# A3G 2018

## Advances in Applied Algebraic Geometry

### University of Bristol, 11-14 December 2018

## Venue

Conference talks are at Engineers House, The Promenade, Clifton Down, Avon, Bristol BS8 3NB (Google Maps).

Colloquium talks are at Mott Lecture Theatre, School of Physics, Tyndall Avenue, Bristol University BS8 1TL (Google Maps).

Organiser: Fatemeh Mohammadi

## Invited Speakers

Spencer Backman (Jerusalem)

Marta Casanellas (Barcelona)

Carlos D'Andrea (Barcelona)

Emanuele Delucchi (Fribourg)

Jan Draisma (Bern)

Robin Evans (Oxford)

Kaie Kubjas (Aalto)

Anna Levina (Tübingen)

Marta Panizzut (TU Berlin)

Johannes Rauh (MPI Leipzig)

Cordian Riener (Tromsø)

Felipe Rincón (Queen Mary)

Kayvan Sadeghi (Cambridge)

Eduardo Sáenz de Cabezón (Logroño)

Anna Seigal (Berkeley)

Bernd Sturmfels (MPI Leipzig)

Timo de Wolff (TU Berlin)

Henry Wynn (LSE)

Piotr Zwiernik (Barcelona)

## Program

### Tuesday, December 11, 2018:

09:30-10:15 Carlos D'Andrea (Barcelona)

10:15-10:45 Coffee Break

10:45-11:30 Cordian Riener (Tromsø)

11:30-12:15 Felipe Rincón (Queen Mary)

12:15-14:00 Lunch

14:00-14:45 Marta Panizzut (TU Berlin)

14:45-15:15 Coffee Break

15:15-16:00 Emanuele Delucchi (Fribourg)

### Wednesday, December 12, 2018:

09:30-10:15 Eduardo Sáenz de Cabezón (Logroño)

10:15-10:45 Coffee Break

10:45-11:30 Kaie Kubjas (Aalto)

11:30-12:15 Spencer Backman (Jerusalem)

12:15-14:00 Lunch

14:00-14:45 Anna Levina (Tübingen)

14:45-15:15 Coffee Break

15:15-16:00 Break

16:00-17:00 Colloquium talk by Marta Casanellas (Barcelona)

17:00-18:00 Reception

18:30-21:00 Dinner

### Thursday, December 13, 2018:

09:30-10:15 Kayvan Sadeghi (Cambridge)

10:15-10:45 Coffee Break

10:45-11:30 Johannes Rauh (MPI Leipzig)

11:30-12:00 Short talks

12:00-13:45 Lunch

13:45-14:30 Dimitra Kosta (Glasgow)

14:30-14:45 Coffee Break

14:45-15:30 Short talks

15:30-17:00 Break

17:00-18:00 Colloquium talk by Bernd Sturmfels (MPI Leipzig)

18:00-19:00 Reception

### Friday, December 14, 2018:

09:30-10:15 Timo de Wolff (TU Berlin)

10:15-10:45 Coffee Break

10:45-11:30 Anna Seigal (Berkeley)

11:30-12:15 Piotr Zwiernik (Barcelona)

12:15-14:00 Lunch

14:00-14:45 Jan Draisma (Bern)

14:45-15:15 Coffee Break

15:15-16:00 Henry Wynn (LSE)

## A3G 2018 Registration

Please register here before November 12th, 2018 at 23:59 UK time. There is limited space available for participation and we will notify accepted participants by 15th of November.

The conference registration fee is £50 and includes lunches and coffee-breaks.

Conference dinner costs an additional £30.

## Titles and Abstracts:

### Tuesday, December 11, 2018:

09:30-10:15 Carlos D'Andrea (Barcelona)

The Rees Algebra of Parametric Curves via liftings

Abstract: This is a joint work with Teresa Cortadellas and David Cox. We study the defining equations of the Rees algebra of ideals arising from curve parametrizations in the plane and in rational normal scrolls, inspired by the work of Madsen and Kustin, Polini and Ulrich. The curves are related by work of Bernardi, Gimigliano and Ida, and we use this framework to relate the defining equations.

10:45-11:30 Cordian Riener (Tromsø)

Vandermonde varieties, mirror spaces, and the homology of symmetric semi-algebraic sets

Abstract: The level sets of the first d (weighted) Newton power sums in R^k for some d ≤ k have been called Vandermonde varieties by Arnold and Giventhal. These varieties have a natural action of the symmetric group, which induces an action on their cohomology groups. By using a formula of Solomon we can study the decomposition of the resulting S_k-module and generalise some of the results obtained by Arnold and Giventhal on the homology modules of such varieties. Since every symmetric polynomial of degree d can be written in terms of Newton power sums, we can infer similar results of all symmetric semi-algebraic sets defined by symmetric polynomials whose degree does not exceed d. (Joint work with Saugata Basu).

