Descartes’ rule of signs for univariate real polynomials is commonly considered as a fundamental result in real algebraic geometry. It was proposed by René Descartes in 1637 in "La Géométrie", an appendix to his "Discours de la méthode". This rule bounds the number of positive roots of a real univariate polynomial by the signvariation of its coefficients. Only recently generalizations to the multivariate case have been obtained. I will present a new optimal Descartes rule for polynomial systems supported on circuits, obtained in collaboration with Alicia Dickenstein and Jens Forsgård.

The development of Gröbner basis theory in the 1960s and Hironaka's results on resolution of singularities contributed to the flourishing and growing importance of methods for the effective study of invariants associated with objects in algebraic geometry. The rapid development of computers made it possible to use the developed algorithms in practice. I will present applications of classical results of counting the number of roots of polynomials (the trace formula - Pedersen, Roy, Szpirglas, Becker, Wörmann) and the local topological degree (Eisenbud, Levine, Khimshiashvili, Szafraniec, Łęcki) using signatures of quadratic forms to calculate some invariants of real polynomial mappings.

In a recent work with A. Renaudineau, we studied the critical locus for the amoeba map along families of curves defined by Viro polynomials. For real curves, the latter locus is a superset of the real part. It is therefore interesting to study this locus from the point of view of real algebraic geometry. Unfortunately, not much is known on its topology besides some bounds on its Betti numbers. We will see that the Log-critical locus admits a Patchworking theorem. We will discuss some constructions and address the sharpness of the bounds mentioned above.

Given a hyperbolic polynomial and a hyperbolicity direction, we define a spectrahedral outer approximation defined by a linear matrix inequality of very small size. We ask whether one can ``extend'' the given hyperbolic polynomial to another hyperbolic polynomial in many more additional variables so that our spectrahedral relaxation coincides on the space corresponding to the original variables with the hyperbolicity cone. A positive answer would solve the Generalized Lax Conjecture saying that every hyperbolicity cone is spectrahedral. We have however only very partial results. We use in many ways the characterization of hyperbolic polynomials in three variables by Helton and Vinnikov.

A widespread principle in real algebraic geometry is to find and use algebraic certificates for geometric statements. This covers for example writing a globally nonnegative polynomial as a sum of squares or expressing a polynomial with only real zeros as the minimal polynomial of a symmetric matrix. In the first part of the talk I will survey some classical results and open problems in this direction. Then I will present a quite general result from a joint work with Christoph Hanselka that implies several of the aforementioned results.

Over the past two decades there has been a surge of activity in the study of stable, hyperbolic and Lorentzian polynomials. Spaces of hyperbolic polynomials were proved to be contractible by Nuij already in 1968. The space of Lorentzian polynomials was recently studied by June Huh and the speaker. It contains all volume polynomials of convex bodies and projective varieties, as well as homogeneous stable polynomials with nonnegative coefficients. It was conjectured by Huh and the speaker that projective spaces of Lorentzian polynomials are homeomorphic to Euclidean balls. I will show how one can use a powerful connection between the symmetric exclusion process in Interacting Particle Systems and the geometry of polynomials to prove this conjecture, and refine Nuij's theorem.

Lorentzian polynomials are a natural generalization of real stable polynomials which have been recently used to derive various log-concavity statements, e.g. on the independent sets of a matroid or on the coefficients of Schur and (conjecturally) Schubert polynomials. In joint work with Petter Brändén and Igor Pak, we use the log-concavity properties of Lorentzian polynomials to derive lower bounds on the volume of transportation polytopes. In particular, we do this by proving that the generating function of the lattice points of such polytopes is Lorentzian up to normalization. Using Gurvits' notion of polynomial capacity, we can then bound the number of such lattice points and obtain a volume bound in the limit. In this talk, we discuss these bounds along with a few key points of their proofs.

I will discuss the problem of estimating the volume of the tube around an algebraic set (possibly singular) as a function of the dimension of the set and its degree. In particular I will show bounds for the volume of neighborhoods of algebraic sets, in the euclidean space or the sphere, in terms of the degree of the defining polynomials, the number of variables and the dimension of the algebraic set, without any smoothness assumption. This problem is crucial in numerical algebraic geometry, where sets of ill-contidioned inputs are typically described by (possibly singular) algebraic sets, and their neighborhoods describe bad-conditioned inputs. Our result generalizes previous work of Lotz on smooth complete intersections in the euclidean space and of Bürgisser, Cucker and Lotz on hypersurfaces in the sphere, and gives a complete solution to Problem 17 in the book titled "Condition" by Bürgisser and Cucker. This is joint work with S. Basu.

In this talk I will introduce the zonoid algebra. Starting from the monoid structure of zonoids in ℝ^d I will explain how to turn this structure into an algebra, where we can “multiply” zonoids. More specifically, I will show that every multilinear map between finite dimensional vector spaces has a unique, continuous, Minkowski multilinear extension to the corresponding space of zonoids. Taking the wedge product of vector spaces as the multilinear map, we get a definition of the wedge of zonoids. This is the definition of the product in our algebra. The motivation for this construction comes from probabilistic intersection theory in a compact homogeneous space, where the zonoid algebra plays the role of a probabilistic cohomology ring. This is joint work with Antonio Lerario, Leo Mathis and Peter Bürgisser.

Determinantal point processes are a very particular kind of random point processes with some valuable properties such as: points exhibit local repulsion and they can be easily sampled. In this talk, we present the basic notions for understanding a determinantal point process, and we study different properties of equidistribution of the points.

A spectrahedron in ℝ³ is the intersection of a 3-dimensional affine linear subspace of dxd real matrices with the cone of positive-semidefinite matrices. Its algebraic boundary is a surface of degree d in ℂ³ called a symmetroid. Generically, symmetroids have (d³-d)/6 nodes over ℂ and the real singularities are partitioned into those which lie on the spectrahedron and those which do not; this gives a coarse combinatorial description of a spectrahedron. For d=3 and 4, the possible partitions are known. In this talk, I will explain how we determined which partitions are possible for d=5, in particular, how we used numerical algebraic geometry to find explicit examples of spectrahedra witnessing each partition. This is joint work with Khazhgali Kozhasov and Mario Kummer.

The largest group that preserves circles are the Möbius transformations and without a notion of distance we propose "celestial surfaces" as two-dimensional analogues of circles, namely surfaces that contain at least two circles through almost each point. We characterize self-intersections of such surfaces using intersection theory and obtain via vector fields the possible shapes of general celestial surfaces in three-space.

A morphism of smooth varieties of the same dimension is called real fibered if the inverse image of the real part of the target is the real part of the source. It goes back to Ahlfors that a real algebraic curve admits a real fibered morphism to the projective line if and only if the real part of the curve disconnects its complex part. Inspired by this result, in a joint work with Mario Kummer and Cédric Le Texier, we are interested in characterising real algebraic varieties of dimension n admitting real fibered morphisms to the n-dimensional projective space. We present a criterion to construct real fibered morphisms that arise as finite surjective linear projections from an embedded variety; this criterion relies on topological linking numbers. We address special attention to real algebraic surfaces. We classify all real fibered morphisms from real del Pezzo surfaces to the projective plane and determine when such morphisms arise as the composition of a projective embedding with a linear projection.