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Topology of cell complexes and manifolds, classification of manifolds (e.g. smoothing, surgery), low dimensional topology (e.g. knot theory, invariants of 4-manifolds), embedding theory, combinatorial and PL topology, geometric group theory, infinite dimensional topology (e.g. Hilbert cube manifolds, theory of retracts).

4 votes
Accepted

Notion of smoothness for set-valued functions

My idea is that if we want to compare $f(p)$ and $f(q)$ for nearby points $p$ and $q$, then we need to be able to put $f(p)$ and $f(q)$ into the same space. To do this, I'm going to assume that $M$ i …
Tom LaGatta's user avatar
  • 8,532
4 votes
1 answer
545 views

Symmetries of the standard probability space

The standard probability space $(I, \mathcal B, \lambda)$ consists of the interval $I = [0,1]$, its Borel $\sigma$-algebra $\mathcal B := \mathcal B(I)$ and Lebesgue measure $\lambda$. In applications …
19 votes
3 answers
3k views

When does the tangent bundle of a manifold admit a flat connection?

Let $M$ be a smooth manifold, and let $TM$ denote its tangent bundle. Under what conditions does $TM$ admit a flat connection $\omega$? Edit: Formerly, I asked about a flat connection on the frame …
Tom LaGatta's user avatar
  • 8,532