Past

We define a set of nonlocal operators and develop a nonlocal vector calculus that mimics the classical differential vector calculus. Included are the definitions of nonlocal divergence, gradient, and curl operators and the derivation of nonlocal integral theorems and identities. We indicate how, through certain limiting processes, the nonlocal operators are connected to their differential counterparts. The nonlocal operators are shown to appear in nonlocal models for diffusion and in the nonlocal, spatial derivative free, peridynamics continuum model for solid mechanics. We show, for example, that unlike elliptic partial differential equations, steady state versions of the nonlocal models do not necessary result in the smoothing of data. We also briefly consider finite element methods for nonlocal problems, focusing on solutions containing jump discontinuities; in this setting, nonlocal models can lead to optimally accurate approximations.

This workshop will probably take place at BP's premises.

I describe recent work on the synchronization of IP3R calcium channels in the interior of cells. Hybrid  models of calcium release couple deterministic equations for diffusion and reactions of calcium ions to stochastic gating transitions of channels. I discuss the validity of such models as well as numerical methods.Hybrid models were used to simulate cooperative release events for clusters of channels. I show that for these so-called puffs the mixing assumption for reactants does not hold. Consequently, useful definitions of averaged calcium concentrations in the cluster are not obvious. Effective reaction kinetics can be derived, however, by separating concentrations for self-coupling of channels and coupling to different channels.

Based on the spatial approach, a Markovian model can be inferred, representing well calcium puffs in neuronal cells. I then describe further reduction of the stochastic model and the synchronization arising for small channel numbers. Finally, the effects of calcium binding proteins on duration of release is discussed.

This talk will be based on arxiv:1106.5254.

This talk will be based on arxiv:1106.5254.

This meeting will mark the 80th birthday of Sir Roger Penrose. Twistor theory is one of his most remarkable discoveries and continues to have applications across pure mathematics and mathematical physics. This meeting will focus on some recent developments with speakers both on geometry and physics.

Speakers:

• Nima Arkani-Hamed (IAS, Princeton): Scattering without space-time
• Mike Eastwood (ANU): CR geometry and conformal foliations
• Nigel Hitchin (Oxford): Twistors and Octonions
• Andrew Hodges (Oxford): Polytopes and amplitudes
• Claude LeBrun (SUNY Stony Brook): On Hermitian, Einstein 4-Manifolds
• David Skinner (Perimeter Institute): Scattering amplitudes from holomorphic linking in twistor space
• Paul Tod (Oxford): Conformal cyclic cosmology

Registration will start at 1.30pm on the 21st with the first lecture at 2.15pm. The meeting will finish by 4.30pm on the 22nd. See the programme for more details.

There will be a reception at 6.30pm on the 21st July (Wadham College) followed by dinner at 7.15 in Wadham College.

• Benjamin Franz - "Hybrid modelling of individual movement and collective behaviour"
• Ingrid Von Glehn - "Image Inpainting on Surfaces"
• Rita Schlackow - "Genome-wide analysis of transcription termination regions in fission yeast"

We consider an active scalar equation with singular drift velocity that is motivated by a model for the geodynamo. We show that the non-diffusive equation is ill-posed in the sense of Hadamard in Sobolev spaces. In contrast, the critically diffusive equation is globally well-posed. This work is joint with Vlad Vicol.

There will be a BP workshop but we are waiting for some suggested alternative dates.

Distortion is an asymptotic invariant of the embeddings

of finitely generated algebras. For group embeddings,

it has been introduced by M.Gromov. The main part of

the talk will be based on a recent work with Yu.Bahturin,

where we consider the behavior of distortion functions

for subalgebras of associative and Lie algebras.

Voiculescu showed how the large N limit of the expected value of the trace of a word on n independent hermitian NxN matrices gives a well known von Neumann algebra. In joint work with Guionnet and Shlyakhtenko it was shown that this idea makes sense in the context of very general planar algebras where one works directly in the large N limit. This allowed us to define matrix models with a non-integral  number of random matrices. I will present this work and some of the subsequent work, together with future hopes for the theory.

We present a dynamic bank run model for liquidity risk where a financial institution finances its risky assets by a mixture of short- and long-term debt. The financial institution is exposed to liquidity risk as its short-term creditors have the possibility not to renew their funding at a finite number of rollover dates. Besides, the financial institution can default due to insolvency at any time until maturity. We compute both insolvency and illiquidity default probabilities in this multi-period setting. We show that liquidity risk is increasing in the volatility of the risky assets and in the ratio of the return that can be earned on the outside market over the return for short-term debt promised by the financial institution. Moreover, we study the influence of the capital structure on the illiquidity probability and derive that illiquidity risk is increasing with the ratio of short-term funding.

