I will recall the Zilber-Pink conjecture for Shimura varieties and give my perspective on current progress towards a proof.
Whilst we all enjoy travelling to exciting and far-off locations, the current climate crisis is making flights less and less attractive. But is there anything we can do about this? By plotting courses that make best use of atmospheric data to minimise aircraft fuel burn, airlines can not only save money on fuel, but also reduce emissions, whilst not significantly increasing flight times. In each case the route between London Heathrow Airport and John F Kennedy Airport in New York is considered. Atmospheric data is taken from a re-analysis dataset based on daily averages from 1st December, 2019 to 29th February, 2020.
Initially Pontryagin’s minimum principle is used to find time minimal routes between the airports and these are compared with flight times along the organised track structure routes prepared by the air navigation service providers NATS and NAV CANADA for each day. Efficiency of tracks is measured using air distance, revealing that potential savings of between 0.7% and 16.4% can be made depending on the track flown. This amounts to a reduction of 6.7 million kg of CO2 across the whole winter period considered.
In a second formulation, fixed time flights are considered, thus reducing landing delays. Here a direct method involving a reduced gradient approach is applied to find fuel minimal flight routes either by controlling just heading angle or both heading angle and airspeed. By comparing fuel burn for each of these scenarios, the importance of airspeed in the control formulation is established.
Finally dynamic programming is applied to the problem to minimise fuel use and the resulting flight routes are compared with those actually flown by 9 different models of aircraft during the winter of 2019 to 2020. Results show that savings of 4.6% can be made flying east and 3.9% flying west, amounting to 16.6 million kg of CO2 savings in total.
Thus large reductions in fuel consumption and emissions are possible immediately, by planning time or fuel minimal trajectories, without waiting decades for incremental improvements in fuel-efficiency through technological advances.
In the late 1980s, Berger and Coburn showed that the Hankel operator $H_f$ on the Segal-Bargmann space of Gaussian square-integrable entire functions is compact if and only if $H_{\bar f}$ is compact using C*-algebra and Hilbert space techniques. I will briefly discuss this and three other proofs, and then consider the question of whether an analogous phenomenon holds for Schatten class Hankel operators.
The concept of path signatures has been widely used in several areas of pure mathematics including in applications to data science. However, we remain unable to answer even the most basic questions about it. For instance, how to fully characterise the set of (untruncated) signatures of bounded variation paths? Can certain norms on signatures be related to the length of a path, like in Fourier isometry? In this talk, we will review some known results, explain the open problems and discuss their difficulties.
Let $\mathcal{B}$ be a (unital) commutative Banach algebra and $\Omega$ the set of non-trivial multiplicative linear functionals $\omega : \mathcal{B} \to \mathbb{C}$. Gelfand theory tells us that the kernels of these functionals are exactly the maximal ideals of $\mathcal{B}$ and, as a consequence, an element $b \in \mathcal{B}$ is invertible if and only if $\omega(b) \neq 0$ for all $\omega \in \Omega$. A generalisation to non-commutative Banach algebras is the local principle of Allan and Douglas, also known as central localisation: Let $\mathcal{B}$ be a Banach algebra, $Z$ a closed subalgebra of the center of $\mathcal{B}$ and $\Omega$ the set of maximal ideals of $Z$. For every $\omega \in \Omega$ let $\mathcal{I}_{\omega}$ be the smallest ideal of $\mathcal{B}$ which contains $\omega$. Then $b \in \mathcal{B}$ is invertible if and only if $b + \mathcal{I}_{\omega}$ is invertible in $\mathcal{B} / \mathcal{I}_{\omega}$ for every $\omega \in \Omega$.
From an operator theory point of view, one of the most important features of the local principle is the application to Calkin algebras. In that case the invertible elements are called Fredholm operators and the corresponding spectrum is called the essential spectrum. Therefore, by taking suitable subalgebras, we can obtain a characterisation of Fredholm operators. Many beautiful results in spectral theory, e.g.~formulas for the essential spectrum of Toeplitz operators, can be obtained in this way. However, the central localisation is often not sufficient to provide a satisfactory characterisation for more general operators. In this talk we therefore consider a generalisation where the ideals $\mathcal{I}_{\omega}$ do not originate from the center of the algebra. More precisely, we will start with general $L^p$-spaces and apply limit operator methods to obtain a Fredholm theory that is applicable to many different settings. In particular, we will obtain characterisations of Fredholmness and compactness in many new cases and also rediscover some classical results.
