And if we know of the existence of some specific solution, how do we determine it explicitly or as accurately as possible? This course develops the foundations required to rigorously establish the existence of solutions to various equations, thereby laying the basis for the study of solutions. Through an understanding of the foundations of analysis, we obtain insight critical in numerous areas of application, such areas ranging across physics, engineering, economics and finance. Topics covered are: sets, functions, metric spaces and normed linear spaces, compactness, connectedness, and completeness.

Banach fixed point theorem and applications, uniform continuity and convergence. General topological spaces, generating topologies, topological invariants, quotient spaces. Introduction to Hilbert spaces and bounded operators on Hilbert spaces. Especially on so called hermitian symmetric spaces which arise through a 1-grading of the Lie algebra of G there exists a complete classification of first order invariant linear differential operators even on more general manifolds that allow a so called almost hermitian structure.

This talk will introduce the notion of an invariant bilinear differential pairing between sections of the aforementioned homogeneous bundles. Moreover we will discuss a classification excluding certain totally degenerate cases of all first order invariant bilinear differential pairings on manifolds with an almost hermitian symmetric structure. The similarities and connections with the linear operator classification will be highlighted and discussed.

In this talk we present a stronger reconstruction guarantee called "adaptive fast convergence", which provides guarantees for the correct reconstruction of all sufficiently long edges of the original tree. We then present a general technique, which unlike previous reconstruction techniques employs dynamic edge-contraction during the reconstruction of the tree. We conclude by demonstrating how this technique is used to achieve adaptive fast convergence.

Thus, the stationarity assumption is the basis for a general asymptotic theory for identification, estimation and forecasting.

## Geometry & Topology Monographs

It guarantees that the increase of the sample size leads to more and more information of the same kind which is basic for an asymptotic theory to make sense. Now non-stationarity modelling has also a long tradition in econometrics. This one is based on the conditional moments of the data generating process. It appears mainly in the heteroscedastic and volatility models, like the GARCH and related models, and stochastic volatility processes Ghysels, Harvey and Renault This non-stationarity appears also in a different way with structural changes models like the switching models Hamilton, , the stopbreak model Diebold and Inoue, , Breidt and Hsu, , Granger and Hyung, and the SETAR models, for instance.

It can also be observed from linear models with time varying coefficients Nicholls and Quinn, , Tsay, Thus, using stationary unconditional moments suggest a global stationarity for the model, but using non-stationary unconditional moments or non-stationary conditional moments or assuming existence of states suggest that this global stationarity fails and that we only observe a local stationary behavior. The growing evidence of instability in the stochastic behavior of stocks, of exchange rates, of some economic data sets like growth rates for instance, characterized by existence of volatility or existence of jumps in the variance or on the levels of the prices imposes to discuss the assumption of global stationarity and its consequence in modelling, particularly in forecasting.

Thus we can address several questions with respect to these remarks. What kinds of non-stationarity affect the major financial and economic data sets? How to detect them? What is the impact of evidence of non-stationarity on the statistics computed from the global non stationary data sets? How can we analyze data sets in the non-stationary global framework? Does the asymptotic theory work in non-stationary framework?

What kind of models create local stationarity instead of global stationarity? How can we use them to develop a modelling and a forecasting strategy? These questions began to be discussed in some papers in the economic literature.

### The Stockholm Logic Seminar

For some of these questions, the answers are known, for others, very few works exist. In this talk I will discuss all these problems and will propose 2 new stategies and modelling to solve them. Several interesting topics in empirical finance awaiting future research will also be discussed.

Both at the cellular and tissular levels, the most important feature of the clock is its ability not simply to keep time, but to adjust its time, or phase, to signals.

We present the parametric impulse phase response curve pIPRC , an analytical analog to the phase response curve PRC used experimentally. We use the pIPRC to understand both the consequences of intercellular signaling and the light entrainment process. Further, we determine which model components determine the phase response behavior of a single oscillator by using a novel model reduction technique.

We reduce the number of model components while preserving the pIPRC and then incorporate the resultant model into a couple SCN tissue model.

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Emergent properties, including the ability of the population to synchronize spontaneously are preserved in the reduction. Finally, we present some mathematical tools for the study of synchronization in a network of coupled, noisy oscillators. The standard diffusion equation is obtained by Einstein's method and its generalisation, Fokker-Plank-Kolmogorov-Feller theory. The time between jumps in Einstein derivation is constant. The rigorous application of the Einstein's method results in new equation, containing the time and the mixed dispersion terms expressing the dispersion of the particle time steps.

