Poster Session

Rida Ait El Manssour
Università degli Studi di Trieste, Italy

On the number of real lines on random invariant cubic surfaces

It is a classical result in algebraic geometry that there are exactly 27 complex lines on a smooth cubic surface in three-dimensional projective space. If the cubic surface is defined by a real equation, the number of real lines on it depends on the coefficients of this equation (generically there could be 3, 7, 15 or 27). In this context, it is natural to ask for the expected number of real lines, by letting the defining equation be a random polynomial. In this direction Basu, Lerario, Lundberg and Peterson have recently proved that if the polynomial is uniformly sampled from the Bombieri-Weil distribution, the expected number of real lines on this cubic is 6\sqrt{2}-3.
The Bombieri-Weil distribution is however just a single instance of a one dimensional family of probability distributions on the space of polynomials which are invariant under orthogonal change of variables (i.e. for which there are no preferred points or directions in the projective space for the corresponding zero set). In this work, we study the average number of real lines for all the elements of this family and prove that the maximal expectation is reached by random purely harmonic polynomials.

Rida Ait El Manssour
Università degli Studi di Trieste, Italy

On the number of real lines on random invariant cubic surfaces

It is a classical result in algebraic geometry that there are exactly 27 complex lines on a smooth cubic surface in three-dimensional projective space. If the cubic surface is defined by a real equation, the number of real lines on it depends on the coefficients of this equation (generically there could be 3, 7, 15 or 27). In this context, it is natural to ask for the expected number of real lines, by letting the defining equation be a random polynomial. In this direction Basu, Lerario, Lundberg and Peterson have recently proved that if the polynomial is uniformly sampled from the Bombieri-Weil distribution, the expected number of real lines on this cubic is 6\sqrt{2}-3.
The Bombieri-Weil distribution is however just a single instance of a one dimensional family of probability distributions on the space of polynomials which are invariant under orthogonal change of variables (i.e. for which there are no preferred points or directions in the projective space for the corresponding zero set). In this work, we study the average number of real lines for all the elements of this family and prove that the maximal expectation is reached by random purely harmonic polynomials.

Carlos Améndola
TU Munich, Germany

Autocovariance Varieties of Moving Average Random Fields

We study the autocovariance functions of moving average random fields over the integer lattice \mathbb{Z}^d from an algebraic perspective. These autocovariances are parametrized polynomially by the moving average coefficients, hence tracing out algebraic varieties. We derive dimension and degree of these varieties and we use their algebraic properties to obtain statistical consequences such as identifiability of model parameters. We connect the problem of parameter estimation to the algebraic invariants known as euclidean distance degree and maximum likelihood degree. Throughout, we illustrate the results with concrete examples. In our computations we use tools from commutative algebra and numerical algebraic geometry.

Mara Belotti
Università degli Studi di Trieste, Italy

On the number of real lines on random invariant cubic surfaces

It is a classical result in algebraic geometry that there are exactly 27 complex lines on a smooth cubic surface in three-dimensional projective space. If the cubic surface is defined by a real equation, the number of real lines on it depends on the coefficients of this equation (generically there could be 3, 7, 15 or 27). In this context it is natural to ask for the expected number of real lines, by letting the defining equation be a random polynomial. In this direction Basu, Lerario, Lundberg and Peterson have recently proved that if the polynomial is uniformly sampled from the Bombieri-Weil distribution, the expected number of real lines on this cubic is
The Bombieri-Weil distribution is however just a single instance of a one dimensional family of probability distributions on the space of polynomials which are invariant under orthogonal change of variables (i.e. for which there are no preferred points or directions in the projective space for the corresponding zero set). In this work we study the average number of real lines for all the elements of this family and prove that the maximal expectation is reached by random purely harmonic polynomials.
(This is the starting case of a more general approach that we plan to investigate in the future, for constructing real hypersurfaces of degree 2n-3 in real projective space of dimension n with many lines on them by simply sampling them randomly from an appropriate distribution.)

Some normalized integral inequalities in Statisitc and Probability

In this poster, we present recent results on some normalized integral inequalities for continuous random variables via Riemann-Liouville integration. Some classical results are generalized and some applications are discussed.

