Schedule

The conference will be held in the Laskey Building in Scarritt-Bennett center on the 2nd Floor in room Laskey C.

1008 19th Ave S, Nashville, TN 37212.

**Monday 12**

1:00 pm Shmuel Weinberger

Title: From aggregation on networks to a problem of Wall.

Abstract: I will start by discussing the problem of aggregation of
preferences in an abstract setting (coming from economics and
engineering) and then move from there towards more abstract problems
and theorems (of other people) regarding operators and quadratic forms
on infinite graphs, and will give applications to problems related to
the meaning of Poincare duality for discrete groups.

(My second talk will fill in gaps in the presentation from the first.)

2:00 pm Stanley Chang

Title: Positive scalar curvature and noncompact manifolds

Abstract: Scalar curvature, the weakest form of curvature that can be assigned pointwise to a manifold, was once imagined to be a possible tool in the classification of higher dimension manifolds. Efforts to understand positive scalar curvature have culminated in the so-called Gromov-Lawson-Rosenberg Conjecture, which is now known to be false. In this talk, we will discuss the progress made in both the compact and noncompact venues to understand the scalar curvature properties of manifolds.

3:30 pm Russell Lyons

Title: Uniform Spanning Forests, the First l^{2}-Betti Number, and
Uniform Isoperimetric Inequalities

Abstract: Uniform spanning trees on finite graphs and their analogues on
infinite graphs are a well-studied area, intimately related to random walks
and electrical networks. It turns out that they are also related to the
first l^{2}-Betti numbers of groups. We illustrate this by proving a
uniform isoperimetric inequality for Cayley graphs.

4:30 pm Simon Thomas

Title: Bratteli Diagrams and the Unitary Duals of Locally Finite Groups

Abstract: I will discuss an attempt to understand the unitary duals of discrete countable groups from the point of view of descriptive set theory.

**Tuesday 13**

9:30 Guoliang Yu

Title: The Novikov conjecture and metric geometry

Abstract: I will give an introductory talk on how metric geometry can be used to study the Novikov conjecture.
If time permits, I will explain a localization technique introduced in my recent joint work with Rufus Willett.

11:00 Mark Sapir

Title: Embedding Cayley graphs into Hilbert spaces and related questions

Abstract: I will talk about the Hilbert space compression functions of finitely generated groups.

12:00 Remi Coulon

Title: Embedding expanders into a group.

Abstract: In 2003, M. Gromov provided a construction to embed certain expanders into a finitely generated groups. He obtained in this way a "monster" with surprising properties. In particular it does not coarsely embed into a Hilbert space. In this talk I will try to explain the main ideas of this construction involving random walks and small cancellation theory.

Lunch

2:30 Russell Lyons

Title: Determinantal Probability Measures

Abstract: It turns out that the uniform spanning tree measure on a finite graph and its extensions to infinite graphs can be described via determinants. More generally, the uniform measure on regular matroids can be described via determinants. But other probability measures on the set of bases of a matroid arise in a similar way via determinants as well. Exterior algebra turns out to provide the proper framework to give easy proofs of various results.

4:00 Steve Ferry

Title: Deforming manifolds in Gromov-Hausdorff space

Abstract: If two manifolds are close to each other in some appropriate sense, must they be homeomorphic? We will discuss situations where this form of "rigidity" holds and other situations where it fails.

5:00 Yves de Cornulier

Title:
On QI-classification of Lie groups

Abstract:
I'll survey results about the quasi-isometry classification of locally compact groups, with an emphasis on the following question: given a Lie group G, which locally compact groups are quasi-isometric to G?

** Wednesday 14**

9:30 Rufus Willett

Title: Expanders and Baum-Connes type conjectures.

Abstract: Thanks to work of Guoliang Yu, it has been known for almost fifteen years that the existence of a coarse embedding into Hilbert space has 'good' consequences for Baum-Connes and Novikov type conjectures. On the other hand, expanders are known not to admit such a coarse embedding, and are known to be counterexamples to some of these conjectures (in some cases). A lot still remains unclear, however. I will discuss what can be said for certain classes of expanders (in particular, those with large girth which are used in the construction of 'exotic' groups), and how this relates to coarse embeddings into Hilbert space. This is joint work with Guoliang Yu.

11:00 Shmuel Weinberger

12:00 Miklós Abért

Title: Groups and graph limits

Abstract: In the talk I will try to explain why the notion of graph
convergence is important in group theory, and more surprisingly, why
group theory is relevant for the theory of graph convergence.

Lunch

2:30 Steve Ferry

Title: Volume Growth, DeRham Cohomology, and the Higson Compactification

Abstract: We construct a variant of DeRham cohomology and use it to
prove that the Higson compactification of R^n has uncountably
generated n^th integral cohmology. We also explain that there is,
nevertheless, a way of using the Higson compactification to prove the
Novikov conjecture for a large class of groups and give a proof, due
to Dranishnikov, that the Higson compactification of n-dimensional
hyperbolic space is acyclic in even dimensions.

4:00 Yuliy Baryshnikov

Title: Packings and coverings

Abstract: I will discuss several natural questions related to packings and coverings in Euclidean spaces:

- does there exist a graph with quadratic ball growth, which cannot be packed in in the plane?

- what is the optimal Holder exponent of a continuous map of a square onto a cube?

- what is the topology of the packing and covering configuration spaces?

5:00 James Lee

Title: Geometric analysis on graphs, algorithms, and complexity

Abstract: I will talk start by discussing some problems in geometric spectral graph theory and their applications to algorithms and computational complexity. I will also touch on some of the relevant analogies between graphs and manifolds, and how these problems show up in other areas like geometric group theory.

**Thursday 15**

9:30 Russell Lyons

Title: Random Complexes via Topologically-Inspired Determinants

Abstract: Uniform spanning trees on finite graphs and their analogues on
infinite graphs have a higher-dimensional analogue on finite and infinite
CW-complexes. On finite complexes, they relate to (co)homology, while on
infinite complexes, they relate to (the higher) l^{2}-Betti numbers. One
use is to get uniform isoperimetric inequalities.

11:00 James Lee

Title: Eigenvalues, flows, and metric uniformization

Abstract: This will be the low-dimensional talk, and will largely address the spectrum of the graph Laplacian on planar graphs and their generalizations. The main theme is: How do we do something like conformal uniformization when we have no conformal structure?

12:00 Alireza Salehi Golsefidy

Title: Expanders and finite quotients of linear groups

Abstract: I will talk about a necessary and sufficient condition for a finitely generated subgroup of SL(n,Q) under which the Cayley graphs of such a group modulo square free integers form a family of expanders (joint with Varju). As an application the fundamental theorem of affine sieve (joint with Sarnak) will be mention.

Lunch

2:30 Yuliy Baryshnikov

4:10 Colloquium talk by Lewis Bowen

5:20 Erik Guentner

Title: Coarse geometry of linear groups

Abstract: I will outline a proof, highlighting the role of permanence properties, that linear groups are coarsely embeddable in Hilbert space. In the first talk, I will focus on the permanence properties themselves, and will explain why, for example, the free product of coarsely embeddable groups is itself coarsely embeddable. The second talk will be devoted to linear groups.

Related links:

6:30 James Lee

Title: Higher-order Cheeger inequalities

Abstract: I will address general graphs, and the connection between higher eigenspaces and expansion of small sets. This turns out to have interesting applications, both to computational complexity, and to the rigorous analysis of certain graph partitioning heuristics.

**Friday 16**

9:30 Erik Guentner

11:00 Alireza Salehi Golsefidy