University of Leeds

Traditionally, stochastic processes are modelled one of two ways: a continuum Fokker-Planck approach, where a PDE is solved to determine the time evolution of the probability density, or a Langevin approach, where the SDE describing the system is sampled, and multiple simulations are used to collect statistics. There is also a third way: the functional or path integral. Originally developed by Wiener in the 1920s to model Brownian motion, path integrals were famously applied to quantum mechanics by Feynman in the 1950s. However, they also have much to offer classical stochastic processes (and statistical physics).  

In this talk I will introduce the formalism at a physicist’s level of rigour, and focus on determining the dominant contribution to the path integral when the noise is weak. There exists a remarkable correspondence between the most-probable stochastic paths and Hamiltonian dynamics in an effective potential [1,2,3]. I will then discuss some applications, including reaction pathways conditioned on finite time [2]. We demonstrate that the most probable pathway at a finite time may be very different from the usual minimum energy path used to calculate the average reaction rate. If time permits, I will also discuss the extremely nonlinear crystal dislocation response to applied stress [4].  

[1] Ge, Hao, and Hong Qian. Int. J. Mod. Phys. B 26.24 1230012 (2012)     

[2] Fitzgerald, Steve, et al. J. Chem. Phys. 158.12 (2023).

[3] Honour, Tom and Fitzgerald, Steve. in press J. Phys. A (2024)

[4] Fitzgerald, Steve. Sci. Rep. 6 (1) 39708 (2016)

For any smooth complex variety Y with an action of a finite group W, Etingof defines the global Cherednik algebra H_c and its spherical subalgebra B_c as certain sheaves of algebras over Y/W. When Y is an n-dimensional abelian variety, the algebra of global sections of B_c is a polynomial algebra on n generators, as shown by Etingof, Felder, Ma, and Veselov. This defines an integrable system on Y. In the case of Y being a product of n copies of an elliptic curve E and W=S_n, this reproduces the usual elliptic Calogero­­--Moser system. Recently, together with P. Argyres and Y. Lu, we proposed that many of these integrable systems at the classical level can be interpreted as Seiberg­­--Witten integrable systems of certain super­symmetric quantum field theories. I will describe our progress in understanding this connection for groups W=G(m, 1, n), corresponding to the case Y=E^n where E is an elliptic curves with Z_m symmetry, m=2,3,4,6. 

We consider stochastic differential equations (SDEs) driven by fractional Brownian motion with Hurst parameter less than 1/2. The drift is a measurable function of time and space which belongs to a certain Lebesgue space. Under subcritical regime, we show that a strong solution exists and is unique in path-by-path sense. When the noise is formally replaced by a Brownian motion, our results correspond to the strong uniqueness result of Krylov and Roeckner (2005). Our methods forgo standard approaches in Markovian settings and utilize Lyons' rough path theory in conjunction with recently developed tools. Joint work with Toyomu Matsuda and Oleg Butkovsky.

The Zilber-Pink conjecture predicts that if V is a proper subvariety of an arithmetic variety S (e.g. abelian variety, Shimura variety, others) not contained in a proper special subvariety of V, then the “unlikely intersections” of V with the proper special subvarieties of S are not Zariski dense in V. In this talk I will present a strong counterpart to the Zilber-Pink conjecture, namely that under some natural conditions, likely intersections are in fact Euclidean dense in V.  This is joint work with Tom Scanlon.

Although tensor products and their symmetrisation have appeared in mathematical literature since at least the mid-nineteenth century, they rarely appear in the function-theoretic operator theory literature. In this talk, I will introduce the symmetric and antisymmetric tensor products from an operator theoretic point of view. I will present results concerning some of the most fundamental operator-theoretic questions in this area, such as finding the norm and spectrum of the symmetric tensor products of operators. I will then work through some examples of symmetric tensor products of familiar operators, such as the unilateral shift, the adjoint of the shift, and diagonal operators.

