Speakers:

Emilie Dufresne, University of York, UK 

Gillian Grindstaff, University of Oxford, UK

Maximilien Gadouleau, Durham University, UK

Olga Kuznetsova, Aalto University, Finland

Gleb Pogudin, Laboratoire d’informatique de l’École Polytechnique, France

Date:

Wednesday 13h April 2022, 10:30am-12:30pm.

Location:

This virtual meeting will be part of the British Applied Mathematics Colloquium, BAMC 2022 that will be hosted at Loughborough University, from 11th-13th April 2022. 

The BAMC 2022 will be a hybrid conference with events mostly in person, but enabling online attendees also.

Registration:

Register for the conference here 

Schedule:

BAMC 2022 Mini-symposium Talks, including questions, are scheduled for 20 minutes.

 10:30  Emilie Dufresne (York) Structural and practical identifiability of ERK kinetics.

 10:50 Gillian Grindstaff (Oxford) Representations of partial leaf sets in phylogenetic tree space.

 11:10 Discussion / Networking.

 11:30  Maximilien Gadouleau (Durham) Linear programming complementation.

 11:50 Olga Kuznetsova (Aalto University) Algebraic Degree of Polynomially Constrained Optimization.

 12:10 Gleb Pogudin (École polytechnique) Exact reductions of dynamical systems.

12:30 Discussion/Networking on zoom 

Abstracts: 

Emilie Dufresne (York) Structural and practical identifiability of ERK kinetics.

Abstract: This talk is based on a paper written in collaboration with Lewis Marsh, Helen Byrne and Heather Harrington, where we explored the algebra, geometry and topology of ERK kinetics.  The MEK/ERK signalling pathway is involved in cell division, cell specialisation, survival and cell death. We studied a polynomial dynamical system describing the dynamics of MEK/ERK proposed by Yeung et al. with their experimental setup, data and known biological information. The experimental dataset is a time-course of ERK measurements in different phosphorylation states following activation of either wild-type MEK or MEK mutations associated with cancer or developmental defects. My focus in this talk will be on identifiability, both structural and practical. Structurally identifiable is concerned with asking whether parameter values can be recovered from perfect data. Practical identifiability addresses the more realistic situation where we assume there is measurement noise. We observe that the original model is structurally but not practically identifiable. We will discuss how algebraic quasi-steady state approximation leads to a smaller simpler model which is both structurally and practically identifiable, while providing a probable explanation for the practical non-identifiability of the original model.

Gillian Grindstaff (Oxford) Representations of partial leaf sets in phylogenetic tree space.

Abstract: The metric space of phylogenetic trees defined by Billera, Holmes, and Vogtmann, which we refer to as BHV space, provides a natural geometric setting for describing collections of trees on the same set of taxa. However, it is sometimes necessary to analyze collections of trees on non-identical taxa sets (i.e., with different numbers of leaves), and in this context it is not evident how to apply BHV space. Ren et al. approached this problem by describing a combinatorial algorithm extending tree topologies to regions in higher dimensional tree spaces, so that one can quickly compute which topologies contain a given tree as partial data. In this work, we refined and adapted their algorithm to work for metric trees to give a full characterization of the linear subspace of extensions of a subtree. We demonstrate how to apply our algorithm to define and search a space of possible supertrees and, for a collection of tree fragments with different leaf sets, to quantify their compatibility.

Maximilien Gadouleau (Durham) Linear programming complementation.

Abstract: In this talk, we introduce a new kind of duality for Linear Programming (LP), that we call LP complementation. We prove that the optimal values of an LP and of its complement are complement pairs (provided that either the original LP or its complement has an optimal value greater than one). The main consequence of the LP complementation theorem is for hypergraphs. We introduce the complement of a hypergraph and we show that the fractional packing numbers of a hypergraph and of its complement are complement pairs; similar results hold for fractional matching, covering and transversal numbers.

This hypergraph complementation theorem has several consequences for fractional graph theory. In particular, we relate the fractional dominating number of a graph to the fractional total dominating number of its complement.

Olga Kuznetsova (Aalto University) Algebraic Degree of Polynomially Constrained Optimization.

Abstract: Consider an optimisation problem whose objective function is in the algebraic closure of the rational function field and the feasible set is an algebraic variety. The algebraic degree of a variety in a given optimisation problem is the number of complex critical points. Using radical parametrisation, we show that the notion of algebraic degree is well-defined for a certain large class of such problems. As a special case, we study the case of the p-th power of the p-norm, for which we generalise the notion of the ED correspondence variety and give explicit formulas for the algebraic degree.

Gleb Pogudin (École polytechnique) Exact reductions of dynamical systems.

Abstract: Dynamical systems are frequently used for modeling in the sciences. Building a detailed model often requires taking into account a large number of factors and, as a result, a model may become quite large. High-dimensional models with dozens or hundreds of state variables are not only challenging computationally but it is also hard to use them to derive mechanistic insights. Exact model reduction is a way to address this issue by finding a self-consistent lower-dimensional projection of the corresponding dynamical system. I will describe recent algorithms for computing such reductions (in particular the CLUE software https://github.com/pogudingleb/CLUE) and demonstrate them on the models from literature.

Sponsors:

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