University of Cambridge

Given a group G acting on a CAT(0) polygonal complex, X, it is natural to ask whether the structure of X allows us to deduce properties of G. We discuss some recent work on local properties that X may possess which allow us to answer these questions - in many cases we can in fact deduce that the group is a linear group over Z.

Computing spectral properties of operators is fundamental in the sciences, with applications in quantum mechanics, signal processing, fluid mechanics, dynamical systems, etc. However, the infinite-dimensional problem is infamously difficult (common difficulties include spectral pollution and dealing with continuous spectra). This talk introduces classes of practical resolvent-based algorithms that rigorously compute a zoo of spectral properties of operators on Hilbert spaces. We also discuss how these methods form part of a broader programme on the foundations of computation. The focus will be computing spectra with error control and spectral measures, for general discrete and differential operators. Analogous to eigenvalues and eigenvectors, these objects “diagonalise” operators in infinite dimensions through the spectral theorem. The first is computed by an algorithm that approximates resolvent norms. The second is computed by building convolutions of appropriate rational functions with the measure via the resolvent operator (solving shifted linear systems). The final part of the talk provides purely data-driven algorithms that compute the spectral properties of Koopman operators, with convergence guarantees, from snapshot data. Koopman operators “linearise” nonlinear dynamical systems, the price being a reduction to an infinite-dimensional spectral problem (c.f. “Koopmania”, describing their surge in popularity). The talk will end with applications of these new methods in several thousand state-space dimensions.

Many scientific questions in biology can be formulated as a direct problem:

given a biochemical system, can one deduce some of its properties? 

For example, one might be interested in deducing equilibria of a given intracellular network.  On the other hand, one might instead be interested in designing an intracellular network with specified equilibria. Such scientific tasks take the form of inverse problems:
given a property, can one design a biochemical system that displays this property? 

Given a biochemical system, can one embed additional molecular species and reactions into the original system to control some of its properties?
These questions are at the heart of the emerging field of synthetic biology, where it has recently become possible to systematically realize dynamical systems using molecules.  Furthermore, addressing these questions for man-made synthetic systems may also shed light on how evolution has overcome similar challenges for natural systems.  In this talk, I will focus on the inverse problems, and outline some of the results and challenges which are important when biochemical systems are designed and controlled.

The Temperley-Lieb algebra is a diagrammatic algebra - defined on a basis of "string diagrams" with multiplication given by "joining the diagrams together".  It first arose as an algebra of operators in statistical mechanics but quickly found application in knot theory (where Jones used it to define his famed polynomial) and the representation theory of $sl_2$.  From the outset, the representation theory of the Temperley-Lieb algebra itself has been of interest from a physics viewpoint and in characteristic zero it is well understood.  In this talk we will explore the representation theory over mixed characteristic (i.e. over positive characteristic fields and specialised at a root of unity).  This gentle introduction will take the listener through the beautifully fractal-like structure of the algebras and their cell modules with plenty of examples.

Chiral fermion anomalies in any spacetime dimension are computed by evaluating an eta-invariant on a closed manifold in one higher dimension. The APS index theorem then implies that both local and global gauge anomalies are detected by bordism invariants, each being classified by certain abelian groups that I will identify. Mathematically, these groups are connected via a short exact sequence that splits non-canonically. This enables one to relate global anomalies in one gauge theory to local anomalies in another, by which we revive (from the bordism perspective) an old idea of Elitzur and Nair for deriving global anomalies. As an example, I will show how the SU(2) anomaly in 4d can be derived from a local anomaly by embedding SU(2) in U(2).

The Drinfeld half-plane is a rigid analytic variety over a p-adic field. In this talk, I will give an overview of the geometric aspects of this space and describe its connection with representation theory.

In this talk, I will present a series of new experimental data, supported by theoretical models, of the transport of ash, aerosols and bubbles in multiphase plumes rising through stratified environments, focussing on the structure of flow and the dispersal of the different phases. The models have relevance for the dispersal of volcanic ash in the atmosphere and ocean, the mixing of aerosols in buildings, and the fate of suspended sediment produced during deep sea mining. 

All five-dimensional non-abelian gauge theories have a U(1)U(1)I​U(1) global symmetry associated with instantonic particles. I will describe a mixed ’t Hooft anomaly between this and other global symmetries of  the theory, namely the one-form center symmetry or ordinary flavor symmetry for theories with fundamental matter. I will explore some general dynamical properties of the candidate phases implied by the anomaly, and apply our results to supersymmetric gauge theories in five dimensions, analysing the symmetry enhancement patterns occurring at their conjectured RG fixed points.

After recalling some motivation for studying highly-connected graphs in the context of operator algebras and large-scale geometry, we will introduce the notion of "asymptotic expansion" recently defined by Li, Nowak, Spakula and Zhang. We will explore some applications of this definition, hopefully culminating in joint work with Li, Vigolo and Zhang.

Pattern formation emerges during development from the interplay between gene regulatory networks (GRNs) acting at the single cell level and cell movements driving tissue level morphogenetic changes. As a result, the timing of cell specification and the dynamics of morphogenesis must be tightly cross-regulated. In the developing zebrafish, mesoderm progenitors will spend varying amounts of time (from 5 to 10hrs) in the tailbud before entering the pre-somitic mesoderm (PSM) and initiating a stereotypical transcriptional trajectory towards a mesodermal fate. In contrast, when dissociated and placed in vitro, these progenitors differentiate synchronously in around 5 hours. We have used a data-driven mathematical modelling approach to reverse-engineer a GRN that is able to tune the timing of mesodermal differentiation as progenitors leave the tailbud’s signalling environment, which also explains our in vitro observations. This GRN recapitulates pattern formation at the tissue level when modelled on cell tracks obtained from live-imaging a developing PSM. Our “live-modelling” framework also allows us to simulate how perturbations to the GRN affect the emergence of pattern in zebrafish mutants. We are now extending this analysis to cichlid fishes in order to explore the regulation of developmental time in evolution.