Togliatti systems join in a beutiful way two seemingly unrelated topics:
1) weak Lefschetz property (WLP) and
2) varieties with degenerate osculating spaces.
The first examples were presented already in 1929 by Togliatti who studied monomial maps from \(\mathbb{P}^2\) to \(\mathbb{P}^5\). However, only recently Mezzetti, Miró-Roig and Ottaviani, using apolarity, proved general results relating projections of Veronese embedding with degenerate general osculating spaces and Artinian ideals that fail the WLP. In our talk we will present the above mentioned algebraic and geometric properties, providing further examples. We will define Togliatti systems and show new results, obtained jointly with Miró-Roig, on their classification, answering a conjecture of Ilardi (corrected by Mezzetti, Miró-Roig and Ottaviani).

We discuss two current projects at the interface of computer vision and algebraic geometry. Work with Joe Kileel, Zuzana Kukelov and Tomas Pajdla introduces the distortion varieties of a given projective variety. These are parametrized by duplicating coordinates and multiplying them with monomials. We study their degrees and defining equations. Exact formulas are obtained for the case of one-parameter distortions, the case of most interest for modeling cameras with image distortion. Work with Jean Ponce and Matthew Trager develops multi-view geometry for algebraic cameras that are represented by congruences in the Grassmannian of lines in 3-space.

I will give a survey talk about the motivations that an everyday person should care about Grothendieck's standard conjectures.

The geometry of the set of restrictions of rank-one tensors to some of their coordinates is studied. This gives insight into the problem of rank-one completion of partial tensors. Particular emphasis is put on the semialgebraic nature of the problem, which arises for real tensors with constraints on the parameters. The algebraic boundary of the completable region is described for tensors parametrized by probability distributions and where the number of observed entries equals the number of parameters. If the observations are on the diagonal of a tensor of format \(d \times \cdots \times d\), the complete semialgebraic description of the completable region is found.

How to ensure an outcome based on collective decisions is "better for all" when a) everyone acts in their personal interest only, and b) we do not know the exact "motives" of each person? Crudely, this is the central problem in mechanism design, a branch of economics. In this talk, I show how insights from tropical geometry can solve such problems.

Abelian varieties are group varieties, that is, loci given by polynomial equations which simultaneously admit a group structure. They are ubiquitous objects in algebraic and arithmetic geometry, as well as in number theory.
It is classically known that general abelian varieties of dimension at most five are Prym varieties associated to covers between algebraic curves. This reduces the study of abelian varieties of small dimension to the beautifully concrete and rich theory of curves.
I will discuss decisive recent progress on finding a structure theorem for abelian varieties of dimension six, and the implications this uniformization result has on the geometry of their moduli space.

We present a large class of groups (no group known to be not in the class) that satisfy the Kervaire- Laudenbach Conjecture about solvability of non-singular equations over groups. We also show that certain singular equations with coefficients over groups in this class are always solvable. Our method is inspired by seminal work of Gerstenhaber-Rothaus, which was the key to prove the Kervaire-Laudenbach Conjecture for residually finite groups. Exploring the structure of the p-local homotopy type of the projective unitary group, we manage to show that many singular equations with coefficients in unitary groups can be solved in the unitary group.
Joint work with Anton Klyachko.

A combinatorial theory of linear systems on graphs and metric graphs has been introduced in analogy with the one on algebraic curves. The interplay is given by the Specialization Lemma. Let \(X\) be a smooth curve over the field of fractions of a complete discrete valuation ring and let \( \mathfrak{X} \) be a strongly semistable regular model of \(X\). It is possible to specialize a divisor on the curve to a divisor on the dual graph of the special fiber of \( \mathfrak{X} \); through this process the rank of the divisor can only increase.
The complete graph \(K_d\) pops up if we take a model of a smooth plane curve of degree \(d\) degenerating to a union of \(d\) lines. Moreover omitting edges from \(K_d\) can be interpreted as resolving singularities of a plane curve. In this talk we present some results on linear systems on complete graphs and complete graphs with a small number of omitted edges, and we will compare them with the corresponding results on plane curves. In particular, we compute the gonality sequence of complete graphs and the gonality of graphs obtained by omitting edges. We explain how to lift these graphs to curves with the same gonality using models of plane curves with nodes and harmonic morphisms. This is partially a joint work with Filip Cools.

The study of cuts in graphs is an interesting topic in discrete mathematics with relations and applications to many other fields, such as algebraic geometry, combinatorial optimization or algebraic statistics. Here we focus on so-called cut polytopes and associated algebraic objects. We present some new results and open questions especially on algebraic properties of cut algebras and cut ideals.
The talk is based on joint work with Sara Saeedi Madani.

How can a discrete space be possibly curved?
Because it shares some properties with Riemannian manifolds with particular curvature structures. In fact, in Riemannian geometry, sectional and Ricci curvature are central notions. They are defined infinitesimally, through first and second derivatives of the metric tensor. Curvature inequalities, like Sec <0, or Ric >0, have geometric consequences, for the angles in geodesic triangles, convexity properties of the distance function, the divergence or convergence of geodesics, the growth of the volume of geodesic balls, the eigenvalues of the Laplace-Beltrami operator, or coupling properties of Brownian motion. In fact, it turns out that some of those local properties are equivalent to certain curvature bounds. Since such properties are also meaningful on metric spaces more general than Riemannian manifolds, we can use them to define corresponding curvature inequalities on such more general spaces, and to explore their consequences and mutual relations. In this talk, I shall introduce or discuss some such curvature inequalities on suitable classes of metric spaces, in particular discrete ones, and explore their geometric consequences.

