Counting $V$-Tangencies and Nodal Domains (19rit271)


(McGill University/Cardiff University)

Igor Wigman (King's College London)


The Banff International Research Station will host the "Counting $V$-tangencies and nodal domains" workshop in Banff from June 30, 2019 to July 7, 2019.

First observed by the physicist and musician Ernst Chladni in the 18th century, nodal sets of Laplace eigenfunctions appear in many mathematical problems as well as those stemming from engineering, physics and natural sciences. Berry in the 1970s predicted that the nodal sets of monochromatic random waves, with a specific example being a Gaussian spherical harmonic of eigenvalue $l(l+1)$, are a good approximation for those coming from high-frequency non-random eigenfunctions in classically chaotic systems.

In this workshop, we plan to address the following question: given a fixed smooth non-zero vector field $V$ on $S^2$, what are the statistics for the number of connected components of the nodal set of a Gaussian spherical harmonic of degree $n \in \mathbb{N}$ that have exactly $k$ tangencies to $V$, for $k \in \mathbb{N}$? A precise enough answer to this question will provide a refinement (following in the spirit of some recent work by Sarnak-Wigman '18) to some existing methods provided by Nazarov-Sodin '08 & '16 for counting the connected components of nodal sets to random spherical harmonics.

The Banff International Research Station for Mathematical Innovation and Discovery (BIRS) is a collaborative Canada-US-Mexico venture that provides an environment for creative interaction as well as the exchange of ideas, knowledge, and methods within the Mathematical Sciences, with related disciplines and with industry. The research station is located at The Banff Centre in Alberta and is supported by Canada's Natural Science and Engineering Research Council (NSERC), the U.S. National Science Foundation (NSF), Alberta's Advanced Education and Technology, and Mexico's Consejo Nacional de Ciencia y Tecnología (CONACYT).