10:00am - 12:00pmRandom geometry and topology
Chair(s): Paul Breiding (Max-Planck Institute for Mathematics in the Sciences, Germany), Lerario Antonio (SISSA), Lundberg Erik (Florida Atlantic University), Kozhasov Khazhgali (Max-Planck Institute for Mathematics in the Sciences, Germany)
This minisymposium is meant to report on the recent activity in the field of random geometry and topology. The idea behind the field is summarized as follows: take a geometric or topological quantity associated to a set of instances, endow the space of instances with a probability distribution and compute the expected value, the variance or deviation inequalities of the quantity. The most prominent example of this is probably Kostlan, Shub and Smales celebrated result on the expected number of real zeros of a real polynomial. Random geometry and topology offers a fresh view on classical mathematical problems. At the same time, since randomness is inherent to models of the physical, biological, and social world, the field comes with a direct link to applications.
More infos at: https://personal-homepages.mis.mpg.de/breiding/siam_ag_2019_RAG.html
(25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise)
Zero-sets of 3D random waves
Federico Dalmao
Universidad de la Republica de Uruguay
We study Berry's model in the three-dimensional case. This model contains as particular cases the monochromatic random waves and the black-body radiation, which are isotropic Gaussian fields that a.s. solve the Helmholtz equation. We generalize it to include more general features as anisotropy. We are interested in the zero-sets of the random waves as they represent lines of darkness, threads of silence, etc. We compute moments and find limit distributions under mild hypothesis and compare it with the well studied 2D-case. This is a joint work with Anne Estrade and José R. León.
Curvature and randomness
Emil Horobet
Sapientia Hungarian University
The Euclidean Distance degree measures the algebraic complexity of writing the optimal solution to the best approximation problem to an algebraic variety as a function of the coordinates of the data point. The number of real-valued critical points of the distance function can be different for different data points. For randomly sampled data the expected number of real valued critical points is of high interest and it is called the average ED degree. In this talk we will see connections between the average ED degree, the ED discriminant and different curvatures of the underlying variety.
Random sections of line bundles over real Riemann surfaces
Michele Ancona
Univ. Claude Bernard Lyon 1
We will explain how to compute the higher moments of the random variable ”number of real zeros of a random polynomial”. More generally, given a line bundle L over a real Riemann surface, we explain how to compute all the moments of the random variable ”number of real zeros of a random section of L”.
On the topology of real components of real sections of vector bundles
Chris Peterson
Colorado State University
This talk will present joint work with Tanner Strunk. At present, the talk will consist of a collection of methods and numerical data concerning the probability of various topologies that arise as the real zeros of real sections of vector bundles. Some of the methods utilized for collecting such data are interesting and might be useful in a general context. However, the main focus of the talk will be on the number of different topologies that can arise in various settings and conjectural relationships between their probabilities. While some initial results are rather intriguing, we are currently only able to provide statistical data rather than theory to support the data. It is the hope that additional insight into the results is obtained by the time of the conference. At the very least, perhaps the data will lead to some interesting discussions.