### Laplace operator and covering spaces

Let M be a closed Riemannian manifold and denote by X its universal covering space equipped with the pulled back Riemannian metric. There is an intimate relation between the Laplace operator on X and the fundamental group of M. One example of this is the result of Brooks from 1981: the fundamental group of M … Continue reading "Laplace operator and covering spaces"

### AMS Special Session on the Mathematics of John Roe

There was a Special Session on Coarse Geometry, Index Theory, and Operator Algebras: Around the Mathematics of  John Roe at the Spring Central and Western Joint Sectional Meeting of the AMS last weekend to which Christopher and I were invited to give talks. John Roe passed away last year. His personal webpage is still online … Continue reading "AMS Special Session on the Mathematics of John Roe"

### Coarse embeddings and non-positive curvature

Let $$(X,d)$$ be a complete, geodesic metric space. $$(X,d)$$ is called an Alexandrov space of global non-positive curvature if for every quadruple of points $$x,y,z,w$$ such that $$w$$ is a metric midpoint of $$x$$ and $$y$$, i.e., $$d(w,x) = d(w,y) = d(x,y)/2$$, we have $d(z,w)^2 + d(x,y)^2/4 \le d(z,x)^2/2 + d(z,y)^2/2.$ If the reverse inequality … Continue reading "Coarse embeddings and non-positive curvature"

### Operators of finite propagation

In the starting post of this series we considered infinite band matrices (with uniformly bounded entries) acting on infinite vectors and asked at the end the question how to determine whether a given matrix, which is not a band matrix, can be approximated by such. Today we provide the setup in order to answer this question properly in … Continue reading "Operators of finite propagation"

### Equivariant band matrices and Fourier series

Recall that in the first post of this series we claimed that there exists an infinite matrix $$T$$ which is in the closure (in operator norm) of the band matrices with uniformly bounded entries, but for which we have $$\|T^{(R)}\| \to \infty$$. Here \[T^{(R)}_{m,n} := \begin{cases} T_{m,n} & \text{ if } |m-n| \le R\\ 0 & … Continue reading "Equivariant band matrices and Fourier series"

### Norms of infinite matrices

This is the first post of a series of posts in which we will eventually venture deep into the realm of coarse geometry. But we will always be motivated by questions which are related to the one that we will discuss here. But our first steps into coarse geometry will be very gently: we will … Continue reading "Norms of infinite matrices"