Carlos Amendola, TU Munich
Jeff Giansiracusa, Swansea University
Eugenie Hunsicker, Loughborough University
Lisa Lamberti, ETH Zurich
Orlando Marigliano, KTH Stockholm
Nelly Villamizar, Swansea University
Thursday 8th April 2021
The conference has now moved fully online, hosted by University of Glasgow
Register for the conference here
BMC BAMC 2021 Mini-symposium Talks, including questions, are scheduled for 20 minutes.
9:00 Orlando Marigliano (KTH Stockholm) Linear spaces of symmetric matrices
9:20 Lisa Lamberti (ETH Zurich) Cluster partitions and fitness landscapes of the Drosophila fly microbiome
9:40 Carlos Amendola (Munich) Conditional Independence in Max-linear Bayesian Networks
10:00 Jeffrey Giansiracusa (Swansea) Tropical differential equations with non-trivial valuations
10.20 Eugenie Hunsicker (Loughborough) Image manifolds and data integration
10:40 Nelly Villamizar (Swansea) Supersmooth splines and ideals of mixed powers of linear forms
Orlando Marigliano (KTH Stockholm): Linear spaces of symmetric matrices
Abstract: Linear spaces of symmetric matrices are interesting in a variety of mathematical contexts. For instance, they come up in optimization, algebraic geometry, and statistics. In the summer of 2020, about 40 researchers gathered on-line over several months to advance the understanding of these objects. In this talk, I report on some of the interesting results that came out of this effort. I also include my own contributions in the context of algebraic statistics. There, the linear spaces play the role of linear Gaussian concentration models. For these models, I discuss geometric and intersection-theoretic formulas for the maximum likelihood degree. I also discuss the more specific example of colored graphical models and their symmetries.
Lisa Lamberti (ETH Zurich): Cluster partitions and fitness landscapes of the Drosophila fly microbiome
Abstract: In applications often data come as high-dimensional point configurations. Properties of such point configurations can be studied via subdivisions of convex polytopes. In this talk, I present how recent advancements in this theory help uncover new biological insights focusing on the case of experimental Drosophila microbiome data. This talk is based on joined work with Eble, Joswig and Ludington arXiv:2009.12277.
Carlos Amendola (Munich): Conditional Independence in Max-linear Bayesian Networks
Abstract: Motivated by extreme value theory, max-linear Bayesian networks have been recently introduced and studied as an alternative to linear structural equation models. However, for max-linear systems the classical independence results for Bayesian networks are far from exhausting valid conditional independence statements. We use tropical linear algebra to derive a compact representation of the conditional distribution given a partial observation, and exploit this to obtain a complete description of all conditional independence relations. We also introduce the notion of an impact graph which describes how extreme events spread deterministically through the network and we give a complete characterization of such impact graphs. Our analysis opens up several interesting questions concerning conditional independence and tropical geometry. Joint work with Claudia Klüppelberg, Steffen Lauritzen and Ngoc Tran.
Jeffrey Giansiracusa (Swansea): Tropical differential equations with non-trivial valuations
Abstract: About 5 years ago Grigoriev introduced a theory of tropicalizing differential equations and their formal power series solutions over a trivially valued field. In this talk I will describe work with my student Stefano Mereta in which we generalize this theory to allow non-trivially valued fields (such as p-adic fields) and relate the theory to a differential version of Berkovich spaces. Payne proved that the Berkovich analytification is homeomorphic to the inverse limit of tropicalizations, and we prove an analogue of this theorem for differential equations.
Eugenie Hunsicker (Loughborough): Image manifolds and data integration
Abstract: Over the past thirty years, mathematical modelling of materials has produced tremendous insights into the self-organisation of matter at the nano scale into complex structures. This can occur through a variety of mechanisms, including evaporation of a solution to leave behind a pattern of nanoparticles on a surface, called evaporative deposition. If we could harness these natural mechanisms, we could create new structured materials and nano-devices in a cost, time, and energy-efficient manner. Current research has produced numerical models of these processes, which explain how the underlying physics generates the variety of structures observed. As yet, however, there is only a general qualitative understanding of how these models relate to experiments and how structures depend on the conditions under which they are produced. Bridging the current gap between model and experiment will require a rigorous quantitative method for fitting the models to experimental data.
If we want to harness these mechanisms for industrial applications and use the models to predict the conditions required to produce desirable structures: How variable are the structures created under the same or similar experimental conditions? How tightly do conditions need to be controlled in order to repeatably produce structures that are similar enough to be interchangeable for industrial uses? How can we determine the correct laboratory conditions to produce specific structures?
To answer these questions, we need to be able fit the model statistically to the data. Fitting models to data involves minimising the error (i.e., distance) between predicted data and real data. Namely, it requires a meaningful way to calculate distance in the data space. As both the models and the data come in the form of complex structures, this involves computing distances between structures using a mathematical description of the geometry of the structure space, which will in turn relate to the geometry of the structures themselves.
This talk will discuss preliminary work on a method to embed simulated image spaces into high dimensional feature spaces in order to permit the development of meaningful image metrics for data integration.
Nelly Villamizar (Swansea): Supersmooth splines and ideals of mixed powers of linear forms
Abstract: The study of multivariate piecewise polynomial functions (or splines) on polyhedral complexes is important in diverse areas of applied mathematics including numerical analysis, approximation theory, and computer aided geometric design.
In the talk we will address various challenges arising in the study of splines with enhanced (super-) smoothness conditions at the vertices or mixed smoothness across the interior the faces of the partition. Such super-smoothness can be imposed but can also appear unexpectedly on certain splines depending on the geometry of the underlying polyhedral partition. Understanding these splines involve the analysis of ideals generated by products of powers of linear forms in several variables. We will present some of the algebraic tools used to study these spaces, as well as some open questions, and various examples to illustrate the approach. This is a joint work with D. Toshniwal.