Past

Optical super-resolution microscopy enables the observations of individual bio-molecules. The arrangement and dynamic behaviour of such molecules is studied to get insights into cellular processes which in turn lead to various application such as treatments for cancer diseases. STFC's Central Laser Facility provides (among other) public access to super-resolution microscope techniques via research grants. The access includes sample preparation, imaging facilities and data analysis support. Data analysis includes single molecule tracking algorithms that produce molecule traces or tracks from time series of molecule observations. While current algorithms are gradually getting away from "connecting the dots" and using probabilistic methods, they often fail to quantify the uncertainties in the results. We have developed a method that samples a probability distribution of tracking solutions using the Metropolis-Hastings algorithm. Such a method can produce likely alternative solutions together with uncertainties in the results. While the method works well for smaller data sets, it is still inefficient for the amount of data that is commonly collected with microscopes. Given the observations of the molecules, tracking solutions are discrete, which gives the proposal distribution of the sampler a peculiar form. In order for the sampler to work efficiently, the proposal density needs to be well designed. We will discuss the properties of tracking solutions and the problems of the proposal function design from the point of view of discrete mathematics, specifically in terms of graphs. Can mathematical theory help to design a efficient proposal function?

Recently, Joyce constructed a Ringel-Hall style graded vertex algebra on the homology of moduli stacks of objects in certain categories of algebro-geometric and representation-theoretic origin. The construction is most natural for 2n-Calabi-Yau categories. We present this construction and explain the geometric reason why it exists. If time permits, we will explain how to compute the homology of the moduli stack of objects in the derived category of a smooth complex projective variety and to identify it with a lattice-type vertex algebra.

We study a class of non linear integro-differential equations on the Wasserstein space related to the optimal control of McKean-Vlasov jump-diffusions. We develop an intrinsic notion of viscosity solutions that does not rely on the lifting to an Hilbert space and prove a comparison theorem for these solutions. We also show that the value function is the unique viscosity solution. Based on a joint work with V. Ignazio, M. Reppen and H. M. Soner

Let n be a random integer (sampled from {1,..,X} for some large X). It is a classical fact that, typically, n will have around (log n)^{log 2} divisors. Must some of these be close together? Hooley's Delta function Delta(n) is the maximum, over all dyadic intervals I = [t,2t], of the number of divisors of n in I. I will report on joint work with Kevin Ford and Dimitris Koukoulopoulos where we conjecture that typically Delta(n) is about (log log n)^c for some c = 0.353.... given by an equation involving an exotic recurrence relation, and then prove (in some sense) half of this conjecture, establishing that Delta(n) is at least this big almost surely.

The dynamics of many physical systems often evolve to asymptotic states that exhibit periodic spatial and temporal variations in their properties such as density, temperature, etc. Such regular patterns look the same when moved by a basic unit and/or rotated by certain special angles. They possess both translational and rotational symmetries giving rise to discrete spatial Fourier transforms. In contrast, an aperiodic crystal displays long range spatial order but no translational symmetry. 

Recently, quasicrystals which are related to aperiodic crystals have been observed to form in diverse physical systems such as metallic alloys (atomic scale) and dendritic-, star-, and block co-polymers (molecular scale). Such quasicrystals lack the lattice symmetries of regular crystals, yet have discrete Fourier spectra. We look to understand the minimal mechanism which promotes the formation of such quasicrystalline structures using a phase field crystal model. Direct numerical simulations combined with weakly nonlinear analysis highlight the parameter values where the quasicrystals are the global minimum energy state and help determine the phase diagram. 

By locating parameter values where multiple patterned states possess the same free energy (Maxwell points), we obtain states where a patch of one type of pattern (for example, a quasicrystal) is present in the background of another (for example, the homogeneous liquid state) in the form of spatially localized dodecagonal (in 2D) and icosahedral (in 3D) quasicrystals. In two dimensions, we compute several families of spatially localized quasicrystals with dodecagonal structure and investigate their properties as a function of the system parameters. The presence of such meta-stable localized quasicrystals is significant as they may affect the dynamics of the crystallisation in soft matter.

The fully coupled numerical simulation of different physical processes, which can typically occur
at variable time and space scales, is often a very challenging task. A common feature of such models is that
their discretization gives rise to systems of linearized equations with an inherent block structure, which
reflects the properties of the set of governing PDEs. The efficient solution of a sequence of systems with
matrices in a block form is usually one of the most time- and memory-demanding issue in a coupled simulation.
This effort can be carried out by using either iteratively coupled schemes or monolithic approaches, which
tackle the problem of the system solution as a whole.

