We are organizing a new two-days AlToGeLiS meeting at the Max Planck Institute of Molecular Cell Biology and Genetics (MPI-CBG) in Dresden on **May 3rd and 4th, 2024**. This event will be a joint meeting between the partner institutions EPFL, KTH, MPI-CBG, MPI-MiS, Oxford University and UiT. There will be 6 talks by invited speakers, one from each institution, and some working sessions lead by the speakers and aimed to promote interactions between PhD students and postdocs.

Instructions on how to get to MPI-CBG can be found here: https://www.mpi-cbg.de/about-us/contact-us.

*Registrations are closed.*

**Friday, May 3rd**

09:00-09:15h Welcome

09:15-10:00h **Wojciech Chacholski** *TDA in the Life Sciences*

10:00-10:15h Coffee break

10:15-11:00h **Liam Solus** *Geometry and Algebra of Complex Causal Networks*

11:00-11:45h **Angélica Torres** *Emergence of oscillations in a two-layer cascade*

12:00-13:30h Lunch

13:30-14:15h **Mario Kummer** *Fourier quasicrystals*

14:15-15:00h **Tobias Boege** *Algebra in probabilistic reasoning*

15:00-15:30h Coffee break

15:30-17:30h Working Session

Social dinner

**Saturday, May 4th**

09:00-09:45h **Henry Kirveslahti** *Geometric problems related to the Euler Characteristic Transform*

09:45-10:00h Coffee break

10:00-10:45h **Otto Sumray** *Quiver Laplacians, Feature Selection, and Chromatin Accessibility*

10:45-12:15h Working Session

12:15-12:30h Closing remarks

The **problems** for the working sessions can be found **here**.

**TDA in the Life Sciences** (Friday 3, 09:15h)

**Wojciech Chacholski**, *KTH Royal Institute of Technology*

For a successful analysis a suitable representation of data by objects amenable for statistical methods is fundamental. There has been an explosion of applications in which homological representations of data played a significant role. I will present one such representation called stable rank and introduce various novel ways of using it to encode geometry, and then analyse, data. I will provide several illustrative examples of how to use stable ranks to find meaningful results in biological data.

**Geometry and Algebra of Complex Causal Networks** (Friday 3, 10:15h) *(slides)*

**Liam Solus**, *KTH Royal Institute of Technology*

The field of causality has recently emerged as a subject of interest in machine learning, largely due to major advances in data collection methods in the biological sciences and tech industries where large-scale observational and experimental data sets can now be efficiently and ethically obtained. The modern approach to causality decomposes the inference process into two fundamental problems: the inference of causal relations between variables in a complex system and the estimation of the causal effect of one variable on another given that such a relation exists. The subject of this talk will be the former of the two problems, commonly called causal discovery, where the aim is to learn a complex causal network from the available data. We will survey recent advances in causal discovery where geometry and algebra have stepped forward to yield new and effective methodology. We will discuss the potential applications of these methods in the life sciences. We end with the proposal of some specific problems in algebra and geometry, whose solutions will expand the applicability of the discussed methodology.

**Emergence of oscillations in a two-layer cascade** (Friday 3, 11h) *(slides)*

**Angélica Torres**, *MPI for Mathematics in the Siences*

The mitogen-activated protein kinase (MAPK) cascades are processes of cell signalling present in all eukaryotic cells. In this work we study the presence of oscillations in the Huang-Ferrell model of these processes. Specifically, we assume mass-action kinetics and focus on the subsystem formed by the first layer of the cascade and the first site of the phosphorylation in the second layer. We use Hurwitz determinants and Newton polytopes to find Hopf bifurcations on the dynamics of the system, which ensures the emergence of oscillatory behavior. This is joint work with Elisenda Feliu.

**Fourier quasicrystals** (Friday 3, 13:30h)

**Mario Kummer**, *TU Dresden*

We will discuss a family of almost periodic point sets called Fourier quasicrystals. We will explain how to construct those from certain real algebraic varieties and translate some of their properties to algebraic invariants of the variety. This is a joint work in progress with Lior Alon, Pavel Kurasov and Cynthia Vinzant.

**Algebra in probabilistic reasoning** (Friday 3, 14:15h) *(slides)*

**Tobias Boege**, *KTH Royal Institute of Technology - UiT The Arctic University of Norway*

Probabilistic reasoning is concerned with the representation and processing of knowledge about a system of random variables. Random variables are used to model a variety of things around us — natural phenomena, outcomes of clinical trials, human decisions — always with uncertainty attached. This talk focuses on the independence relation of random variables which is the basis of many statistical models like Bayesian networks. I will show how to phrase reasoning tasks as algebraic geometry problems and how to approach them computationally.

**Geometric problems related to the Euler Characteristic Transform** (Saturday 4, 9h) *(slides)*

**Henry Kirveslahti**, *EPFL*

Statistical analysis of shapes dates back to the work of Kendall in 1970s. The advent of high fidelity computer representation of shapes called for digital for shape representation. One such digital structure is based on diffeomorphisms between the shapes. But not all shapes are diffeomorphic. A Topological-Data-Analysis inspired alternative to the diffeomorphism based representation has been achieved by integrating against the Euler characteristic. However, in practice this transform is seldom treated in a digital manner. In this talk we discuss some problems related digitalization of this idea.

**Quiver Laplacians, Feature Selection, and Chromatin Accessibility** (Saturday 4, 10h) *(slides)*

**Otto Sumray**, *University of Oxford*

The challenge of selecting the most relevant features of a given dataset arises ubiquitously in data analysis and dimensionality reduction. However, features found to be of high importance for the entire dataset may not be relevant to subsets of interest, and vice versa. Given a feature selector and a fixed decomposition of the data into subsets, we describe a method for identifying selected features which are compatible with the decomposition into subsets. We achieve this by re-framing the problem of finding compatible features to one of finding sections of a suitable quiver representation. In order to approximate such sections, we then introduce a Laplacian operator for quiver representations valued in Hilbert spaces. We provide explicit bounds on how the spectrum of a quiver Laplacian changes when the representation and the underlying quiver are modified in certain natural ways. Finally, we apply this machinery to the study of peak-calling algorithms which measure chromatin accessibility in single-cell data. We demonstrate that eigenvectors of the associated quiver Laplacian yield locally and globally compatible features.

**Identifiability in Mixtures of Discrete Decomposable Graphical Models** (Saturday 4, 10h) Cancelled

**Jane Coons**, *University of Oxford*

In a discrete graphical model, an underlying graph encodes conditional independences among a set of discrete random variables that label the vertices of the graph. Mixtures of these models allow us to incorporate the assumption that the population is split into subpopulations, each of which may follow a different distribution in the graphical model. From the perspective of algebraic statistics, a discrete graphical model is the real positive piece of a toric variety and the mixture model is part of one of its secant variety. In this talk, we use discrete geometry to investigate the dimensions of the second mixtures of decomposable discrete graphical models. We show that when the underlying graph is not a “clique star”, the mixture model has the maximal dimension. This in turn allows us to prove results on the local identifiability of the parameters.

We are looking forward to meeting you in Dresden!

**Advisory Board**

Kathryn Hess Bellwald (EPFL), Sandra Di Rocco (KTH), Heather Harrington (MPI-CBG and University of Oxford), Bernd Sturmfels (MPI-MiS), Cordian Riener (UiT).

**Organizing committee**

Kelly Maggs (EPFL), Kemal Rose (KTH), Vincenzo Galgano (MPI-CBG), Marina Garrote-López (MPI-MiS), Gillian Grindstaff (University of Oxford) and Ettore Turatti (UiT).