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The motion of ideal shallow water vortices near piecewise flat topography is studied without background rotation for two choices of bottom topography. First, topography is chosen in the form of a rectilinear step change in depth. Finite area monopolar vortices which propagate steadily, without change in shape, parallel to the step are computed numerically. Next, the general motion of a pair of point vortices of arbitrary circulation near an escarpment is found explicitly using Hamiltonian techniques. Some vortex paths are periodic and for specific initial conditions each vortex comprising the pair translates parallel to the step. Comparisons to point vortex trajectories are made with vortex patch trajectories computed using contour dynamics. Agreement between the two trajectories is close provided a vortex patch is sufficiently away from the escarpment. Then, the scattering at a rectilinear step change in depth of a shallow water vortex pair consisting of two patches of equal, but oppositely signed vorticity is studied. Using the constants of motion an explicit relationship is derived relating the angle of incidence to the refracted angle after crossing. It is found that for certain initial conditions a pair can be totally internally reflected by the escarpment. For large depth changes numerical computations show the coherence of the vortex pair is lost on encountering the escarpment. The second part of this work concerns the dynamics of shallow water vortices near circular topography. Finite area monopolar vortices which translate without change in shape around the topography arc computed near a seamount or well including the limiting cases of each: an island or deep well. The behaviour of a vortex pair propagating toward circular topography is examined. Using Hamiltonian techniques, trajectories of point vortices exterior to the topography are found and are compared to trajectories of vortex patches computed using contour dynamics. Point vortex trajectories can be periodic and, for specific initial conditions, each vortex orbits the topography with the same frequency. Finally, laboratory experiments arc performed to find the behaviour of a dipole propagating toward a step change in height. Dipoles approaching the step from either deep or shallow water at normal and oblique incidence are considered. Qualitative observations agree well with theoretical predictions: a dipole increases its separation when crossing from deep water and decreases for a dipole crossing from shallow water. Furthermore, for dipoles approaching from shallow water with a sufficiently large incident angle the dipole was observed to perform total internal reflection.

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This thesis is concerned with approximation on compact homogeneous spaces. The first part of the research involves a particular kind of compact homogeneous space, the hypersphere, S ͩˉ¹ embedded in R ͩ. It is a calculation of three integrals associated with approximation using radial basis functions, calculating the Fourier-Gegenbauer coefficients for two such functions. The latter part of the research is a calculation of an error bound for compact homogeneous spaces when interpolating with a G-invariant kernel, a generalisation of a result already known for spheres.

This thesis studies some aspects of the homotopy type of function spaces X^Y where X,Y are topological spaces. The thesis is in two parts. Part A (Chapters I-IV) contains a discussion of some known facts on the homotopy type of function spaces under the heads of homology (Chapter II), homotopy groups (Chapter III) and Postnikov systems (Chapter IV). Also, in Chapter II, a theorem on duality is given which is useful in determining the low-dimensional homotopy type of (Sⁿ)^X when X = S^ru e^r+q (r + q < n). Chapter IV contains the statements of the problems whose solution is the motivation of the theory of Part B. These problems, which occur naturally in attempting to find the Postnikov system of X^Y by induction on the Postnikov system of X, are roughly of the type of determining k^Y : X^Y→A^Y when X,Y are spaces, A is a topological abelian group and k : X→A is a map. This problem we call here the "k-invariant problem". It is a commonplace that the most important property of function spaces is the "exponential law" which states that under certain restrictions the spaces X^Z&Y and (X^Y)^Z are homeomorphic. In fact it is usually the case that the only properties of the function space required are that as a set X^Y is the set of maps Y→X, and that the exponential law holds. In Chapter I, as preparation for the work of Part B, a brief discussion of the exponential law in a general category is given. The rest of the chapter shows how the well-known weak-topological product may be used to obtain an exponential law for all (Hausdorff) spaces. The weak product is also shown to be convenient in the theory of the identification maps. The theory of Part B is given in terns of css-complexes (complete semi-simplical complexes) with base point. In Chapter V the well-known css-exponential law is extended to the category of css-M-ads, and the exponential law for complexes with base point obtained. The relation between the topological and css-function spaces is discussed, and it is shown that the singular functor preserves the exponential law. The further theoretical work of Part B is initially of two kinds. First, the function complex A^Y where A is an FD-complex, is related, by means of maps and functors, with mapping objects in the category of FD-complexes and chain complexes. This is done in such a way to preserve the exponential law. Second, a generalised cohomology of a complex is introduced, with the coefficient group replaced by an arbitrary chain complex (or FD-complex). The theories of cohomology operations and of Eilenberg-Maclane complexes are correspondingly generalised. Using these two sets of constructions, a solution of the k-invariant problem is given in terms of chain complexes (Chapter IX, S.2). The rest of Part B is concerned with obtaining the cohomological solution of the k-invariant problem, putting the results in a form suitable for computation, and obtaining applications.

This thesis investigates statistical approaches to interpreting the output of persistent homology, a multi-resolution algorithm for discovering topological structure in data. We provide a brief introduction to the theory of topology and homology. The output is a set of intervals, visualised either as a 'barcode' or as a set of points called a persistence diagram. We discuss suitable metrics for persistence diagrams. The following chapter demonstrates how to compute persistent homology using R. Following this foundational work, we find a confidence set for the true persistence diagram of the underlying space using a sample diagram. Such sets aid with the interpretation of persistence diagrams by identifying points that are likely representative of true topological features, and those points that are noise due to sampling. We present two methods of constructing confidence sets. The first assumes that the support of the sampling density is not too 'spiky'. The second method uses a stronger assumption that the data are a realisation of a homogeneous Poisson process, which leads to a less conservative confidence set. In the middle section of this thesis, we investigate further sampling properties of persistence diagrams. Sampling on the circle leads us to propose a barcode test of sampling uniformity. We look at the diagrams of samples from the unit square, which is topologically simple, and propose these as a model for the noise in diagrams from other spaces. We propose density corrected persistent homology that makes sample diagrams less sensitive to the geometry of the underlying space and the sampling density. In the last section of this thesis, we demonstrate how persistent homology can be used to identify topological structure in correlation and partial correlation matrices. This relates to the problem of structure learning in graphical models.

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In this thesis we introduce the notion of a cdp-functor on the category of proper schemes over a Noetherian base, and we show that cdp-functors to Waldhausen categories extend to factors that satisfy the excision property. This allows us to associate with a cdp-functor an Euler-Poincaré characteristic that sends the class of a proper scheme to the class of its image. Applying this construction to the Yoneda embedding yields a monoidal proper-fibred Waldhausen category over Noetherian schemes, with canonical cdp-functors to its fibres. Also, we deduce a motivic measure to the Grothendieck ring of finitely presented simplicially stable motivic spaces with the cdh-topology.