# Algebraic Geometry

### Varieties, Polyhedra, Computation

##### Winter Semester 2019/2020

# Abstracts of the opening conference

Below there are the abstracts of the talks of the opening conference (7th-11th October 2019) of the thematic semester Algebraic Geometry.

### Felipe Cucker, TU Berlin

#### Condition and semialgebraic geometry

We overview several geometric features of semialgebraic sets for which a quantitative aspect relates to the condition of the descriotion of the set.

### Jan Draisma, U Bern

#### The coarse structure of GL_inf-varieties

A (n affine) GL_inf-variety over C is a closed, GL_inf-stable subset of the inverse limit P_inf of the P(C^n), where P is a fixed Schur functor. For example, for P=S^2, P_inf is the space of symmetric infinite-by-infinite matrices. In this example, a matrix either has finite rank k, in which case it lies in the image of the smaller Schur functor (S^1)^k under the morphism (v_1,...,v_k) -> v_1^2 + ... + v_k^2 of GL_inf-varieties; or else its orbit is dense in P_inf. In particular, every GL_inf-variety in S^2 is an orbit closure. In ongoing work with Arthur Bik, Rob Eggermont, and Andrew Snowden we generalise this dichotomy to other Schur functors. Among other things, we prove that any GL_inf-variety in any P_inf is the closure of a finite-dimensional family of orbits. We also prove a version of Chevalley's theorem on constructible sets, which implies that membership in the image of a fixed GL_inf-morphism can be tested deterministically in polynomial time.

### Mathias Drton, U Washington

#### Maximum likelihood thresholds for covariance matrices with Kronecker product structure

The matrix normal model is a statistical model that assumes multivariate data to be generated from a Gaussian distribution whose covariance matrix is the Kronecker product of two positive definite matrices. Prior work shows that maximum likelihood estimators (MLEs) in these models may exist for surprisingly small sample sizes. We formulate algebraic conditions for existence of the MLE and show that in special cases the normal form for matrix pencils can be leveraged to derive the precise sample size needed for almost sure existence. (joint work with Satoshi Kuriki)

### Michael Joswig, TU Berlin

#### The Schläfli Fan

Smooth tropical cubic surfaces are parametrized by maximal cones in the unimodular secondary fan of the triple tetrahedron. There are 344,843,867 such cones, organized into a database of 14,373,645 symmetry classes. The Schläfli fan gives a further refinement of these cones. It reveals all possible patterns of the 27 or more lines on tropical cubic surfaces, thus serving as a combinatorial base space for the universal Fano variety. This article develops the relevant theory and offers a blueprint for the analysis of big data in tropical algebraic geometry. We conclude with a sparse model for cubic surfaces over a field with valuation. Joint work with Marta Panizzut and Bernd Sturmfels.

### Kaie Kubjas, Aalto U Helsinki

#### Maximum likelihood estimation of toric Fano varieties

Maximum likelihood estimation aims to find a point of a statistical model that best explains observational data. We study the maximum likelihood (ML) estimation problem for toric Fano varieties. First we show that with one exception for all $2$-dimensional Gorenstein toric Fano varieties the ML degree equals the degree of the variety and we provide expressions that allow to compute the maximum likelihood estimate in the closed form. We then explore the reasons for ML degree drop using A-discriminants and intersection theory. Finally we show that varieties associated to $3$-valent phylogenetic trees and known from the work of Buczynska and Wisniewski have ML degree one. This follows from a more general result on the multiplicativity of ML degrees of codimension zero toric fiber products.

### Pierre Lairez, Inria

#### Numerical periods in effective algebraic geometry

Thanks to several recent progress, we can now compute the periods of quartic surfaces to arbitrary precision, and consequently many algebraic invariants: Picard group, endomorphism ring, number of embedded smooth rational curve of a given degree, etc. We start compiling a database of K3 surfaces with their invariants. This talk will aim at a hands-on presentation of the tools involved and a presentation of several examples of the database. Joint work with Emre Sertöz.

### Laurent Manivel, CNRS Toulouse

#### On the geometry of skew-symmetric three-forms

There has been a huge activity in the last decades around the geometry of tensors, with all kinds of potential applications. In this talk I will concentrate on skew-symmetric three-forms. In low dimensions, up to eight, there exist only finitely many types of such forms. The critical dimensions are nine and ten, for which extremely rich geometries emerge.

