Geometry Seminar / Graduate Lectures

Rigidity for metric reduced products under forcing axioms

Forcing axioms are powerful set theoretic statements that decide many questions left open by the usual axioms of set theory. Certain mild consequences of these forcing axioms imply general lifting theorems for homomorphisms between reduced products of countable structures. These allow to characterise precisely those homomorphisms, between such quotients of products, which have a lift that factorises into homomorphisms between the factor structures. We will discuss extensions of such lifting theorems to the metric setting and ensuing rigidity phenomena for particular (metric) reduced product structures of interest.
This is based on joint work with Ilijas Farah and Alessandro Vignati, with Alessandro Vignati, and with Andreas Thom.

There is a well-known hierarchy of regularity for continuous Z-periodic functions on Rⁿ,

C⁰⊃C¹⊃. . .⊃C^∞⊃analytic⊃trigonometric polynomials,

and the decay of Fourier coefficients precisely reflects this hierarchy. In particular, the Fourier transform ^f has finite support if and only if f is a trigonometric polynomial.

Periodic hypersurfaces in Rⁿ exhibit a similar hierarchy of regularity, yet no direct analogue of this Fourier characterization is known.

In this talk, I will present joint work with Mario Kummer in which we give a surprising sufficient Fourier criterion ensuring that a C^1+ε periodic hypersurface Σ⊂Rⁿ is the zero set of a trigonometric polynomial of the form

p(e^2πix₁, . . . , e^2πix_n),

where p is a Lee--Yang polynomial.

The criterion is formulated using a notion recently introduced by Yves Meyer: a periodic positive Radon measure m on Rⁿ is called a lighthouse measure if supp(m) has zero Lebesgue measure and supp(^m) is contained in a proper double cone.

Our proof produces explicit examples of such lighthouse measures and establishes an inverse theorem. It relies on recent breakthroughs concerning the existence and classification of one-dimensional Fourier quasicrystals. No field-specific background is assumed. This work builds on collaborations with Alex Cohen, Pavel Kurasov, and Cynthia Vinzant.