Past

Arctic sea ice is one of the most sensitive components of the Earth’s climate system. The underlying ocean plays an important role in the evolution of the ice cover through its heat flux at the ice-ocean interface. Despite its importance, the spatio-temporal variations of this heat flux are not well understood. In this talk, I will take the following approach to study the variations in the heat flux. First, I will consider the problem of classical Rayleigh-Bénard convection and systematically explore the effects of fractal boundaries on heat transport using direct numerical simulations. And second, I will analyze time-series data from the Surface Heat Budget of the Arctic Ocean (SHEBA) program using Multifractal Detrended Fluctuation Analysis (MFDFA) to understand the nature of fluctuations in the heat flux. I will also discuss developing simple stochastic ODEs using results from these studies.

Would you like to meet some of your fellow students, and some graduate students and postdocs, in an informal and relaxed atmosphere, while building your communication skills?  In this Friday@2 session, you'll be able to play a selection of board games, meet new people, and practise working together.  What better way to spend the final Friday afternoon of term?!  We'll play the games in the south Mezzanine area of the Andrew Wiles Building, outside L3.

This challenge relates to problems (of a mathematical nature) in generating optimal solutions for natural flood management.  Natural flood management involves large numbers of small scale interventions in a much larger context through exploiting natural features in place of, for example, large civil engineering construction works. There is an optimisation problem related to the catchment hydrology and present methods use several unsatisfactory simplifications and assumptions that we would like to improve on.

Stevenhagen conjectured that the density of d such that the negative Pell equation x^2-dy^2=-1 is solvable over the integers is 58.1% (to the nearest tenth of a percent), in the set of positive squarefree integers having no prime factors congruent to 3 modulo 4. In joint work with Peter Koymans, Djordjo Milovic, and Carlo Pagano, we use a recent breakthrough of Smith to prove that the infimum of this density is at least 53.8%, improving previous results of Fouvry and Klüners, by studying the distribution of the 8-rank of narrow class groups of quadratic number fields.

The problem of a bubble moving steadily in a Hele-Shaw cell goes back to Taylor and Saffman in 1959.  It is analogous to the well-known selection problem for Saffman-Taylor fingers in a Hele-Shaw channel.   We apply techniques in exponential asymptotics to study the bubble problem in the limit of vanishing surface tension, confirming previous numerical results, including a previously predicted surface tension scaling law.  Our analysis sheds light on the multiple tips in the shape of the bubbles along solution branches, which appear to be caused by switching on and off exponentially small wavelike contributions across Stokes lines in a conformally mapped plane. 

In the talk we will define higher K-groups, and explain some of their relations to number theory

We present some regularity estimates for viscosity solutions to a class of possible degenerate and singular integro-differential equations whose leading operator switches between two different types of fractional elliptic phases, according to the zero set of a modulating coefficient a = a(·, ·). The model case is driven by the following nonlocal double phase operator,

$$\int \frac{|u(x) − u(y)|^{p−2} (u(x) − u(y))} {|x − y|^{n+sp}} dy+ \int a(x, y) \frac{|u(x) − u(y)|^{ q−2} (u(x) − u(y))} {|x − y|^{n+tq}} dy$$

where $q ≥ p$ and $a(·, ·) = 0$. Our results do also apply for inhomogeneous equations, for very general classes of measurable kernels. By simply assuming the boundedness of the modulating coefficient, we are able to prove that the solutions are Hölder continuous, whereas similar sharp results for the classical local case do require a to be Hölder continuous. To our knowledge, this is the first (regularity) result for nonlocal double phase problems.

   It is a long-standing open problem whether the ring of integers Z has an existential first-order definition in Q, the field of rational numbers. A few years ago, Jochen Koenigsmann proved that Z has a universal first-order definition in Q, building on earlier work by Bjorn Poonen. This result was later generalised to number fields by Jennifer Park and to global function fields of odd characteristic by Kirsten Eisenträger and Travis Morrison, who used classical machinery from number theory and class field theory related to the behaviour of quaternion algebras over global and local fields.

