Edinburgh University

(EFC-DGAs) lead to an algebraic approach to derived analytic geometry, pioneered for more general Fermat theories by Carchedi and Roytenberg.
 
They are well-suited to modelling finite-dimensional analytic spaces, and classical theorems in analysis ensure they give a largely equivalent theory to Lurie's more involved approach via pregeometries. DG dagger affinoid spaces provide a well-behaved class of geometric building blocks whose homotopy theory is governed by the underlying EFC-DGAs. 

Time permitting, I might also say a little about non-commutative generalisations.
 

Optimization problems subject to PDE constraints constitute a mathematical tool that can be applied to a wide range of scientific processes, including fluid flow control, medical imaging, option pricing, biological and chemical processes, and electromagnetic inverse problems, to name a few. These problems involve minimizing some function arising from a particular physical objective, while at the same time obeying a system of PDEs which describe the process. It is necessary to obtain accurate solutions to such problems within a reasonable CPU time, in particular for time-dependent problems, for which the “all-at-once” solution can lead to extremely large linear systems.

In this talk we consider iterative methods, in particular Krylov subspace methods, to solve such systems, accelerated by fast and robust preconditioning strategies. In particular, we will survey several new developments, including block preconditioners for fluid flow control problems, a circulant preconditioning framework for solving certain optimization problems constrained by fractional differential equations, and multiple saddle-point preconditioners for block tridiagonal linear systems. We will illustrate the benefit of using these new approaches through a range of numerical experiments.

This talk is based on work with Santolo Leveque (Scuola Normale Superiore, Pisa), Spyros Pougkakiotis (Yale University), Jacek Gondzio (University of Edinburgh), and Andreas Potschka (TU Clausthal).

Supersymmetric localization allows one to reduce the computation of the partition function of a supersymmetric theory to a finite-dimensional integral, but the space over which one integrates is often singular. In this talk I will explain how one can use shifted symplectic geometry to get rigorous definitions of partition functions and state spaces in theories with extended supersymmetry. For instance, this gives a field-theoretic origin of DT invariants of CY4 manifolds. This is a report on joint work with Brian Williams.

Recent years have seen many new ideas appearing in the solution theories of singular stochastic partial differential equations. An exciting application of SPDEs that is beginning to emerge is to the construction and analysis of quantum field theories. In this talk, I will describe how stochastic quantisation of Parisi–Wu can be used to study QFTs, especially those arising from gauge theories, the rigorous construction of which, even in low dimensions, is largely open.

The classification of anti de Sitter black holes is an open problem of central importance in holography. In this talk, I will present new advances in classification of supersymmetric solutions to five-dimensional minimal gauged supergravity. In particular, we prove a black hole uniqueness theorem within a ‘Calabi-type’ subclass of solutions with biaxial symmetry. This subclass includes all currently known black hole solutions within this theory.

Khovanov-Rozansky homology is a link invariant that categorifies the HOMFLY-PT polynomial. I will describe a geometric model for this invariant, living in the monodromic Hecke category. I will also explain how it allows to identify objects representing graded pieces of Khovanov-Rozansky homology, using a remarkable family of character sheaves. Based on joint works with Roman Bezrukavnikov.

I will discuss the use of partitioned schemes for neural networks. This work is in the tradition of multrate numerical ODE methods in which different components of system are evolved using different numerical methods or with different timesteps. The setting is the training tasks in deep learning in which parameters of a hierarchical model must be found to describe a given data set. By choosing appropriate partitionings of the parameters some redundant computation can be avoided and we can obtain substantial computational speed-up. I will demonstrate the use of the procedure in transfer learning applications from image analysis and natural language processing, showing a reduction of around 50% in training time, without impairing the generalization performance of the resulting models. This talk describes joint work with Tiffany Vlaar.

In many machine learning tasks, it is crucial to extract low-dimensional and descriptive features from a data set. In this talk, I present a method to extract features from multi-dimensional space-time signals which is motivated, on the one hand, by the success of path signatures in machine learning, and on the other hand, by the success of models from the theory of regularity structures in the analysis of PDEs. I will present a flexible definition of a model feature vector along with numerical experiments in which we combine these features with basic supervised linear regression to predict solutions to parabolic and dispersive PDEs with a given forcing and boundary conditions. Interestingly, in the dispersive case, the prediction power relies heavily on whether the boundary conditions are appropriately included in the model. The talk is based on the following joint work with Andris Gerasimovics and Hendrik Weber: https://arxiv.org/abs/2108.05879

 In this talk, we will present the tools of regularity structures to deal with singular stochastic PDEs that involve non-translation invariant differential operators. We describe in particular the renormalized equation for a very large class of spacetime dependent renormalization schemes. Our approach bypasses the previous approaches in the translation-invariant setting. This is joint work with Ismael Bailleul.