Critical sets of random smooth functions on compact manifolds

Abstract

ABSTRACT. Given a compact, connected Riemann manifold without boundary (M, g) of dimension m and a large positive constant L we denote by U L the subspace of C

Fluctuations of the total number of critical points of random spherical harmonics

“…It is known [16,8] that, as ℓ → ∞, the expected total number of critical points N c (f ℓ ) is asymptotic to…”

Critical points of multidimensional random Fourier series: Variance estimates

“…The computations in [29] also show that for any symmetric polyhedron P there exists an explicit constant C = C m (P) > 0 such that Z νP ∼ C m (P)ν m as ν → ∞ inside N.…”

“…In [29] we have computed Z P for symmetric polyhedra P, i.e., polyhedra that are invariant under the obvious action of the symmetric group S m on R m and we have observed that for the polyhedra P considered by Arnold the lower bound Z P is so close to the upper bound K m (P) that we can conclude via simple topological arguments that N P max = K m (P). The computations in [29] also show that for any symmetric polyhedron P there exists an explicit constant C = C m (P) > 0 such that Z νP ∼ C m (P)ν m as ν → ∞ inside N.…”

Complexity of random smooth functions on compact manifolds

“…The many technical assumptions in Adler-Taylor Theorem are trivially satisfied in this case. In [23] we proved this theorem in the case B = R and ω = 0. The strategy used there can be modified to yield the more general Theorem 2.1.…”

“…In [23] we showed that there exists a universal (explicit) constant C m that depends only on the dimension m such that…”