3:00pm - 5:00pmRandom geometry and topology
Chair(s): Paul Breiding (Max-Planck Institute for Mathematics in the Sciences, Germany), Antonio Lerario (SISSA), Erik Lundberg (Florida Atlantic University), Khazhgali Kozhasov (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 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.
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.
Degree of Random Monomial Ideals
Jay Yang
University of Minnesota
In joint work with Lily Silverstein and Dane Wilburne, we investigate the behavior of the standard pairs of a random monomial ideal. We then use this to explore the degree and arithmetic degree of random monomial ideals.
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)}.
