Past

This is joint work with Dani Wise and builds on his earlier

work. Let M be a compact oriented irreducible 3-manifold which is neither a

graph manifold nor a hyperbolic manifold. We prove that the fundamental

group of M is virtually special. This means that it virtually embeds in a

right angled Artin group, and is in particular linear over Z.

The Schramm-Loewner evolution (SLE(\kappa)) is a family of random fractal curves that arise in a natural way as scaling limits of interfaces in critical models in statistical physics. The SLE curves are constructed by solving the Loewner differential equation driven by a scaled Brownian motion. We will give an overview of some of the almost sure properties of SLE curves, concentrating in particular on properties that can be derived by studying the the geometry of growing curve locally at the tip. We will discuss a multifractual spectrum of harmonic measure at the tip, regularity in the capacity parameterization, and continuity of the curves as the \kappa-parameter is varied while the driving Brownian motion sample is kept fixed.

This is based on joint work with Greg Lawler, and with Steffen Rohde and Carto Wong.

An intriguing, and largely open, question in mathematical fluid dynamics is whether solutions of the Navier-Stokes equations converge in some sense to a solution of the Euler equations in the zero viscosity limit. In fact this convergence could conceivably fail due to oscillations and concentrations occuring in the sequence.

In the late 1980s, R. DiPerna and A. Majda extended the classical concept of Young measure to obtain a notion of measure-valued solution of the Euler equations, which records precisely these oscillation and concentration effects. In this talk I will present a result recently obtained in joint work with L. Székelyhidi, which states that any such measure-valued solution is generated by a sequence of distributional solutions of the Euler equations.

The result is interesting from two different viewpoints: On the one hand, it emphasizes the huge flexibility of the concept of weak solution for Euler; on the other hand, it provides an example of a characterization theorem for Young measures in the tradition of D. Kinderlehrer and P. Pedregal where the differential constraint on the generating sequence does not satisfy the constant rank condition.

There are two main statistical mechanical models of ferromagnetism: the simpler and better-understood Ising model, and the more realistic and more challenging classical Heisenberg model, where the spins are in the 2-sphere instead of in {-1,+1}. In dimensions one and two, the classical Heisenberg model with nearest-neighbor interactions has no phase transition, but in three dimensions it has been intractable.

To shed some light on the qualitative behavior of the 3D Heisenberg model, we use the versatile tools of mean-field theory and Stein's method in recent work with Elizabeth Meckes, studying the Heisenberg model on a complete graph with the number of vertices going to infinity. Our results include detailed descriptions of the magnetization, the empirical spin distribution, the free energy, and a second-order phase transition.

A map betweem metric spaces is a bilipschitz homeomorphism if it

is Lipschitz and has a Lipschitz inverse; a map is a bilipschitz embedding

if it is a bilipschitz homeomorphism onto its image. Given metric spaces

X and Y, one may ask if there is a bilipschitz embedding X--->Y, and if

so, one may try to find an embedding with minimal distortion, or at least

estimate the best bilipschitz constant. Such bilipschitz embedding

problems arise in various areas of mathematics, including geometric group

theory, Banach space geometry, and geometric analysis; in the last 10

years they have also attracted a lot of attention in theoretical computer

science.

The lecture will be a survey bilipschitz embedding in Banach spaces from

the viewpoint of geometric analysis.

Abstract: In this talk, I will discuss the proportionality between tree amplitudes and the ultraviolet divergences in their one-loop corrections in Yang-Mills and (N < 4) Super Yang-Mills theories in four-dimensions. From the point of view of local perturbative quantum field theory, i.e. Feynman diagrams, this proportionality is straightforward: ultraviolet divergences at loop-level are absorbed into coefficients of local operators/interaction vertices in the original tree-amplitude. Ultraviolet divergences in loop amplitudes are also calculable through on-shell methods. These methods ensure manifest gauge-invariance, even at loop-level (no ghosts), at the expense of manifest locality. From an on-shell perspective, the proportionality between the ultraviolet divergences the tree amplitudes is thus not guaranteed. I describe systematic structures which ensure proportionality, and their possible connections to other recent developments in the field.

• Savina Joseph - Current generation in solar cells
• Shengxin Zhu - Spectral distribution, smoothing effects and smoothness matching for radial basis functions
• Ingrid von Glehn - Solving surface PDEs with the closest point method

The section conjecture of Grothendieck's anabelian geometry speculates about a description of the set of rational

points of a hyperbolic curve over a number field entirely in terms of profinite groups and Galois theory.

In the talk we will discuss local to global aspects of the conjecture, and what can be achieved when sections with

additional group theoretic properties are considered.

