Past

A model of a financial market is complete if any payoff can be obtained as the terminal value of a self-financing trading strategy. It is well known that numerous models, for example stochastic volatility models, are however incomplete. We present conditions, which, in a general diffusion framework, guarantee that in such cases the market of primitive assets enlarged with an appropriate number of traded derivative contracts is complete. From a purely mathematical point of view we prove an integral representation theorem which guarantees that every local Q-martingale can be represented as a stochastic integral with respect to the vector of primitive assets and derivative contracts.

Evolution by natural selection has resulted in a remarkable diversity of organism morphologies. But is it possible for developmental processes to create “any possible shape?” Or are there intrinsic constraints? I will discuss our recent exploration into the shapes of bird beaks. Initially, inspired by the discovery of genes controlling the shapes of beaks of Darwin's finches, we showed that the morphological diversity in the beaks of Darwin’s Finches is quantitatively accounted for by the mathematical group of affine transformations. We have extended this to show that the space of shapes of bird beaks is not large, and that a large phylogeny (including finches, cardinals, sparrows, etc.) are accurately spanned by only three independent parameters -- the shapes of these bird beaks are all pieces of conic sections. After summarizing the evidence for these conclusions, I will delve into our efforts to create mathematical models that connect these patterns to the developmental mechanism leading to a beak. It turns out that there are simple (but precise) constraints on any mathematical model that leads to the observed phenomenology, leading to explicit predictions for the time dynamics of beak development in song birds. Experiments testing these predictions for the development of zebra finch beaks will be presented.

Based on the following papers:

http://www.pnas.org/content/107/8/3356.short

http://www.nature.com/ncomms/2014/140416/ncomms4700/full/ncomms4700.html

Ice sheets are among the key controls on global climate and sea-level change. A detailed understanding of ice sheet dynamics is crucial so to make accurate predictions of their mass balance into the future. Ice streams are the dominant negative component in this balance, accounting for up to 90$\%$ of the Antarctic ice flux into ice shelves and ultimately into the sea. Despite their importance, our understanding of ice-stream dynamics is far from complete.

A range of observations associate ice streams with meltwater. Meltwater lubricates the ice at its bed, allowing it to slide with less internal deformation. It is believed that ice streams may appear due to a localization feedback between ice flow, basal melting and water pressure in the underlying sediments. I will present a model of subglacial water flow below ice sheets, and particularly below ice streams. This hydrologic model is coupled to a model for ice flow. I show that under some conditions this coupled system gives rise to ice streams by instability of the internal dynamics.

We are dwelling in the Big Data age. The diversity of the uses of Big Data unleashes limitless possibilities. Many people are talking about ways to use Big Data to track the collective human behaviours, monitor electoral popularity, and predict financial fluctuations in stock markets, etc. Big Data reveals both challenges and opportunities, which are not only related to technology but also to human itself. This talk will cover various current topics and trends in Big Data research. The speaker will share his relevant experiences on how to use analytics tools to obtain key metrics on online social networks, as well as present the challenges of Big Data analytics.

Bio: Ning Wang (Ph.D) works as Researcher at the Oxford Internet Institute. His research is driven by a deep interest in analysing a wide range of sociotechnical problems by exploiting Big Data approaches, with the hope that this work could contribute to the intersection of social behavior and computational systems.

An important military task in a high-technology environment is to understand the set of radars present in it, since the radars will be, to a greater or lesser extent, indicative of the ships, aircraft and other military units which are present.

The transmissions of the different radars typically overlap in most of the dimensions which characterise then, such as frequency and bearing, and their pulses are interleaved in time. If, however, we are able to separate the individual pulse trains which are present then not only does this allow us to know how many different radars are present, but the characteristics of the pulse train are indicative of the type of the radar.

The problem of recognising the pulse trains is not trivial, because many radars 'jitter' their transmissions and pulses may be missing or two pulses may occur together, causing the characteristics of the pulse to be 'garbled.' The jittering may be used as a way to reject mutual interference between the radars, to resolve ambiguities in measurements of range or velocity or to make it harder to jam the radar.