Hermann Schubert developed in the 19th century a calculus for answering enumerative questions in algebraic geometry, e.g., ''How many lines intersect four curves of degrees \(d_1, \ldots ,d_4\) in three-dimensional space in general position?''. In his 15th problem, Hilberts asked for a rigorous foundation of Schubert's enumerative calculus, which led to important progress in algebraic geometry and topology (intersection theory of the Grassmannians). However, Schubert calculus only yields the typical number of complex solutions. Is there a meaningful way to speak about the typical number of REAL solutions?
We shall outline a way to do so, by assuming that the given objects (the four curves in the above example) are randomly rotated and to inquire about the expected number of real solutions. The approach blends ideas from real algebraic geometry with integral geometry and the theory of random polytopes.
Joint work with Antonio Lerario.

Given a finite morphism between smooth projective curves one can associate to it a Prym variety (an abelian variety, not necessarily principal). The corresponding Prym map is the map between the moduli space of coverings and the moduli space of abelian varieties with some fixed polarization type. By dimension reasons, only in very few cases one can expect the Prym map to be generically finite over its image.
In this talk I will recall some notorious examples where the Prym map is finite and the geometrical interpretation of the fiber. I also explain the case of étale cyclic coverings of degree 7 over a genus 2-curve. We show that the Prym map is generically finite over a special subvariety of the moduli space of 6-dimensional abelian varieties with polarization type (1,1,1,1,1,7). By extending the map to a proper map on a partial compactification of the space of coverings and performing a local analysis we compute that the degree of this Prym map is 10.
This a joint work with Herbert Lange.

In this talk I present a complete solution to the dimension question for restricted Boltzmann machines (RBMs) that was first conjectured by Cueto, Morton, and Sturmfels in 2010. The RBM always has the expected dimension. This result comes from recent work with Jason Morton on the dimension and other properties of a much larger class of models called Kronecker product models: Take two sufficient statistics matrices \(A, B\) (in the RBM case, these are independence models, so Segre matrices). Take the Kronecker product of these matrices to obtain a new matrix \(F = A \otimes B\)Why care about the standard conjectures? (representing a new sufficient statistic), then marginalize the hidden variables corresponding to one of the original matrices. As this construction suggests, one can allow arbitrary exponential families (toric varieties) in place of the two independence models (Segre varieties) from the RBM.

Model reduction of bio-chemical networks relies on the knowledge of slow and fast variables. We provide a geometric method, based on the Newton polytope, to identify slow variables of a bio-chemical network with polynomial rate functions. The gist of the method is the notion of tropical equilibration that provides approximate descriptions of slow invariant manifolds. Compared to extant numerical algorithms such as the intrinsic low dimensional manifold method, our approach is symbolic and utilizes orders of magnitude instead of precise values of the model parameters. Application of this method to a large collection of biochemical network models supports the idea that the number of dynamical variables in minimal models of cell physiology can be small, in spite of the large number of molecular regulatory actors. Moreover, the the computed solution polytopes can be related to metastable regimes allowing to describe the qualitative dynamics of chemical reactions networks.

We consider the problem of determining the real rank of a real homogeneous polynomial f of degree d, i.e. the smallest possible number r such that f can be written as a sum of r d-th powers of real linear forms. In geometric complexity theory the real rank of a polynomial is considered — among other notions of rank — as a measure of the computational complexity of evaluating the polynomial. After some general preliminaries we will present upper and lower bounds on the real rank of a monomial and we characterize those monomials for which the lower bound is attained.
This is joint work with Enrico Carlini, Alessandro Oneto and Emanuele Ventura.

In this talk I will discuss how computational algebraic geometry can be useful for studying questions arising in systems biology. In particular I will focus on the problem of comparing models and data through the lens of computational algebraic geometry and statistics. I will provide two concrete examples of biological signalling systems that are better understood with the developed methods.

This talk deals with the interplay between tropical varieties and non-archimedean analytic varieties in the sense of Berkovich. The Berkovich space associated to a variety over a field with a non-archimedean valuation contains a lot of points which are not visible in algebraic geometry. Under certain conditions these additional points are helpful to find tropical varieties inside Berkovich spaces. After an introduction explaining tropicalizations and Berkovich spaces, we will discuss some results from joint work with Joseph Rabinoff and Walter Gubler.

The real rank two locus of an algebraic variety is the closure of the union of all secant lines spanned by real points. We seek a semi-algebraic description of this set. Its algebraic boundary consists of the tangential variety and the edge variety. Our study of Segre and Veronese varieties yields a characterization of tensors of real rank two.
Joint with Anna Seigal.

A beautiful theorem of J. Thas asserts that the ten points of a Desargues configuration of lines and points formed by two perspective triangles in the projective plane can serve as the ten nodes of a unique rational plane curve of degree 6. I will give an algebraic-geometric proof of this theorem by using the geometry of a special Coble surfaces obtained as blow-ups of ten nodes of a rational sextic. We will also discuss an interesting dynamics of automorphisms of these special Coble surfaces.

We consider the discrete logarithm problem of non-hyperelliptic curves. Observing that the computation can be sped up with the computation of special linear systems, we turn to the efficient computation of these. We then explore several techniques to do so. Finally, we come back to the discrete logarithm problem and present some theorems for curves of a fixed genus.

Verbitsky's Global Torelli theorem has been one of the most important advances in the theory of holomorphic symplectic manifolds in the last years. In a joint work with Ben Bakker (University of Georgia) we prove a version of the Global Torelli theorem for singular symplectic varieties and discuss applications. Symplectic varieties have interesting geometric as well as arithmetic properties, their birational geometry is particularly rich. We focus on birational contractions of symplectic varieties and generalize a number of known results for moduli spaces of sheaves to general deformations.
Our results are obtained through the interplay of Hodge theory, deformation theory, and a further instance of the concept "how to deduce beautiful consequences from ugly behavior of moduli spaces".