This talk aims to discuss recent advances in the monolithic solution of coupled multi-physics problems, with
application to poromechanical simulations in fractured porous media. The problem is addressed either by proper
sparse approximations of the Schur complements or special splittings that can partially uncouple the variables
related to different physical processes. The selected approaches can be included in a more general preconditioning
framework that can help accelerate the convergence of Krylov subspace solvers. The generalized preconditioner
relies on approximately decoupling the different processes, so as to address each single-physics problem
independently of the others. The objective is to provide an algebraic framework that can be employed as a
general ``black-box'' tool and can be regarded as a common starting point to be later specialized for the
particular multi-physics problem at hand.

Numerical experiments, taken from real-world examples of poromechanical problems and fractured media, are used to
investigate the behaviour and the performance of the proposed strategies.

In the twentieth century we leant that the theory of communication is a mathematical theory. Mathematics is able to add to the value of data, for example by removing redundancy (compression) or increasing robustness (error correction). More famously mathematics can add value to data in the presence of an adversary which is my personal definition of cryptography. Cryptographers talk about properties of confidentiality, integrity, and authentication, though modern cryptography also facilitates transparency (distributed ledgers), plausible deniability (TrueCrypt), and anonymity (Tor).
Modern cryptography faces new design challenges as people demand more functionality from data. Some recent requirements include homomorphic encryption, efficient zero knowledge proofs for smart contracting, quantum resistant cryptography, and lightweight cryptography. I'll try and cover some of the motivations and methods for these.
 

In the seminar, I will talk about a parabolic toy-model for the incompressible Navier-Stokes equations, that satisfies the same energy inequality, same scaling symmetry and which is also super-critical in dimension 3. I will present some partial regularity results that this model shares with the incompressible model and other results that occur only for our model.

When Gromov defined non-positively curved cube complexes no one knew what they would be useful for.
Decades latex they played a key role in the resolution of the Virtual Haken conjecture.
In one of the early forays into experimenting with cube complexes, Aitchison, Matsumoto, and Rubinstein produced some nice results about certain "cubed" manifolds, that in retrospect look very prescient.
I will define non-positively curved cube complexes, what it means for a 3-manifold to be cubed, and discuss what all this Haken business is about.
 

This session will discuss how Douglas Hartree and Arthur Porter used Meccano — a child’s toy and an engineer’s tool — to build an analogue computer, the Hartree Differential Analyser in 1934. It will explore the wider historical and social context in which this model computer was rooted, before providing an opportunity to engage with the experiential aspects of the 'Kent Machine,' a historically reproduced version of Hartree and Porter's original model, which is also made from Meccano.

The 'Kent Machine' sits at a unique intersection of historical research and educational engagement, providing an alternative way of teaching STEM subjects, via a historic hands-on method. The session builds on the work and ideas expressed in Otto Sibum's reconstruction of James Joule's 'Paddle Wheel' apparatus, inviting attendees to physically re-enact the mathematical processes of mechanical integration to see how this type of analogue computer functioned in reality. The session will provide an alternative context of the history of computing by exploring the tacit knowledge that is required to reproduce and demonstrate the machine, and how it sits at the intersection between amateur and professional science.

Donaldson-Thomas invariants of a Calabi-Yau 3-fold Y are related to Pandharipande-Thomas invariants via a wall-crossing formula known as the DT/PT correspondence, proved by Bridgeland and Toda. The same relation holds for the “local invariants”, those encoding the contribution of a fixed smooth curve in Y. We show how to lift the local DT/PT correspondence to the motivic level and provide an explicit formula for the local motivic invariants, exploiting the critical structure on certain Quot schemes acting as our local models. Our strategy is parallel to the one used by Behrend, Bryan and Szendroi in their definition and computation of degree zero motivic DT invariants. If time permits, we discuss a further (conjectural) cohomological upgrade of the local DT/PT correspondence.
Joint work with Ben Davison.
 

We describe the main geometric tools required to work on the manifold of fixed-rank symmetric positive-semidefinite matrices: we present expressions for the Riemannian logarithm and the injectivity radius, to complement the already known Riemannian exponential. This manifold is particularly relevant when dealing with low-rank approximations of large positive-(semi)definite matrices. The manifold is represented as a quotient of the set of full-rank rectangular matrices (endowed with the Euclidean metric) by the orthogonal group. Our results allow understanding the failure of some curve fitting algorithms, when the rank of the data is overestimated. We illustrate these observations on a dataset made of covariance matrices characterizing a wind field.

