Real Applied Algebraic Geometry

TU Berlin, September 19/20, 2019


Phaseless rank, semidefinite lifts and amoebas (Joao Gouveia)
We consider the problem of minimizing the rank of a complex matrix where the absolute values of the entries are given. We call this minimum the phaseless rank of the matrix of entrywise absolute values. In this talk we will present this quantity and some of its properties, including some extensions to a classic result of Camion and Hoffman from 1966. We will proceed to connect this to the study of amoebas of algebraic varieties and of semidefinite representations of convex sets. As a consequence, we attain several new results in both these subjects, among them a counterexample for a recent conjecture of Nisse and Sottile on the existence of amoeba bases, and a new upper bound on the complex semidefinite extension complexity of polytopes, dependent only on their number of vertices and facets. This talk is based on joint work with António Pedro Goucha.

Rational quartic spectrahedra (Martin Helsø)
A spectrahedron is called rational if its boundary admits a rational parametrisation. The Zariski closure in complex projective space of the boundary is a symmetroid. Rational quartic symmetroids in 3-space have been classified in previous work; they have a triple point, an elliptic double point or are singular along a curve. In this talk, I will give bounds on the number of real singularities on the algebraic and topological boundary of rational quartic spectrahedra in 3-space. This is joint work with Kristian Ranestad.

Describing the Jelonek set of a polynomial map via Newton polytopes (Boulos el Hilany)
A polynomial map f=(f_1,...,f_n): C^n --> C^n is said to be non-proper at a point p if the preimage of any compact neighbourhood of p is not compact. The set of all such points, called the Jelonek set of f, measures how far such a map is from being proper. I will restrict to a certain large class of maps f above and present a new method for computing the non-properness set that depends on the geometry of the corresponding Newton polytopes. This method is mostly combinatorial, leads to significantly faster computations, and holds true for the real case. With this approach, previously-known results can be deduced in an easier fashion as well as new applications arise.

Chebyshev polynomials and best rank-one approximation ratio (Khazhgali Kozhasov)
We establish a new extremal property of the classical Chebyshev polynomials in the context of best rank-one approximation of tensors. We also give some necessary conditions for a tensor to be a minimizer of the ratio of spectral and Frobenius norms.

Mathematics of 3D genome reconstruction in diploid organisms (Kaie Kubjas)
The 3D organization of the genome plays an important role for gene regulation. Chromosome conformation capture techniques allow one to measure the number of contacts between genomic loci that are nearby in the 3D space. In this talk, we study the problem of reconstructing the 3D organization of the genome from whole genome contact frequencies in diploid organisms, i.e. organisms that contain two indistinguishable copies of each genomic locus. In particular, we study the identifiability of the 3D organization of the genome and optimization methods for reconstructing it. Since every possible 3D organization is a solution to a system of polynomial equations, the identifiability question reduces to a question in algebraic geometry. For reconstruction, we use convex optimization methods. This talk is based on joint work with Anastasiya Belyaeva, Lawrence Sun and Caroline Uhler.

Classification of real algebraic curves in real minimal del Pezzo surfaces (Matilde Manzaroli)
The study of the topology of real algebraic varieties dates back to the work of Harnack, Klein and Hilbert in the 19th century; in particular, the isotopy type classification of real algebraic curves with a fixed degree in the real projective plane is a classical subject that has undergone considerable evolution. On the other hand, apart from studies concerning Hirzebruch surfaces and at most degree 3 surfaces in the real projective space, not much is known for more general ambient surfaces. In particular, this is because varieties constructed using the patchworking method are hypersurfaces of toric varieties. However, there are many other real algebraic surfaces. Among these are the real rational surfaces, and more particularly the real minimal rational surfaces. In this talk, we present some results about the classification of topological types realized by real algebraic curves of "small class" in real minimal del Pezzo surfaces which are real non-toric surfaces with non-connected real parts. We will explain how combine variations of classical methods with degeneration methods, that have found recent applications in real enumerative geometry, and the exploitation of Welschinger invariants to get through such classifications.

On the semidefinite extension degree of convex sets (Claus Scheiderer)
A spectrahedron is the solution set of a linear matrix inequality (LMI), or equivalently, an affine-linear section of the cone of positive semidefinite matrices of some size. Semidefinite programming allows to efficiently optimize linear functions over spectrahedra, or more generally over their images under linear maps. Following Averkov we define, for a convex set K, the semidefinite extension degree sxdeg(K) as the smallest integer d such that K is a linear image of an intersection of finitely many spectrahedra that are all described by LMIs of size at most d. This defines a strict hierarchy among all spectrahedral shadows, as Fawzi and Averkov have recently shown. We relate the semidefinite extension degree to sums of squares representations, and use this to show for all convex subsets of the plane that sxdeg(K) is at most 2. As a consequence, all semidefinite programs in the plane can be performed as second order cone programming.

Tensors under congruence action (Anna Seigal)
Matrix congruence extends naturally to the setting of tensors. We apply methods from tensor decomposition, algebraic geometry and numerical optimization to this group action. Given a tensor in the orbit of another tensor, we compute a matrix which transforms one to the other. Our primary application is an inverse problem from stochastic analysis: the recovery of paths from their third order signature tensors. This talk is based on joint work with Max Pfeffer and Bernd Sturmfels.

An SOS counterexample to an inequality of symmetric functions (Alexander Heaton)
It is known that differences of symmetric functions corresponding to various bases are nonnegative on the nonnegative orthant exactly when the partitions defining them are comparable in dominance order. The only exception is the case of homogeneous symmetric functions where it is only known that dominance of the partitions implies nonnegativity of the corresponding difference of symmetric functions. It was conjectured by Cuttler, Greene, and Skandera in 2011 that the converse also holds, as in the cases of the monomial, elementary, power-sum, and Schur bases. In this talk, we describe a counterexample, showing that homogeneous symmetric functions break the pattern. We use a semidefinite program to find a positive semidefinite matrix whose factorization provides an explicit sums of squares decomposition of the polynomial H44 − H521 as a sum of 41 squares. This certificate of nonnegativity disproves the conjecture, since a polynomial which is a sum of squares of other polynomials cannot be negative, and since the partitions 44 and 521 are incomparable in dominance order. This is joint work with Isabelle Shankar of UC-Berkeley.

Towards the existence of maximal hypersurfaces in toric varieties (Kristin Shaw)
Given an algebraic variety defined over the real numbers, the Smith-Thom inequality bounds the sum of the Betti numbers of the real part by the sum of the Betti numbers of the complexification. In the case of plane curves, this inequality reduces to Harnack’s bound on the number of connected components of the real part of the curve. Around 2007, Itenberg and Viro announced that this inequality is tight for hypersurfaces in projective space, that is to say for every dimension and any degree there exists a real algebraic hypersurface for which the Smith-Thom inequality is tight. In this talk, I will explain how recent applications of tropical geometry to the study of Viro’s patchworking method in the unimodular case can provide a proof of this statement in projective space as well as for hypersurfaces of other toric varieties. This is joint work in progress with Brugallé, Bertrand, and Renaudineau.

Lorentzian polynomials (Petter Brändén)
I will give an overview of my work with June Huh on Lorentzian polynomials, polynomials that generalize hyperbolic polynomials as well as Minkowski volume polynomials of convex bodies.