# ASGARD Math 2019

The 4th Scandinavian Gathering Around Remarkable Discrete Mathematics

Check out the meeting on Cubic Surfaces to be held May 13th in Oslo as well http://cubics.wikidot.com/event

## Mini Courses

Paolo Aluffi (Florida State University) - **Characteristic classes of singular varieties**

We will give a motivated introduction to Fulton-MacPherson intersection theory,

with emphasis on the notion of Segre class. We will focus on techniques for computations

of these classes in template situations such as blow-ups and excess and residual

intersection formulas. We will discuss applications of these notions to the definition and

computation of characteristic classes generalizing to singular varieties standard

invariants such as Chern classes of nonsingular varieties.

Cynthia Vinzant (North Carolina State University) - **Real rootedness, log-concavity, and matroids**

Matroids are combinatorial structures that model independence, such as among edges in a graph and vectors in a linear space. I will introduce the theory of matroids along with classes of real polynomials capturing many of their important features. Real rooted univariate polynomials are ubiquitous in combinatorics and there are several interesting multivariate generalizations. In increasing order of generality, we will discuss determinantal, stable, and completely log-concave polynomials, their real and combinatorial properties, and their applications to matroids.

## Program

All talks and minicourses will be held in Vilhelm Bjerknes' hus Auditorium 1.

Lunch each day will be in Abels utsikt (12th floor of Niels Henrik Abels hus).

Tuesday | Wednesday | Thursday | |
---|---|---|---|

9:30-11:00 | Minicourse - Vinzant | Minicourse - Vinzant | Minicourse - Vinzant |

11:00-11:30 | Break | Break | Break |

11:30-12:00 | Solus | Kohn | Seynaeve |

12:00-12:30 | Weinstein | Görlach | Yuen |

12:30-14:00 | Lunch | Lunch | Lunch |

14:00-15:30 | Minicourse - Aluffi | Minicourse - Aluffi | Minicourse - Aluffi |

15:30-16:00 | Break | Break | Break |

16:00-16:30 | Bunnett | Dinu | Brugallé * |

16:30-17:00 | Monin |

A **conference dinner** will be organized for Wednesday evening. We will eat at **Vippa **at 19:00.

## Contributed talks

Erwan Brugallé |
A classical problem in enumerative geometry is to determine the number of plane algebraic rational curves of degree d passing through 3d-1 points (eg the number of lines passing through two points). We will address several generalizations of this problem in real, tropical, and symplectic geometries. The first part will mainly be devoted to tropical refined invariants defined by Block and Göttsche, and generalized by Göttsche and Schroeter. Relations discussed there are suggested by a formula relating enumerative invariants of two real symplectic 4-manifolds differing by a surgery along a real Lagrangian sphere. This formula, whose origin can be traced back to a work by Abramovich and Bertam, will be discussed in the second part. We will give as well applications to Welschinger invariants of rational symplectic 4-manifolds. This is the first part of a series of two lectures given by Brugallé as part of his BFS Invited Professorship. |

Dominic Bunnett | Stability of hypersurfaces in toric varieties and Newton polytopesThe stability of a hypersurface Z in projective space is determined, via the Hilbert-Mumford criterion, by checking if the origin lies within the Newton polytope P of the hypersurface. Given that the geometry of Z is also encapsulated by P, one finds a rich connection between stability of hypersurfaces and their geometry. The moduli of hypersurfaces in toric varieties is constructed using non-reductive geometric invariant theory and an analogous analysis using the Newton polytope uncovers discrete geometry as a powerful tool to study the moduli of such hypersurfaces. After introducing the fundamentals of non-reductive GIT, we show how one relates stability of hypersurfaces and certain discrete geometric conditions on their Newton Polytopes. |

Rodica Dinu | Gorenstein t-spread Veronese algebrasThe aim of this talk is to present the t-spread Veronese algebras which have the Gorenstein property. This talk is based on the paper https://arxiv.org/abs/1901.01561. |

