Differential Geometry Working Seminar

Wednesday, July 6, 2022 9:30 am - 9:30 am EDT

Talk #1 (9:30am - 10:15 am): Prerecorded talk by Sven Hirsch, Duke University

"Spacetime harmonic functions and applications to relativity" (2021)

Talk #2 (10:30am - 11:30 am): Prerecorded talk by Vladimir Arnold, Steklov Mathematical Institute of Russian Academy of Sciences

"Statistics of the Morse theory of smooth functions" (2008)

Abstract: [translated from Russian]

On a two-dimensional sphere there are exactly 17746 topologically different Morse functions with 4 saddles. This result, based on combinatorics of random graphs, was obtained only a couple of years ago (during the research of Hilbert's 16th problem in real algebraic geometry on topological classification of polynomials).

[In 2007], the American mathematician L. Nikolaescu proved the Arnold conjecture that the number of such functions with T saddles on a two-dimensional sphere grows T to degree 2T (using methods going back to quantum field theory and to the mirror symmetry theory of physics).

The paper describes these studies and their counterparts for the theory of smooth functions on other manifolds, e.g. for functions on a torus and for trigonometric polynomials with a fixed Newton diagram of a given Coxter affine group.

MC 5403