A theorem of Chatzidakis, van den Dries and Macintyre, stemming ultimately from the Lang-Weil estimates, asserts, roughly, that if $\phi(x,y)$ is a formula in the language of rings (where $x,y$ are tuples) then the size of the solution set of $\phi(x,a)$ in any finite field $F_q $(where $a$ is a parameter tuple from $F_q$) takes one of finitely many dimension-measure pairs as $F_q$ and $a$ vary: for a finite set $E$ of pairs $(\mu,d)$ ($\mu$ rational, $d$ integer) dependent on $\phi$, any set $\phi(F_q,a)$ has size roughly $\mu q^d$ for some $(\mu,d) \in E$.

This led in work of Elwes, Steinhorn and myself to the notion of 'asymptotic class’ of finite structures (a class satisfying essentially the conclusion of Chatzidakis-van den Dries-Macintyre). As an example, by a theorem of Ryten, any family of finite simple groups of fixed Lie type forms an asymptotic class. There is a corresponding notion for infinite structures of  'measurable structure’ (e.g. a pseudofinite field, by the Chatzidakis-van den Dries-Macintyre theorem, or certain pseudofinite difference fields).

I will discuss a body of work with Sylvy Anscombe, Charles Steinhorn and Daniel Wolf which generalises this, incorporating a richer range of examples with fewer model-theoretic constraints; for example, the corresponding infinite 'generalised measurable’ structures, for which the definable sets are assigned values in some ordered semiring, need no longer have simple theory. I will also discuss a variant in which sizes of definable sets in finite structures are given exactly rather than asymptotically.

Deep-marine volcanism drives Earth's most energetic transfers of heat and mass between the crust and the oceans. Yet little is known of the primary source and intensity of the energy release that occurs during seafloor volcanic events owing to the lack of direct observations. Seafloor magmatic activity has nonetheless been correlated in time with the appearance of massive plumes of hydrothermal fluid known as megaplumes. However, the mechanism by which megaplumes form remains a mystery. By utilising observations of pyroclastic deposits on the seafloor, we show that their dispersal required an energy discharge that is sufficiently powerful (1-2 TW) to form a hydrothermal discharge with characteristics that align precisely with those of megaplumes observed to date. The result produces a conclusive link between tephra production, magma extrusion, tephra dispersal and megaplume production. However, the energy flux is too high to be explained by a purely volcanic source (lava heating), and we use our constraints to suggest other more plausible mechanisms for megaplume creation. The talk will cover a combination of new fluid mechanical fundamentals in volcanic transport processes, inversion methods and their implications for volcanism in the deep oceans.

The Kelvin wave is perhaps the most important of the equatorially trapped waves in the terrestrial atmosphere and ocean, and plays a role in various phenomena such as tropical convection and El Nino. Theoretically, it can be understood from the linear dynamics of a stratified fluid on an equatorial beta plane, which, with simple assumptions about the disturbance structure, leads to wavelike solutions propagating along the equator, with exponential decay in latitude. However, when the simplest possible background flow is added (with uniform latitudinal shear), the Kelvin wave (but not the other equatorial waves) becomes unstable. This happens in an extremely unusual way: there is instability for arbitrarily small nondimensional shear p, and the growth rate is proportional to exp(-1/p^2) as p->0. This in contrast to most hydrodynamic instabilities, in which the growth rate typically scales as a positive power of p-p_c as the control parameter p passes through a critical value p_c.

This Kelvin wave instability has been established numerically by Natarov and Boyd, who also speculated as to the underlying mathematical cause. Here we show how the growth rate and full spatial structure of the instability may be derived using matched asymptotic expansions applied to the (linear) equations of motion. This involves an adventure with Whittaker functions in the exponentially-decaying tails of the Kelvin waves, and a trick to reveal the exponentially small growth rate from a formulation that only uses regular perturbation expansions. Numerical verification of the analysis is also interesting and challenging, since special high-precision solutions of the governing ODE are required even when the nondimensional shear is not that small (circa 0.5).

 I will give a short survey on classical inequalities for Laplace eigenvalues, tell about related history and questions. I will then discuss the so-called Korevaar method, and new results generalising to higher eigenvalues a number of classical inequalities known for the first Laplace eigenvalue only.