This talk will describe some work on the bouncing of particle-like (“kink”) solutions to a nonlinear wave equation, called the sine-Gordon equation, against a fixed boundary. Away from the boundary, this equation has a property known as integrability, making the scattering of the kinks particularly simple. However, if this integrability is broken at the boundary, then the scattering becomes surprisingly complicated, in ways that will be outlined in the talk with the help of some movies.

This talk addresses the low energy physics of the Sachdev-Ye-Kitaev model, a paradigm of strongly interacting (Majorana) quantum matter. A salient feature of this system is its exceptionally high degree of symmetry under reparameterizations of physical time. At low energies this symmetry is spontaneously broken and the ensuing infinite dimensional Goldstone mode manifold takes strong influence on all physical observables. We will discuss the effects of these fluctuations on the example of the so-called out of time ordered correlation functions, diagnostic tools to describe both manifestations of quantum chaos in the system and its conjectured duality to an AdS2 gravitational bulk. While previous work predicts exponential decay of these correlations in time our main finding is that at large time scales non-perturbative Goldstone mode fluctuations generate a crossover to power law behavior. This phenomenon must have ramifications in the physics of the holographic bulk which, however, we do not understand at present.

A total set of states for which we have no resolution of the identity (a 'pre-basis'), is considered in a finite dimensional Hilbert space. A dressing formalism renormalizes them into density matrices which resolve the identity, and makes them a 'generalized basis', which is practically useful. The dresssing mechanism is inspired by Shapley's methodology in cooperative game theory, and it uses Moebius transforms. There is non-independence and redundancy in these generalized bases, which is quantified with a Shannon type of entropy. Due to this redundancy, calculations based on generalized bases, are sensitive to physical changes and robust in the presence of noise. For example, the representation of an arbitrary vector in such generalized bases, is robust when noise is inserted in the coefficients. Also in a physical system with ground state which changes abruptly at some value of the coupling constant, the proposed methodology detects such changes, even when noise is added to the parameters in the Hamiltonian of the system.

I discuss one and two-parameter solutions of sigma models on symmetric spaces contained in E11. Embedding one-parameter sigma model solutions in space-time give a metric which depends on harmonic functions typical in general relativity, supergravity and M-theory. Embedding two-parameter sigma model solutions in space-time give a metric which depends on general travelling wave functions in M* and M’-theory (theories which have space-time signatures with more than one time). Weyl reflection allows the latter solutions to be mapped to M-theory solutions where the wave functions depend explicitly on extra co-ordinates contained in the fundamental representation of E11. I will also give an example of two-time physics realisable in the laboratory

While algebraic topology is now well established as an applicable branch of mathematics, its emergence in neuroscience is surprisingly recent. In this talk I will present a summary of an ongoing joint project with mathematician and neuroscientists. I will start with some basic facts on neuroscience and the digital reconstruction of a rat’s neocortex by the Blue Brain Project in EPFL. I will then explain how data emerging from this reconstruction can be mapped into abstract graphs that in turn give rise to certain mathematical objects in the realm of algebraic and combinatorial topology. Following a short introduction to some of the basic tools of algebraic topology, I will explain how they can potentially be used in the context of neuroscience. Having set up the scene, I will proceed by presenting the results of an ongoing collaboration with the Blue Brain Project team. In particular I shall demonstrate how the topological techniques give new insights on the behaviour of neural systems and inspire new directions in neuroscience research.

In this talk we consider holographic duals of F-theory solutions to 2d SCFT's. We approach the problem by classifying a particular class of solutions of type IIB supergravity with AdS_3 factors and varying axio-dilaton. The class of solutions we discuss consist of D3 and 7-brane configurations and naturally fall into the realm of F-theory. We prove that for (0,4) supersymmetry in 2d the solutions are essentially unique and we match the holographic central charges to field theory results. We comment on future directions, including AdS_3 solutions of F-theory, preserving different amounts of supersymmetry.

"The Role of Elementary Particle Accelerators" http://www.city.ac.uk/events/2017/may/edwards-lecture-the-role-of-elementary-particle-accelerators (Public lecture and Colloquium)

I will discuss some novel algebraic structures and how they lead to the quantum field theories that arise on the world volumes of 2-branes and 5-branes in M-theory.

I will review recent developments related to the holographic reconstruction of the black hole interior in AdS/CFT and I will discuss the implications for the black hole information paradox.

