Random Geometry and Topology Minisymposium at SIAM AG 2019

This is the website for the minisymposium on Random Geometry and Topology to be held at the SIAM Conference on Applied Algebraic Geometry 2019.

Location: University of Bern, Bern, Switzerland.

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.


Organizers

Paul Breiding, Max-Planck-Institute for Mathematics in the Sciences Leipzig.

Khazhgali Kozhasov, Max-Planck-Institute for Mathematics in the Sciences Leipzig.

Antonio Lerario, SISSA.

Erik Lundberg, Florida Atlantic University.


Speakers and Schedule

Thursday 10:00-12:00 Friday 10:00-12:00 Saturday 10:00-12:00
Federico Dalmao Leo Mathis Andrew Newman
Emil Horobet Michele Stecconi Peter Bürgisser
Michele Ancona Raffaella Mulas Christian Lehn
Chris Peterson Orlando Marigliano Hanieh Keneshlou

All talks will take place in room Unitobler, F006.


Titles and Abstracts

Peter Bürgisser: The real tau-conjecture is true on average

Koiran's real tau-conjecture claims that the number of real zeros of a structured polynomial given as a sum of m products of k real sparse polynomials, each with at most t monomials, is bounded by a polynomial in mkt. This conjecture has a major consequence in complexity theory since it would lead to superpolynomial bounds for the arithmetic circuit size of the permanent. We confirm the conjecture in a probabilistic sense by proving that if the coefficients involved in the description of f are independent standard Gaussian random variables, then the expected number of real zeros of f is O(mkt), which is linear in the number of parameters.

Chris Peterson: On the topology of real components of real sections of vector bundles

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.

Emil Horobet: Curvature and randomness

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.

Michele Ancona: Random sections of line bundles over real Riemann surfaces

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”.

Federico Dalmao: Zero-sets of 3D random waves

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.

Leo Mathis: Grassmann Integral Geometry

In this talk, I will give an overview on the probabilistic intersection theory on the real Grassmanniann. In particular I will focus on Schubert cells. We will show how Young Tableaux can be used to describe not only the cell itself but also its tangent and normal at one point. This description allows us to use the integral geometry formula to compute the average volume of intersection of Schubert cells. Moreover we will see how one can associate to any Schubert cell - and more generally to any submanifold of a homogeneous space - a particular convex body (zonoid) on the cotangent bundle together with a law of multiplication that form a probabilistic graded ring. I will present all the numerous questions that arise with this new point of view on integral geometry in homogeneous spaces.

Michele Stecconi: Topology of Gaussian Random Fields

We present the space of smooth Gaussian Random Fields on a smooth manifold and discuss the notion of narrow convergence (or convergence in law) for sequences of such fields, which provides a good language to investigate a class of common problems in stochastic geometry. We will describe how narrow convergence is equivalent to smooth convergence of the covariance functions and see an application of this result to Kostlan polynomials. (This is a joint work with Antonio Lerario)

Raffaella Mulas: Spectrum of the Laplace Operator for Random Geometric Graphs

We investigate some properties of the spectrum of the normalized Laplace operator for random geometric graphs in the thermodynamic regime. This is a joint work with Antonio Lerario.

Andrew Newman: The integer homology threshold for random simplicial complexes

The very first problem considered in the now-classic Linial-Meshulam model was to generalize the connectivity threshold from Erdős-Rényi random graphs to higher dimensions as homological connectivity. Early work by Linial, Meshulam, and Wallach had established this homology-vanishing threshold for finite field coefficients, however this a priori does not establish the threshold for integer coefficients. In joint work with Elliot Paquette discussed here, we establish this threshold for homology with integer coefficients to vanish.

Orlando Marigliano: Sampling from the uniform distribution on a variety

This talk presents the problem of sampling from the uniform distribution on a real affine variety with finite volume, given just its defining polynomials. We can choose a point on such a variety by choosing first a hyperplane of the right codimension and then one of its intersection points with the variety. In this talk, I explain how to do this such that the chosen point is uniformly distributed. I show examples of the corresponding algorithm for sampling in action and highlight a connection to topological data analysis. This is joint work with Paul Breiding.

Christian Lehn: Geometric limit theorems in topological data analysis

In a joint work with V. Limic and S. Kalisnik Verosek we generalize the notion of barcodes in topological data analysis in order to prove limit theorems for point clouds sampled from an unknown distribution as the number of points goes to infinity. We also investigate rate of convergence questions for these limiting processes.

Hanieh Keneshlou: Quantitative Singularity theory for Random Polynomials

In this talk, based on a joint work with A. Lerario and P. Breiding, I will present some probabilistic approximations of singularity type of a polynomial. The case of special interest is the zero set of a polynomial. We will show with an overwhelming probability, the set of real zeros of a polynomial of degree d can be realized as the zero set of a polynomial of degree sqrt{d log(d)}.


Information for speakers

See also the website of the SIAM AG Activity Group.