## Abstracts of Talks

### Pierre-Emmanuel Caprace, University of Louvain

### Isometry Groups of Proper CAT(0) Spaces

CAT(0) spaces provide a unified framework for the study of large families of geometric objects, including simply connected Riemannian manifolds of non-positive sectional curvature and Euclidean buildings. Numerous other examples are known and have attracted much attention in geometric group theory. While any attempt of getting an exhaustive understanding of locally compact CAT(0) spaces seems unrealistic, it appears that the higher rank symmetric spaces and Bruhat-Tits buildings play a very special and prominent role in the theory, perhaps because of their remarkable rigidity properties. The talk will illustrate this fact by means of recent results based on a detailed study of the full isometry group of a locally compact CAT(0) space.

### Benson Farb, University of Chicago

### Some Universality Phenomena for Pseudo-Anosov Dilations

Attached to a pseudo-Anosov
homeomorphism *f* of a surface *S* is an algebraic integer
called the *dilatation* of *f*. This number determines
the entropy of *f*, the growth rate of lengths of curves
under iteration by *f*, the length of the corresponding
loop in moduli space, and much more.

The list of numbers occuring as dilatations as both *f* and *S* vary is quite mysterious. In this talk I will
explain the discovery of some "universality phenomena" concerning this list of numbers. This is joint work with
Chris Leininger and Dan Margalit.

### Tom Farrell, Binghamton University

### Bundles with Negatively Curved Fibers

This talk is a report on joint work with Pedro Ontaneda. Let *M* be
a closed smooth manifold which can support a Riemannian metric with sectional
curvatures all negative; e.g. a hyperbolic metric. We are interested in
smooth *M*-bundles *p* :* E *→ *B* whose
abstract fiber is *M*; but all of whose specific fibers *p*^{–1}(*x*), *x* in *B*,
are equipped with negatively curved Riemannian metrics *b _{x}*,
which vary continuously with

*x*. This is called a bundle with negatively curved fibers. Two such bundles

*p*,

_{i}*i*= 0 or 1, are equivalent if there exists a smooth

*M*-bundle

*E*with negatively curved fibers and base space

*Bx*[0,1] whose restriction to

*Bxi*,

*i*= 0 or 1, is isomorphic to

*E*(via a bundle map which is an isometry between fibers). Consider the forget structure map

_{i}*F*from

*M*-bundles with negatively curved fibers to smooth

*M*-bundles. One can ask the following three natural questions about

*F*.

- Is
*F*non constant, or does the image of*F*consist only of the trivial bundle? - Is
*F*onto? - Is
*F*one-to-one?

In the case that *B* is the *n*-dimensional sphere with *n* < (1/3) dim(*M*),
some hopefully interesting information on these questions will be presented.

### Cameron Gordon, University of Texas

### Surface Subgroups of Doubles of Free Groups

A well-known question of Gromov asks whether every one-ended
word hyperbolic group *G* contains a surface subgroup. We consider the
special case where *G* is the double of a free group *F* along the cyclic
subgroup generated by an element *w* in *F*, and give various conditions on *w* under which Gromov's question has an affirmative answer. This is joint
work with Henry Wilton.

### Bruce Kleiner, Courant Institute

### A New Proof of Gromov's Theorem on Groups of Polynomial Growth

In 1981 Gromov showed that any finitely generated group of polynomial growth contains a finite index nilpotent subgroup. This has a variety of applications, ranging from dynamics to probability theory. Gromov's proof was based in part on a beautiful rescaling argument, and the Montgomery-Zippin solution to Hilbert's fifth problem on topological groups.

The purpose of the lecture is to describe a new, much shorter, proof of Gromov's theorem, based on harmonic maps instead of the Montgomery-Zippin theory. I will begin by reviewing the history of Gromov's theorem, and some of its applications.

### Seonhee Lim, Cornell University

### Volume Entropy of Buildings

Volume entropy of a building is the exponential growth rate of the volumes of metric balls. We will give several characterizations of the volume entropy, analogous to the ones for trees, useful for finding some lower bound on volume entropy for certain hyperbolic buildings. A conjecture of Katok, related to volume entropy rigidity, says that for manifolds (of nonpositive curvature), the volume entropy is minimized when the Liouville measure is a measure of maximal entropy. We show that this is not the case for some hyperbolic buildings. (This is a joint work with Francois Ledrappier.)

### Robert MacPherson, Institute for Advanced Study

### The Geometry of Crystal Decompositions in Materials

Individual crystals in a metal or ceramic are cells of a three-dimensional cell decomposition of the material. The geometry and combinatorics of this cell complex influence the properties of the material. This cell complex evolves over time. In 1952, von Neumann gave a simple and useful formula for the growth rate of an individual crystal in 2 dimensions. This formula is generalized to 3 (and higher) dimensions in joint work with David Srolovitz.

The cell complex of crystals is expected to be topologically and metrically universal, in a statistical sense, independent of the detailed chemical nature of the material itself. There are many interesting mathematical questions about the combinatorial topology of this universal cell complex structure.

### Kasra Rafi, University of Chicago

### Counting Closed Geodesics in a Stratum

We obtain an asymptotic formula for the number of closed geodesics in
a stratum of Teichmüller space.
In particular, we compute the asymptotics, as *R* tends to infinity, of
the number of pseudo-Anosov elements
of the mapping class group which have translation length less than *R* and orientable invariant foliations.
This is joint work with Alex Eskin and Maryam Mirzakhani.

### Bertrand Rémy, University of Lyon

### Rigidity and Quasi-isomorphism Classes of Simple Twin Building Lattices

The main subject matter of the talk will be twin building
lattices
that have been shown to be simple in a previous work. Simplicity
holds *generically* when the buildings under consideration are
not
Euclidean; the most well-understood class of such groups is provided
by Kac-Moody theory. These groups are often finitely presented and
enjoy Kazhdan's property (T). The latter property suggests to
investigate the rigidity properties of the natural action of the
groups: we have a *higher-rank versus hyperbolic* rigidity result.
One point is that *higher-rank* has to be given a suitable sense;
the
statement can probably be improved. In the same spirit, our most
recent result is the fact that we can obtain infinitely many quasi-isometry
classes of finitely presented simple groups. This is joint work with P.-E.
Caprace.

### Anna Wienhard, Princeton University

### Domains of Discontinuity

For a big class of homomorphisms of the fundamental group of a
closed surface into a semisimple Lie group *G* satisfying some dynamical
properties, we construct domains of discontinuity in flag varieties *G*/*P* for
the
action of a surface group given by a homomorphism into *G* satisfying
some dynamical property. The important and quite surprising
property is that these domains of discontinuity have compact quotient. The
homomorphisms we consider include in particular all *higher
Teichmueller spaces* and generalizations of quasi-fuchsian
representations.

This is joint work with Olivier Guichard.