As with conventional Higgs bundles, calculating Betti numbers of twisted Higgs bundle moduli spaces through Morse theory requires us to

study holomorphic chains. For the case when the base is P^1, we present a recursive method for constructing all the possible stable chains of a given type and degree by representing a family of chains by a quiver. We present the Betti numbers when the twists are O(1) and O(2), the latter of which coincides with the co-Higgs bundles on P^1. We offer some open questions. In doing so, we mention how these numbers have appeared elsewhere recently, namely in calculations of Mozgovoy related to conjectures coming from the physics literature (Chuang-Diaconescu-Pan).

Many radar designs transmit trains of pulses to estimate the Doppler shift from moving targets, in order to distinguish them from the returns from stationary objects (clutter) at the same range. The design of these waveforms is a compromise, because when the radar's pulse repetition frequency (PRF) is high enough to sample the Doppler shift without excessive ambiguity, the range measurements often also become ambiguous. Low-PRF radars are designed to be unambiguous in range, but are highly ambiguous in Doppler. High-PRF radars are, conversely unambiguous in Doppler but highly ambiguous in range. Medium-PRF radars have a moderate degree of ambiguity (say five times) in both range and Doppler and give better overall performance.

The ambiguities mean that multiple PRFs must be used to resolve the ambiguities (using the principle of the Chinese Remainder Theorom). A more serious issue, however, is that each PRF is now 'blind' at certain ranges, where the received signal arrives at the same time as the next pulse is transmitted, and at certain Doppler shifts (target speeds), when the return is 'folded' in Doppler so that it is hidden under the much larger clutter signal.

A practical radar therefore transmits successive bursts of pulses at different PRFs to overcome the 'blindness' and to resolve the ambiguities. Analysing the performance, although quite complex if done in detail, is possible using modern computer models, but the inverse problems of synthesing waveforms with a given performance remains difficult. Even more difficult is the problem of gaining intuitive insights into the likely effect of altering the waveforms. Such insights would be extremely valuable for the design process.

This problem is well known within the radar industry, but it is hoped that by airing it to an audience with a wider range of skills, some new ways of looking at the problem might be found.

I will discuss the application of Zariski geometries to Mordell Lang, and review the main ideas which are used in the interpretation of a field, given the assumption of non local modularity. I consider some open problems in adapting Zilber's construction to the case of minimal types in separably closed fields.

If two L-functions are added together, the Euler product is destroyed.

Thus the linear combination is not an L-function, and hence we should

not expect a Riemann Hypothesis for it. This is indeed the case: Not

all the zeros of linear combinations of L-functions lie on the

critical line.

However, if the two L-functions have the same functional equation then

almost all the zeros do lie on the critical line. This is not seen

when they have different functional equations.

We will discuss these results (which are due to Bombieri and Hejhal)

during the talk, and demonstrate them using characteristic polynomials

of random unitary matrices, where similar phenomena are observed. If

the two matrices have the same determinant, almost all the zeros of

linear combinations of characteristic polynomials lie on the unit

circle, whereas if they have different determinants all the zeros lie

off the unit circle.

Systems with delays frequently appear in engineering. The presence of delays makes system analysis and control design very complicated. In this talk, the standard H-infinity control problem of time-delay systems will be discussed. The emphasis will be on systems having an input or output delay. The problem is solved in the frequency domain via reduction to a one-block problem and then further to an extended Nehari problem using a simple and intuitive method. After solving the extended Nehari problem, the original problem is solved. The solvability of the extended Nehari problem (or the one-block problem) is equivalent to the nonsingularity of a delay-dependent matrix and the solvability conditions of the standard H-infinity control problem with a delay are then formulated in terms of the existence of solutions to two delay-independent algebraic Riccati equations and a delay-dependent nonsingular matrix.

In this talk, I will discuss various aspects of approximation by radial basis functions on spheres. After a short introduction to the subject of scattered data approximation on spheres and optimal recovery, I will particularly talk about error analysis, a hybrid approximation scheme involving polynomials and radial basis functions and, if time permits, solving nonlinear parabolic equations on spheres.

