Sam Howison's 60th birthday workshop June 2018

• Sam Howison's 60th birthday workshop will take place at the Mathematical Institute, Oxford University on June 27-28th 2018.

Programme

All talks will take place in lecture room 3 (L3) and all refreshments (tea/coffee/lunches) will be in the north crystal of the mezzanine (tbc).

Titles and Abstracts (alphabetical by speaker)

• David Abrahams:  Improved convergence of modal expansions for waveguide problems

Modal expansions are the natural representation of the solution of a very wide range of waveguide problems in acoustics, elasticity and electromagnetics. However, it has long been recognised that the convergence of such modal sums can be very poor indeed, due to (weakly) singular behaviour of the field at points of discontinuity or change in geometry. Many methods have been derived to enhance convergence, but few take into consideration the precise form of this singular wave-field. We show that, by careful consideration of the latter, a remarkable increase in convergence can be achieved. This new approach is demonstrated by way of several specific examples.

• Chris Budd:  Things that go bang in the night

Many physical systems appear to develop singularities and blow up in a finite time. Such systems are hard both to analyse and compute on. However a touch of the self similar methods I learned whilst I was at Oxford with Sam can not only transform the analysis by also give vastly improved numerical methods. However some nasty surprises as ever lurk.just below the surface. In this talk I will briefly review some of the ideas behind the analysis and computation of blowing up solutions. I will finish with some recent results on blow up in the generalized KdV equation. ( Joint work with Vivi Rottschafer, Leiden)

• Rene Carmona:  Incentivizing Players in a Mean Field Game

Joint work with Francois Delarue. The talk will attempt to quantify the efficiency (or lack thereof) of Mean Field Games (MFGs) equilibria, and propose ways to incentivize players, in the spirit of the theory of mechanism design,  for them to adopt an optimal behavior or generate an optimal value.  On the one hand, the social cost of MFG Nash equilibria is higher than the social cost of an equilibrium resulting from a central planner optimization. This property is well known. It has been documented and quantified for many models under the name of Price of Anarchy (PoA). On the other hand, while MFG Nash equilibria are stable by their very nature, We show that equilibria resulting from a central planner optimization are unstable and we propose  a notion of price of instability (PoI) for the social welfare equilibrium as identified by the solution of a central planner optimization problem. We compute (analytically and numerically) this PoI measure for linear quadratic models.

• John Chadam:  An Optimal Stopping Problem Arising from Hedge Fund Investing

We analyze the optimal withdrawal time for an investor in a hedge fund with a first-loss or shared-loss fee structure, given as the solution of an optimal stopping problem on the fund's assets with a piecewise linear payoff function. Assuming that the underlying follows a geometric Brownian motion, we present a complete solution of the problem in the infinite horizon case, showing that the continuation region is a finite interval, and that the smooth- fit condition may fail to hold at one of the endpoints. In the finite horizon case, we show the existence of a pair of optimal exercise boundaries and analyze their properties, including smoothness and convexity.

• John Chapman:  Do the experts know what they are talking about? 

• Michael Coughlan:  Network models for melt ponds on sea ice

Arctic sea ice forms a thin layer of ice at the ocean surface, which mediates key climate feedbacks. During summer, surface melting produces considerable volumes of water, which collect on the ice surface in ponds in low-lying ice topography. These ponds begin to connect as the melt season progresses. Connected ponds have a complicated, fractal geometry and vary in area from tens to thousands of square meters. Their collective area contributes strongly to changes in the ice-albedo feedback, where the area and depth of melt ponds decreases the average surface albedo of the Arctic and leads to an increase in overall melting. The areal coverage of ponds in the Arctic is difficult to estimate, and they are currently poorly represented in climate models.  Recently, network models have been used to understand the evolution of these ponds and to predict their extent, however these models have lacked important physics. In this talk I will discuss preliminary attempts to model the growth of single ponds and the modelling of a network of ponds accounting for the different ways in which they interconnect and affect each other.

• Linda Cummings:  Slow viscous flows in doubly-connected domains

We discuss the problem of two-dimensional Stokes flow with two free boundaries, at each of which a constant surface tension acts.  An integration of the stress boundary conditions leads to (time-dependent) constants of integration associated with each boundary and, while these can be chosen for convenience at one of the free boundaries, they remain arbitrary at the second and must be determined as part of the solution.  Following early work of Hopper and Richardson for the simply-connected problem, and of Crowdy & Tanveer and Richardson for the doubly-connected one (as well as subsequent work by these and other authors), conformal mapping methods from complex analysis are used to reformulate the problem. By means of analytic continuation of the complex form of the boundary conditions at each free boundary, two globally-valid equations governing the motion are obtained: these must be consistent in order for solutions to exist. We discuss the implications of this consistency condition, together with some open questions. Our results are illustrated by means of specific examples. 

