Past

A subset of the integers larger than 1 is called $\textit{primitive}$ if no member divides another. Erdős proved in 1935 that the sum of $1/(n \log n)$ over $n$ in a primitive set $A$ is universally bounded for any choice of $A$. In 1988, he famously asked if this universal bound is attained by the set of prime numbers. In this talk we shall discuss some recent progress towards this conjecture and related results, drawing on ideas from analysis, probability, & combinatorics.

The Gibbons-Hawking ansatz is a powerful method for constructing a large family of hyperkaehler 4-manifolds (which are thus Ricci-flat), which appears in a variety of contexts in mathematics and theoretical physics. I will describe work in progress to understand the theory of minimal surfaces and mean curvature flow in these 4-manifolds. In particular, I will explain a proof of a version of the Thomas-Yau Conjecture in Lagrangian mean curvature flow in this setting. This is joint work with G. Oliveira.

The smooth mapping class group of the 4-sphere is pi_0 of the space of orientation preserving self-diffeomorphisms of S^4. At the moment we have no idea whether this group is trivial or not. Watanabe has shown that higher homotopy groups can be nontrivial. Inspired by Watanabe's constructions, we'll look for interesting self-diffeomorphisms of S^4. Most of the talk will be an outline for a program to find a nice geometric generating set for this mapping class group; a few small steps in the program are actually theorems. The point of finding generators is that if they are explicit enough then you have a hope of either showing that they are all trivial or finding an invariant that is well adapted to obstructing triviality of these generators.

We reconsider the approximations of the Black-Scholes model by discrete time models such as the binominal or the trinominal model.

We show that for continuous and bounded claims one may approximate the replication in the Black-Scholes model by trading in the discrete time models. The approximations holds true in measure as well as "with bounded risk", the latter assertion being the delicate issue. The remarkable aspect is that this result does not apply to the well-known binominal model, but to a much wider class of discrete approximating models, including, eg.,the trinominal model. by an example we show that we cannot do the approximation with "vanishing risk".

We apply this result to portfolio optimization and show that, for utility functions with "reasonable asymptotic elasticity" the solution to the discrete time portfolio optimization converge to their continuous limit, again in a wide class of discretizations including the trinominal model. In the absence of "reasonable asymptotic elasticity", however, surprising pathologies may occur.

Joint work with David Kreps (Stanford University)

We revisit portfolio selection models by considering a distributionally robust version, where the region of distributional uncertainty is around the empirical measure and the discrepancy between probability measures is dictated by optimal transport costs. In many cases, this problem can be simplified into an empirical risk minimization problem with a regularization term. Moreover, we extend a recently developed inference methodology in order to select the size of the distributional uncertainty in a data-driven way. Our formulations allow us to inform the distributional uncertainty region using market information (e.g. via implied volatilities). We provide substantial empirical tests that validate our approach.
(This presentation is based on the following papers: https://arxiv.org/pdf/1802.04885.pdf and https://arxiv.org/abs/1810.024….)

In this talk I will discuss a new proposal for constructing quantizations of holomorphic Poisson structures, and generalized complex manifolds more generally, which is based on using the A model of an associated symplectic manifold known as a Morita equivalence. This construction will be illustrated through the example of toric Poisson structures.

https://arxiv.org/pdf/1906.11251.pdf

The puzzling phenonenon of wave localization refers to unexpected confinement of waves triggered by disorder in the propagating medium. Localization arises in many physical and mathematical systems and has many important implications and applications. A particularly important case is the Schrödinger equation of quantum mechanics, for which the localization behavior is crucial to the electrical properties of materials. Mathematically it is tied to exponential decay of eigenfunctions of operators instead of their expected extension throughout the domain. Although localization has been studied by physicists and mathematicians for the better part of a century, many aspects remain mysterious. In particular, the sort of deterministic quantitative results needed to predict, control, and exploit localization have remained elusive. This talk will focus on major strides made in recent years based on the introduction of the landscape function and its partner, the effective potential. We will describe these developments from the viewpoint of a computational mathematician who sees the landscape theory as a completely unorthodox sort of a numerical method for computing spectra.

In my talk I will discuss the use of topological methods in the analysis of neural data. I will show how to obtain good state spaces for Head Direction Cells and Grid Cells. Topological decoding shows how neural firing patterns determine behaviour. This is a local to global situation which gives rise to some reflections.

This week's Fridays@2 will be a panel discussion focusing on what it is like to pursue a research degree. The panel will share their thoughts and experiences in a question-and-answer session, discussing some of the practicalities of being a postgraduate student, and where a research degree might lead afterwards. Participants include:

Jono Chetwynd-Diggle (Smith Institute)

Victoria Patel (PDE CDT, Mathematical Institute)

Robin Thompson (Christ Church)

Rosemary Walmsley (DPhil student Health Economics Research Centre, Oxford) 

The spatial coordination of cellular differentiation enables functional organogenesis. How coordination results in specific patterns of differentiation in a robust manner is a fundamental question for all developmental systems in biology. Theoreticians such as Turing and Wolpert have proposed the importance of specific mechanisms that enable certain types of patterns to emerge, but these mechanisms are often difficult to identify in natural systems. Therefore, we have started using synthetic biology to ask whether specific mechanisms of pattern formation can be engineered into a simple cellular background. In this talk, I will show several examples of emergent spatial patterning that results from the insertion of synthetic signalling pathways and transcriptional logic into E. coli. In all cases, we use computational modelling to initially design circuits with a desired outcome, and improve the selection of biological components (DNA sub-sequences) that achieve this outcome according to a quantifiable measure. In the specific case of Turing patterns, we have yet to produce a functional system in vivo, but I will describe new analytical tools that are helping to guide the design of synthetic circuits that can produce a Turing instability.

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.