Discrete problems have a habit of being beautiful but difficult. This can be true even of discrete problems whose continuous analogues are easy. For example: computing the surface area of a sphere of radius N^{1/2} in k-dimensional Euclidean space (easy). Counting the number of representations of an integer N as a sum of k squares (historically hard). In this talk we'll survey a menagerie of discrete analogues of operators arising in harmonic analysis, including singular integral operators (such as the Hilbert transform), maximal functions, and fractional integral operators. In certain cases we can learn everything we want to know about the discrete operator immediately, from its continuous analogue. In other cases the discrete operator requires a completely new approach. We'll see what makes a discrete operator easy/hard to treat, and outline some of the methods that are breaking new ground, key aspects of which come from number theory. In particular, we will highlight the roles played by theta functions, exponential sums, Waring's problem, and the circle method of Hardy and Littlewood. No previous knowledge of singular integral operators or the circle method will be assumed.

In euclidean space there is a well-known parallelogram law relating the

length of vectors a, b, a+b and a-b. In the talk I give a similar formula

for translation lengths of isometries of CAT(0)-spaces. Given an action of

the automorphism group of a free product on a CAT(0)-space, I show that

certain elements can only act by zero translation length. In comparison to

other well-known actions this leads to restrictions about homomorphisms of

these groups to other groups, e.g. mapping class groups.

This talk will summarize some of the problems and conjectures that I haven't managed to solve (although I have tried to) while spending my three years in this job. It will cover the areas of group theory, representation theory, both of general finite groups and of symmetric groups, and fusion systems.

In joint work with Tim Perutz, we give a complete characterization of the Fukaya category of the punctured torus, denoted by $Fuk(T_0)$. This, in particular, means that one can write down an explicit minimal model for $Fuk(T_0)$ in the form of an A-infinity algebra, denoted by A, and classify A-infinity structures on the relevant algebra. A result that we will discuss is that no associative algebra is quasi-equivalent to the model A of the Fukaya category of the punctured torus, i.e., A is non-formal. $Fuk(T_0)$ will be connected to many topics of interest: 1) It is the boundary category that we associate to a 3-manifold with torus boundary in our extension of Heegaard Floer theory to manifolds with boundary, 2) It is quasi-equivalent to the category of perfect complexes on an irreducible rational curve with a double point, an instance of homological mirror symmetry.

(Note change in time and location)

The purpose of this talk is to give an introduction to the theory and

practice of integer factorization. More precisely, I plan to talk about the

p-1 method, the elliptic curve method, the quadratic sieve, and if time

permits the number field sieve.

Bacteria are ubiquitous on Earth and perform many vital roles in addition to being responsible for a variety of diseases. Locomotion allows the bacterium to explore the environment to find nutrient-rich locations and is also crucial in the formation of large colonies, known as biofilms, on solid surfaces immersed in the fluid. Many bacteria swim by turning corkscrew-shaped flagella. This can be studied computationally by considering hydrodynamic forces acting on the bacterium as the flagellum rotates. Using a boundary element method to solve the Stokes flow equations, it is found that details of the shape of the cell and flagellum affect both swimming efficiency and attraction of the swimmer towards flat no-slip surfaces. For example, simulations show that relatively small changes in cell elongation or flagellum length could make the difference between an affinity for swimming near surfaces and a repulsion. A new model is introduced for considering elastic behaviour in the bacterial hook that links the flagellum to the motor in the cell body. This model, based on Kirchhoff rod theory, predicts upper and lower bounds on the hook stiffness for effective swimming.

There have been significant progress in the calculation of scattering amplitudes in N=4 SYM. In this talk I will consider `form factors', which are defined not only with on-shell asymptotic states but also with one off-shell operator inserted. Such quantities are in some sense the hybrid of on-shell quantities (such

as scattering amplitudes) and off-shell quantities (such as correlation functions). We will see that form factors inherit many nice properties of scattering amplitudes, in particular we will consider their supersymmetrization and the dual picture.

In many applications it is of interest to compute minimizers of

a functional I(u) which is the of the form $J(u)=\Phi(u)+R(u)$,

with $R(u)$ quadratic. We describe a stochastic algorithm for

this problem which avoids explicit computation of gradients of $\Phi$;

it requires only the ability to sample from a Gaussian measure

with Cameron-Martin norm squared equal to $R(u)$, and the ability

to evaluate $\Phi$. We show that, in an appropriate parameter limit,

a piecewise linear interpolant of the algorithm converges weakly to a noisy

gradient flow. \\

Joint work with Natesh Pillai (Harvard) and Alex Thiery (Warwick).

In the 1990's Haagerup discovered a new subfactor, and hence a new topological quantum field theory, that has so far proved inaccessible by the methods of quantum groups and conformal field theory. It was the subfactor of smallest index beyond 4. This led to a classification project-classify all subfactors to as large an index as possible. So far we have gone as far as index 5. It is known that at index 6 wildness phenomena occur which preclude a simple listing of all subfactors of that index. It is possible that wildness occurs at a smaller index value, the main candidate being approximately 5.236.