Oliver Gäfvert
KTH Royal Institute of Technology, Sweden

Complexity of variety learning

Extracting structure from a point cloud is a fundamental problem in data analysis, and when this structure is coming from polynomial equations we can use the machinery of algebraic geometry. We study the set of closest real algebraic hypersurfaces to a finite set of points, and investigate methods to approximate the closest hypersurface of a given degree. We investigate the complexity of this problem by considering the Euclidean Distance Degree (EDD) of the variety of point configurations lying on a hypersurface of a given degree and how the complexity depends on the number of points in the configuration.

Alexandros Grosdos Koutsoumpelias
Universität Osnabrück, Germany

Algebraic statistics of local Dirac mixture moments

Leo Mathis
SISSA, Italy

Real Schubert Problems and the Segre Zonoid

I will present recent work in integral geometry in the real Grassmannian. I will explain how random real Schubert problem can be linked with the volume of special convex bodies, namely zonoids. I will show how this allows us to establish asymptotic formulae.

This is a joint work with Antonio Lerario and Peter Bürgisser.

Chiara Meroni
Università degli Studi di Trieste, Italy

On the number of real lines on random invariant cubic surfaces

It is a classical result in algebraic geometry that there are exactly 27 complex lines on a smooth cubic surface in three-dimensional projective space. If the cubic surface is defined by a real equation, the number of real lines on it depends on the coefficients of this equation (generically there could be 3, 7, 15 or 27). In this context it is natural to ask for the expected number of real lines, by letting the defining equation be a random polynomial. In this direction Basu, Lerario, Lundberg and Peterson have recently proved that if the polynomial is uniformly sampled from the Bombieri-Weil distribution, the expected number of real lines on this cubic is 6\sqrt{2}-3.
The Bombieri-Weil distribution is however just a single instance of a one dimensional family of probability distributions on the space of polynomials which are invariant under orthogonal change of variables (i.e. for which there are no preferred points or directions in the projective space for the corresponding zero set). In this work we study the average number of real lines for all the elements of this family and prove that the maximal expectation is reached by random purely harmonic polynomials.

Frank Röttger
OVGU Magdeburg, Germany

Asymptotics of a locally dependent statistic on finite reflection groups

We discuss the asymptotic behaviour of the number of descents in a random signed permutation and its inverse, which was posed as an open problem by Chatterjee and Diaconis in a recent publication. For that purpose, we generalize their result for the asymptotic normality of the number of descents in a random permutation and its inverse to other finite reflection groups. This is achieved by applying their proof scheme on signed permutations, so elements of Coxeter groups of type B, which is also known as the hyperoctahedral group. Furthermore, a similar central limit theorem for elements of Coxeter groups of type D is derived via Slutsky's Theorem and a bound on the Wasserstein distance of certain normalized statistics with local dependency structures and bounded local components is proven for both types of Coxeter groups.

Liam Solus
KTH Royal Institute of Technology, Sweden

Interventional Markov Equivalence for Mixed Graph Models

We will discuss the problem of characterizing Markov equivalence of graphical models under general interventions. Recently, Yang et al. (2018) gave a graphical characterization of interventional Markov equivalence for DAG models that relates to the global Markov properties of DAGs. Based on this, we extend the notion of interventional Markov equivalence using global Markov properties of loopless mixed graphs and generalize their graphical characterization to ancestral graphs. On the other hand, we also extend the notion of interventional Markov equivalence via modifications of factors of distributions Markov to acyclic directed mixed graphs. We prove these two generalizations coincide at their intersection; i.e., for directed ancestral graphs. This yields a graphical characterization of interventional Markov equivalence for causal models that incorporate latent confounders and selection variables under assumptions on the intervention targets that are reasonable for biological applications.

Michele Stecconi
SISSA, Italy

The expected topology of the singular locus of Kostlan random polynomials

We study the set of points on the
m-dimensional Sphere at which a homogenous polynomial attains a given
nondegenerate singularity, i.e. points where the r-jet prolongation of
the polynomial meets transversally some given semialgebraic
submanifold of the space of r-jets. The topology of the singular locus is thought as the data of the m+1 vector of its Betti numbers.
I will present a recent result which establishes that in the
Kostlan case, under reasonable hypothesis on the singularity, the
expected value of each Betti number grows like d^(m/2), namely like the square root of the
corresponding deterministic upper bound, as the degree d of the
polynomial tends to infinity. This phenomena is justified by the fact that the restriction of a Kostlan polynomial to an affine disk of radius ~d^(-1/2) converges as a random field in a very strong way.