### Contractible 3-manifolds and positive scalar curvature

It is known that $$\mathbb{R}^3$$ admits a complete metric of uniformly positive scalar curvature. In fact, for any closed manifold $$X$$ and any $$k \ge 3$$ the manifold $$X \times \mathbb{R}^k$$ admits a complete metric of uniformly positive scalar curvature by a result of Rosenberg and Stolz (link). Now there exist contractible, open 3-manifolds which are not … Continue reading "Contractible 3-manifolds and positive scalar curvature"

### Prizes, prizes, prizes

Several prizes have been awarded in the past few weeks to mathematicians. Kyoto Prize The Kyoto Prize 2018 in the category Basic Sciences was awarded to Masaki Kashiwara from the RIMS at Kyoto University. (announcement) The Kyoto Prize is awarded annually to “those who have contributed significantly to the scientific, cultural, and spiritual betterment of mankind” … Continue reading "Prizes, prizes, prizes"

### Peter Scholze a new director at MPI Bonn

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"