Madsen–Weiss for geometrically minded topologists

Madsen-Weiss for geometrically minded topologists

We give an alternative proof of Madsen-Weiss’ generalized Mumford conjecture. Our proof is based on ideas similar to Madsen-Weiss’ original proof, but it is more geometrical and less homotopy theoretical in nature. At the heart of the argument is a geometric version of Harer stability, which we formulate as a theorem about folded maps.

Interpreter for topologists

Let M be a transitive model of set theory. There is a canonical interpretation functor between the category of regular Hausdorff, continuous open images of Čech-complete spaces of M and the same category in V , preserving many concepts of topology, functional analysis, and dynamics. The functor can be further canonically extended to the category of Borel subspaces. This greatly simplifies and e...

Analytic Novikov for topologists

We explain for topologists the “dictionary” for understanding the analytic proofs of the Novikov conjecture, and how they relate to the surgery-theoretic proofs. In particular, we try to explain the following points: (1) Why do the analytic proofs of the Novikov conjecture require the introduction of C-algebras? (2) Why do the analytic proofs of the Novikov conjecture all use Ktheory instead of...

When Topologists Are Politicians...

Topologists Take Scalpel to Brain Scans

In recent years, magnetic resonance imaging has revolutionized the ability of doctors to understand the organization and the functioning of the human brain. But if you look at a brain scan close up—really, really close—you’ll see some things that nature never intended. Because of glitches in the digital imaging process, nonexistent connec-tions can appear between adjacent folds of the cerebral ...