• Ian Frigaard:  Stuck on Fluids  

• Martin Gould:  Counterparty Credit Limits: An Effective Tool for Mitigating Counterparty Risk?

A counterparty credit limit (CCL) is a limit imposed by a financial institution to cap its maximum possible exposure to a specified counterparty. Although CCLs are designed to help institutions mitigate counterparty risk by selective diversification of their exposures, their implementation restricts the set of trading opportunities that institutions can access. We address the question of how this mechanism impacts trade prices and volatility, both empirically and via a new model of trading with CCLs. Empirically, we find that CCLs cause little impact on the trades that we study. However, our model highlights that in extreme situations, CCLs could serve to destabilize prices and thereby influence systemic risk.  

• Ben Hambly:  Simple models for systemic risk

We formulate a simple model for the interaction between banks and, by taking a large population limit, derive a stochastic McKean-Vlasov equation for the evolution of the banking system as a whole. Solutions to this equation can break down, depending on the parameters, and we could interpret these breakdowns as financial crises. We will discuss the mathematical results for these models as well as their connections to other applications.  

• John King:  High-order diffusion and moving-boundary problems

Some rather specific, but to me interesting, mathematical aspects will be noted. 

• Andrew Lacey:  A Coupled Fluid-Layer and Plate Model for Micro-Electro-Mechanical Systems

An electrostatic micro-electro-mechanical system (MEMS) is modelled as a flexible plate or membrane moving under the influence of electrical attraction towards a fixed base plate. Motion is resisted by pressure of a gas in the thin gap between the plates which results in a damping effect.  A two-ODE version of the coupled problem is derived and considered, to gain a qualitative understanding of possible behaviour. In particular, it is of interest as to whether the solution to the mathematical model tends to a steady state or instead ceases to exist after a finite time. The latter corresponds to "touch-down" in the MEMS device, when the flexible plate comes into contact with the base plate. 

• Terry Lyons:  Signatures, unparameterised paths, and the classification of streams 

• Hilary Ockendon:  The Director and the delta shock

Sam took over the Directorship of OCIAM from me in 2002 and soon after we were involved in some work on delta shocks.  This talk will combine some ideas on delta shocks with some reminiscences of Sam as Director. 

• John Ockendon:  Sam's Way

This talk will describe a few of the many adventures that I have had with Sam over the past 40 years, both mathematical and otherwise. 

• James Oliver:  Some open free boundary problems 

• Roxana Pamfil:  Analysing consumer preference in grocery stores using annotated networks

Transactions in a store can be represented as a bipartite network of customers connected to products that they previously purchased. Dense clusters or "communities" in such networks reveal sets of customers with similar shopping patterns, and sets of products that are preferred by the same customer types. Once customers and products are partitioned in this way, one can fill in "missing links" within communities to obtain new product recommendations.  For a given network, there may be several different partitions of the nodes into communities that are close to optimal. As shown previously in the literature, using additional data in the form of node annotations can improve the quality and interpretability of the output partition. For our application, we annotate product nodes with their corresponding category (e.g., "berries" or "plain yoghurt"). Detecting communities in this network reveals sets of customers with distinctive shopping patterns, including a group of price-sensitive customers who prefer snacks and frozen meals and a group of people who are more likely to buy produce, fresh meat, and other ingredients for cooking at home. 

• Colin Please:  Some open problems inspired by Sam

This talk will outline some simple mechanics problems inspired by Sam that pose some interesting questions. A recent free boundary problem arising from an industrial problem that Sam has worked on will also be  considered. Observations on Sam's style will also be discussed.  