The problems caused by pulses overlapping are likely to become more severe in the future because the pulses of the individual radars are becoming longer.

Although solutions currently exist which can cope, to at least some extent, with most of these issues, the purpose of bringing this topic to the seminar is to allow a fresh look at the problem from first principles.

The most valuable asset that people in a sovereign state can have is good, sustainable governance. Setting up a system of good, sustainable governance is not easy. The big and well-known problem is time inconsistency of optimal policies. A mechanism that has proven valuable in mitigating the time inconsistency problem is rule by law. The too-big-to-fail problem in banking is the result of the time inconsistency problem. In this lecture I will argue there is an alternative financial system that is not subject to the too-big-to-fail problem. The alternative arrangement I propose is a pure transaction banking system. Transaction banks are required to hold 100$\%$ interest bearing reserves and can pay tax-free interest on demand deposits. With this system, there cannot be a bank run as there is no place to run to. Mutual arrangements would finance all business investment, which is not currently the case.

I will give a short introduction to geometric stability theory and independence relations, focussing on the tree properties. I will then introduce one of the main examples for general measureable structures, the two sorted structure of a vector space over a field with a bilinear form. I will state some results for this structure, and give some open questions. This is joint work with William Anscombe.

We give an overview of Kitaev's lattice model in the setting of an arbitrary finite group G (where $G = Z_{2}$ is the famous Toric Code). We also exhibit the connection this model has with so-called 123-TQFTs (topological quantum field theories), making use of ideas coming from higher gauge theory and Hopf algebra representations.

The synthesis of complex-shaped colloids and nanoparticles has recently undergone unprecedented advancements. It is now possible to manufacture particles shaped as dumbbells, cubes, stars, triangles, and cylinders, with exquisite control over the particle shape. How can particle geometry be exploited in the context of capillarity and surface-tension phenomena? This talk examines this question by exploring the case of complex-shaped particles adsorbed at the interface between two immiscible fluids, in the small Bond number limit in which gravity is not important. In this limit, the "Cheerio's effect" is unimportant, but interface deformations do emerge. This drives configuration dependent capillary forces that can be exploited in a variety of contexts, from emulsion stabilisation to the manufacturing of new materials. It is an opportunity for the mathematics community to get involved in this field, which offers ample opportunities for careful mathematical analysis. For instance, we find that the mathematical toolbox provided by 2D potential theory lead to remarkably good predictions of the forces and torques measured experimentally by tracking particle pairs of cylinders and ellipsoids. New research directions will also be mentioned during the talk, including elasto-capillary interactions and the simulation of multiphase composites.

For the last few years Soundararajan and I have been developing an alternative "pretentious" approach to analytic number theory. Recently Harper established a more intuitive proof of Halasz's Theorem, the key result in the area, which has allowed the three of us to provide new (and somewhat simpler) proofs to several difficult theorems (like Linnik's Theorem), as well as to suggest some new directions. We shall review these developments in this talk.

This talk will be a brief introduction to some standard conjectures surrounding motivic L-functions, which might be viewed as the arithmetic motivation for Langlands reciprocity.

I will discuss the well-posedness of a new nonlinear model for nematic

elastomers. The main novelty is that the Frank energy penalizes

spatial variations of the nematic director in the deformed, rather

than in the reference configuration, as it is natural in the case of

large deformations.

Riley and Dison's hydra groups are a family of group and subgroup pairs $(G_k, H_k)$ for which the subgroup $H_k$ has distortion like the $k$-th Ackermann function. One wants to know if finite quotients can distinguish elements that are not in $H_k$, as a positive answer would allow you to construct a hands-on family of finitely presented, residually finite groups with arbitrarily large Dehn functions. I'll explain why we get a negative answer.