   In this talk, I will sketch a variation on the techniques used to obtain the aforementioned results. It allows for a relatively short and uniform treatment of global fields of all characteristics that is significantly less dependent on class field theory. Instead, a central role is played by Hilbert's Reciprocity Law for quaternion algebras. I will conclude with an example of a non-global set-up where the existence of a reciprocity law similarly yields universal definitions of certain subrings.

Knotoids are a generalisation of knots that deals with open curves. In the past few years, they’ve been extensively used to classify entanglement in proteins. Through a double branched cover construction, we prove a 1-1 correspondence between knotoids and strongly invertible knots. We characterise forbidden moves between knotoids in terms of equivariant band attachments between strongly invertible knots, and in terms of crossing changes between theta-curves. Finally, we present some applications to the study of the topology of proteins. This is based on joint works with D.Buck, H.A.Harrington, M.Lackenby and with D. Goundaroulis.

Finitely presented groups are a natural algebraic generalisation of the collection of finite groups. Unlike the finite case there is almost no hope of any kind of classification.

The goal of random groups is therefore to understand the properties of the "typical" finitely presented group. I will present a couple of models for random groups and survey some of the main theorems and open questions in the area, demonstrating surprising correlations between these probabilistic models, geometry and analysis.

The hard Lefschetz theorem is a fundamental statement about the symmetry of the cohomology of algebraic varieties. In nearly all cases that we systematically understand it, it comes with a geometric meaning, often in form of Hodge structures and signature data for the Hodge-Riemann bilinear form.

Nevertheless, similar to the role the standard conjectures play in number theory, several intriguing combinatorial problems can be reduced to hard Lefschetz properties, though in extreme cases without much geometric meaning, lacking any existence of, for instance,  an ample cone to do Hodge theory with.

I will present a way to prove the hard Lefschetz theorem in such a situation, by introducing biased pairing and perturbation theory for intersection rings. The price we pay is that the underlying variety, in a precise sense, has itself to be sufficiently generic. For instance, we shall see that any quasismooth, but perhaps nonprojective toric variety can be "perturbed" to a toric variety with the same equivariant cohomology, and that has the hard Lefschetz property.

Finally, I will discuss how this applies to prove some interesting theorems in geometry, topology and combinatorics. In particular, we shall see a generalization of a classical result due to Descartes and Euler: We prove that if a simplicial complex embeds into euclidean 2d-space, the number of d-simplices in it can exceed the number of (d-1)-simplices by a factor of at most d+2.

We consider estimation of ordinary differential equation (ODE) models from noisy observations. For this problem, one conventional approach is to fit numerical solutions (e.g., Euler, Runge–Kutta) of ODEs to data. However, such a method does not account for the discretization error in numerical solutions and has limited estimation accuracy. In this study, we develop an estimation method that quantifies the discretization error based on data. The key idea is to model the discretization error as random variables and estimate their variance simultaneously with the ODE parameter. The proposed method has the form of iteratively reweighted least squares, where the discretization error variance is updated with the isotonic regression algorithm and the ODE parameter is updated by solving a weighted least squares problem using the adjoint system. Experimental results demonstrate that the proposed method improves estimation accuracy by accounting for the discretization error in a data-driven manner. This is a joint work with Yuto Miyatake (Osaka University).

In this talk, I will report about a joint work with D. Ciubotaru, in which we investigate the Dunkl version of the classical Howe-duality (O(k),spo(2|2)). Similar Fischer-type decompositions were studied before in the works of Ben-Said, Brackx, De Bie, De Schepper, Eelbode, Orsted, Soucek and Somberg for other Howe-dual pairs. Our work builds on the notion of a Dirac operator for Drinfeld algebras introduced by Ciubotaru, which was inspired by the analogous theory for Lie algebras, as well as the work of Cheng and Wang on classical Howe dualities.

