Traditionally, most studies of quantum many-body systems have been mainly concerned with properties of the states of low-lying energy. Recently, however, fascinating features of the full energy spectrum have been uncovered. Among these are eigenstate phase transitions, where sharp transitions occur not only in the ground state, but in all the states. I describe a simple example of such, a transition for a strong zero mode in the XYZ spin chain. The strong zero mode is an operator that pairs states in different symmetry sectors, resulting in identical spectra up to exponentially small finite-size corrections. Such pairing occurs in the Ising/Majorana fermion chain and possibly in parafermionic systems and strongly disordered many-body localized phases. My proof here shows that the strong zero mode occurs in a clean interacting system, and that it possesses some remarkable structure – despite being a rather elaborate operator, it squares to the identity.

Deformations of cell sheets are ubiquitous in early animal development, yet they arise from an intricate interplay of cell shape changes, cell migration, cell intercalation, and cell division. We combine theory and experiment to explore what is perhaps the simplest instance of cell sheet folding, the "inversion" process in the green alga Volvox: at the end of cell division, a Volvox embryo consists of several thousand cells arrayed to form a thin spherical sheet, but those cell poles whence will emanate the flagella point into the sphere. In a process hypothesised to arise from cell shape changes alone, the embryos therefore turn themselves inside out to acquire the ability to swim. We have recently acquired the first three-dimensional time-lapse visualisations of this inversion, using light sheet microscopy to reveal the intriguing dynamics of the process. A theoretical model, in which cell shape changes correspond to local variations of intrinsic curvature and stretches of an elastic shell, sheds light on the underlying mechanics of inversion and reproduces the shapes and dynamics of inversion qualitatively. This is joint work with Stephanie Höhn, Aurelia Honerkamp-Smith, Philipp Khuc Trong and Ray Goldstein.

Neurotransmitter-gated ion channels are complex neuroreceptors located in the membrane of nerve cells that control the rapid transmission of nerve impulses. Their malfunction is linked to serious neurological disorders, including Alzheimer’s disease, and they are major therapeutic targets; in invertebrates they are involved in insecticide resistance. However, we have little idea of how they function at the molecular level due to their complexity and limited experimental information. In particular it is not clear how the binding of a small molecule (the neurotransmitter) triggers a series of events culminating into the opening (gating) of the transmembrane channel: ions can then flood across the cell membrane modifying the cell activity. State-of-the-art and novel computational techniques are therefore crucial to build an accurate picture at the atomic level of the mechanisms that drive the activation of these ion channels, complementing the available experimental data. We have used a range of simulation techniques, including metadynamics (a method for accelerating rare events and sample free energy landscapes), to explore the mechanisms of neurotransmitter binding and a potential molecular switch for channel gating. As prototypical examples, we have focussed on the insect GABA-activated RDL receptor and on the serotonin-activated 5-HT3 receptor.

Systems biology aims to understand the molecular interactions that turn genes on/off, ultimately regulating cellular decisions. These interactions may be described by a mathematical model that is a polynomial dynamical system. Generally these interactions are unknown, leading to multiple models; therefore it is desirable to compare models with experimental data (e.g., steady-state concentrations of proteins). Often model parameter values are unknown, and data is limited (subset of measurable variables, often with noise). An emerging field, `algebraic systems biology', offers algebraic approaches to study problems systems biology. We present an algebro-geometric method for ruling out models with limited information and apply it to a biological system known to dysfunction in many colorectal cancers. We are currently extending the framework to include dynamics (i.e. time course data) using differential algebra elimination and will present preliminary results.

**** POLYGON SEMINAR**** Permutation groups and related algebras have proved to be powerful tools for understanding the counting and correlators of gauge invariant operators in 1-Matrix and multi-matrix models. Mathematical structures such as Belyi maps underlying the mixing of trace structures have been uncovered and finite N effects have been encoded using Young diagram data. These results have found applications in studies of BPS, near-BPS and non-BPS operators in N=4 SYM and quiver gauge theories. I will review some of this work and describe some open problems.

I would like to explain, with emphasis on computational details, how one starts from the refined topological vertex, constructs a topological partition function, and chooses the parameters of the latter to obtain conformal blocks of Virasoro minimal models.

ps. Prof. Foda will give an introduction to topological string theory and minimal models in the first hour or so before launching into his more technical topic.

The conjectured relation between higher spin theories on anti de-Sitter (AdS) spaces and weakly coupled conformal field theories is reviewed. I shall then outline the evidence in favour of a concrete duality of this kind, relating a specific higher spin theory on AdS3 to a family of 2d minimal model CFTs. Finally, I shall explain how this relation fits into the framework of the familiar stringy AdS/CFT correspondence.

I shall explain why there is symmetry enhancement near black hole and brane horizons. I shall also present some applications which include the classification of AdS backgrounds in supergravity.

I will discuss the idea that a black hole in string theory is made up of a large number of microscopic constituents. For a class of black holes and black strings with extended supersymmetry, one has an exact counting formula for the number of states. I will sketch the idea and derivation of some of these formulas. I will then discuss applications to the AdS_2/CFT_1 and the AdS_3/CFT_2 correspondences.