### Francisco Santos, U de Cantabria

#### Width of lattice polytope

Hollow polytopes (that is, polytopes with no interior lattice points) are important both in algebraic geometry and integer optimization.
One of their most important invariants is their lattice width which, by the "flatness theorem" is bounded in fixed dimension.
We will review several recent results related to the width of lattice polytopes. Among them:

- we look at how to construct hollow polytopes of width larger than their dimension, trying to improve lower bounds on the flatness constant.

- we conjecture that the tight upper bound for the width of hollow convex bodies in dimension three is 2+\sqrt{2}, attained by a certain tetrahedron.

- we show how width can be used as a tool to classify lattice polytopes. Eg: In dimension three there are finitely many (perhaps non-hollow) lattice polytopes of width larger than one for each number of lattice points. In dimension four, there are finitely many empty simplices of width larger than two, which has been used to completely classify empty 4-simplices (equivalently, to classify terminal quotient singularities of dimension four).

### Reinhold Schneider, TU Berlin

#### Variational Monte Carlo - theoretical bridge between numerics and statistical learning

For solving solve high-dimensional PDE's, which can be casted into a variational framework. For computational purpose the objective functional is restricted to appropriate possibly non-linear and even non-convex model classes. In Variational Monte Carlo we replace the objective functional by an empirical (surrogate) functional, in a similar way as for risk minimization or loss functions in statistical learning.For the optimization we need only computable gradients at sample points.

At a price, we have to consider "convergence in probability", i.e. error estimates holds with high probability. A first analysis has been carried out in the spirit of Cucker and Smale. A new modification based an restricted isometry property (RIP) can provide exact reconstruction for solutions out of the model classes and further quasi-optimal convergence. We have in mind algebraic concepts for the model classes like hierarchical tensors (HT tensors or tree-based tensor networks) etc., but the approach applies also to deep neural networks.

### Rainer Sinn, FU Berlin

#### Sums of Squares and Projective Varieties

Writing a real polynomial p as a sum of squares of polynomials is a certificate of positivity for p that is exploited in polynomial optimization because of its connections to the feasibility problem in semidefinite programming via the Gram map. The Gram map naturally generalizes to the context of real projective varieties, where sums of squares certify nonnegativity of homogeneous elements of even degree in the homogeneous coordinate rings. I will report on surprising connections between sums of squares and invariants of (embedded) projective varieties.

### Orsola Tommasi, U Padova

#### Local systems on M_2 and the top weight cohomology of M_{2,n}

The moduli space M_{g,n} of smooth n-pointed complex curves of genus g is a classical object of study in algebraic geometry. However, its topological invariants are still not well-known. If one looks at the cohomology of M_{g,n}, the best known part is the so-called tautological subring, which is generated by geometrically natural classes. However, recent work by Chan, Galatius and Payne highlighted the importance of a completely different part of cohomology: its top weight part, defined using mixed Hodge structures. This top weight part turns out to be quite combinatorial in nature because of its relationship with tropical geometry. In this talk, I would like to present an alternative approach to the study of the top weight cohomology for genus g=2, using local systems. This is joint work (in progress) with Dan Petersen.

### Timo de Wolff, Technische Universität Braunschweig

#### Nonnegativity, Discriminants, and Tropical Geometry

Certifying nonnegativity of real, multivariate polynomials is a key problem in real algebraic geom-etry since the 19th century. In the 21st century, the problem gained signicant momentum due toits relevance in polynomial optimization. Let R^A denote the space of all real polynomials with support A \subset N^n. We study certificates called (sums of) nonnegative circuit polynomials (SONC) or agiforms, which can be obtained from the inequality of arithmetic and geometric means. SONCs form a full dimensional subcone S of the cone of nonnegative polynomials in R^A. In particular, S is not contained the cone of sums of squares. We describe the boundary of the cone S as a space stratified in real semi-algebraic varieties. In order to describe the single strata, we will go on an exploration through the mathematical universe on which we will encounter discriminants, polytopes and their triangulations, and tropicalgeometry. This is joint work with Jens Forsgård.