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The Bogomolov-Miyaoka-Yau inequality for minimal compact complex surfaces of general type was proved in 1977 independently by Miyaoka, using methods of algebraic geometry, and by Yau, as an outgrowth of his proof of the Calabi conjectures. In this talk, we outline our program to prove the conjecture that symplectic 4-manifolds with $b^+>1$ obey the Bogomolov-Miyaoka-Yau inequality. Our method uses Morse theory on the gauge theoretic moduli space of non-Abelian monopoles, where the Morse function is a Hamiltonian for a natural circle action and natural two-form.  We shall describe generalizations of Donaldson’s symplectic subspace criterion (1996) from finite to infinite dimensions. These generalized symplectic subspace criteria can be used to show that the natural two-form is non-degenerate and thus an almost symplectic form on the moduli space of non-Abelian monopoles. This talk is based on joint work with Tom Leness and the monographs https://arxiv.org/abs/2010.15789  (to appear in AMS Mathematical Surveys and Monographs), https://arxiv.org/abs/2206.14710 and https://arxiv.org/abs/2410.13809. 

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Arham Farid (MI EDI Officer) will join us to chat about current EDI initiatives and to hear our thoughts about ways EDI can improve in the Maths Institute.

Rare and extreme events are notoriously hard to handle in any complex stochastic system: They are simultaneously too rare to be reliably observable in experiments or numerics, but at the same time often too impactful to be ignored. Large deviation theory provides a classical way of dealing with events of extremely small probability, but generally only yields the exponential tail scaling of rare event probabilities. In this talk, I will discuss theory, and algorithms based upon it, that improve on this limitation, yielding sharp quantitative estimates of rare event probabilities from a single computation and without fitting parameters. Notably, these estimates require the computation of determinants of differential operators, which in relevant cases are not traceclass and require appropriate renormalization. We demonstrate that the Carleman--Fredholm operator determinant is the correct choice. Throughout, I will demonstrate the applicability of these methods to high-dimensional real-world systems, for example coming from atmosphere and ocean dynamics.

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Everything you have been taught about Turing patterns is wrong! (Well, not everything, but qualifying statements tend to weaken a punchy first sentence). Turing patterns are universally used to generate and understand patterns across a wide range of biological phenomena. They are wonderful to work with from a theoretical, simulation and application point of view. However, they have a paradoxical problem of being too easy to produce generally, whilst simultaneously being heavily dependent on the details. In this talk I demonstrate how to fix known problems such as small parameter regions and sensitivity, but then highlight a new set of issues that arise from usually overlooked issues, such as boundary conditions, initial conditions, and domain shape. Although we’ve been exploring Turing’s theory for longer than I’ve been alive, there’s still life in the old (spotty) dog yet.

Dr Steph Foulds will talk about; 'Lazy Quantum Walks with Native Multiqubit Gates'

Quantum walks, the quantum analogue to the classical random walk, have been shown to deliver the Dirac equation in the continuum limit. Recent work has shown that 'lazy', open quantum walks can be mapped to computational methods for fluid simulation such as lattice Boltzmann method, quantum fluid dynamics, and smoothed-particle hydrodynamics. This work concerns evaluating the ability of near-term hardware to perform small, proof-of-concept quantum walks - but crucially with the inclusion of a rest state to encompass 'lazy' quantum walks, providing an integral step towards quantum walks for fluid simulation.

Neutral atom hardware is a promising choice of platform for implementing quantum walks due to its ability to implement native multiqubit gates and to dynamically re-arrange qubits. Using detail realistic modelling for near-term multiqubit Rydberg gates via two-photon adiabatic rapid passage, SPAM, and passive error, we present the gate sequences and final state fidelities for quantum walks with and without a rest state on 4 to 16-node rings. This, along with results of an error model with improved two- and three-qubit gate fidelities, leads us to conclude that a native four-qubit gate is required for the near-term implementation of interesting quantum walks on neutral atom hardware.

Please note; this talk is hosted by Rutherford Appleton Laboratory, Harwell Campus, Didcot, OX11 0QX

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Speaker Prof Dr Maria Lukacova will talk about 'Numerical analysis of oscillatory solutions of compressible flows'

Oscillatory solutions of compressible flows arise in many practical situations.  An iconic example is the Kelvin-Helmholtz problem, where standard numerical methods yield oscillatory solutions. In such a situation,  standard tools of numerical analysis for partial differential equations are not applicable. 

We will show that structure-preserving numerical methods converge in general to generalised solutions, the so-called dissipative solutions. 
The latter describes the limits of oscillatory sequences. We will concentrate on the inviscid flows, the Euler equations of gas dynamics, and mention also the relevant results obtained for the viscous compressible flows, governed by the Navier-Stokes equations.

We discuss a concept of K-convergence that turns a weak convergence of numerical solutions into the strong convergence of
their empirical means to a dissipative solution. The latter satisfies a weak formulation of the Euler equations modulo the Reynolds turbulent stress.  We will also discuss suitable selection criteria to recover well-posedness of the Euler equations of gas dynamics. Theoretical results will be illustrated by a series of numerical simulations.  

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Professor Luis Nunes Vicente will talk about 'Reducing Sample Complexity in Stochastic Derivative-Free Optimization via Tail Bounds and Hypothesis Testing';

We introduce and analyze new probabilistic strategies for enforcing sufficient decrease conditions in stochastic derivative-free optimization, with the goal of reducing sample complexity and simplifying convergence analysis. First, we develop a new tail bound condition imposed on the estimated reduction in function value, which permits flexible selection of the power used in the sufficient decrease test, q in (1,2]. This approach allows us to reduce the number of samples per iteration from the standard O(delta^{−4}) to O(delta^{-2q}), assuming that the noise moment of order q/(q-1) is bounded. Second, we formulate the sufficient decrease condition as a sequential hypothesis testing problem, in which the algorithm adaptively collects samples until the evidence suffices to accept or reject a candidate step. This test provides statistical guarantees on decision errors and can further reduce the required sample size, particularly in the Gaussian noise setting, where it can approach O(delta^{−2-r}) when the decrease is of the order of delta^r. We incorporate both techniques into stochastic direct-search and trust-region methods for potentially non-smooth, noisy objective functions, and establish their global convergence rates and properties. 

This is joint work with Anjie Ding, Francesco Rinaldi, and Damiano Zeffiro.

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Dr Lindon Roberts will talk about: 'Optimization Algorithms for Bilevel Learning with Applications to Imaging'

Many imaging problems, such as denoising or inpainting, can be expressed as variational regularization problems. These are optimization problems for which many suitable algorithms exist. We consider the problem of learning suitable regularizers for imaging problems from example (training) data, which can be formulated as a large-scale bilevel optimization problem. 

In this talk, I will introduce new deterministic and stochastic algorithms for bilevel optimization, which require no or minimal hyperparameter tuning while retaining convergence guarantees. 

This is joint work with Mohammad Sadegh Salehi and Matthias Ehrhardt (University of Bath), and Subhadip Mukherjee (IIT Kharagpur).