Past

There is a well-acknowledged analogy between mapping class
groups and lattices in higher rank groups. I will discuss to which
extent does Margulis's superrigidity hold for mapping class groups:
examples, very partial results and questions.

Counting nodal curves in linear systems $|L|$ on smooth projective surfaces $S$ is a problem with a long history. The G\"ottsche conjecture, now proved by several people, states that these counts are universal and only depend on $c_1(L)^2$, $c_1(L)\cdot c_1(S)$, $c_1(S)^2$ and $c_2(S)$. We present a quite general definition of reduced Gromov-Witten and stable pair invariants on S. The reduced stable pair theory is entirely computable. Moreover, we prove that certain reduced Gromov-Witten and stable pair invariants with many point insertions coincide and are both equal to the nodal curve counts appearing in the Göttsche conjecture. This can be seen as version of the MNOP conjecture for the canonical bundle $K_S$. This is joint work with R. P. Thomas.

We consider a broker who has to place a large order which consumes a sizable part of average daily trading volume. By contrast to the previous literature, we allow the liquidity parameters of market depth and resilience to vary deterministically over the course of the trading period. The resulting singular optimal control problem is shown to be tractable by methods from convex analysis and, under

minimal assumptions, we construct an explicit solution to the scheduling problem in terms of some concave envelope of the resilience adjusted market depth.

When ice is raised to a temperature above its usual melting temperature
of 273 K, small cylindrical discs of water form within the bulk of the
ice. Subsequent internal melting of the ice causes these liquid discs to
grow radially outwards. However, many experimentalists have observed
that the circular interface of these discs is unstable and eventually
the liquid discs turn into beautiful shapes that resemble flowers or
snowflakes. As a result of their shape, these liquid figures are often
called liquid snowflakes. In this talk I'll discuss a simple
mathematical model of liquid snowflake formation and I'll show how a
combination of analytical and numerical methods can yield much insight
into the dynamics which govern their growth.

Given a form $F(x)$, the circle method is frequently used to provide an asymptotic for the number of representations of a fixed integer $N$ by $F(x)$. However, it can also be used to prove results of a different flavor, such as showing that almost every number (in a certain sense) has at least one representation by $F(x)$. In joint work with Roger Heath-Brown, we have recently considered a 2-dimensional version of such a problem. Given two quadratic forms $Q_1$ and $Q_2$, we ask whether almost every integer (in a certain sense) is simultaneously represented by $Q_1$ and $Q_2$. Under a modest geometric assumption, we are able to prove such a result if the forms are in $5$ variables or more. In particular, we show that any two such quadratic forms must simultaneously attain prime values infinitely often. In this seminar, we will review the circle method, introduce the idea of a Kloosterman refinement, and investigate how such "almost all" results may be proved.

Abstract: By using the Skorohod equation we derive an
iteration procedure which allows us to solve a class of reflected backward
stochastic differential equations with non-linear resistance induced by the
reflected local time. In particular, we present a new method to study the
reflected BSDE proposed first by El Karoui et al. (1997).

Abstract: In the talk we first introduce the level set equation for the evolution by mean curvature flow, explaining the main difference between the standard Euclidean case and the horizontal evolution.

Then we will introduce a stochastic representation formula for the viscosity solution of the level set equation related to the value function of suitable associated stochastic controlled ODEs which are motivated by a concept of intrinsic Brownian motion in Carnot-Caratheodory spaces.

In an incomplete continuous-time securities market with uncertainty generated by Brownian motions, we derive closed-form solutions for the equilibrium interest rate and market price of risk processes. The economy has a finite number of heterogeneous exponential utility investors, who receive partially unspanned income and can trade continuously on a finite time-interval in a money market account and a single risky security. Besides establishing the existence of an equilibrium, our main result shows that if the investors' unspanned income has stochastic countercyclical volatility, the resulting equilibrium can display both lower interest rates and higher risk premia compared to the Pareto efficient equilibrium in an otherwise identical complete market. This is joint work with Peter Ove Christensen.

• Matt Webber - ‘Stochastic neural field theory’
• Yohan Davit - ‘Multiscale modelling of deterministic problems with applications to biological tissues and porous media’
• Patricio Farrell - ‘An RBF multilevel algorithm for solving elliptic PDEs’

I shall present a very general class of virtual elements in a structure, ultraimaginaries, and analyse their model-theoretic properties.

The use of formal mathematical models in sociology started in the 1940s and attracted mathematicians such as Frank Harary in the 1950s. The idea is to take the rather intuitive ideas described in social theory and express these in formal mathematical terms. Social network analysis is probably the best known of these and it is the area which has caught the imagination of a wider audience and has been the subject of a number of popular books. We shall give a brief over view of the field of social networks and will then look at three examples which have thrown up problems of interest to the mathematical community. We first look at positional analysis techniques and give a formulation that tries to capture the notion of social role by using graph coloration. We look at algebraic structures, properties, characterizations, algorithms and applications including food webs. Our second and related example looks at core-periphery structures in social networks. Our final example relates to what the network community refer to as two-mode data and a general approach to analyzing networks of this form. In all cases we shall look at the mathematics involved and discuss some open problems and areas of research that could benefit from new approaches and insights.

The derivatives of PDE models are key ingredients in many

important algorithms of computational science. They find applications in

diverse areas such as sensitivity analysis, PDE-constrained

optimisation, continuation and bifurcation analysis, error estimation,

and generalised stability theory.

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These derivatives, computed using the so-called tangent linear and

adjoint models, have made an enormous impact in certain scientific fields

(such as aeronautics, meteorology, and oceanography). However, their use

in other areas has been hampered by the great practical

difficulty of the derivation and implementation of tangent linear and

adjoint models. In his recent book, Naumann (2011) describes the problem

of the robust automated derivation of parallel tangent linear and

adjoint models as "one of the great open problems in the field of

high-performance scientific computing''.

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In this talk, we present an elegant solution to this problem for the

common case where the original discrete forward model may be written in

variational form, and discuss some of its applications.

This talk will discuss the notion of a Nahm transform in differential geometry, as a way of relating solutions to one differential equation on a manifold, to solutions of another differential equation on a different manifold. The guiding example is the correspondence between solutions to the Bogomolny equations on $\mathbb{R}^3$ and Nahm equations on $\mathbb{R}$. We extract the key features from this example to create a general framework.

Nematic elastomers are rubbery elastic solids made of cross-linked polymeric chains with embedded nematic mesogens. Their mechanical behaviour results from the interaction of electro-optical effects typical of nematic liquid crystals with the elasticity of a rubbery matrix. We show that the geometrically linear counterpart of some compressible models for these materials can be justified via Gamma-convergence. A similar analysis on other compressible models leads to the question whether linearised elasticity can be derived from finite elasticity via Gamma-convergence under weak conditions of growth (from below) of the energy density. We answer to this question for the case of single well energy densities.

We discuss Ogden-type extensions of the energy density currently used to model nematic elastomers, which provide a suitable framework to study the stiffening response at high imposed stretches.

Finally, we present some results concerning the attainment of minimal energy for both the geometrically linear and the nonlinear model.