L3

We will discuss shock reflection phenomena, mathematical formulation of shock reflection problem, structures of  shock reflection configurations, and von Neumann conjectures on transition between regular and Mach reflections. Then we will describe the results on existence and properties of regular reflection solutions for potential flow equation. The approach is to reduce the shock reflection problem to a free boundary problem for a nonlinear  elliptic equation in self-similar coordinates, where the reflected shock is the free boundary, and ellipticity degenerates near a part of a fixed boundary. We will discuss the techniques and methods used in the study of such free boundary problems.

 

I will discuss the free compact boson CFT thrown out of equilibrium by marginal deformations, modeled by quenching or periodically driving the compactification radius of the free boson between two different values. All the dynamics will be shown to be crucially dependent on the ratio of the compactification radii via the Zamolodchikov distance in the space of marginal deformations. I will present various exact analytic results for the Loschmidt echo and the time evolution of energy density for both the quench and the periodic drive. Finally, I will present a non-perturbative computation of the  Rényi divergence, an information-theoretic distance measure, between two marginally deformed thermal density matrices.

The talk will be based on the recent preprint: arXiv:2206.11287

Can the S-matrix be complexified in a way consistent with causality? Since the 1960's, the affirmative answer to this question has been well-understood for 2→2 scattering of the lightest particle in theories with a mass gap at low momentum transfer, where the S-matrix is analytic everywhere except at normal-threshold branch cuts. We ask whether an analogous picture extends to realistic theories, such as the Standard Model, that include massless fields, UV/IR divergences, and unstable particles. Especially in the presence of light states running in the loops, the traditional iε prescription for approaching physical regions might break down, because causality requirements for the individual Feynman diagrams can be mutually incompatible. We demonstrate that such analyticity problems are not in contradiction with unitarity. Instead, they should be thought of as finite-width effects that disappear in the idealized 2→2 scattering amplitudes with no unstable particles, but might persist at higher multiplicity. To fix these issues, we propose an iε-like prescription for deforming branch cuts in the space of Mandelstam invariants without modifying the analytic properties. This procedure results in a complex strip around the real part of the kinematic space, where the S-matrix remains causal. To help with the investigation of related questions, we introduce holomorphic cutting rules, new approaches to dispersion relations, as well as formulae for local behavior of Feynman integrals near branch points, all of which are illustrated on explicit examples.

In space dimension 2 we present a finite element complex for the deformation operator acting on vectorfields and the linearized curvature operator acting on symmetric 2 by 2 matrices. We also present the tools that were used in the construction, namely the BGG diagram chase and the framework of finite element systems. For this general framework we can prove a de Rham theorem on cohomology groups in the flat case and a Bianchi identity in the case with curvature.

Dr Oliver Sheridan-Methven from Citadel Securities, (an InFoMM and MScMCF alumni), will be talking about his experiences from studying at the Mathematical Institute, interviewing for jobs, to working in finance. Now in Zurich, Oliver is a quantitative developer in the advanced scientific computing team at Citadel Securities, a world leading market maker. Citadel Securities specialises in ultra high frequency trading, low latency execution, and their researchers tackle cutting edge machine learning and data science problems on colossal data sets with humongous computational resources. Oliver will be talking about his own experiences, and also how mathematicians are naturally great fits for a huge number of roles at Citadel Securities.

Hilbert complexes are chains of spaces linked by operators, with properties that are crucial to establishing the well-posedness of certain systems of partial differential equations. Designing stable numerical schemes for such systems, without resorting to nonphysical stabilisation processes, requires reproducing the complex properties at the discrete level. Finite-element complexes have been extensively developed since the late 2000's, in particular by Arnold, Falk, Winther and collaborators. These are however limited to certain types of meshes (mostly, tetrahedral and hexahedral meshes), which limits options for, e.g., local mesh refinement.

In this talk we will introduce the Discrete De Rham complex, a discrete version of one of the most popular complexes of differential operators (involving the gradient, curl and divergence), that can be applied on meshes consisting of generic polytopes. We will use a simple magnetostatic model to motivate the need for (continuous and discrete) complexes, then give a presentation of the lowest-order version of the complex and sketch its links with the CW cochain complex on the mesh. We will then briefly explain how this lowest-order version is naturally extended to an arbitrary-order version, and briefly present the associated properties (Poincaré inequalities, primal and adjoint consistency, commutation properties, etc.) that enable the analysis of schemes based on this complex.

Tensor structured linear operators play an important role in matrix equations and low-rank modelling. Motivated by this we consider the problem of approximating a matrix by a sum of Kronecker products. It is known that an optimal approximation in Frobenius norm can be obtained from the singular value decomposition of a rearranged matrix, but when the goal is to approximate the matrix as a linear map, an operator norm would be a more appropriate error measure. We present an alternating optimization approach for the corresponding approximation problem in spectral norm that is based on semidefinite programming, and report on its practical performance for small examples.
This is joint work with Venkat Chandrasekaran and Mareike Dressler.

That the linear systems arising upon the discretization of elliptic PDEs can be solved efficiently is well-known, and iterative solvers that often attain linear complexity (multigrid, Krylov methods, etc) have proven very successful. Interestingly, it has recently been demonstrated that it is often possible to directly compute an approximate inverse to the coefficient matrix in linear (or close to linear) time. The talk will argue that such direct solvers have several compelling qualities, including improved stability and robustness, the ability to solve certain problems that have remained intractable to iterative methods, and dramatic improvements in speed in certain environments.

After a general introduction to the field, particular attention will be paid to a set of recently developed randomized algorithms that construct data sparse representations of large dense matrices that arise in scientific computations. These algorithms are entirely black box, and interact with the linear operator to be compressed only via the matrix-vector multiplication.

 We develop a probabilistic framework for analysing model-based reinforcement learning in the episodic setting. We then apply it to study finite-time horizon stochastic control problems with linear dynamics but unknown coefficients and convex, but possibly irregular, objective function. Using probabilistic representations, we study regularity of the associated cost functions and establish precise estimates for the performance gap between applying optimal feedback control derived from estimated and true model parameters. We identify conditions under which this performance gap is quadratic, improving the linear performance gap in recent work [X. Guo, A. Hu, and Y. Zhang, arXiv preprint, arXiv:2104.09311, (2021)], which matches the results obtained for stochastic linear-quadratic problems. Next, we propose a phase-based learning algorithm for which we show how to optimise exploration-exploitation trade-off and achieve sublinear regrets in high probability and expectation. When assumptions needed for the quadratic performance gap hold, the algorithm achieves an order (N‾‾√lnN) high probability regret, in the general case, and an order ((lnN)2) expected regret, in self-exploration case, over N episodes, matching the best possible results from the literature. The analysis requires novel concentration inequalities for correlated continuous-time observations, which we derive.

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Dr Lukasz Szpruch

We investigate the maximum distance of a rank-two tensor to rank-one tensors. An equivalent problem is given by the minimal ratio of spectral and Frobenius norm of a tensor. For matrices the distance of a rank k matrix to a rank r matrices is determined by its singular values, but since there is a lack of a fitting analog of the singular value decomposition for tensors, this question is more difficult in the regime of tensors.