Quantum fluctuations of the string worldsheet lead to important effects in gauge-string duality. One example is the Lüscher term is the QQ-bar potential in QCD, important for matching the lattice data. I will discuss non-conformal holography of the N=2* theory which provides a controllable setup where the effects of string fluctuations are explicitly calculable in field theory, using localization, even though the problem is intrinsically strongly-coupled. The results obtained by direct quantization of the dual string theory perfectly match the field-theory predictions, after subtle effects like dilaton coupling to the string worldsheet are properly taken into account.

The gauge-gravity correspondence identifies a field theory with a gravitational theory. The gravitational theory is weakly coupled when the field theory has large coupling and vice versa, which mostly prevents matching nontrivial results between the two descriptions. I will discuss cases when the field theory calculation can be reduced to a finite dimensional matrix integral, representing some counting problems. I will then evaluate the integral exactly and reexpand the exact result, which is valid for all coupling, at strong coupling. The resulting expression should match a weak coupling gravitational (or string theoretic) calculation and I’ll comment on what is known from that direction.

I will discuss connections between supersymmetric gauge theories in three dimensions and an exciting development in representation theory known as symplectic duality. I will focus on the simplest example of this phenomenon, which arises from a U(1) gauge theory with N hypermultiplets.

The Quantum Hall Effect is one of the richest phenomena in condensed matter (now featured in no fewer than three Nobel prizes!). I will introduce and explain it, insofar as we understand it. Towards the end, I will say a few words about work to try and rationalise and relate the many models of the effect.

String theory in the tensionless limit is expected to have a large gauge symmetry. By recasting string theory on the AdS background as a generalization of Vasiliev's theory of massless higher-spin fields, it has become possible to understand the nature of this symmetry. In this talk, I will first give an overview of three-dimensional Vasiliev theory and its dual CFT. I will then discuss the current understanding of the symmetry algebra of string theory as well as open problems related to it.

The Riemann hypothesis asserts that the nontrivial zeros of the Riemann zeta function should be of the form 1/2 + i E_n, where the set of numbers {E_n} are real. The so-called Hilbert-Pólya conjecture assumes that {E_n} should correspond to the eigenvalues of an operator that is Hermitian. The discovery of such an operator, if it exists, thus amounts to providing a proof of the Riemann hypothesis. In 1999 Berry and Keating conjectured that such an operator should correspond to a quantisation of the classical Hamiltonian H = xp. Since then, the Berry-Keating conjecture has been investigated intensely in the literature, but its validity has remained elusive up to now. In this talk I will derive a “Hamiltonian” (a differential operator), whose classical counterpart is H = xp, having the property that with a suitable boundary condition on its eigenstates, the eigenvalues {E_n} correspond to the nontrivial zeros of the Riemann zeta function. This Hamiltonian is not Hermitian, but is symmetric under space-time reflection (PT symmetric) in a special way. A formal argument will be given for the construction of the metric operator to define an inner-product space for the eigenstates, and the formally “Hermitian" counterpart Hamiltonian. The talk is based on the work carried out in collaboration with Carl M. Bender (Washington University) and Markus P. Müller (University of Western Ontario).

In recent years, the discovery of integrable sectors of string theory has expanded the range of models which are solvable by the techniques of the Bethe ansatz and by the general theory of quantum groups. In this talk, we will give an overview of the S-matrix theory involved in the solution of these models, with mention to the problem of massless particles emerging in a particular instance of the AdS/CFT correspondence.

We establish a precise correspondence between the ABC Conjecture and N=4 super-Yang-Mills theory. This is achieved by combining three ingredients: (i) Elkies' method of mapping ABC-triples to elliptic curves in his demonstration that ABC implies Mordell/Faltings; (ii) an explicit pair of elliptic curve and associated Belyi map given by Khadjavi-Scharaschkin; and (iii) the fact that the bipartite brane-tiling/dimer model for a gauge theory with toric moduli space is a particular dessin d'enfant in the sense of Grothendieck. We explore this correspondence for the highest quality ABC-triples as well as large samples of random triples. The Conjecture itself is mapped to a statement about the fundamental domain of the toroidal compactification of the string realization of N=4 SYM.

Sleep is core to our ability to function and there is increasing evidence that poor or mis-timed sleep increases our risk of cardio vascular disease, type II diabetes, obesity and cognitive decline. In this talk I will discuss the mechanisms that are believed to underlie sleep wake regulation and review some recent mathematical models. In particular, I will focus on age-related changes to sleep and explain how the mathematical models can be used to bring insight into the possible mechanisms that could cause these changes.