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 3: Random geometry and topology
Time:
Saturday, 13/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)

 

The integer homology threshold for random simplicial complexes

Andrew Newman
TU Berlin

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.

 

The real tau-conjecture is true on average

Peter Bürgisser
TU Berlin

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.

 

Geometric limit theorems in topological data analysis

Christian Lehn
Universität Chemnitz

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.

 

Quantitative Singularity theory for Random Polynomials

Hanieh Keneshlou
MPI MiS Leipzig

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)}.