Gromov and Piatetski-Shapiro proved the existence of finite volume non-arithmetic hyperbolic manifolds of any given dimension. In dimension four and higher, we show that there are about $v^v$ such manifolds of volume at most $v$, considered up to commensurability. Since the number of arithmetic ones tends to be polynomial, almost all hyperbolic manifolds are non-arithmetic in an appropriate sense. Moreover, by restricting attention to non-compact manifolds, our result implies the same growth type for the number of quasi- isometry classes of lattices in $SO(n,1)$. Our method involves a geometric graph-of-spaces construction that relies on arithmetic properties of certain quadratic forms.

A joint work with Arie Levit.

We will give an idea of derived algebraic geometry and sketch a number of more or less recent directions, including derived symplectic geometry, derived Poisson structures, quantizations of moduli spaces, derived analytic geometry, derived logarithmic geometry and derived quadratic structures.

Abstract: We propose a term structure power price model that, in contrast to widely accepted no-arbitrage based approaches, accounts for the non-storable nature of power. It belongs to a class of equilibrium game theoretic models with players divided into producers and consumers. Consumers' goal is to maximize a mean-variance utility function subject to satisfying inelastic demand of their own clients (e.g households, businesses etc.) to whom they sell the power on. Producers, who own a portfolio of power plants each defined by a running fuel (e.g. gas, coal, oil...) and physical characteristics (e.g. efficiency, capacity, ramp up/down times, startup costs...), would, similarly, like to maximize a mean-variance utility function consisting of power, fuel, and emission prices subject to production constraints. Our goal is to determine the term structure of the power price at which production matches consumption. In this talk we outline that such a price exists and develop conditions under which it is also unique. Under condition of existence, we propose a tractable quadratic programming formulation for finding the equilibrium term structure of the power price. Numerical results show performance of the algorithm when modeling the whole system of UK power plants.

We consider the market for a risky asset for which agents have interdependent private valuations. We study competitive rational expectations equilibria under the standard CARA-normal assumptions. Equilibrium is partially revealing even though there are no noise traders. Complementarities in information acquisition arise naturally in this setting. We characterize stable equilibria with endogenous information acquisition. Our framework encompasses the classical REE models in the CARA-normal tradition.

Inexact hardware trades reduced numerical precision against a reduction

in computational cost. A reduction of computational cost would allow

weather and climate simulations at higher resolution. In the first part

of this talk, I will introduce the concept of inexact hardware and

provide results that show the great potential for the use of inexact

hardware in weather and climate simulations. In the second part of this

talk, I will discuss how rounding errors can be assessed if the forecast

uncertainty and the chaotic behaviour of the atmosphere is acknowledged.

In the last part, I will argue that rounding errors do not necessarily

degrade numerical models, they can actually be beneficial. This

conclusion will be based on simulations with a model of the

one-dimensional Burgers' equation.

Biharmonic maps are the solutions of a variational problem for maps

between Riemannian manifolds. But since the underlying functional

contains nonlinear differential operators that behave badly on the usual

Sobolev spaces, it is difficult to study it with variational methods. If

the target manifold has enough symmetry, however, then we can combine

analytic tools with geometric observations and make some statements

about existence and regularity.

Pancakes.

Abstract: We introduce a new framework for defining integration against rough path. This framework generalizes rough integral, and gives a natural explanation of some of the regularity requirements in rough path theory.

The filtration on the infinite symmetric product of spheres by number of

factors provides a sequence of spectra between the sphere spectrum and

the integral Eilenberg-Mac Lane spectrum. This filtration has received a

lot of attention and the subquotients are interesting stable homotopy

types.

In this talk I will discuss the equivariant stable homotopy types, for

finite groups, obtained from this filtration for the infinite symmetric

product of representation spheres. The filtration is more complicated

than in the non-equivariant case, and already on the zeroth homotopy

groups an interesting filtration of the augmentation ideal of the Burnside

rings arises. Our method is by `global' homotopy theory, i.e., we study

the simultaneous behaviour for all finite groups at once. In this context,

the equivariant subquotients are no longer rationally trivial, nor even

concentrated in dimension 0.

