16th meeting, Edinburgh

Speakers:

Oliver Clarke, University of Edinburgh

Marina Garrote López, Max Planck Institute for Mathematics in the Sciences, Leipzig

Sam Martin, Earlham Institute

Vasiliki Petrotou, The Hebrew University of Jerusalem

Roan Talbut, Imperial College London

Beihui Yuan, Swansea University

Date:

July 20-21, 2023

Location:

In person at Bayes Centre, University of Edinburgh, 5th floor, room: 5.46 Bayes Centre .

Local organiser:

Dimitra Kosta, University of Edinburgh

Registration :

If you wish to attend the meeting, please register by sending an e-mail 📬 to Dimitra Kosta (D.Kosta@ed.ac.uk).

Schedule:

Thursday 20th July 2023

10:30-11:00 Welcome and coffee

11:00-12:00 Vasiliki Petrotou, Lefschetz Properties via anisotropy for lattice polytopes

12:00-13:00 Beihui Yuan, An algebraic framework for Geometric continuous splines

13:00-14:30 Lunch

14:30-15:30 Roan Talbut, Wasserstein Distances on Tropical Projective Torii of Different Dimensions

15:30-16:30 Oliver Clarke, From toric degenerations of the Grassmannian to its algebraic matroid

18:30 Dinner

Friday 21st July 2023

9:30-10:00 Morning Coffee

10:00-11:00 Marina Garrote López, Identifiability of level-1 species networks from gene tree quartets

11:00-12:00 Sam Martin, Dimensions of phylogenetic network varieties

12:30 Lunch

Abstracts collection

Oliver Clarke, University of Edinburgh

Title: From toric degenerations of the Grassmannian to its algebraic matroid

Abstract: In this talk, based on upcoming work with S. Tanigawa, I will introduce the algebraic matroid of an affine variety, which encodes information about its coordinate projections. In the case of the Grassmannian, I will show how its algebraic matroid can be constructed from simple pieces: the algerbaic matroid of matching fields. Matching fields naturally arise in the construction of toric degenerations of Grassmannian. We will see that this is no coincidence, with a common link of tropical geometry.

Marina Garrote López, Max Planck Institute for Mathematics in the Sciences, Leipzig

Title: Identifiability of level-1 species networks from gene tree quartets

Abstract: The inference of phylogenetic networks, essential for understanding evolutionary relationships involving hybridization and horizontal gene transfer, presents formidable challenges in both theory and practice. While standard phylogenetic methods can infer gene trees from genetic data, these trees only indirectly reflect the species network topology due to horizontal inheritance and incomplete lineage sorting. Current algorithms, such as SNAQ and NANUQ, summarize gene trees using quartet counts and pseudolikelihood computations to obtain phylogenetic networks. Although these quartet-based methods seem powerful, there hasn't been a comprehensive analysis of the identifiable features of level-1 phylogenetic networks under the Network Multispecies Coalescent (NMSC) model.

Previous research has shown that certain level-1 network topologies and numerical parameters can be identified, but gaps remain in understanding the full topology. In this talk, we will aim to fill these gaps and address both, the identifiability of the full topology of the network, including edge directions, as well as the numerical parameters of edge lengths and hybridization probabilities. Our approach involves studying the ideals defined by quartet concordance factors for topological semi-directed networks.

This is joint work with Elizabeth S. Allman, Hector Baños and John A. Rhodes.

Sam Martin, Earlham Institute

Title: Dimensions of phylogenetic network varieties

Abstract: Phylogenetic networks provide a means of describing the evolutionary history of taxa that have undergone “horizontal” evolution events such as hybridization or lateral gene transfer. The mutation process of a single site in shared DNA sequence for a set of such taxa can be modelled as a Markov process on a phylogenetic network, and the site-pattern probability distributions from such a model can be viewed as a projective variety. In our recent work, we have given an explicit description of the dimension of such varieties for level-1 phylogenetic networks under any group-based model of evolution. I will give an introduction to phylogenetic networks and an overview of the models from an algebraic perspective. Then I will describe our recent dimension results, focussing on the toric fiber product of two ideals, and finish with some applications to identifiability problems. Joint work with Elizabeth Gross and Robert Krone.

Vasiliki Petrotou, The Hebrew University of Jerusalem

Title: Lefschetz Properties via anisotropy for lattice polytopes

Abstract: The concept of generic anisotropy is a useful tool for proving Lefschetz properties of graded commutative algebras. The aim of the talk is to describe the connection between anisotropy and Lefschetz properties and present results related to the characteristic 2 anisotropy of certain algebras associated to Gorenstein IDP lattice polytopes.

This is based on joint work with Karim Adiprasito, Stavros Papadakis and Johanna Steinmeyer.

Roan Talbut, Imperial College London

Title: Wasserstein Distances on Tropical Projective Torii of Different Dimensions

Abstract: Phylogenetic Trees are a fundamental tool for summarising the mutation structure of many diseases. In practice, we cannot identify the exact tree, but instead identify a posterior distribution on the space of all trees with the observed leaf set. Wasserstein distances are a popular tool for comparing distributions, but require our measures to live in the same space; this is generally not the case when studying the phylogenetic trees of independent evolutionary processes. To this end, we define Wasserstein distances between measures on tree spaces of different dimensions. Using the tropical interpretation of tree space [1], we study optimal transport between general measures on tropical projective torii. Our method mirrors the semi-orthogonal map methodology used by Cai and Lim [2] in the Euclidean setting, establishing the same powerful behaviour in a more complex tropical setting. We also establish an algorithm for computing this tropical Wasserstein distance across different dimensions in practice.

This is joint work with Daniele Tramontano , Mathias Drton and Anthea Monod.

References

[1] David Speyer and Bernd Sturmfels. “The Tropical Grassmannian”. In: Adv. Geom. 4.3 (2004), pp. 389–411. doi: doi:10.1515/advg.2004.023.

[2] Yuhang Cai and Lek-Heng Lim. “Distances Between Probability Distributions of Different Dimensions”. In: IEEE Transactions on Information Theory 68.6 (2022), pp. 4020–4031. doi: 10.1109/TIT.2022.3148923.

Beihui Yuan, Swansea University

Title: An algebraic framework for Geometric continuous splines

Abstract: Geometrically continuous splines (G-splines) are piecewise polynomial functions defined on a collection of patches which are “stitched together”. Compared to the traditional parametric splines, G-splines are more flexible in construction of shapes with complicated topology. In this talk, we provide an algebraic framework for studying G-splines. An immediate consequence of introducing this framework is the application of algebraic methods to estimate the dimension of G-spline spaces and to construct a basis for given G-spline space.

This talk is based on a program joint with Angelos Mantzaflaris, Bernard Mourrain and Nelly Villamizar.

Sponsors:

We are grateful for the financial support from the Glasgow Mathematical Journal Learning and Research Support Fund, from the Edinburgh Mathematical Society and the London Mathematical Society.

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