Oxford University

In this seminar I will begin by giving an overview of some problems in stochastic simulation and uncertainty quantification. I will then outline the Multilevel Monte Carlo for situations in which accurate simulations are very costly, but it is possible to perform much cheaper, less accurate simulations.  Inspired by the multigrid method, it is possible to use a combination of these to achieve the desired overall accuracy at a much lower cost.

AI tools like ChatGPT, Microsoft Copilot, GitHub Copilot, Claude and even older AI-enabled tools like Grammarly and MS Word, are becoming an everyday part of our research environment.  This last-minute opening up of a seminar slot due to the unfortunate illness of our intended speaker (who will hopefully re-schedule for next term) gives us an opportunity to discuss what this means for us as researchers; what are good helpful uses of AI, and are there uses of AI which we might view as inappropriate?  Please come ready to participate with examples of things which you have done yourselves with AI tools.

Quantum computers are becoming a reality and current generations of machines are already well beyond the 50-qubit frontier. However, hardware imperfections still overwhelm these devices and it is generally believed the fault-tolerant, error-corrected systems will not be within reach in the near term: a single logical qubit needs to be encoded into potentially thousands of physical qubits which is prohibitive.

Due to limited resources, in the near term, hybrid quantum-classical protocols are the most promising candidates for achieving early quantum advantage and these need to resort to quantum error mitigation techniques. I will explain the basic concepts and introduce hybrid quantum-classical protocols are the most promising candidates for achieving early quantum advantage. These have the potential to solve real-world problems---including optimisation or ground-state search---but they suffer from a large number of circuit repetitions required to extract information from the quantum state. I will finally identify the most likely areas where quantum computers may deliver a true advantage in the near term.

Bálint Koczor

Associate Professor in Quantum Information Theory

Mathematical Institute, University of Oxford

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In this talk we explore structural results about sets with small doubling in k dimensions. We start in the continuous world with a sharp stability result for the Brunn-Minkowski inequality conjectured by Figalli and Jerison and work our way to the discrete world, where we discuss the natural extension: we show that non-degenerate sets in Z^k with doubling close to 2^k are close to convex progressions i.e. convex sets intersected with a sub-lattice. This talk is based on joint work with Peter van Hintum and Hunter Spink.

Detectors in collider experiments are modeled by light-ray operators in Quantum Field Theory. For example, energy detectors are certain null integrals of the stress-energy tensor, localized at an angle on the celestial sphere, where they collect quanta that escape in their direction. In this talk, I will discuss a series of work developing a nonperturbative, convergent operator product expansion (OPE) for light-ray operators in Conformal Field Theories (CFTs). Objects appearing in the expansion are more general light-ray operators, whose matrix elements can be computed by the generalized Lorentzian inversion formula. An important application is to event shapes in collider physics, which correspond to correlation functions of light-ray operators within the state created by the incoming particles. I will discuss some applications of the light-ray OPE in CFT, and mention some extensions to QCD which make contact with measurements at the LHC. Talk based primarily on [1905.01311] and [2010.04726].

We consider close-packed tiling models of geometric objects -- a mixture of hardcore dimers and plaquettes -- as a generalisation of the familiar dimer models. Specifically, on an anisotropic cubic lattice, we demand that each site be covered by either a dimer on a z-link or a plaquettein the x-y plane. The space of such fully packed tilings has an extensive degeneracy. This maps onto a fracton-type `higher-rank electrostatics', which can exhibit a plaquette-dimer liquid and an ordered phase. We analyse this theory in detail, using height representations and T-duality to demonstrate that the concomitant phase transition occurs due to the proliferation of dipoles formed by defect pairs. The resultant critical theory can be considered as a fracton version of the Kosterlitz-Thouless transition. A significant new element is its UV-IR mixing, where the low energy behavior of the liquid phase and the transition out of it is dominated by local (short-wavelength) fluctuations, rendering the critical phenomenon beyond the renormalization group paradigm.

The integral of mean curvature squared is a conformal invariant of surfaces reintroduced by Willmore in 1965 whose study exercised a tremendous influence on geometric analysis and most notably on minimal surfaces in the last years.

On the other hand, the Loewner energy is a conformal invariant of planar curves introduced by Yilin Wang in 2015 which is notably linked to SLE processes and the Weil-Petersson class of (universal) Teichmüller theory.

In this presentation, after a brief historical introduction, we will discuss some recent developments linking the Willmore energy to the Loewner energy and mention several open problems.

Joint work with Yilin Wang (MIT/MSRI)

I will discuss a string theory way to study G_2 instanton moduli space and explain how to compute the instanton partition function for a certain G_2 manifold. An important insight comes from the twisted M-theory on the G_2 manifold. Building on the example, I will explain a possibility to extend the story to a large set of conjectural G_2 manifolds and a possible connection to 4d N=1 SCFT via geometric engineering. This talk is based on https://arxiv.org/abs/2109.01110 and a series of works in progress with Michele Del Zotto and Yehao Zhou.

I will discuss the regularity of solutions to quasilinear systems satisfying a Legendre-Hadamard ellipticity condition. For such systems it is known that weak solutions may which fail to be C^1 in any neighbourhood, so we cannot expect a general regularity theory. However if we assume an a-priori regularity condition of the solutions we can rule out such counterexamples. Focusing on solutions to Euler-Lagrange systems, I will present an improved regularity results for solutions whose gradient satisfies a suitable BMO / VMO condition. Ideas behind the proof will be presented in the interior case, and global consequences will also be discussed.