Model reduction is one of the most used tools to characterize real-world complex systems. A large realistic model is approximated by a simpler model on a smaller state space, capturing what is considered by the user as the most important features of the larger model. In this talk we will introduce a new information-theoretic criterion, called "autoinformation", that aggregates states of a Markov chain and provide a reduced model as Markovian (small memory of the past) and as predictable (small level of noise) as possible. We will discuss the connection of autoinformation to widely accepted model reduction techniques in network science such as modularity or degree-corrected stochastic block model inference. In addition to our theoretical results, we will validate such technique with didactic and real-life examples. When applied to the ocean surface currents, our technique, which is entirely data-driven, is able to identify the main global structures of the oceanic system when focusing on the appropriate time-scale of around 6 months.
arXiv link: https://arxiv.org/abs/2005.00337

“The Royal Swedish Academy of Sciences has today decided to award the 2021 Nobel Prize in Physics for ground-breaking contributions to our understanding of complex physical systems”

Last Tuesday this announcement got many in our community very excited: never before had the Nobel prize been awarded to a topic so closely related to Network Science. We will try to understand the contributions that have led to this Nobel Prize announcement and their ties with networks science. The presentation will be held by Erik Hörmann, who has been lucky enough to have had the honour and pleasure of studying and working with one of the awardees, Professor Giorgio Parisi, before joining the Mathematical Institute.

Twisted K-theory is an enrichment of topological K-theory that allows local coefficient systems called twists. For spaces and twists equipped with an action by a group, equivariant twisted K-theory provides an even finer invariant. Equivariant twists over Lie groups gained increasing importance in the subject due to a result by Freed, Hopkins and Teleman that relates the corresponding K-groups to the Verlinde ring of the associated loop group. From the point of view of homotopy theory only a small subgroup of all possible twists is considered in classical treatments. In this talk I will discuss a construction that is joint work with David Evans and produces interesting examples of non-classical twists over the Lie groups SU(n) and over tori constructed from exponential functors. They arise naturally as Fell bundles and are equivariant with respect to the conjugation action of the group on itself. For the determinant functor our construction reproduces the basic gerbe over SU(n) used by Freed, Hopkins and Teleman.

 In this talk, we show some recent results related to the study of mechanical instabilities in slender structures. First, we propose a model of metamaterial sheets inspired by the pellicle of Euglenids, unicellular organisms capable of swimming due to their ability of changing their shape. These structures are composed of interlocking elastic rods which can freely slide along their edges. We characterize the kinematics and the mechanics of these structures using the special Cosserat theory of rods and by assuming axisymmetric deformations of the tubular assembly. We also characterize the mechanics of a single elastic beam constrained to smoothly slide along a rigid support, where the distance between the rod midline and the constraint is fixed and finite. In the presence of a straight support, the rod can deform into shapes exhibiting helices and perversions, namely transition zones connecting together two helices with opposite chirality.

Finally, we develop a mathematical model of damaged axons based on the theory of continuum mechanics and nonlinear elasticity. In several pathological conditions, such as coronavirus infections, multiple sclerosis, Alzheimer's and Parkinson's diseases, the physiological shape of axons is altered and a periodic sequence of bulges appears. The axon is described as a cylinder composed of an inner passive part, called axoplasm, and an outer active cortex, composed mainly of F-actin and able to contract thanks to myosin-II motors. Through a linear stability analysis, we show that, as the shear modulus of the axoplasm diminishes due to the disruption of the cytoskeleton, the active contraction of the cortex makes the cylindrical configuration unstable to axisymmetric perturbations, leading to a beading pattern.

The complexity of biological systems necessitates that we develop mathematical models to further our understanding of these systems. Mathematical models of these systems are generally based on heterogeneous sets of experimental data, resulting in a seemingly heterogeneous collection of models that ostensibly represent the same system. To understand the system, and to reveal underlying design principles, we therefore need to understand how the different models are related to each other with a view to obtaining a unified mathematical description. This goal is complicated by the number of distinct mathematical formalisms that may be employed to represent the same system, making direct comparison of the models very difficult. In this talk I will discuss two general methodologies, namely comparison by distance and comparison by equivalence, that allow us to compare model structures in a systematic way by representing models as labelled simplicial complexes. The distance can be obtained either directly from the simplicial complexes, or from the persistence intervals obtained by employing persistent homology with a flat filtration. Model equivalence is used to determine the conceptual similarity of models and can be automated by using group actions on the simplicial complexes. We apply our methodology for model comparison to demonstrate a particular equivalence between a positional-information model and a Turing-pattern model from developmental biology, which constitutes a novel observation for two classes of models that were previously regarded as unrelated. We also discuss an alternative framework for model comparison by representing models as groups, which allows for the application of group-theoretic techniques within our model comparison methodology.