• Mason Porter:  Punctuating Literature with Time-Series Analysis

Whether enjoying the lucid prose of Samuel D. Howison or slogging through the cumbersome, seemingly mindless, heavy-set prattle (full of parentheses, em-dashes, compound adjectives, and Oxford commas) of Mason A. Porter, readers will notice stylistic signatures not only in word choice and grammar, but also in punctuation itself. Indeed, visual sequences of punctuation from different authors produce marvelously different (and visually striking) sequences.  After spending several years discussing the trivial subtleties of grammar and punctuation --- much to the dismay of our students, to whom we were supposed to be making substantive comments about their academic work --- Sam and I encountered a blog entry by Adam Calhoun with striking visualizations of punctuation, and we asked if we could quantify punctuation sequences. In this talk, I'll discuss where we are with this project. Are the properties of such sequences a distinguishing feature of different authors? Is it possible to distinguish literary genres based on their punctuation sequences?  Do the punctuation styles of authors evolve over time? Are Sam and I on to something interesting in trying to do writing analysis without words, or are we full of sound and fury (signifying nothing)?  This project is joint work with Alexandra Darmon, Marya Bazzi, and (obviously) Sam Howison. 

• Christoph Reisinger:  The Asymptotics of Penalisation

This talk straddles three areas Sam has made instrumental contributions to: free boundary problems, matched asymptotic expansions, and mathematical finance. Our starting point is the optimal stopping problem associated with American option pricing, where penalisation is a key tool for both the theoretical analysis and efficient computations. In joint work with Sam and Jan-Hendrik Witte (2013), we gave a precise asymptotic description of the penalised value function and the resulting approximate exercise boundary. Recently, with Yufei Zhang (2018), we extended the penalty approach to mixed optimal stopping and control problems in infinite activity jump models, where the performance is evaluated by nonlinear expectations. Although we do not have a fine-grained picture as in the earlier case, the high-level results are still valid. 

• Caoimhe Rooney:  Homogenisation Applied to Electrical Calcination of Carbon Materials

Calcination describes the heat treatment of anthracite particles in a furnace to produce a partially-graphitised material which is suitable for use in electrodes and for other met- allurgical applications. Electric current is passed through a bed of anthracite particles, here referred to as a coke bed, causing Ohmic heating and high temperatures which result in the chemical and structural transformation of the material.  Understanding the behaviour of such mechanisms on the scale of a single particle is often dealt with through the use of computational models such as DEM (Discrete Element Methods). However, because of the great discrepancy between the length scale of the particles and the length scale of the furnace, we can exploit asymptotic homogenisation theory to simplify the problem.  In this talk, we will present some results relating to the electrical and thermal conduction through granular material which define effective quantities for the conductivities by considering a microscopic representative volume within the material. The effective quantities are then used as parameters in the homogenised macroscopic model to describe calcination of anthracite. 

• William Shaw:  Differential equations in finance and statistics

Many applications to finance of differential equations have been developed by Sam Howison and his colleagues. I had cause to utilise his ideas in the industrial context of building a path-dependent convertible bond model, and in the research context of using asymptotic methods to develop an accurate and computationally straightforward approach to Asian options. That work was done with Jeff Dewynne and extended by my student Sergei Siyanko. More recent work has centred on the elucidation and solution of the non-linear ODEs that govern quantile functions. Classical series methods may be deployed generating intriguing non-linear recurrence equations, but more ground has been taken using changes of variable to recycle quantiles from one base distribution into another. This has lead to efficient methods for the simulation of random variables in GPU environments. I’ll finish by touching briefly on the integro-differential systems that map a characteristic function directly to the associated quantile function. 

• Ronnie Sircar:  Trading, Market Impact and Nonlinear Systems

We discuss problems where impact from optimal or equilibrium trading leads to challenging nonlinear systems and fixed point problems. These may arise from: (i) Market impact from a significant group of portfolio optimizers in a constrained market with clearing conditions; (ii) High frequency trading to a target; (iii) Oligopolies with a small number of influential players, or a continuum of players with aggregate impact; (iv) Optimal execution where trading speed is penalized.  These are addressed with computational and analytical methods, and well-posedness of the problems is crucial in the absence of general theory. 

• Stephen Wilson:  Evaporating Droplets

Accommodation

June is a rather busy time of the year in Oxford. We advise conference participants to book accommodation as early as they can to avoid disappointment. Here is a selection of hotels, but there are many more around the area.

http://www.cotswoldlodgehotel.co.uk/
http://www.galaxie.co.uk/
http://www.cotswoldhouse.co.uk/
http://www.parklandsoxford.co.uk/
http://macdonald-randolph.hotel-rn.com/?lbl=ggl
http://www.keble.ox.ac.uk/bed-breakfast

http://www.universityrooms.com  (for college accommodation)