When $\partial D$ is spatially homogeneous, we show that we can recover the lower and upper Minkowski dimensions of $\partial D$ from the sort time behaviour of $E(s)$. Furthermore, when the Minkowski dimension exists, finer geometric fluctuations can be recovered and $E(s)$ is controlled by $s^\alpha e^{f(\log(1/s))}$ for small $s$, for some $\alpha \in (0, \infty)$ and some regularly varying function $f$. The function $f$ is not constant is general and carries some geometric information.

When $\partial D$ is statistically self-similar, the Minkowski dimension and content of $\partial D$ typically exist and can be recovered from $E(s)$. Furthermore, $E(s)$ has an almost sure expansion $E(s) = c s^{\alpha} N_\infty + o(s^\alpha)$ for small $s$, for some $c$ and $\alpha \in (0, \infty)$ and some positive random variable $N_\infty$ with unit expectation arising as the limit of some martingale. In some cases, we can show that the fluctuations around this almost sure behaviour are governed by a central limit theorem, and conjecture that this is true more generally.

This is based on joint work with David Croydon and Ben Hambly.

We are dwelling in the Big Data age. The diversity of the uses

of Big Data unleashes limitless possibilities. Many people are talking

about ways to use Big Data to track the collective human behaviours,

monitor electoral popularity, and predict financial fluctuations in

stock markets, etc. Big Data reveals both challenges and opportunities,

which are not only related to technology but also to human itself. This

talk will cover various current topics and trends in Big Data research.

The speaker will share his relevant experiences on how to use analytics

tools to obtain key metrics on online social networks, as well as

present the challenges of Big Data analytics.

\\

Bio: Ning Wang (Ph.D) works as Researcher at the Oxford Internet

Institute. His research is driven by a deep interest in analysing a wide

range of sociotechnical problems by exploiting Big Data approaches, with

the hope that this work could contribute to the intersection of social

behavior and computational systems.

We consider a large class of three dimensional continuous dynamic fluctuation models, and show that they all rescale and converge to the stochastic Allen-Cahn equation, whose solution should be interpreted after a suitable renormalization procedure. The interesting feature is that, the coefficient of the limiting equation is different from one's naive guess, and the renormalization required to get the correct limit is also different from what one would naturally expect. I will also briefly explain how the recent theory of regularity structures enables one to prove such results. Joint work with Martin Hairer.

Cardinal characteristics of the continuum are (definitions for) cardinals that are provably uncountable and at most the cardinality c of the reals, but which (if the continuum hypothesis fails) may be strictly less than c.  Cichon's diagram is a standard diagram laying out all of the ZFC-provable inequalities between the most familiar cardinal characteristics of the continuum.  There is a natural analogy that can be drawn between these cardinal characteristics and highness properties of Turing oracles in computability theory, with implications taking the place of inequalities.  The diagram in this context is mostly the same with a few extra equivalences: many of the implications were trivial or already known, but there remained gaps, which in joint work with Brendle, Ng and Nies we have filled in.

Topological insulators are a type of system in condensed matter physics that exhibit a robustness that physicists like to call topological. In this talk I will give a definition of a subclass of such systems: gapped, free fermions. We will look at how such systems, as shown by Kitaev, can be classified in terms of topological K-groups by using the Clifford module model for K-theory as introduced by Atiyah, Bott and Shapiro. I will be using results from Wednesday's JTGT, where I'll give a quick introduction to topological K-theory.

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Molecular modelling has become a valuable tool and is increasingly part of the standard methodology of chemistry, physics, engineering and biology. The power of molecular modelling lies in its versatility: as potential energy functions improve, a broader range of more complex phenomena become accessible to simulation, but much of the underlying methodology can be re-used. For example, the Verlet method is still the most popular molecular dynamics scheme for constant energy molecular dynamics simulations despite it being one of the first to be proposed for the purpose.

