Rice University Department of Mathematics Colloquium

Complexity of triviality in topology with applications to geometric variational problems

4:00 pm Thursday, November 8, 2017
Alexander Nabutovsky (University of Toronto)

Abstract:The first theme of the talk is that some geometric objects have trivial topology, but it is very difficult to see that this, indeed, is the case. The second theme is that this phenomenon implies the ruggedness of some moduli spaces in differential and combinatorial geometry, and that it even forces the existence of non-trivial solutions of some problems in geometric calculus of variations. These phenomena were previously known for dimensions >4 (joint work with Shmuel Weinberger). However, recently we managed to prove that they already exist in dimension 4 (joint work with Boris Lishak). In particular, I will show that (1) it is very difficult to untie some trivial 2-knots in the four-dimensional space; (2) there exist ``many" contractible 2-dimensional complexes, which are extremely difficult to contract and for each pair of them it is also very difficult to see that they are homotopy equivalent to each other; (3) for each n>3 there exist infinitely many distinct local minima of curvature-pinching sup |K| diam^2 (C^0 norm of the sectional curvature normalized by the square of the diameter) on the space of Riemannian structures on the n-sphere; (4) in high-dimensions there exist many non-trivial ``thick" knots of codimension one (unlike the usual knot theory).

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