Coming soon...
-
Have you ever wanted to go on a treasure hunt? Then come to PSS this week as I go through a list of clues that led me to find the elusive treasure (vector) that I require. We will traverse through black holes (singularities, no physics involved!), steer clear of stacks, and blow up some obstacles in the way. I will explain quiver varieties, and what I've been thinking about for the last couple of years, no knowledge of algebraic geometry required!
-
With an ulterior motive of trying to get into the flow ahead of an upcoming confirmation viva, I will be talking about my work from the first year of my PhD. I am looking for steady solutions to the free boundary Navier—Stokes equations on an inclined plane, by use of bifurcation theory techniques... i.e. yes, I'm literally trying to make waves with my research. Come along to hear about what a Fredholm operator is, how bifurcation can (hopefully!) be used to find rigorous solutions to PDEs, and possibly another bad fluid mechanics pun.
-
In this talk, I go over my passion project on the surprisingly non-trivial strategy of the kid's card game Top Trumps, featuring very accessible levels of probability, analysis and programming. The main focus is on the Lord of the Rings collectors edition decks and I will have a thing or two to say about some of the questionable choices they have made!
-
Underdamped Langevin dynamics (ULD) is a stochastic differential equation of great interest to those in the molecular dynamics and machine learning communities. Interest from the latter is due to ergodic properties of ULD, meaning that under assumptions on the potential function, the process can be used as proxy for generating samples from an unnormalized (log-concave) target density. Due to the presence of a non-linear term, one cannot simulate ULD exactly and must apply a numerical discretisation.
In this talk, we introduce the numerical method “QUICSORT”, which is a third order convergent method for simulating ULD. Our method is based off discretising the “Shifted ODE” method of Foster, Lyons and Oberhauser, where Brownian motion is replaced by a piecewise linear path matching higher order stochastic integrals. During the talk we will motivate the construction of QUICSORT, discuss the key ideas in the proof of third order convergence (in the 2-Wasserstein metric) and highlight the key role played by numerical contractivity. To the best of our knowledge, this is the first method to achieve third order convergence over the infinite-time horizon for strongly convex MCMC problems. We conclude with a numerical experiment (courtesy of Andraž Jelinčič) illustrating the performance of our method against the current state of the art sampler for high dimensional problems.
Joint work with James Foster.
-
In 1637, Fermat posed a seemingly simple problem in the margin of his copy of arithmetica, and added the following note: "I have discovered a truly marvellous proof of this, which this margin is too narrow to contain", a sentence that would haunt (Halloween pun intended) mathematicians for centuries.
For 358 years an army of number theorists worked together desperately trying to prove this infamous problem. A proof was first announced in 1994 (then corrected and thus completed in 1995). In this talk we give an overview of how this proof was constructed over the years for this elusive problem. In particular we study a very special class of curves named elliptic curves, which have uses in all sorts of areas of pure mathematics but are also known for their use in cryptography.
As nearly all the number theory department is absent, I have written this talk for non-number theorists and so I will not assume much number theory specific knowledge.
-
In this talk, we'll embark on a journey through the world of rough path theory—a framework for understanding complex signals driven by irregular, noisy inputs. At the heart of our exploration is the inverse problem of noise recovery: can we reconstruct the hidden noise driving a system, using only observations of the resulting path?
There's something for everyone on this journey: numerical analysis for the approximation aficionados, probability for the lovers of randomness, and a dash of algebra for those curious about compact ways to represent these rough paths. Join us as we navigate the noise to uncover the hidden forces guiding rough dynamics!
-
The majority of research in changepoint detection has a focus on a change in the mean of data. However, for financial data for example, it is often necessary to detect changes in variance and in the tail distribution. This talk will cover some underlying extreme value theory and will apply these methods to changepoint detection algorithms with an application in finance.
-
In this talk, we will discuss the regularity theory of the Navier-Stokes equations connected to the millennium prize problem: it is not known whether solutions to the 3D incompressible Navier-Stokes equations starting from "nice" initial data stay "nice" forever. Supposing that there is a solution that does not, certain norms/quantities of the solution approach infinity as the solution loses its smoothness. I will introduce you to a certain new, interesting, local quantity that does this, talk about some methods in the proof, and explain why we care about it!
-
Splitting methods are widely used for solving initial value problems (IVPs) due to their ability to simplify complicated evolutions into more manageable subproblems which can be solved efficiently and accurately. Traditionally, these methods are derived using analytic and algebraic techniques from numerical analysis, including truncated Taylor series and their Lie algebraic analogue, the Baker-Campbell-Hausdorff formula. These tools enable the development of high-order numerical methods that provide exceptional accuracy for small timesteps. Moreover, these methods often (nearly) conserve important physical invariants, such as mass, unitarity, and energy. However, in many practical applications the computational resources are limited. Thus, it is crucial to identify methods that achieve the best accuracy within a fixed computational budget, which might require taking relatively large timesteps. In this regime, high-order methods derived with traditional methods often exhibit large errors since they are only designed to be asymptotically optimal. Machine Learning techniques offer a potential solution since they can be trained to efficiently solve a given IVP with less computational resources. However, they are often purely data-driven, come with limited convergence guarantees in the small-timestep regime and do not necessarily conserve physical invariants. In this work, we propose a framework for finding machine learned splitting methods that are computationally efficient for large timesteps and have provable convergence and conservation guarantees in the small-timestep limit. We demonstrate numerically that the learned methods, which by construction converge quadratically in the timestep size, can be significantly more efficient than established methods for the Schrödinger equation if the computational budget is limited.
-
In this PSS, I will introduce you to the card game of Poker, focusing on the no-limit Texas Hold'em variety. I will also provide a brief overview of basic poker strategy, supported by some mathematical insights. Finally, we’ll run a quick poker tournament to put this theory into practice. The goal is to conduct the tournament during the full 2-hour PSS session, but if you need to leave partway through, that’s completely fine.
N.B. The tournament is free to enter, and the prize is chocolates. So, feel free to come along and play. But most importantly, remember: you’ve got to know when to hold ’em, know when to fold ’em, know when to walk away, and know when to run.