University of Nottingham

In this talk I will provide an overview of our current research at the interface of quantum field theory (QFT), Lorentzian geometry and higher categorical structures. I will present operads which encode the rich algebraic structure of QFTs on Lorentzian manifolds and show that in low dimensions their algebras relate to familiar algebraic structures. Our operads share certain similarities with the little disk operads from topology, in particular they involve a homotopical localization at geometric embeddings related to ‘time evolution’. I will show that, in contrast to the topological context, this homotopical localization can be strictified in many important classes of examples, which is loosely speaking due to the 1-dimensional nature of time evolution in Lorentzian geometry. I will conclude by explaining how simple examples of such Lorentzian QFTs can be constructed from a homotopical generalization of the concept of Green’s operators for hyperbolic partial differential equations, which we call Green hyperbolic complexes. Throughout this talk, I will frequently comment on the similarities and differences between our approach, factorization algebras and functorial field theories.

I will present recent results on the growth of entanglement between two adjacent regions in a tripartite, one-dimensional many-body system after a quantum quench. Combining a replica trick with a space-time duality transformation a universal relation between the entanglement negativity and Renyi-1/2 mutual information can be derived, which holds at times shorter than the sizes of all subsystems. The proof is directly applicable to any local quantum circuit, i.e., any lattice system in discrete time characterised by local interactions, irrespective of the nature of its dynamics. The derivation indicates that such a relation can be directly extended to any system where information spreads with a finite maximal velocity. The talk is based on Phys. Rev. Lett. 129, 140503 (2022).

The translation method for constructing quasiconvex lower bound of a given function in the calculus of variations and the notion of compensated convex transforms for tightly approximate functions in Euclidean spaces will be briefly reviewed. By applying the upper compensated convex transform to the finite maximum function we will construct computable quasiconvex functions with finitely many point wells contained in a subspace with rank-one matrices. The complexity for evaluating the constructed quasiconvex functions is O(k log k) with k the number of wells involved. If time allows, some new applications of compensated convexity will be briefly discussed.

Progress in developing a consistent theory that describes physical phenomena
at scales where quantum and general relativistic effects are large is
hindered by the lack of experiments. In this talk, we present a proposal
that would overcome this experimental obstacle by using a Bose-Einstein
condensate (BEC) to test for possible conflicts between quantum theory and
general relativity. Recent developments in large BEC systems allows us to
verify if gravitationally-induced wave function collapse occurs at the
timescales predicted by Roger Penrose. BECs with high particle numbers
(N>10^9) can also be used to demonstrate quantum field theory in curved
spacetime by observing how changes in the spacetime affect the phononic
quantum field of a BEC. These effects will enable the development of a new
generation of instruments that will be able to probe scales where new
physics might emerge, with applications including gravitational wave
detectors, gravimeters, gradiometers and dark energy probes.

Algebraic quantum field theories (AQFTs) are traditionally described as functors that assign algebras (of observables) to spacetime regions. These functors are required to satisfy a list of physically motivated axioms such as commutativity of the multiplication for spacelike separated regions. In this talk we will show that AQFTs can be described as algebras over a colored operad. This operad turns out to be interesting as it describes an interpolation between non-commutative and commutative algebraic structures. We analyze our operad from a homotopy theoretical perspective and determine a suitable resolution that describes the commutative behavior up to coherent homotopies. We present two concrete constructions of toy-models of algebras over the resolved operad in terms of (i) forming cochains on diagrams of simplicial sets (or stacks) and (ii) orbifoldization of equivariant AQFTs.

In laboratories around the world, scientists use magnetic stirrers to mix solutions and dissolve powders. It is well known that at high drive rates the stir bar jumps around erratically with poor mixing, leading to its nick-name 'flea'. Investigating this behaviour, we discovered a state in which the flea levitates stably above the base of the vessel, supported by magnetic repulsion between flea and drive magnet. The vertical motion is oscillatory and the angular motion a superposition of rotation and oscillation. By solving the coupled vertical and angular equations of motion, we characterised the flea’s behaviour in terms of two dimensionless quantities: (i) the normalized drive speed and (ii) the ratio of magnetic to viscous forces. However, Earnshaw’s theorem states that levitation via any arrangement of static magnets is only possible with additional stabilising forces. In our system, we find that these forces arise from the flea’s oscillations which pump fluid radially outwards, and are only present for a narrow range of Reynold's numbers. At slower, creeping flow speeds, only viscous forces are present, whereas at higher speeds, the flow reverses direction and the flea is no longer stable. We also use both the levitating and non-levitating states to measure rheological properties of the system.

In recent years, ideas from the world of partial differential equations (PDEs) have found their way into the arena of graph and network problems. In this talk I will discuss how techniques based on nonlinear PDE models, such as the Allen-Cahn equation and the Merriman-Bence-Osher threshold dynamics scheme can be used to (approximately) detect particular structures in graphs, such as densely connected subgraphs (clustering and classification, minimum cuts) and bipartite subgraphs (maximum cuts). Such techniques not only often lead to fast algorithms that can be applied to large networks, but also pose interesting theoretical questions about the relationships between the graph models and their continuum counterparts, and about connections between the different graph models.

The Ginzburg-Landau functional serves as a model for the formation of vortices in many physical contexts. The natural gradient flow, the parabolic Ginzburg-Landau equation, converges in the limit of small vortex size and finite number of vortices to a system of ODEs. Passing to the limit of many vortices in this ODE, one can derive a mean field PDE, similar to the passage from point vortex systems to the 2D Euler equations. In the talk, I will present quantitative estimates that allow us to directly connect the parabolic GL equation to the limiting mean field PDE. In contrast to recent work by Serfaty, our work is restricted to a fairly low number of vortices, but can handle vortex sheet initial data in bounded domains. This is joint work with Daniel Spirn (University of Minnesota).

I will talk in down to earth terms about several main features of this theory.

Neural field models describe the coarse-grained activity of populations of

interacting neurons. Because of the laminar structure of real cortical

tissue they are often studied in two spatial dimensions, where they are well

known to generate rich patterns of spatiotemporal activity. Such patterns

have been interpreted in a variety of contexts ranging from the

understanding of visual hallucinations to the generation of

electroencephalographic signals. Typical patterns include localised

solutions in the form of travelling spots, as well as intricate labyrinthine

structures. These patterns are naturally defined by the interface between

low and high states of neural activity. Here we derive the equations of

motion for such interfaces and show, for a Heaviside firing rate, that the

normal velocity of an interface is given in terms of a non-local Biot-Savart

type interaction over the boundaries of the high activity regions. This

exact, but dimensionally reduced, system of equations is solved numerically

and shown to be in excellent agreement with the full nonlinear integral

equation defining the neural field. We develop a linear stability analysis

for the interface dynamics that allows us to understand the mechanisms of

pattern formation that arise from instabilities of spots, rings, stripes and

fronts. We further show how to analyse neural field models with

linear adaptation currents, and determine the conditions for the dynamic

instability of spots that can give rise to breathers and travelling waves.

We end with a discussion of amplitude equations for analysing behaviour in

the vicinity of a bifurcation point (for smooth firing rates). The condition

for a drift instability is derived and a center manifold reduction is used

to describe a slowly moving spot in the vicinity of this bifurcation. This

analysis is extended to cover the case of two slowly moving spots, and

establishes that these will reflect from each other in a head-on collision.