Conference Agenda

Overview and details of the sessions of this conference. Please select a date or location to show only sessions at that day or location. Please select a single session for detailed view (with abstracts and downloads if available).

 
Session Overview
Session
MS146, part 2: Random geometry and topology
Time:
Friday, 12/Jul/2019:
10:00am - 12:00pm

Location: Unitobler, F006
30 seats, 57m^2

Presentations
10:00am - 12:00pm

Random 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)

 

Grassmann Integral Geometry

Leo Mathis
SISSA

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.

 

Topology of Gaussian Random Fields

Michele Stecconi
SISSA

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)

 

Spectrum of the Laplace Operator for Random Geometric Graphs

Raffaella Mulas
MPI MiS Leipzig

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.

 

Sampling from the uniform distribution on a variety

Orlando Marigliano
MPI MiS Leipzig

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.