevant
constructions in some detail, concentrating on basic ideas and geometric
input.
Holomorphic geometry on compact complex manifolds
I will explain some of the techniques that went into the classification of
various types of holomorphic geometric structure (2nd order ODEs in
particular) on compact Kaehler manifolds.
Two scalar functions define all axi-symmetric transverse tracefree
tensors on flat space
A TT tensor is a symmetric \(3 \times 3\) matrix for which each row is
divergence-free and the tensor is also tracefree. This means that one has 6
independent components satisfying 4 equations. Therefore one expects 2
degrees of freedom per space point. Such TT tensors arise naturally in
several areas in General Relativity. I will show that all axisymmetric TT
tensors on flat space can be constructed from 2 axi-symmetric scalar
potentials.
Manifolds with almost nonnegative curvature operator
The curvature tensor of a Riemannian manifold also gives rise to a symmetric
operator on two-forms, the curvature operator. Closed manifolds admitting
metrics with almost nonnegative curvature operator play an important role in
the emerging theory of collapsing under a lower bound on the curvature
operator. I will discuss constructions of spaces with as well as global
obstructions to this property.
Loop Spaces and Positive Scalar Curvature
Recently, there have been a number of interesting developments
concerning the problem of understanding the space of metrics of positive
scalar curvature on a smooth manifold. In this talk, I will first provide
some background to this problem. I will then discuss a new result which
shows that in the case when the underlying manifold is a sphere of
dimension $n$, where $n$ is at least three but not equal to four, the
space of metrics of positive scalar curvature is weakly homotopy
equivalent to an $n$-fold loop space. This result makes considerable use
of a recent theorem of Botvinnik on concordance and isotopy in the space
of metrics of positive scalar curvature.
Milnor fibres and non-crossing partitions
Each arrangement of hyperplanes in a complex vector space has an associated
fibration whose base space is the once-punctured complex plane and whose
fibre is called the Milnor fibre. In this talk, non-crossing partitions are
used to describe a new triangulated space which can be used to shed new
light on the Milnor fibre when the arrangement arises from a finite
reflection group.
This is joint work with Thomas Brady and Michael Falk (Northern Arizona
University).