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.  

Spreading processes on geometric networks are often influenced by a network’s underlying spatial structure, and it is insightful to study the extent to which a spreading process follows that structure. In particular, considering a threshold contagion on a network whose nodes are embedded in a manifold and which has both 'geometric edges' that respect the geometry of the underlying manifold, as well as 'non-geometric edges' that are not constrained by the geometry of the underlying manifold, one can ask whether the contagion propagates as a wave front along the underlying geometry, or jumps via long non-geometric edges to remote areas of the network. 
Taylor et al. developed a methodology aimed at determining the spreading behaviour of threshold contagion models on such 'noisy geometric networks' [1]. This methodology is inspired by nonlinear dimensionality reduction and is centred around a so-called 'contagion map' from the network’s nodes to a point cloud in high dimensional space. The structure of this point cloud reflects the spreading behaviour of the contagion. We apply this methodology to a family of noisy-geometric networks that can be construed as being embedded in a torus, and are able to identify a region in the parameter space where the contagion propagates predominantly via wave front propagation. This consolidates contagion map as both a tool for investigating spreading behaviour on spatial network, as well as a manifold learning technique. 
[1] D. Taylor, F. Klimm, H. A. Harrington, M. Kramar, K. Mischaikow, M. A. Porter, and P. J. Mucha. Topological data analysis of contagion maps for examining spreading processes on networks. Nature Communications, 6(7723) (2015)

In this talk I will explain (a) what observations speak for the
hypothesis of dark matter, (b) what observations speak for
the hypothesis of modified gravity, and (c) why it is a mistake
to insist that either hypothesis on its own must
explain all the available data. The right explanation, I will argue,
is instead a suitable combination of dark matter and modified
gravity, which can be realized by the idea that dark matter
has a superfluid phase.

Given a prime p and an elliptic curve E (say over Q), one can associate a "mod p Galois representation" of the absolute Galois group of Q by considering the natural action on p-torsion points of E.

In 1972, Serre showed that if the endomorphism ring of E is "minimal", then there exists a prime P(E) such that for all p>P(E), the mod p Galois representation is surjective. This raised an immediate question (now known as Serre's uniformity conjecture) on whether P(E) can be bounded as E ranges over elliptic curves over Q with minimal endomorphism rings.

I'll sketch a proof of this result, the current status of the conjecture, and (time permitting) some extensions of this result (e.g. to abelian varieties with appropriately analogous endomorphism rings).

The translation method for constructing quasiconvex lower bound of a given function in the calculus of variations and the notion of compensated convex transforms for tightly approximate functions in Euclidean spaces will be briefly reviewed. By applying the upper compensated convex transform to the finite maximum function we will construct computable quasiconvex functions with finitely many point wells contained in a subspace with rank-one matrices. The complexity for evaluating the constructed quasiconvex functions is O(k log k) with k the number of wells involved. If time allows, some new applications of compensated convexity will be briefly discussed.

We will introduce the Witt vectors of a ring with coefficients in a bimodule and use them to calculate the components of the Hill-Hopkins-Ravenel norm for cyclic p-groups. This algebraic construction generalizes Hesselholt's Witt vectors for non-commutative rings and Kaledin's polynomial Witt vectors over perfect fields. We will discuss applications to the characteristic polynomial over non-commutative rings and to the Dieudonné determinant. This is all joint work with Krause, Nikolaus and Patchkoria.

The Keller Segel model for chemotaxis is a two-dimensional system of parabolic or elliptic PDEs.
Motivated by the study of the fully parabolic model using probabilistic methods, we give rise to a non linear SDE of McKean-Vlasov type with a highly non standard and singular interaction. Indeed, the drift of the equation involves all the past of one dimensional time marginal distributions of the process in a singular way. In terms of approximations by particle systems, an interesting and, to the best of our knowledge, new and challenging difficulty arises: at each time each particle interacts with all the past of the other ones by means of a highly singular space-time kernel.

In this talk, we will analyse the above probabilistic interpretation in $d=1$ and $d=2$.

We discuss a realizationwise correspondence between a Brownian  excursion (conditioned to reach height one) and a triple consisting of

(1) the local time profile of the excursion,

(2) an array of independent time-homogeneous Poisson processes on the real line, and

(3) a fair coin tossing sequence,  where (2) and (3) encode the ordering by height respectively the left-right ordering of the subexcursions.

