14:30
CDT in Mathematics of Random Systems October Workshop 2022
Abstract
2:30 -3.00 Will Turner (CDT Student, Imperial College London)
Topologies on unparameterised path space
The signature of a path is a non-commutative exponential introduced by K.T. Chen in the 1950s, and appears as a central object in the theory of rough paths developed by T. Lyons in the 1990s. For continuous paths of bounded variation, the signature may be realised as a sequence of iterated integrals, which provides a succinct summary for multimodal, irregularly sampled, time-ordered data. The terms in the signature act as an analogue to monomials for finite dimensional data: linear functionals on the signature uniformly approximate any compactly supported continuous function on unparameterised path space (Levin, Lyons, Ni 2013). Selection of a suitable topology on the space of unparameterised paths is then key to the practical use of this approximation theory. We present new results on the properties of several candidate topologies for this space. If time permits, we will relate these results to two classical models: the fixed-time solution of a controlled differential equation, and the expected signature model of Levin, Lyons, and Ni. This is joint work with Thomas Cass.
3.05 -3.35 Ross Zhang (CDT Student, University of Oxford)
Random vortex dynamics via functional stochastic differential equations
The talk focuses on the representation of the three-dimensional (3D) Navier-Stokes equations by a random vortex system. This new system could give us new numerical schemes to efficiently approximate the 3D incompressible fluid flows by Monte Carlo simulations. Compared with the 2D Navier-Stokes equation, the difficulty of the 3D Navier-Stokes equation lies in the stretching of vorticity. To handle the stretching term, a system of stochastic differential equations is coupled with a functional ordinary differential equation in the 3D random vortex system. Two main tools are developed to derive the new system: the first is the investigation of pinned diffusion measure, which describes the conditional distribution of a time reversal diffusion, and the second is a forward-type Feynman Kac formula for nonlinear PDEs, which utilizes the pinned diffusion measure to delicately overcome the time reversal issue in PDE. Although the main focus of the research is the Navier-stokes equation, the tools developed in this research are quite general. They could be applied to other nonlinear PDEs as well, thereby providing respective numerical schemes.
3.40 - 4.25pm Dr Cris Salvi (Imperial College London)
Signature kernel methods
Kernel methods provide a rich and elegant framework for a variety of learning tasks including supervised learning, hypothesis testing, Bayesian inference, generative modelling and scientific computing. Sequentially ordered information often arrives in the form of complex streams taking values in non-trivial ambient spaces (e.g. a video is a sequence of images). In these situations, the design of appropriate kernels is a notably challenging task. In this talk, I will outline how rough path theory, a modern mathematical framework for describing complex evolving systems, allows to construct a family of characteristic kernels on pathspace known as signature kernels. I will then present how signature kernels can be used to develop a variety of algorithms such as two-sample hypothesis and (conditional) independence tests for stochastic processes, generative models for time series and numerical methods for path-dependent PDEs.
4.30 Refreshments
CDT in Mathematics of Random Systems April Workshop 2022
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Abstract
1:30pm Julian Meier, University of Oxford
Interacting-Particle Systems with Elastic Boundaries and Nonlinear SPDEs
We study interacting particle systems on the positive half-line. When we impose an elastic boundary at zero, the particle systems give rise to nonlinear SPDEs with irregular boundaries. We show existence and uniqueness of solutions to these equations. To deal with the nonlinearity we establish a probabilistic representation of solutions and regularity in L2.
2:15pm Dr Omer Karin, Imperial College London
Mathematical Principles of Biological Regulation
Modern research in the life sciences has developed remarkable methods to measure and manipulate biological systems. We now have detailed knowledge of the molecular interactions inside cells and the way cells communicate with each other. Yet many of the most fundamental questions (such as how do cells choose and maintain their identities? how is development coordinated? why do homeostatic processes fail in disease?) remain elusive, as addressing them requires a good understanding of complex dynamical processes. In this talk, I will present a mathematical approach for tackling these questions, which emphasises the role of control and of emergent properties. We will explore the application of this approach to various questions in biology and biomedicine, and highlight important future directions.
CDT in Mathematics of Random Systems February Workshop
For remote access please contact lydia.noa@imperial.ac.uk
13.20 – 13.50 Alessandro Micheli (CDT Student, Imperial College London)
Closed-loop Nash competition for liquidity
13.50 – 14.20 Terence Tsui (CDT Student, University of Oxford)
Uncovering Genealogies of Populations with Local Density Regulation
14.25 - 15:10 Dr Barbara Bravi (Lecturer in Biomathematics, Department of Mathematics, Imperial College London)
Path integral approaches to model reduction in biochemical networks
November CDT in Maths of Random Systems Seminars
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Abstract
High-dimensional approximation of Hamilton-Jacobi-Bellman PDEs – architectures, algorithms and applications
Hamilton-Jacobi Partial Differential Equations (HJ PDEs) are a central object in optimal control and differential games, enabling the computation of robust controls in feedback form. High-dimensional HJ PDEs naturally arise in the feedback synthesis for high-dimensional control systems, and their numerical solution must be sought outside the framework provided by standard grid-based discretizations. In this talk, I will discuss the construction novel computational methods for approximating high-dimensional HJ PDEs, based on tensor decompositions, polynomial approximation, and deep neural networks.
(HoRSe seminar at Imperial College) Moduli of Calabi-Yau 3-folds and instantons on $G_2$ manifolds
Abstract
This talk will be largely speculative. First we consider the formal properties that could be expected of a "topological field theory" in 6+1 dimensions defined by $G_2$ instantons. We explain that this could lead to holomorphic bundles over moduli spaces of Calabi-Yau 3-folds whose ranks are the DT-invariants. We also discuss in more detail the compactness problem for $G_2$ instantons and associative submanifolds.
The talk will be held in Room 408, Imperial College Maths Department, Huxley Building, 180 Queen’s Gate, London.
(HoRSe seminar at Imperial college) Gauge theory and exceptional holonomy
Abstract
This talk will review material, well-known to specialists, on calibrated geometry and Yang-Mills theory over manifolds with holonomy $SU(3)$, $G_2$ or $Spin(7)$. We will also describe extensions of the standard set-up, modelled on Gromov's "taming forms" for almost-complex structures.
The talk will be held in Room 408, Imperial College Maths Department, Huxley Building, 180 Queen’s Gate, London.
10:00
16:30
15:00
Geometric Transitions and Typical Black Hole Microstates
Abstract
Triangular Seminar