Today: Geometric Structures seminar - Michele Stecconi - Tuesday June 4th, 2024 , 14:00 (Rome time) - room 133 at SISSA

[Samuel Borza]

Dear All,

This is to remind you of today's seminar from the series "Geometric Structures".

Title: Sobolev-Malliavin regularity of the nodal volume
Time: Tuesday June 4th, 2024 , 14:00 (Rome time)
Venue: room 133 - SISSA main building

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Abstract:

Consider the (d-1)-volume V(f) of the level set of a smooth function f on a compact Riemannian manifold of arbitrary dimension d. We show that, if restricted to a generic finite dimensional vector space of smooth functions, the functional f-->V(f) belongs to an appropriate Sobolev space. A fundamental ingredient is to understand the Sobolev regularity of the function t-->V(f-t) that expresses the volume of the level t of a ``typical'' Morse function.

This result can be stated more naturally in the language of a Gaussian random field f, in which case V(f) is a random variable and being Sobolev (Malliavin) implies that its law has an absolutely continuous component.
This was an open question in the 2 dimensional case: both the differentiability and the regularity of the law of the nodal length were unknown.

The result I will present completes the picture in that we describe what happens for d=2: in short, V(f) is Sobolev only if the topology of the zero set is constant for all f in the given vector space. Nevertheless, the law of V(f) has an absolutely continuous component.

(A joint work with Giovanni Peccati.)
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More information can be found here: https://sites.google.com/view/geometric-structures/

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Samuël Borza