The three components turn out to be independent,  with (1) giving a time change that is responsible for the time-homogeneity of the Poisson processes.

 By the Ray-Knight theorem, (1) is the excursion of a Feller branching diffusion;  thus the metric structure associated with (2), which generates the so-called lookdown space, can be seen as representing the genealogy underlying the Feller branching diffusion. 

Because of the independence of the three components, up to a time change the distribution of this genealogy does not change under a conditioning on the local time profile. This gives also a natural access to genealogies of continuum populations under competition,  whose population size is modeled e.g. by the Fellerbranching diffusion with a logistic drift.

The lecture is based on joint work with Stephan Gufler and Goetz Kersting.

The Green's function of the Laplace operator has been widely studied in geometric analysis. Manifolds admitting a positive Green's function are called nonparabolic. By Li and Yau, sharp pointwise decay estimates are known for the Green's function on nonparabolic manifolds that have nonnegative Ricci
curvature. The situation is more delicate when curvature is not nonnegative everywhere. While pointwise decay estimates are generally not possible in this
case, we have obtained sharp integral decay estimates for the Green's function on manifolds admitting a Poincare inequality and an appropriate (negative) lower bound on Ricci curvature. This has applications to solving the Poisson equation, and to the study of the structure at infinity of such manifolds.

We present a first-principles derivation of a weak-strong duality between the four-dimensional fishnet theory in the planar limit and a discretized string-like model living in AdS5. At strong coupling, the dual description becomes classical and we demonstrate explicitly the classical integrability of the model. We test our results by reproducing the strong coupling limit of the 4-point correlator computed before non-perturbatively from the conformal partial wave expansion. Next, by applying the canonical quantization procedure with constraints, we show that the model describes a quantum integrable chain of particles propagating in AdS5. Finally, we reveal a discrete reparametrization symmetry of the model and reproduce the spectrum when known analytically. Due to the simplicity of our model, it could provide an ideal playground for holography. Furthermore, since the fishnet model and N=4 SYM theory are continuously linked our consideration could shed light on the derivation of AdS/CFT for the latter. This talk is based on recent work with Amit Sever.

Speaker: Joseph Keir (North)
Title: Dispersion (or not) in nonlinear wave equations
Abstract: Wave equations are ubiquitous in physics, playing central roles in fields as diverse as fluid dynamics, electromagnetism and general relativity. In many cases of these wave equations are nonlinear, and consequently can exhibit dramatically different behaviour when their solutions become large. Interestingly, they can also exhibit differences when given arbitrarily small initial data: in some cases, the nonlinearities drive solutions to grow larger and even to blow up in a finite time, while in other cases solutions disperse just like the linear case. The precise conditions on the nonlinearity which discriminate between these two cases are unknown, but in this talk I will present a conjecture regarding where this border lies, along with some conditions which are sufficient to guarantee dispersion.

Speaker: Priya Subramanian (South)
Title: What happens when an applied mathematician uses algebraic geometry?
Abstract: A regular situation that an applied mathematician faces is to obtain the equilibria of a set of differential equations that govern a system of interest. A number of techniques can help at this point to simplify the equations, which reduce the problem to that of finding equilibria of coupled polynomial equations. I want to talk about how homotopy methods developed in computational algebraic geometry can solve for all solutions of coupled polynomial equations non-iteratively using an example pattern forming system. Finally, I will end with some thoughts on what other 'nails' we might use this new shiny hammer on.

Lines and planes can be fitted to data by minimising the sum of squared distances from the data to the geometric object.  But what about fitting objects from topology such as simplicial complexes?  I will present a method of fitting topological objects to data using a maximum likelihood approach, generalising the sum of squared distances.  A simplicial mixture model (SMM) is specified by a set of vertex positions and a weighted set of simplices between them.  The fitting process uses the expectation-maximisation (EM) algorithm to iteratively improve the parameters.

Remarkably, if we allow degenerate simplices then any distribution in Euclidean space can be approximated arbitrarily closely using a SMM with only a small number of vertices.  This theorem is proved using a form of kernel density estimation on the n-simplex.

How can it be that so many clever, competent and capable people can feel that they are just one step away from being exposed as a complete fraud? Despite evidence that they are performing well they can still have that lurking fear that at any moment someone is going to tap them on the shoulder and say "We need to have a chat". If you've ever felt like this, or you feel like this right now, then this Friday@2 session might be of interest to you. We'll explore what "Imposter Feelings" are, why we get them and steps you can start to take to help yourself and others. This event is likely to be of interest to undergraduates and MSc students at all stages. 

