UBC

From each point of a Poisson point process start growing a balloon at rate 1. When two balloons touch, they pop and disappear. Will balloons reach the origin infinitely often or not? We answer this question for various underlying spaces. En route we find a new(ish) 0-1 law, and generalize bounds on independent sets that are factors of IID on trees. Joint work with Omer Angel and Gourab Ray.

Transient or unstable behavior is often ignored in considering long time dynamics in the deterministic world. However, stochastic effects can change the picture dramatically, so that the transients can dominate the long range behavior.

Coherence resonance is one relatively simple example of this transformation, and we consider others such as noise-driven synchronization in networks, recurrence of diseases, and stochastic stabilization in systems with delay.

The challenge is to identify common features in these phenomena, leading to new approaches for other systems of this type. Some recurring themes include the influence of multiple time scales, cooperation of both discrete and continuous aspects in the dynamics, and the remnants of underlying bifurcation structure visible through the noise.

A 2-dimensional Topological Quantum Field Theory (TQFT) is a symmetric monoidal functor from the category of 2-dimensional cobordisms to the category of vector spaces. A classic result states that 2d TQFTs are classified by commutative Frobenius algebras.  I show how to extend this result to open-closed TQFTs using a class of 2-manifolds with corners, how to use the Moore-Segal relations in order to find a canonical form and a complete set of invariants for our cobordisms and how to classify open-closed TQFTs algebraically.  Open-closed TQFTs can be used to find algebraic counterparts of Bar-Natan's topological extension of Khovanov homology from links to tangles and in order to get hold of the braided monoidal 2-category that governs this aspect of Khovanov homology. I also sketch what open-closed TQFTs reveal about the categorical ladder of combinatorial manifold invariants according to Crane and Frenkel.

references:

1] A. D. Lauda, H. Pfeiffer:

Open-closed strings: Two-dimensional extended TQFTs and Frobenius algebras,

Topology Appl. 155, No. 7 (2008) 623-666, arXiv:math/0510664

2] A. D. Lauda, H. Pfeiffer: State sum construction of two-dimensional open-closed Topological Quantum Field Theories,

J. Knot Th. Ramif. 16, No. 9 (2007) 1121-1163,arXiv:math/0602047

3] A. D. Lauda, H. Pfeiffer: Open-closed TQFTs extend Khovanov homology from links to tangles, J. Knot Th. Ramif., in press, arXiv:math/0606331.