A mechanism is described to symmetrize the ultraspherical spectral method for self-adjoint problems. The resulting discretizations are symmetric and banded. An algorithm is presented for an adaptive spectral decomposition of self-adjoint operators. Several applications are explored to demonstrate the properties of the symmetrizer and the adaptive spectral decomposition.

Topology optimisation finds the optimal material distribution of a fluid or solid in a domain, subject to PDE and volume constraints. There are many formulations and we opt for the density approach which results in a PDE, volume and inequality constrained, non-convex, infinite-dimensional optimisation problem without a priori knowledge of a good initial guess. Such problems can exhibit many local minima or even no minima. In practice, heuristics are used to obtain the global minimum, but these can fail even in the simplest of cases. In this talk, we will present an algorithm that solves such problems and systematically discovers as many of these local minima as possible along the way.

This talk will explain how Lie polynomials underpin the structure of the so-called double copy relationship between gauge and gravity theories (and a network of other theories besides).  ABHY have recently shown that Lie polynomials arise naturally also in the geometry of the space K_n of momentum invariants, Mandelstams, and can be expressed in the space of n-3-forms dual to certain associahedral (n-3)-planes. They also arise in the moduli space M_{0,n} of n points on a Riemann sphere up to Mobius transformations in the n-3-dimensional homology.  The talk goes on to give a natural correspondendence between K_n and the cotangent bundle of M_{0.n} through which the relationships of some of these structures can be expressed.  This in particular gives a natural framework for expressing the CHY and ambitwistor-string formulae for scattering amplitudes of gauge and gravity theories and goes some way to expressing their double copy relations.   This is part of joint work in progress with Hadleigh Frost.

We aim to generate gene coexpression networks from gene expression data. In our networks, nodes represent genes and edges depict high positive correlation in their expression across different samples. Methods based on Pearson correlation are the most commonly used to generate gene coexpression networks. We propose the use of distance correlation as an effective alternative to Pearson correlation when constructing gene expression networks. Our methodology pipeline includes a thresholding step which allows us to discriminate which pairs of genes are coexpressed. We select the value of the threshold parameter by studying the stability of the generated network, rather than relying on exogenous biological information known a priori.

Charles Babbage (1791–1871) was Lucasian Professor of mathematics in Cambridge from 1828–1839. He displayed a fertile curiosity that led him to study many contemporary processes and problems in a way which emphasised an analytic, data driven view of life.

In popular culture Babbage has been celebrated as an anachronistic Victorian engineer. In reality, Babbage is best understood as a figure rooted in the enlightenment, who had substantially completed his core investigations into 'mechanisation of thought' by the mid 1830s: he is thus an anachronistic Georgian: the construction of his first difference engine design is contemporary with the earliest public railways in Britain.

A fundamental question that must strike anybody who examines Babbage's precocious designs is: how could one individual working alone have synthesised a workable computer design, designing an object whose complexity of behaviour so far exceeded that of contemporary machines that it would not be matched for over a hundred years?

We shall explore the extent to which the answer lies in the techniques Babbage developed to reason about complex systems. His Notation which shows the geometry, timing, causal chains and the abstract components of his machines, has a direct parallel in the Hardware Description Languages developed since 1975 to aid the design of large scale electronics. In this presentation, we shall provide a basic tutorial on Babbage's notation showing how his concepts of 'pieces' and 'working points' effectively build a graph in which both parts and their interactions are represented by nodes, with edges between part-nodes and interaction-nodes denoting ownership, and edges between interaction-nodes denoting the transmission of forces between individual assemblies within a machine. We shall give examples from Babbage's Difference Engine 2 for which a complete set of notations was drawn in 1849, and compare them to a design of similar complexity specified in 1987 using the Inmos HDL.

L-functions have become a vital part of modern number theory over the past century, allowing comparisons between arithmetic objects with seemingly very different properties. In the first part of this talk, I will give an overview of where they arise, their properties, and the mathematics that has developed in order to understand them. In the second part, I will give a sketch of the beautiful result of Herbrand-Ribet concerning the arithmetic interpretations of certain special values of the Riemann zeta function, the prototypical example of an L-function.