Complex networks have been used to model almost any

real-world complex systems. An especially important

issue regards how to related their structure and dynamics,

which contributes not only for the better understanding of

such systems, but also to the prediction of important

dynamical properties from specific topological features.

In this talk I revise related research developed recently

in my group. Particularly attention is given to the concept

of accessibility, a new measurement integrating topology

and dynamics, and the relationship between frequency of

visits and node degree in directed modular complex

networks. Analytical results are provided that allow accurate

prediction of correlations between structure and dynamics

in systems underlain by directed diffusion. The methodology

is illustrated with respect to the macaque cortical network.

Uncertainty quantification can begin by specifying the initial state of a system as a probability measure. Part of the state (the 'parameters') might not evolve, and might not be directly observable. Many inverse problems are generalisations of uncertainty quantification such that one modifies the probability measure to be consistent with measurements, a forward model and the initial measure. The inverse problem, interpreted as computing the posterior probability measure of the states, including the parameters and the variables, from a sequence of noise-corrupted observations, is reviewed in the talk. Bayesian statistics provides a natural framework for a solution but leads to very challenging computational problems, particularly when the dimension of the state space is very large, as when arising from the discretisation of a partial differential equation theory.

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In this talk we show how the Bayesian framework leads to a new algorithm - the 'Variational Smoothing Filter' - that unifies the leading techniques in use today. In particular the framework provides an interpretation and generalisation of Tikhonov regularisation, a method of forecast verification and a way of quantifying and managing uncertainty. To deal with the problem that a good initial prior may not be Gaussian, as with a general prior intended to describe, for example a geological structure, a Gaussian mixture prior is used. This has many desirable properties, including ease of sampling to make 'numerical rocks' or 'numerical weather' for visualisation purposes and statistical summaries, and in principle can approximate any probability density. Robustness is sought by combining a variational update with this full mixture representation of the conditional posterior density.

I will discuss methods for spatio-temporal modelling in cellular and molecular biology. Three classes of models will be considered: (i) microscopic (molecular-based, individual-based) models which are based on the simulation of trajectories of individual molecules and their localized interactions (for example, reactions); (ii) mesoscopic (lattice-based) models which divide the computational

domain into a finite number of compartments and simulate the time evolution of the numbers of molecules in each compartment; and (iii) macroscopic (deterministic) models which are written in terms of reaction-diffusion-advection PDEs for spatially varying concentrations. In the first part of my talk, I will discuss connections between the modelling frameworks (i)-(iii). I will consider chemical reactions both at a surface and in the bulk. In the second part of my talk, I will present hybrid (multiscale) algorithms which use models with a different level of detail in different parts of the computational domain. The main goal of this multiscale methodology is to use a detailed modelling approach in localized regions of particular interest (in which accuracy and microscopic detail is important) and a less detailed model in other regions in which accuracy may be traded for simulation efficiency. I will also discuss hybrid modelling of chemotaxis where an individual-based model of cells is coupled with PDEs for extracellular chemical signals.

We study minimizers of the Ginzburg-Landau (GL) functional \[E_\epsilon(u):=\frac{1}{2}\int_A |\nabla u|^2 + \frac{1}{4\epsilon^2} \int_A(1-|u|^2)^2\] for a complex-valued order parameter $u$ (with no magnetic field). This functional is of fundamental importance in the theory of superconductivity and superuidity; the development of these theories led to three Nobel prizes. For a $2D$ domain $A$ with holes we consider “semistiff” boundary conditions: a Dirichlet condition for the modulus $|u|$, and a homogeneous Neumann condition for the phase $\phi = \mathrm{arg}(u)$. The principal

result of this work (with V. Rybalko) is a proof of the existence of stable local minimizers with vortices (global minimizers do not exist). These vortices are novel in that they approach the boundary and have bounded energy as $\epsilon\to0$.

In contrast, in the well-studied Dirichlet (“stiff”) problem for the GL PDE, the vortices remain distant from the boundary and their energy blows up as

$\epsilon\to 0$. Also, there are no stable minimizers to the homogeneous Neumann (“soft”) problem with vortices.

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Next, we discuss more recent results (with V. Rybalko and O. Misiats) on global minimizers of the full GL functional (with magnetic field) subject to semi-stiff boundary conditions. Here, we show the existence of global minimizers with vortices for both simply and doubly connected domains and describe the location of their vortices.

The aim of this talk is to give an introduction to Geometric Invariant Theory (GIT) for reductive groups over the complex numbers. Roughly speaking, GIT is concerned with constructing quotients of group actions in the category of algebraic varieties. We begin by discussing what properties we should like quotient varieties to possess, highlighting so-called `good' and `geometric' quotients, and then turn to search for these quotients in the case of affine and projective varieties. Here we shall see that the construction runs most smoothly when we assume our group to be reductive (meaning it can be described as the complexification of a maximal compact subgroup). Finally, we hope to say something about the Hilbert-Mumford criterion regarding semi-stability and stability of points, illustrating it by constructing the rough moduli space of elliptic curves.

