# A3G 2018

## Advances in Applied Algebraic Geometry

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

## 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 Piotr Zwiernik (Barcelona)
- 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
- 12:00-13:45 Lunch
- 13:45-14:30 Robin Evans (Oxford)
- 14:30-15:00 Coffee Break
- 15:00-15:45
- 15:45-17:00 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 Kaie Kubjas (Aalto)
- 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)

- 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)

### Wednesday, December 12, 2018:

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

- 10:45-11:30 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.

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

- 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 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)

- 13:45-14:30 Robin Evans (Oxford)

**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.

- 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 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 and Caroline Uhler.

- 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.

## Organiser

Fatemeh Mohammadi (Bristol)