University of Leeds

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. 

There are many recent points of contact of model theory and other 
parts of mathematics: o-minimality and Diophantine geometry, geometric group 
theory, additive combinatorics, rigid geometry,...  I will probably 
emphasize  long-standing themes around stability, Diophantine geometry, and 
analogies between ODE's and bimeromorphic geometry.