In this talk I will analyse ERK time course data by developing mathematical models of enzyme kinetics. I will present how we can use differential algebra and geometry for model identifiability, and topological data analysis to study these the dynamics of ERK. This work is joint with Lewis Marsh, Emilie Dufresne, Helen Byrne and Stanislav Shvartsman.

We will present three problems that we are interested in:

Forecast of volatility both at the instrument and portfolio level by combining a model based approach with data driven research
We will deal with additional complications that arise in case of instruments that are highly correlated and/or with low volumes and open interest.
Test if volatility forecast improves metrics or can be used to derive alpha in our trading book.

Price predication using physical oil grades data
Hypothesis:
Physical markets are most reflective of true fundamentals. Derivative markets can deviate from fundamentals (and hence physical markets) over short term time horizons but eventually converge back. These dislocations would represent potential trading opportunities.
The problem:
Can we use the rich data from the physical market prices to predict price changes in the derivative markets?
Solution would explore lead/lag relationships amongst a dataset of highly correlated features. Also explore feature interdependencies and non-linearities.
The prediction could be in the form of a price target for the derivative (‘fair value’), a simple direction without magnitude, or a probabilistic range of outcomes.

Modelling oil balances by satellite data
The flow of oil around the world from being extracted, refined, transported and consumed, forms a very large dynamic network. At both regular and irregular intervals, we can make noisy measurements of the amount of oil at certain points in the network.
In addition, we have general macro-economic information about the supply and demand of oil in certain regions.
Based on that information, with general information about the connections between nodes in the network i.e. the typical rate of transfer, one can build a general model for how oil flows through the network.
We would like to build a probabilistic model on the network, representing our belief about the amount of oil stored at each of our nodes, which we refer to as balances.
We want to focus on particular parts of the network where our beliefs can be augmented by satellite data, which can be done by focusing on a sub network containing nodes that satellite measurements can be applied to.

Let G be a finite abelian group, let k be a number field, and let x be an element of k. We count Galois extensions K/k with Galois group G such that x is a norm from K/k. In particular, we show that such extensions always exist. This is joint work with Christopher Frei and Daniel Loughran.

The talk will begin with an introduction to the science of what determines the behaviour of a liquid on a on a surface and giving an overview of some of the different theories that can be used to describe the shape and structure of the liquid in the drop. These include microscopic density functional theory (DFT), which describes the liquid structure on the scale of the individual liquid molecules, and mesoscopic thin film equation (PDE) and kinetic Monte-Carlo models. A DFT based method for calculating the binding potential ?(h) for a film of liquid on a solid surface, where h is the thickness of the liquid film, will be presented. The form of ?(h) determines whether or not the liquid wets the surface. Calculating drop profiles using both DFT and also from inputting ?(h) into the mesoscopic theory and comparing quantities such as the contact angle and the shape of the drops, we find good agreement between the two methods, validating the coarse-graining. The talk will conclude with a discussion of some recent work on modelling evaporating drops with applications to inkjet printing.

First, we will give a brief overview of the asymptotic analysis results in the context of optimal investment. Then, we will focus on the sensitivity of the expected utility maximization problem in a continuous semimartingale market with respect to small changes in the market price of risk. Assuming that the preferences of a rational economic agent are modeled by a general utility function, we obtain a second-order expansion of the value function, a first-order approximation of the terminal wealth, and construct trading strategies that match the indirect utility function up to the second order. If a risk-tolerance wealth process exists, using it as numeraire and under an appropriate change of measure, we reduce the approximation problem to a Kunita–Watanabe decomposition. Then we discuss possible extensions and special situations, in particular, the power utility case and models that admit closed-form solutions. The central part of this talk is based on the joint work with Mihai Sirbu.

Domain decomposition methods are widely employed for the numerical solution of partial differential equations on parallel computers. We develop an adjoint-based a posteriori error analysis for overlapping multiplicative Schwarz domain decomposition and for overlapping additive Schwarz. In both cases the numerical error in a user-specified functional of the solution (quantity of interest), is decomposed into a component that arises due to the spatial discretization and a component that results from of the finite iteration between the subdomains. The spatial discretization error can be further decomposed in to the errors arising on each subdomain. This decomposition of the total error can then be used as part of a two-stage approach to construct a solution strategy that efficiently reduces the error in the quantity of interest.

