## September 17, 2011

### QVEST, Autumn 2011

#### Posted by Urs Schreiber

This October we have the next QVEST meeting

• Quarterly Seminar on Topology and Geometry

7th of October, 2011

Utrecht University

seminar website

The speakers are

• Laurent Bartholdi (Göttingen)

Growth and Poisson boundaries of groups

Abstract: Let $G$ be a finitely generated group. A rich interplay between algebra and geometry arises by viewing G as a metric space, or as a metric measured space. I will describe two invariants of finitely generated groups, namely growth and Poisson boundary, and explain by new examples that their relationship is deep, but still mysterious.

Its growth function, $\gamma(n)$, counts the number of group elements that can be written as a product of at most $n$ generators. This function depends on the choice of generators, but only mildly: say $\gamma$ is equivalent to $\delta$ if $\gamma(n) \leq \delta(C n) \leq \delta(C^2 n)$ for some positive $C$; then the equivalence class of $\gamma$ is independent of the choice of generators.

For example, the growth of $\mathbb{Z}^d$ is asymptotic to $n^d$, while the growth of a free group is asymptotic to $2^n$. There are groups whose growth function is known to lie strictly between polynomials and exponentials; I will describe the first examples for which the asymptotic growth is known. I will also describe an example of a group of exponential growth, whose Poisson boundary is trivial for all finitely-supported random walks. Perhaps surprisingly, both examples come from the same general construction, permutational wreath products.

• Chenchang Zhu (Göttingen)

Higher extensions of Lie algebroids, integration of Courant algebroids and string Lie 2-algebras

Abstract: Recently, many efforts have been made to integrate a Courant algebroid, namely to find a global object associated to a Courant algebroid (For example, a global object corresponds to a Lie algebra is a Lie group). One of the reasons is probably that the standard Courant algebroid serves as the generalized tangent bundle of a generalized complex manifold of Hichin and Gualtieri. Thus the integration will help to understand the global symmetry of such manifolds. Our idea is that we first view an exact Courant algebroid as an extension of the tangent bundle by its coadjoint representation (up to homotopy) a la Abad-Crainic, then we perform the integration by the usual method of integration of an extension. We find that such higher extensions of Lie algebroids also include the example of string Lie 2-algebras.

• Chris Rogers (Göttingen)

Higher Symplectic Geometry and Geometric Quantization

Higher analogues of algebraic and geometric structures studied in symplectic geometry naturally arise on manifolds equipped with a closed non-degenerate form of degree greater than or equal to 2. In this talk, I will first explain how such a manifold gives an L-∞ algebra of “Hamiltonian” differential forms, just as a symplectic manifold gives a Poisson algebra of functions. I will then describe how to prequantize these manifolds and, within this context, sketch the relationship between the $L_\infty$ algebra of Hamiltonian forms and the $L_\infty$ structure on a Courant algebroid. Finally, I’ll discuss generalizations of real polarizations, and describe how twisted vector bundles play the role of the “quantum states” in higher geometric quantization.