This talk is based on joint work with Christian Seifert.
Lipschitz (or H\"older) spaces $C^\delta, \, k< \delta <k+1$, $k\in\mathbb{N}_0$, are the set of functions that are more regular than the $\mathcal{C}^k$ functions and less regular than the $\mathcal{C}^{k+1}$ functions. The classical definitions of H\"older classes involve pointwise conditions for the functions and their derivatives. This implies that to prove regularity results for an operator among these spaces we need its pointwise expression. In many cases this can be a rather involved formula, see for example the expression of $(-\Delta)^\sigma$ in (Stinga, Torrea, Regularity Theory for the fractional harmonic oscilator, J. Funct. Anal., 2011.)
In the 60's of last century, Stein and Taibleson, characterized bounded H\"older functions via some integral estimates of the Poisson semigroup, $e^{-y\sqrt{-\Delta}},$ and of the Gauss semigroup, $e^{\tau{\Delta}}$. These kind of semigroup descriptions allow to obtain regularity results for fractional operators in these spaces in a more direct way.
In this talk we shall see that we can characterize H\"older spaces adapted to other differential operators $\mathcal{L}$ by means of semigroups and that these characterizations will allow us to prove the boundedness of some fractional operators, such as $\mathcal{L}^{\pm \beta}$, Riesz transforms or Bessel potentials, avoiding the long, tedious and cumbersome computations that are needed when the pointwise expressions are handled.
The variational data assimilation (VarDA) problem is usually solved using a method equivalent to Gauss-Newton (GN) to obtain the initial conditions for a numerical weather forecast. However, GN is not globally convergent and if poorly initialised, may diverge such as when a long time window is used in VarDA; a desirable feature that allows the use of more satellite data. To overcome this, we apply two globally convergent GN variants (line search & regularisation) to the long window VarDA problem and show when they locate a more accurate solution versus GN within the time and cost available.
Joint work with Coralia Cartis, Amos S. Lawless, Nancy K. Nichols.
Stochastic differential equations have Taylor expansions in terms of iterated Wiener integrals. The convergence of such expansion depends on the limiting behavior of the order-N iterated integrals as N tends to infinity. Recently, there has been increased interests in processes stopped at a random time. A breakthrough in the study of the iterated integrals of Brownian motion up to the exit time of a domain was included in the work of Lyons-Ni (2012). The paper leaves open an interesting question: what is the sharp rate of decay for the expected iterated integrals up to the exit time. We will review the state of the art in this problem and report some recent progress. Joint work with Ni Hao (UCL).
In 2016, Habegger and Pila published a proof of the Zilber-Pink conjecture for curves in abelian varieties (all defined over $\mathbb{Q}^{\rm alg}$). Their article also contained a proof of the same conjecture for a product of modular curves that was conditional on a strong arithmetic hypothesis. Both proofs were extensions of the Pila-Zannier strategy based in o-minimality that has yielded many results in this area. In this talk, we will explain our generalisation of the strategy to the Zilber-Pink conjecture for any Shimura variety. This is joint work with J. Ren.
Work with Jemima Tabeart, Sarah Dance, Nancy Nichols, Joanne Waller (University of Reading) and Stefano Migliorini, Fiona Smith (Met Office).
In environmental prediction variational data assimilation (DA) is a method for using observational data to estimate the current state of the system. The DA problem is usually solved as a very large nonlinear least squares problem, in which the fit to the measurements is balanced against the fit to a previous model forecast. These two terms are weighted by matrices describing the correlations of the errors in the forecast and in the observations. Until recently most operational weather and ocean forecasting systems assumed that the errors in the observations are uncorrelated. However, as we move to higher resolution observations then it is becoming more important to specify observation error correlations. In this work we look at the effect this has on the conditioning of the optimization problem. In the context of a linear system we develop bounds on the condition number of the problem in the presence of correlated observation errors. We show that the condition number is very dependent on the minimum eigenvalue of the observation error correlation matrix. We then present results using the Met Office data assimilation system, in which different methods for reconditioning the correlation matrix are tested. We investigate the effect of these different methods on the conditioning and the final solution of the problem.