11:30-12:15 Felipe Rincón (Queen Mary)

An introduction to Tropical Geometry

Abstract: I will give an introduction to tropical geometry, with examples of some of the techniques that have been successful in approaching classical algebraic geometry problems.

12:15-14:00 Lunch

14:00-14:45 Marta Panizzut (TU Berlin)

K3 Polytopes and their quartic surfaces

Abstract: Tropical geometry is a recent area of mathematics at the interface between algebraic geometry, polyhedral geometry and combinatorics. The tropical counterpart of algebraic varieties are tropical varieties, which are polyhedral complexes satisfying certain combinatorial properties. The closure of the connected components of the complement of a tropical hypersurface are called regions. They have the structure of convex polyhedra. A 3-dimensional polytope is a K3 polytope if it is the closure of the bounded region of a smooth tropical quartic surface. In this talk we begin by studying properties of K3 polytopes. In particular we exploit their duality to regular unimodular central triangulations of reflexive polytopes in the fourth dilation of the standard tetrahedron. Then we focus on quartic surfaces that tropicalize to K3 polytopes, and we look at them through the lenses of Geometric Invariant Theory. We will highlight the computational aspect of this project. This is a joint work with Gabriele Balletti and Bernd Sturmfels.

14:45-15:15 Coffee Break

15:15-16:00 Emanuele Delucchi (Fribourg)

On stability properties of phylogenetic consensus methods

Abstract: Consensus methods are data-aggregation algorithms that should satisfy some desirable and reasonable properties (like “fairness” in social choice theory, where this field of study was initiated and drew some of the terminology). Perhaps the most well-known result is K. Arrow’s theorem on the (non)existence of certain desirable vote-evaluation systems. In the last 30 years (deterministic) consensus methods have been studied in the context of aggregating phylogenetic data. Here, among the “desirable” features there has been recent focus on properties that would allow for efficient computations under scaling of the available data set. In recent joint work with L. Hoessly and G. Paolini we answered (in the negative) some questions by Bryant, Francis and Steel about the existence of such algorithms with a specific set of desired features.

### Wednesday, December 12, 2018:

09:30-10:15 Eduardo Sáenz de Cabezón (Logroño)

Depolarization of monomial ideals

Abstract: Polarization is an important operation that takes monomial ideals to squarefree monomial ideals. It has been used in many situations such as Hartshorne's proof of the conenctedness of the Hilbert scheme or the study of associated primes. We study the converse operation, depolarization and investigate the set of ideals that have the same polarization and the propeties they share.

10:45-11:30 Kaie Kubjas (Aalto)

Mathematics of 3D Genome Reconstruction in Diploid Organisms

Abstract: 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. This talk is based on joint work with Mohab Safey El Din, Lawrence Sun, Caroline Uhler.

11:30-12:15 Spencer Backman (Jerusalem)

On the combinatorics of zonotopes

Abstract: I will discuss some recent results on the combinatorics of zonotopes.

14:00-14:45 Anna Levina (Tübingen)

Mathematical modeling of neuronal avalanches.

Abstract: The human brain is a huge complex system of many interacting elements. It was shown that many computational properties in such systems are optimized close to the so-called critical state. This observation of optimality has led to a hypothesis that the brain should also operate close to the criticality. In recent years, many experimental studies found signatures of criticality in the recordings from different neuronal systems. Most prominently, the observation of power-law scaling in the activity propagation cascades termed neuronal avalanches. In my presentation, I will discuss challenges in the modeling of critical systems and solutions to some of them. I will present a simple mathematical model that allows capturing central features of neuronal avalanches. The most basic version of the model uses all-to-all connectivity. It allows for a full description of the network model as a skew-product random transformation on the simple support we called “inhabited set”. Restricting topology to a ring network results in a surprising complexity of the inhabited set and unexpected links to the algebraic combinatorics and G-parking functions.

16:00-17:00 Colloquium talk by Marta Casanellas (UPC Universitat Politècnica de Catalunya)

From phylogenetics to algebraic geometry

Abstract: Many of the evolutionary models used in phylogenetics can be viewed as algebraic varieties. In this expository talk we will explain the main goals of phylogenetics, introduce evolutionary Markov models on trees, and show how algebraic varieties arise in this context. Moreover, we will see how an in-depth geometric study of these varieties leads to improvements on phylogenetic reconstruction methods. We shall illustrate these improvements by showing results on simulated and real data and by comparing them to widely used methods in phylogenetics.

### Thursday, December 13, 2018:

09:30-10:15 Kayvan Sadeghi (Cambridge)

On Exchangeability in Network Models

Abstract: We derive representation theorems for exchangeable distributions on finite and infinite graphs using elementary arguments based on geometric and graph-theoretic concepts. Our results elucidate some of the key differences, and their implications, between statistical network models that are finitely exchangeable and models that define a consistent sequence of probability distributions on graphs of increasing size. We also show that, for finitely exchangeable network models, the empirical subgraph densities are maximum likelihood estimates of their theoretical counterparts. We then characterize all possible conditional independence structures for finitely exchangeable random graphs.