Paul Görlach | Injection dimensions of projective varietiesWe consider the problem of finding the smallest dimension of a projective space to which a given complex algebraic variety $X$ admits an injective morphism. Based on connectedness theorems, we prove that a linear subsystem of a strict power of a line bundle can only give rise to an injective morphism if its dimension is at least $2\dim X$. We explore the ramifications of this result, with a focus on products of projective spaces, in which case there are close connections to separating invariants of torus actions, projections of Segre-Veronese varieties and the rank 2 geometry of partially symmetric tensors. |

Kathlén Kohn | Moment Varieties of Measures on PolytopesThis talk brings many areas together: discrete geometry, statistics, algebraic geometry, invariant theory, geometric modeling, symbolic and numerical computations. We introduce moments of polytopes and study the algebraic relations among them. This is already a non-trivial matter for quadrangles in the plane. In fact, we need to combine invariant theory of the affine group with numerical algebraic geometry to compute first relevant relations. Moreover, we show that the numerator of the generating function of all moments of a fixed polytope is the adjoint of the polytope, which is known from geometric modeling. This talk is based on joint work with Boris Shapiro and Bernd Sturmfels. |

Leonid Monin | Cohomology of toric bundles and rings of conditions of horospherical varieties.Ring of conditions is a version of intersection theory defined by De Concini and Procesi for spherical homogeneous spaces. In the case of an algebraic torus (C^*)^n, the ring of conditions has a convex geometric description as a ring generated by the volume polynomial on the space of polytopes. One can extend this description to the case of horospherical homogeneous spaces using an analogue of Bernstein-Kouchnirenko theorem for toric bundles. In my talk I will explain these results. |

Tim Seynaeve | Flag matroids: Algebra and GeometryMatroids are ubiquitous in modern combinatorics. As discovered by Gelfand, Goresky, MacPherson and Serganova there is a beautiful connection between matroid theory and the geometry of Grassmannians: realizable matroids correspond to torus orbits in Grassmannians. Further, as observed by Fink and Speyer general matroids correspond to classes in the K-theory of Grassmannians. This yields in particular a geometric description of the Tutte polynomial. I will describe these constructions, and how to generalise some of them to polymatroids. More precisely, we study the class of flag matroids and their relations to flag varieties. In this way, we obtain an analogue of the Tutte polynomial for flag matroids. This is joint work with A. Cameron, R. Dinu, and M. Michalek. See https://arxiv.org/abs/1811.00272. |

Liam Solus | Real zeros and the alternatingly increasing property in algebraic combinatoricsA central theme in algebraic, geometric, and topological combinatorics is the investigation of distributional properties of combinatorial generating polynomials, such as symmetry, log-concavity, and unimodality. Recently, new questions in the field ask when such polynomials possess another distributional property, called the alternatingly increasing property, which implies unimodality. The alternatingly increasing property for a given polynomial is equivalent to a unique pair of symmetric polynomials both being unimodal with nonnegative coefficients. We will discuss a systematic approach to proving the alternatingly increasing property using real zeros of these symmetric polynomials. We will then look at some applications of these methods to recent questions and conjectures in algebraic combinatorics. |

Madeleine Weinstein | Voronoi Cells of VarietiesEvery real algebraic variety determines a Voronoi decomposition of its ambient Euclidean space. Each Voronoi cell is a convex semialgebraic set in the normal space of the variety at a point. We compute the algebraic boundaries of these Voronoi cells. Using intersection theory, we give a formula for the degrees of the algebraic boundaries of Voronoi cells of curves and surfaces. We discuss an application to low-rank matrix approximation. This is joint work with Diego Cifuentes, Kristian Ranestad, and Bernd Sturmfels. |

Chi Ho Yuen | The dimension of an amoebaAn amoeba is the image of a subvariety X of an algebraic torus under the logarithmic moment map. Nisse and Sottile conjectured that the (real) dimension of an amoeba is smaller than the expected one, namely, two times the complex dimension of X, precisely when X has certain symmetry with respect to torus actions. We prove their conjecture and derive a formula for the dimension of an amoeba. We also provide a connection with tropical geometry. This is joint work with Jan Draisma and Johannes Rau. |

## Organizers

Corey Harris (University of Oslo)

Felipe Rincón (Queen Mary University of London)

Kristin Shaw (University of Oslo)