Joint work with: Sina Greenwood, Brian Raines and Casey Sherman

Abstract: We say a space $X$ with property $\C P$ is \emph{universal} for orbit spectra of homeomorphisms with property $\C P$ provided that if $Y$ is any space with property $\C P$ and the same cardinality as $X$ and $h:Y\to Y$ is any (auto)homeomorphism then there is a homeomorphism$g:X\to X$ such that the orbit equivalence classes for $h$ and $g$ are isomorphic. We construct a compact metric space $X$ that is universal for homeomorphisms of compact metric spaces of cardinality the continuum. There is no universal space for countable compact metric spaces. In the presence of some set theoretic assumptions we also give a separable metric space of size continuum that is universal for homeomorphisms on separable metric spaces.

Actin polymerization in vivo is regulated spatially and temporally by a web of signalling proteins. We developed detailed physico-chemical, stochastic models of lamellipodia and filopodia, which are projected by eukaryotic cells during cell migration, and contain dynamically remodelling actin meshes and bundles. In a recent work we studied how molecular motors regulate growth dynamics of elongated organelles of living cells. We determined spatial distributions of motors in such organelles, corresponding to a basic scenario when motors only walk along the substrate, bind, unbind, and diffuse. We developed a mean field model, which quantitatively reproduces elaborate stochastic simulation results as well as provides a physical interpretation of experimentally observed distributions of Myosin IIIa in stereocilia and filopodia. The mean field model showed that the jamming of the walking motors is conspicuous, and therefore damps the active motor flux. However, when the motor distributions are coupled to the delivery of actin monomers towards the tip, even the concentration bump of G-actin that they create before they jam is enough to speed up the diffusion to allow for severalfold longer filopodia. We found that the concentration profile of G-actin along the filopodium is rather non-trivial, containing a narrow minimum near the base followed by a broad maximum. For efficient enough actin transport, this non-monotonous shape is expected to occur under a broad set of conditions. We also find that the stationary motor distribution is universal for the given set of model parameters regardless of the organelle length, which follows from the form of the kinetic equations and the boundary conditions.

I’ll report on my recent work (with co-authors Holt and Ciobanu) on Artin

groups of large type, that is groups with presentations of the form

G = hx1, . . . , xn | xixjxi · · · = xjxixj · · · , 8i 3. (In fact, our results still hold when some, but not all

possible, relations with mij = 2 are allowed.)

Recently, Holt and I characterised the geodesic words in these groups, and

described an effective method to reduce any word to geodesic form. That

proves the groups shortlex automatic and gives an effective (at worst quadratic)

solution to the word problem. Using this characterisation of geodesics, Holt,

Ciobanu and I can derive the rapid decay property for most large type

groups, and hence deduce for most of these that the Baum-Connes conjec-

ture holds; this has various consequence, in particular that the Kadison-

Kaplansky conjecture holds for these groups, i.e. that the group ring CG

contains no non-trivial idempotents.

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This talk will be about various ways in which a variety can be "connected by higher genus curves", mimicking the notion of rational connectedness. At least in characteristic zero, the existence of a curve with a large deformation space of morphisms to a variety implies that the variety is in fact rationally connected. Time permitting I will discuss attempts to show this result in positive characteristic.

Cloaking recently attracts a lot of attention from the scientific community due to the progress of advanced technology. There are several ways to do cloaking. Two of them are based on transformation optics and negative index materials. Cloaking based on transformation optics was suggested by Pendry and Leonhardt using transformations which blow up a point into the cloaked regions. The same transformations had previously used by Greenleaf et al. to establish the non-uniqueness for Calderon's inverse problem. These transformations are singular and hence create a lot of difficulty in analysis and practical applications. The second method of cloaking is based on the peculiar properties of negative index materials. It was proposed by Lai et al. and inspired from the concept of complementary media due to Pendry and Ramakrishna. In this talk, I will discuss approximate cloaking using these two methods. Concerning the first one, I will consider the situation, first proposed in the work of Kohn et al., where one uses transformations which blow up a small ball (instead of a point) into cloaked regions. Many interesting issues such as finite energy and resonance will be mentioned. Concerning the second method, I provide the (first) rigorous analysis for cloaking using negative index materials by investigating the situation where the loss (damping) parameter goes to 0. I will also explain how the arguments can be used not only to establish the rigor for other interesting related phenomena using negative index materials such as superlenses and illusion optics but also to enlighten the mechanism of these phenomena.