One of the most important challenges in molecular modelling remains the computation of averages with respect to the canonical Gibbs (constant temperature) distribution, for which the Verlet method is not appropriate. Whereas constant energy molecular dynamics prescribes a set of equations (Newton's equations), there are many alternatives for canonical sampling with quite different properties. The challenge is therefore to identify formulations and numerical methods that are robust and maximally efficient in the computational setting.

One of the simplest and most effective methods for sampling is based on Langevin dynamics which mimics coupling to a heat bath by the incorporation of random forces and an associated dissipative term. Schemes for Langevin dynamics simulation may be developed based on the familiar principle of splitting. I will show that the invariant measure ('long term') approximation may be strongly affected by a simple re-ordering of the terms of the splitting. I will describe a transition in weak numerical order of accuracy that occurs (in one case) in the t->infty limit.

I will also entertain some more radical suggestions for canonical sampling, including stochastic isokinetic methods that enable the use of greatly enlarged timesteps for expensive but slowly-varying force field components.

We will define the Ran space as well as Ran space versions of some of the prestacks we've already seen, and explain what is meant by the homology of a prestack. Following Gaitsgory and possibly Drinfeld, we'll show how the Ran space machinery can be used to prove that the space of rational maps is homologically contractible.

For many questions of scientific interest, all-atom molecular simulations are still out of reach, in particular in materials engineering where large numbers of atoms, and often expensive force fields, are required. A long standing challenge has been to construct concurrent atomistic/continuum coupling schemes that use atomistic models in regions of space where high accuracy is required, with computationally efficient continuum models (e.g., FEM) of the elastic far-fields.

Many different mechanisms for achieving such atomistic/continuum couplings have been developed. To assess their relative merits, in particular accuracy/cost ratio, I will present a numerical analysis framework. I will use this framework to analyse in more detail a particularly popular scheme (the BQCE scheme), identifying key approximation parameters which can then be balanced (in a surprising way) to obtain an optimised formulation.

Finally, I will demonstrate that this analysis shows how to remove a severe bottlenecks in the BQCE scheme, leading to a new scheme with optimal convergence rate.

Algebraically closed fields, and in general varieties are among the first examples
of Zariski Geometries.
I will consider specialisations of algebraically closed fields and varieties.
In the case of an algebraically closed field K, I will show that a specialisation
is essentially a residue map, res from K to a residue field k.  
In both cases I will show universality of the specialisation is controlled by the
transcendence degree of K over k.  

In the late 70s -- early 80s Makanin came up with a very simple, but very powerful idea to approach solving equations in free groups. This simplicity makes Makanin-like procedures ubiquitous in mathematics: in dynamical systems, geometric group theory, 3-dimensional topology etc. In this talk I will explain loosely how Makanin's algorithm works.

In 1878, Jordan showed that if $G$ is a finite group of complex $n \times n$ matrices, then $G$ has a normal subgroup whose index in $G$ is bounded by a function of $n$ alone. He showed only existence, and early actual bounds on this index were far from optimal. In 1985, Weisfeiler used the classification of finite simple groups to obtain far better bounds, but his work remained incomplete when he disappeared. About eight years ago, I obtained the optimal bounds, and this work has now been extended to subgroups of all (complex) classical groups. I will discuss this topic at a “colloquium” level – i.e., only a rudimentary knowledge of finite group theory will be assumed.

Auctioneers may wish to sell related but different indivisible goods in

a single process. To develop such techniques, we study the geometry of

how an agent's demanded bundle changes as prices change. This object

is the convex-geometric object known as a `tropical hypersurface'.

Moreover, simple geometric properties translate directly to economic

properties, providing a new taxonomy for economic valuations. When

considering multiple agents, we study the unions and intersections of

the corresponding tropical hypersurfaces; in particular, properties of

the intersection are deeply related to whether competitive equilibrium

exists or fails. This leads us to new results and generalisations of

existing results on equilibrium existence. The talk will provide an

introductory tour to relevant economics to show the context of these

applications of tropical geometry. This is joint work with Paul

Klemperer.