Usually, adding the second time derivative results in additional initial data. For the equation derived, the condition of limited solution when time tends to infinity provides with uniqueness of the Caushy problem solution. The solution of the pulse injection problem describing a common tracer injection experiment is studied in greater detail. The new theory predicts delay of the maximum of the tracer, compared to the velocity of the flow, while its forward "tail" contains much more particles than in the solution of the classical parabolic advection-dispersion equation.

This is in agreement with the experimental observations and predictions of the direct simulation. Tissue engineering could be called 'the science of spare parts'. Although currently in its infancy, its long-term aim is to grow functional tissues and organs in vitro to replace those which have become defective through age, trauma or disease. Recent experiments have shown that mechanical interactions between cells and the materials in which they are grown have an important influence on tissue architecture, but in order to understand these effects, we first need to understand the mechanics of the gels themselves.

Many biological gels e.

I will present a simple continuum model of gel mechanics, based on treating the gel as a transversely isotropic viscous material. Two canonical problems are considered involving thin two-dimensional films: extensional flow, and squeezing flow of the fluid between two rigid plates.

- Iemanjá Claus.
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Neglecting inertia, gravity and surface tension, in each regime we can exploit the thin geometry to obtain a leading-order problem which is sufficiently tractable to allow the use of analytical methods. I discuss how these results could be exploited practically to determine the mechanical properties of real gels.

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If time permits, I will also talk about work currently in progress which explores the interaction between gel mechanics and cell behaviour. Data assimilation aims to solve one of the fundamental problems ofnumerical weather prediction - estimating the optimal state of the atmosphere given a numerical model of the dynamics, and sparse, noisy observations of the system.

A standard tool in attacking this filtering problem is the Kalman filter.

We consider the problem when only partial observations are available. In particular we consider the situation where the observational space consists of variables which are directly observable with known observational error, and of variables of which only their climatic variance and mean are given. We derive the corresponding Kalman filter in a variational setting.

We analyze the variance constraining Kalman filter VCKF filter for a simple linear toy model and determine its range of optimal performance. We explore the variance constraining Kalman filter in an ensemble transform setting for the Lorenz system, and show that incorporating the information on the variance on some un-observable variables can improve the skill and also increase the stability of the data assimilation procedure.

## Fragments of geometric topology from the sixties

Using methods from dynamical systems theory we then systems where the un-observed variables evolve deterministically but chaotically on a fast time scale. The aim of the Hmm Sessions is for people to get together to solve puzzles as a group. There will be lots of time to solve puzzles in groups and to celebrate the clever solutions of others.

- Freshly-baked pastries.
- An ICMI Study.
- The Biotechnology of Ethanol.

The lunchbreak provides time to socialise, play games or to continue solving puzzles bring your own lunch, or go out to nearby Rundle Mall to buy lunch on the day. Our ability to accurately predict and explain the spread of an infectious disease is a significant factor in our ability to implement effective interventions. Our ability to accurately model disease spread depends on how accurately we capture the various effects. This is complicated by the fact that infectious disease spread involves a number of time scales. Four that are particularly relevant are: duration of infection in an individual, duration of partnerships between individuals, the time required for an epidemic to spread through the population, and the time required for the population structure to change demographic or otherwise.

Mathematically simple models of disease spread usually make the implicit assumption that the duration of partnerships is by far the shortest time scale in the system.

Thus they miss out on the tendency for infected individuals to deplete their local pool of susceptibles. Depending on the details of the disease in question, this effect may be significant. I will then discuss early results in applying these methods to diseases such as HIV in which the population time scales are relevant. Congratulations to the following members of the School for their success in the ARC Discovery Grants which were announced recently. More information Algebraic topology.

Time Series III. Topology and Analysis III. Time-periodic shear layers occur naturally in a wide range of applications from engineering to physiology. Transition to turbulence in such flows is of practical interest and there have been several papers dealing with the stability of flows composed of a steady component plus an oscillatory part with zero mean.

In such flows a possible instability mechanism is associated with the mean component so that the stability of the flow can be examined using some sort of perturbation-type analysis. This strategy fails when the mean part of the flow is small compared with the oscillatory component which, of course, includes the case when the mean part is precisely zero. This difficulty with analytical studies has meant that the stability of purely oscillatory flows has relied on various numerical methods.

Until very recently such techniques have only ever predicted that the flow is stable, even though experiments suggest that they do become unstable at high enough speeds. In this talk I shall expand on this discrepancy with emphasis on the particular case of the so-called flat Stokes layer. This flow, which is generated in a deep layer of incompressible fluid lying above a flat plate which is oscillated in its own plane, represents one of the few exact solutions of the Navier-Stokes equations.