Federer’s characterization, which is a central result in the theory of functions of bounded variation, states that a set is of finite perimeter if and only if n−1-dimensional Hausdorff measure of the set's measure-theoretic boundary is finite. The measure-theoretic boundary consists of those points where both the set and its complement have positive upper density. I show that the characterization remains true if the measure-theoretic boundary is replaced by a smaller boundary consisting of those points where the lower densities of both the set and its complement are at least a given positive constant.

I will present Pila's Modular Zilber-Pink with Derivatives (MZPD) conjecture, which is a Zilber-Pink type statement for the j-function and its derivatives, and discuss some weak and functional/differential analogues. In particular, I will define special varieties in each setting and explain the relationship between them. I will then show how one can prove the aforementioned weak/functional/differential MZPD statements using the Ax-Schanuel theorem for the j-function and its derivatives and some basic complex analytic geometry. Note that I gave a similar talk in Oxford last year (where I discussed a differential MZPD conjecture and proved it assuming an Existential Closedness conjecture for j), but this talk is going to be significantly different from that one (the approach presented in this talk will be mostly complex analytic rather than differential algebraic, and the results will be unconditional).

A graph of groups decomposition is a way of splitting a group into smaller and hopefully simpler groups. A natural thing to try and do is to keep splitting until you can't split anymore, and then argue that this decomposition is unique. This is the idea behind JSJ decompositions, although, as we shall see, the strength of the uniqueness statement for such a decomposition varies depending on the class of groups that we restrict our edge groups to

I will discuss some recent results around Hilbert schemes of points on singular surfaces, obtained in joint work with Craw, Gammelgaard and Gyenge, and their connection to combinatorics (of coloured partitions) and representation theory (of affine Lie algebras and related algebras such as their W-algebra). 

The study of 'moments' of random matrices (expectations of traces of powers of the matrix) is a rich and interesting subject, with fascinating connections to enumerative geometry, as discovered by Harer and Zagier in the 1980’s. I will give some background on this and then describe some recent work which offers some new perspectives (and new results). This talk is based on joint work with Fabio Deelan Cunden, Francesco Mezzadri and Nick Simm.

Ocean biogeochemical models used in climate change predictions are very computationally expensive and heavily parameterised. With derivatives too costly to compute, we optimise the parameters within one such model using derivative-free algorithms with the aim of finding a good optimum in the fewest possible function evaluations. We compare the performance of the evolutionary algorithm CMA-ES which is a stochastic global optimization method requiring more function evaluations, to the Py-BOBYQA and DFO-LS algorithms which are local derivative-free solvers requiring fewer evaluations. We also use initial Latin Hypercube sampling to then provide DFO-LS with a good starting point, in an attempt to find the global optimum with a local solver. This is joint work with Coralia Cartis and Samar Khatiwala.
 

A mathematical structure is `axiomatizable' if it is completely determined by some family of sentences in a suitable first-order language. This idea has been explored for various kinds of structure, but I will concentrate on groups. There are some general results (not many) about which groups are or are not axiomatizable; recently there has been some interest in the sharper concept of 'finitely axiomatizable' or FA - that is, when only a finite set of sentences (equivalently, a single sentence) is allowed.

While an infinite group cannot be FA, every finite group is so, obviously. A profinite group is kind of in between: it is infinite (indeed, uncountable), but compact as a topological group; and these groups share many properties of finite groups, though sometimes for rather subtle reasons. I will discuss some recent work with Andre Nies and Katrin Tent where we prove that certain kinds of profinite group are FA among profinite groups. The methods involve a little model theory, and quite a lot of group theory.

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.

We consider the problem of global minimization with bound constraints. The problem is known to be intractable for large dimensions due to the exponential increase in the computational time for a linear increase in the dimension (also known as the “curse of dimensionality”). In this talk, we demonstrate that such challenges can be overcome for functions with low effective dimensionality — functions which are constant along certain linear subspaces. Such functions can often be found in applications, for example, in hyper-parameter optimization for neural networks, heuristic algorithms for combinatorial optimization problems and complex engineering simulations.

Extending the idea of random subspace embeddings in Wang et al. (2013), we introduce a new framework (called REGO) compatible with any global min- imization algorithm. Within REGO, a new low-dimensional problem is for- mulated with bound constraints in the reduced space. We provide probabilistic bounds for the success of REGO; these results indicate that the success is depen- dent upon the dimension of the embedded subspace and the intrinsic dimension of the function, but independent of the ambient dimension. Numerical results show that high success rates can be achieved with only one embedding and that rates are independent of the ambient dimension of the problem.