For a fixed n we will investigate homomorphisms Out(F_n) to
Out(F_m) (i.e. free representations) and Out(F_n) to
GL_m(K) (i.e. K-linear representations). We will
completely classify both kinds of representations (at least for suitable
fields K) for a range of values $m$.
Abstract: In this talk we will introduce a new particle approximation scheme to solve the stochastic filtering problem. This new scheme makes use of the Kusuoka-Lyons-Victoir (KLV) method to approximate the dynamics of the signal. In order to control the computational cost, a partial sampling procedure based on the tree based branching algorithm (TBBA) is performed. The novelty of the method lies in the fact that the weights used in the TBBA are computed combining the cubature weights and the filtering weights. In this way, we can avoid the sample degeneracy problem inherent to particle filters. We will also present some simulations showing the performance of the method.
It is a general belief that the heat kernels on fractals should exhibit highly oscillatory behaviors as opposed to the classical case of Riemannian manifolds.
For example, on a class of finitely ramified fractals, called (affine) nested fractals, a canonical ``Brownian motion" has been constructed and its transition density (heat kernel) $p_{t}(x,y)$ satisfies $c_{1} \leq t^{d_{s}/2} p_{t}(x,x) \leq c_{2}$ for $t \leq 1$ for any point $x$ of the fractal; here $d_{s}$ is the so-called spectral dimension. Then it is natural to ask whether the limit of this quantity as $t$ goes to 0 exists or not, and it has been conjectured NOT to exist by many people.
In this talk, I will present partial affirmative answers to this conjecture. First, for a general (affine) nested fractal, the non-existence of the limit is shown to be true for a ``generic" (in particular, almost every) point. Secondly, the same is shown to be valid for ANY point of the fractal in the particular cases of the $d$-dimensional standard Sierpinski gasket with $d\geq 2$ and of the $N$-polygasket with $N\geq 3$ odd, e.g. the pentagasket ($N=5$) and the heptagasket ($N=7$).
In this talk we will review M-theory dualities and recent attempts to make these dualities manifest in eleven-dimensional supergravity. We will review the work of Berman and Perry and then outline a prescription, called non-linear realisation, for making larger duality symmetries manifest. Finally, we will explain how the local symmetries are described by generalised geometry, which leads to a duality-covariant constraint that allows one to reduce from generalised space to physical space.
Image segmentation and a number of other problems from image processing and computer vision can be regarded
as interface problems. Recently, diffusive and sharp interface techniques have been used for these problems.
In this talk, we will first briefly explain these models and compare the advantages and disadvantages of these models. Numerically, these models can be solved through some PDEs. In the end, we will show some recent results on how to use graph cut to solve these interface problems. Moreover, the global minimizer can be guaranteed even the problem is nonconex and nonlinear. The use of max-flow in a network setting and also in an infinite dimensional setting will be explained.
We will first motivate and review some implicit schemes that arises from the discretization of non linear PDEs in finance or in optimal control problems - when using finite differences methods or finite element methods.
For the american option problem, we are led to compute the solution of a discrete obstacle problem, and will give some results for the convergence of nonsmooth Newton's method for solving such problems.
Implicit schemes are interesting for their stability properties, however they can be too costly in practice.
We will then present some novel schemes and ideas, based on the semi-lagrangian approach and on discontinuous galerkin methods, trying to be as much explicit as possible in order to gain practical efficiency.
Algebraic theories, locally presentable categories and their application to type theories. The seminar will take place in Lecture Theatre A of the Department of Computer Science.
A SMEC device is an array of aerofoil-shaped parallel hollow vanes forming linear venturis, perforated at the narrowest point where the vanes most nearly touch. When placed across a river or tidal flow, the water accelerates through the venturis between each pair of adjacent vanes and its pressure drops in accordance with Bernoulli’s Theorem. The low pressure zone draws a secondary flow out through the perforations in the adjacent hollow vanes which are all connected to a manifold at one end. The secondary flow enters the manifold through an axial flow turbine.
SMEC creates a small upstream head uplift of, say 1.5m – 2.5m, thereby converting some of the primary flow’s kinetic energy into potential energy. This head difference across the device drives around 80% of the flow between the vanes which can be seen to act as a no-moving-parts venturi pump, lowering the head on the back face of the turbine through which the other 20% of the flow is drawn. The head drop across this turbine, however, is amplified from, say, 2m up to, say, 8m. So SMEC is analogous to a step-up transformer, converting a high-volume low-pressure flow to a higher-pressure, lower-volume flow. It has all the same functional advantages of a step-up transformer and the inevitable transformer losses as well.
The key benefit is that a conventional turbine (or Archimedes Screw) designed to work efficiently at a 1.5m – 2.5m driving head has to be of very large diameter with a large step-up gearbox. In many real-World locations, this makes it too expensive or simply impractical, in shallow water for example.
The work we did in 2009-10 for DECC on a SMEC across the Severn Estuary concluded that compared to a conventional barrage, SMEC would output around 80% of the power at less than half the capital cost. Crucially, however, this greatly superior performance is achieved with minimal environmental impact as the tidal signal is preserved in the upstream lagoon, avoiding the severe damage to the feeding grounds of migratory birdlife that is an unwelcome characteristic of a conventional barrage.