10:45-11:30 Johannes Rauh (MPI Leipzig)

Conditional independence ideals with hidden variables

Abstract: We study a class of determinantal ideals that are related to conditional independence (CI) statements with hidden variables. Such CI statements correspond to determinantal conditions on flattenings of marginals of the tensor P of joint probabilities of the observed random variables. We focus on an example that generalizes the CI ideals of the intersection axiom. In this example, the minimal primes are again determinantal ideals, which is not true in general. This is a joint work with Oliver Clarke and Fatemeh Mohammadi.

13:45-14:30 Dimitra Kosta (Glasgow)

Unboundedness of Markov complexity of monomial curves in A^n for n≥ 4

Abstract: Computing the complexity of Markov bases is an extremely challenging problem; no formula is known in general and there are very few classes of toric ideals for which the Markov complexity has been computed. A monomial curve C in A^3 has Markov complexity m(C) two or three. Two if the monomial curve is complete intersection and three otherwise. Our main result shows that there is no d in N such that m(C) ≤ d for all monomial curves C in A^4. The same result is true even if we restrict to complete intersections. We extend this result to all monomial curves in A^n ,n ≥ 4.

17:00-18:00 Colloquium talk by Bernd Sturmfels (MPI Leipzig)

Varieties of Signature Tensors

Abstract: We discuss recent developments in computational algebraic geometry that were motivated by the study of rough paths in stochastic analysis. Every path in a real vector space is encoded in a signature tensor whose entries are iterated integrals. As the path varies over a nice family we obtain an algebraic variety with interesting properties.

### Friday, December 14, 2018:

09:30-10:15 Timo de Wolff (TU Berlin)

Nondegenerate multistationarity in small reaction networks

Abstract: Much attention has been focused in recent years on the following algebraic problem arising from applications: which chemical reaction networks, when taken with mass-action kinetics, admit multiple positive steady states? The interest behind this question is in steady states that are stable. As a step toward this difficult question, here we address the question of multiple nondegenerate positive steady states. Mathematically, this asks whether certain families of parametrized, real, sparse polynomial systems ever admit multiple positive real roots that are simple. Our main results settle this problem for certain types of small networks, and our techniques point the way forward for larger networks. This is joint work with Anne Shiu.

10:15-10:45 Coffee Break

10:45-11:30 Anna Seigal (Berkeley)

Learning Paths from Signature Tensors

Abstract: 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 signature tensors of order three. We establish identifiability results and recovery algorithms for piecewise linear paths, polynomial paths, and generic dictionaries. This is joint work with Max Pfeffer and Bernd Sturmfels.

11:30-12:15 Piotr Zwiernik (Barcelona)

Maximum likelihood under total positivity

Abstract: Probability distributions that are multivariate totally positive of order 2 (MTP2) appeared in the theory of positive dependence and in statistical physics through the celebrated FKG inequality. The MTP2 property is stable under marginalization, conditioning and it appears naturally in various probabilistic graphical models with hidden variables. In real algebraic geometry it appears, for example, in the description of the tensors of nonnegative rank at most two. Maximum likelihood estimation is the most natural way to estimate parameters of a model in statistics. In this talk I want to explain some of the fascinating mathematics behind the maximum likelihood estimation problem under the MTP2 constraint. These results show that the MTP2 property can be particularly useful in high-dimensional settings in statistics.

12:15-14:00 Lunch

14:00-14:45 Jan Draisma (Bern)

On Strength

Abstract: An infinite-by-infinite matrix A either has infinite rank, in which case its orbit under left and right multiplication by invertible matrices is dense its ambient space, or else A factors as B*C where B and C are infinite-by-k and k-by-infinite, respectively. This dichotomy turns out to hold in much more general ambient spaces, namely, limits of polynomial functors. There, an element either has a dense orbit, or else lies in the image of a polynomial transformation from a smaller polynomial functor. This dichotomy is at the heart of recent progress in commutative algebra, e.g. the resolution of Stillman's conjecture by Ananyan-Hochster and by Erman-Sam-Snowden, and the Noetherianity of polynomial functors. But it also leads to practical algorithmic questions about algebraic mathematical models. This talk is based on ongoing joint work with Bik-Eggermont-Snowden and on joint work with Lason-Leykin.

14:45-15:15 Coffee Break

15:15-16:00 Henry Wynn (LSE)

Algebraic methods for sparse grids

Abstract: Sparse grids, sometimes called Smolyak grids, are used for polynomial quadrature particularly in the context of solving partial differential equations. They give exact evaluation of moments up to certain polynomial degree order. A certain subclass of nested grids maps very well into the theory of monomial ideals and the coefficients in the formulae used in the underlying interpolation can be evaluated in terms of Betti numbers. The algebra leads naturally to a broad definition of the concept of nesting and some standard examples fall under the definition.