I will present a four-term exact sequence relating the cohomology of a fibration to the cohomology of an open set obtained by removing the preimage of a general linear section of the base. This exact sequence respects three filtrations, the Hodge, weight, and perverse Leray filtrations, so that it is an exact sequence of mixed 
Hodge structures on the graded pieces of the perverse Leray filtration. I claim that this sequence should be thought of as a mirror to the Clemens-Schmid sequence describing the structure of a degeneration and formulate a "mirror P=W" conjecture relating the filtrations on each side. Finally, I will present evidence for this conjecture coming from the K3 surface setting. This is joint work with Charles F. Doran.

Sela proved in 2006 that the (non abelian) free groups are stable. This implies the existence of a well-behaved forking independence relation, and raises the natural question of giving an algebraic description in the free group of this model-theoretic notion. In a joint work with Rizos Sklinos we give such a description (in a standard fg model F, over any set A of parameters) in terms of the JSJ decomposition of F over A, a geometric group theoretic tool giving a group presentation of F in terms of a graph of groups which encodes much information about its automorphism group relative to A. The main result states that two tuples of elements of F are forking independent over A if and only if they live in essentially disjoint parts of such a JSJ decomposition.

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.