To help successfully commercialise the technology, however, we will eventually want to build a reliable (CFD?) computer model of SMEC which even if partly parametric, would benefit hugely from an improved understanding of the small-scale turbulence and momentum transfer mechanisms in the mixing section.
This talk will be accessible to non-specialists and in particular details how model theory naturally leads to specific representations of abelian group rings as rings of global sections. The model-theoretic approach is motivated by algebraic results of Amitsur on the Semisimplicity Problem, on which a brief discussion will first be given.
A central challenge in socio-physics is understanding how groups of self-interested agents make collective decisions. For humans many insights in the underlying opinion formation process have been gained from network models, which represent agents as nodes and social contacts as links. Over the past decade these models have been expanded
to include the feedback of the opinions held by agents on the structure of the network. While a verification of these adaptive models in humans is still difficult, evidence is now starting to appear in opinion formation experiments with animals, where the choice that is being made concerns the direction of movement. In this talk I show how analytical insights can be gained from adaptive networks models and how predictions from these models can be verified in experiments with swarming animals. The results of this work point to a similarity between swarming and human opinion formation and reveal insights in the dynamics of the opinion formation process. In particular I show that in a population that is under control of a strongly opinionated minority a democratic consensus can be restored by the addition of
uninformed individuals.
A number is said to be $y$-smooth if all of its prime factors are
at most $y$. A lot of work has been done to establish the (equi)distribution
of smooth numbers in arithmetic progressions, on various ranges of $x$,$y$
and $q$ (the common difference of the progression). In this talk I will
explain some recent results on this problem. One ingredient is the use of a
majorant principle for trigonometric sums to carefully analyse a certain
contour integral.
Basic and mild introduction to Generalized Geometry from the very beginning: the generalized tangent space, generalized metrics, generalized complex structures... All topped with some Lie type B flavour. Suitable for vegans. May contain traces of spinors.
In this talk, we first study the Mean Curvature Flow, an evolution equation for submanifolds of some Euclidean space. We review a famous monotonicity formula of Huisken and its application to classifying so-called Type I singularities. Then, we discuss the Ricci Flow, which might be seen as the intrinsic analog of the Mean Curvature Flow for abstract Riemannian manifolds. We explain how Huisken's classification of Type I singularities can be adopted to this intrinsic setting, using monotone quantities found by Perelman.
The first group known to be finitely presented but having infinitely generated 3rd homology was constructed by Stallings. Bieri extended this to a series of groups G_n such that G_n is of type F_{n-1} but not of type F_n. Finally, Bestvina and Brady turned it into a machine that realizes prescribed finiteness properties. We will discuss some of these examples.
Mathematical models of chemotactic movement of bacterial populations are often written as systems of partial differential equations for the densities of bacteria and concentrations of extracellular signaling chemicals. This approach has been employed since the seminal work of Keller and Segel in the 1970s [Keller and Segel, J. Theor. Biol., 1971]. The system has been shown to permit travelling wave solutions which correspond to travelling band formation in bacterial colonies, yet only under specific criteria, such as a singularity in the chemotactic sensitivity function as the signal approaches zero. Such a singularity generates infinite macroscopic velocities that ar biologically unrealistic. Here we present a microscopic model that takes into consideration relevant details of the intracellular processes while avoiding the singularity in the chemotactic sensitivity. We show that this model permits travelling wave solutions and predicts the formation of other bacterial patterns such as radial and spiral streams. We also present connections of this microscopic model with macroscopic models of bacterial chemotaxis. This is joint work with Radek Erban, Benjamin Franz, Hyung Ju Hwang, and Kevin J.
Painter.
Saturated fusion systems are a relatively new class of objects that are often described as the correct 'axiomatisation' of certain p-local phenomena in algebraic topology. Despite these geometric beginnings however, their structure is sufficiently rigid to afford its own local theory which in some sense mimics the local theory of finite groups. In this talk, I will briefly motivate the definition of a saturated fusion system and discuss a remarkable result of Michael Aschbacher which proves that centralisers of normal subsystems of a saturated fusion system, F, exist and are themselves saturated. I will then attempt to justify his definition in the case where F is non-exotic by appealing to some classical group theoretic results. If time permits I will speculate about a topological characterisation of the centraliser as the set of homotopy fixed points of a certain action on the classifying space of F.
A dot product representation of a graph assigns to each vertex $s$ a vector $v(s)$ in ${\bf R}^k$ in such a way that $v(s)^T v(t)$ is greater than $1$ if and only $st$ is an edge. Similarly, in a distance representation $|v(s)-v(t)|$ is less than $1$ if and only if $st$ is an edge.
I will discuss the solution of some open problems by Spinrad, Breu and Kirkpatrick and others on these and related geometric representations of graphs. The proofs make use of a connection to oriented pseudoline arrangements.
(Joint work with Colin McDiarmid and Ross Kang)