MPI Bonn

We prove that for closed surfaces of fixed topological type, orientable or non-orientable, there exists a unit volume metric, smooth away from finitely many conical singularities, that
maximizes the first eigenvalue of the Laplace operator among all unit volume metrics. The key ingredient are several monotonicity results, which have partially been conjectured to hold before. This
is joint work with Henrik Matthiesen.

Spectra provide a way of understanding cohomology theories in terms of homotopy theory. Spectra are a bit like CW-complexes, they have homotopy groups which may be used to characterize homotopy equivalences. However, a spectrum has homotopy groups in negative degrees, too, and they are abelian groups in all degrees. We will discuss spectra representing ordinary cohomology, bordism, and K-theory.

I will survey recent advances in our understanding of extended
3-dimensional topological field theories. I will describe recent work (joint
with B. Bartlett, C. Douglas, and J. Vicary) which gives an explicit
"generators and relations" classification of partially extended 3D TFTS
(assigning values only to 3-manifolds, surfaces, and 1-manifolds). This will
be compared to the fully-local case (which has been considered in joint work
with C. Douglas and N. Snyder).

 Parametrized spectra are topological objects that represent
twisted forms of cohomology theories.  In this talk I will describe a theory
of parametrized spectra as highly structured bundle-like objects.  In
particular, we can make sense of the structure "group" of a bundle of
spectra.  This point of view leads to new examples and a good framework for
twisted equivariant cohomology theories.  

Topological field theories give a connection between

topology and algebra. This connection can be exploited in both

directions: using algebra to construct topological invariants, or

using topology to prove algebraic theorems. In this talk, I will

explain an interesting example of the latter phenomena. Radford's

theorem, as generalized by Etingof-Nikshych-Ostrik, says that in a

finite tensor category the quadruple dual functor is easy to

understand. It's somewhat mysterious that the double dual is hard to

understand but the quadruple dual is easy. Using topological field

theory, we show that Radford's theorem is exactly the consequence of

the Dirac belt trick in topology. That is, the double dual

corresponds to the generator of $\pi_1(\mathrm{SO}(3))$ and so the

quadruple dual is trivial in an appropriate sense exactly because

$\pi_1(\mathrm{SO}(3)) \cong \mathbb{Z}/2$. This is part of a large

project, joint with Chris Douglas and Chris Schommer-Pries, to

understand local field theories with values in the 3-category of

tensor categories via the cobordism hypothesis.

The goal of this talk is to construct new examples of hyperbolic

aspherical complexes. More precisely, given an aspherical simplicial

complex P and a subcomplex Q of P, we are looking for conditions under

which the complex obtained by attaching a cone of base Q on P remains

aspherical. If Q is a set of loops of a 2-dimensional complex, J.H.C.

Whitehead proved that this new complex is aspherical if and only if the

elements of the fundamental group of P represented by Q do not satisfy

any identity. To deal with higher dimensional subcomplexes we use small

cancellation theory and extend the geometric point of view developed by

T. Delzant and M. Gromov to rotation families of groups. In particular

we obtain hyperbolic aspherical complexes obtained by attaching a cone

over the "real part" of a hyperbolic complex manifold.