Microscopic green algae show great diversity in structural complexity, and successfully evolved efficient swimming strategies at low Reynolds numbers. Gonium is one of the simplest multicellular algae, with only 16 cells arranged in a flat plate. If the swimming of unicellular organisms, like Chlamydomonas, is nowadays widely studied, it is less clear how a colony made of independent Chlamydomonas-like cells performs coordinated motion. This simple algae is therefore a key organism to model the evolution from single-celled to multicellular locomotion.

In the absence of central communication, how can each cell adapt its individual photoresponse to efficiently reorient the whole algae? How crucial is the distinctive Gonium squared structure?

In this talk, I will present experiments investigating the shape and the phototactic swimming of Gonium, using trajectory tracking and micro-pipette techniques. I will explain our model linking the individual flagella response to the colony trajectory. This eventually emphasises the importance of biological noise for efficient swimming.

 Mathematics for the mind: network dynamical systems for neurodegenerative disease pathology

Travis Thompson

Can mathematics understand neurodegenerative diseases?  The modern medical perspective on neurological diseases has evolved, slowly, since the 20th century but recent breakthroughs in medical imaging have quickly transformed medicine into a quantitative science.  Today, mathematical modeling and scientific computing allow us to go farther than observation alone.  With the help of  computing, experimental and data-informed mathematical models are leading to new clinical insights into how neurodegenerative diseases, such as Alzheimer's disease, may develop in the human brain.  In this talk, I will overview my work in the construction, analysis and solution of data and clinically-driven mathematical models related to AD pathology.  We will see that mathematical modeling and scientific computing are indeed indispensible for cultivating a data-informed understanding of the brain, AD and for developing potential treatments.

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Simplified battery models via homogenisation  

Robert Timms

Lithium-ion batteries (LIBs) are one of the most popular forms of energy storage for many modern devices, with applications ranging from portable electronics to electric vehicles. Improving both the performance and lifetime of LIBs by design changes that increase capacity, reduce losses and delay degradation effects is a key engineering challenge. Mathematical modelling is an invaluable tool for tackling this challenge: accurate and efficient models play a key role in the design, management, and safe operation of batteries. Models of batteries span many length scales, ranging from atomistic models that may be used to predict the rate of diffusion of lithium within the active material particles that make up the electrodes, right through to models that describe the behaviour of the thousands of cells that make up a battery pack in an electric vehicle. Homogenisation can be used to “bridge the gap” between these disparate length scales, and allows us to develop computationally efficient models suitable for optimising cell design.

The organized movement of intracellular material is part of the functioning of cells and the development of organisms. These flows can arise from the action of molecular machines on the flexible, and often transitory, scaffoldings of the cell. Understanding phenomena in this realm has necessitated the development of new simulation tools, and of new coarse-grained mathematical models to analyze and simulate. In that context, I'll discuss how a symmetry-breaking "swirling" instability of a motor-laden cytoskeleton may be an important part of the development of an oocyte, modeling active material in the spindle, and what models of active, immersed polymers tell us about chromatin dynamics in the nucleus.

The organized movement of intracellular material is part of the functioning of cells and the development of organisms. These flows can arise from the action of molecular machines on the flexible, and often transitory, scaffoldings of the cell. Understanding phenomena in this realm has necessitated the development of new simulation tools, and of new coarse-grained mathematical models to analyze and simulate. In that context, I'll discuss how a symmetry-breaking "swirling" instability of a motor-laden cytoskeleton may be an important part of the development of an oocyte, modeling active material in the spindle, and what models of active, immersed polymers tell us about chromatin dynamics in the nucleus.

Some proteins (in their folded form) are classified as being knotted.

The function of the knotting is mysterious since knotting seemingly

would make the folding process unnecessarily complicated.  To

function, proteins need to fold quickly and reproducibly, and

misfolding can have catastrophic results.  For example, Mad Cow

disease and the human analog, Creutzfeldt-Jakob disease, come from

misfolded proteins.

Traditionally, knotting is only defined for closed curves, where the

topology is trapped in the loop.  However, proteins have free ends, as

well as most of the objects that humans consider as being knotted

(like shoelaces and strings of lights).  Defining knotting in open

curves is tricky and ambiguous.  We consider some definitions of

knotting in open curves and see how one of these definitions is used

to characterize the knotting in proteins.