QMUL

The line operators in a 2+1D TQFT form an algebraic structure called a modular tensor category (MTC). There is a natural action of a Galois group on MTCs which maps a given TQFT to other 'Galois conjugate' TQFTs. I will describe this Galois action and give several examples of Galois conjugate TQFTs. Galois action on a unitary TQFT can result in a non-unitary TQFT. I will derive a sufficient condition under which unitarity is preserved. Finally, I will describe the invariance of 0-form and 1-form symmetries of TQFTs under Galois action.    

Synchronization is a collective phenomenon that pervades the natural systems from neurons to fireflies. In a network, synchronization of the dynamical variables associated to the nodes occurs when nodes are coupled to their neighbours as captured by the Kuramoto model. However many complex systems include also higher-order interactions among more than two nodes and sustain dynamical signals that might be related to higher-order simplices such as nodes of triangles. These dynamical topological signals include for instance fluxes which are dynamical variables associated to links.

In this talk I present a new topological approach [1] to synchronization on simplicial complexes. Here the theory of synchronization is combined with topology (specifically Hodge theory) for formulating the higher-order Kuramoto model that uses the higher-order Laplacians and provides the main synchronization route for topological signals. I will show that the dynamics defined on links can be projected to a dynamics defined on nodes and triangles that undergo a synchronization transition and I will discuss how this procedure can be immediately generalized for topological signals of higher dimension. Interestingly I will show that when the model includes an adaptive coupling of the two projected dynamics, the transition becomes explosive, i.e. synchronization emerges abruptly.

This model can be applied to study synchronization of topological signals in the brain and in biological transport networks as it proposes a new set of topological transformations that can reveal collective synchronization phenomena that could go unnoticed otherwise.

[1] Millán, A.P., Torres, J.J. and Bianconi, G., 2019. Explosive higher-order Kuramoto dynamics on simplicial complexes. Physical Review Letters (in press) arXiv preprint arXiv:1912.04405.

I will talk about the implications of UV completion of quantum gravity on the low energy spectrums. I will introduce the constraints on low-energy effective theory due to unitarity and analyticity of scattering amplitudes, in particular an infinite set of new unitarity constraints on the forward-limit limit of four-point scattering amplitudes due to the work of Arkani-Hamed et al. In three dimensions, we find the constraints imply that light states with charge-to-mass ratio z greater than 1 can only be consistent if there exists other light states, preferably neutral. Applied to the 3D Standard Model like spectrum, where the low energy couplings are dominated by the electron with z \sim 10^22, this provides a novel understanding of the need for light neutrinos.

 I will describe recent progress in the study of scattering amplitudes in gauge theory and gravity at loop level, using the formalism of the scattering equations. The scattering equations relate the kinematics of the scattering of massless particles to the moduli space of the sphere. Underpinned by ambitwistor string theory, this formalism provides new insights into the relation between tree-level and loop-level contributions to scattering amplitudes. In this talk, I will describe results up to two loops on how loop integrands can be constructed as forward-limits of trees. One application is the loop-level understanding of the colour-kinematics duality, a symmetry of perturbative gauge theory which relates it to perturbative gravity.

Non-abelian gauge theories underly particle physics, including collision processes at particle accelerators. Recently, quantum scattering probabilities in gauge theories have been shown to be closely related to their counterparts in gravity theories, by the so-called double copy. This suggests a deep relationship between two very different areas of physics, and may lead to new insights into quantum gravity, as well as novel computational methods. This talk will review the double copy for amplitudes, before discussing how it may be extended to describe exact classical solutions such as black holes. Finally, I will discuss hints that the double copy may extend beyond perturbation theory. 

In this talk, we consider the question of exact (boundary) controllability of wave equations: whether one can steer their solutions from any initial state to any final state using appropriate boundary data. In particular, we discuss new and fully general results for linear wave equations on time-dependent domains with moving boundaries. We also discuss the novel geometric Carleman estimates that are the main tools for proving these controllability results

I will review the concept of duality in quantum systems from the 2D Ising model to superconformal field theories in higher dimensions. Using some of these latter theories, I will explain how a generalized concept of duality emerges: these are dualities not between full theories but between algebraically well-defined sub-sectors of strikingly different theories.

NOTE CHANGE OF TIME AND PLACE

It is known by results of Macintyre and Chatzidakis-Hrushovski that the theory ACFA of existentially closed difference fields is decidable. By developing techniques of difference algebraic geometry, we view quantifier elimination as an instance of a direct image theorem for Galois formulae on difference schemes. In a context where we restrict ourselves to directly presented difference schemes whose definition only involves algebraic correspondences, we develop a coarser yet effective procedure, resulting in a primitive recursive quantifier elimination. We shall discuss various algebraic applications of Galois stratification and connections to fields with Frobenius.

The study of difference algebraic geometry stems from the efforts of Macintyre and Hrushovski to

count the number of solutions to difference polynomial equations over fields with powers of Frobenius.

We propose a notion of multiplicity in the context of difference algebraic schemes and prove a first principle

of preservation of multiplicity. We shall also discuss how to formulate a suitable intersection theory of difference schemes.