A bivariant theory for the Cuntz semigroup is introduced and analysed. This is used to define a Cuntz-analogue of K-homology, which turns out to provide a complete invariant for compact Hausdorff spaces. Furthermore, a classification result for the class of unital and stably finite C*-algebras is proved, which can be considered as a formal analogue of the Kirchberg-Phillips classification result for purely infinite C*-algebras by means of KK-theory, i.e. bivariant K-theory. An equivariant extension of the bivariant Cuntz semigroup proposed in this thesis is also presented, and some well-known classification results are derived within this new theory, thus showing that it can be applied successfully to the problem of classification of some actions by compact groups over certain C*-algebras of the stably finite type.

This thesis is concerned with the well-posedness of solutions to certain linear and nonlinear parabolic PDEs on evolving spaces. We first present an abstract framework for the formulation and well-posedness of linear parabolic PDEs on abstract evolving Hilbert spaces. We introduce new function spaces and a notion of a weak time derivative called the weak material derivative for this purpose. We apply this general theory to moving hypersurfaces and Sobolev spaces and study four different linear problems including a coupled bulk-surface system and a dynamical boundary problem. Then we formulate a Stefan problem itself on an evolving surface and consider weak solutions given integrable data through the enthalpy approach, using a generalisation to the Banach space setting of the function spaces introduced in the abstract framework. We finish by studying a nonlocal problem: a porous medium equation with a fractional diffusion posed on an evolving surface and we prove well-posedness for bounded initial data.

In this thesis, we investigate three different phenomena in uniformly hyperbolic dynamics. First, we study entry time statistics for -mixing actions. More specifically, given a -mixing dynamical system (X ,T, BX,µ) we find conditions on a family of sets {Hn ⇢ X : n 2 N} so that µ(Hn)⌧n tends in law to an exponential random variable, where ⌧n is the entry time to Hn. We apply this to hyperbolic toral automorphisms, and we obtain that µ(Hn)⌧n tends in law to an exponential random variable when {Hn ⇢ X : n 2 N} are shrinking sets along the unstable direction. Second, we prove escape rate results for special flows over subshifts of finite type, over conformal repellers and over Axiom A diffeomorphisms. Finally, we study escape rates for Axiom A flows. Our results are based on a discretisation of the flow and the application of the results in [39]. Third, we study the smoothness of the stationary measure with respect to smooth perturbations of the iterated function scheme and the weight functions that define it. Our main theorems relate the smoothness of the perturbation of: the iterated function scheme and the weight functions; to the smoothness of the perturbation of the stationary measure. The results depend on the smoothness of: the iterated function scheme and the weights functions; and the space on which the stationary measure acts as a linear operator.

The KPZ universality class is expected to contain a large class of random growth processes. In some of these models, there is an additional structure provided by multiple non-intersecting paths and utilisation of this additional structure has led to derivations of exact formulae for the distribution of quantities of interest. Motivated by this we study the multi-layer extension of the stochastic heat equation introduced by O'Connell and Warren in [OW11] which is the continuum analogue of the above mentioned structure. We also show that a multi-layer Cole-Hopf solution to the KPZ equation is well defined.

The first part of this thesis is concerned with the question of global uniqueness of solutions to the initial value problem in general relativity. In 1969, Choquet-Bruhat and Geroch proved, that in the class of globally hyperbolic Cauchy developments, there is a unique maximal Cauchy development. The original proof, however, has the peculiar feature that it appeals to Zorn’s lemma in order to guarantee the existence of this maximal development; in particular, the proof is not constructive. In the first part of this thesis we give a proof of the above mentioned theorem that avoids the use of Zorn’s lemma. The second part of this thesis investigates the behaviour of so-called Gaussian beam solutions of the wave equation - highly oscillatory and localised solutions which travel, for some time, along null geodesics. The main result of this part of the thesis is a characterisation of the temporal behaviour of the energy of such Gaussian beams in terms of the underlying null geodesic. We conclude by giving applications of this result to black hole spacetimes. Recalling that the wave equation can be considered a “poor man’s” linearisation of the Einstein equations, these applications are of interest for a better understanding of the black hole stability conjecture, which states that the exterior of our explicit black hole solutions is stable to small perturbations, while the interior is expected to be unstable. The last part of the thesis is concerned with the wave equation in the interior of a black hole. In particular, we show that under certain conditions on the black hole parameters, waves that are compactly supported on the event horizon, have finite energy near the Cauchy horizon. This result is again motivated by the investigation of the conjectured instability of the interior of our explicit black hole solutions.

No description available.

The main objects of study of the thesis are two classes of special vector configurations appeared in the geometry and the theory of integrable systems. In the first part we consider a special class of vector configurations known as the V-systems, which appeared in the theory of the generalised Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equations. Several families of V-systems are known, but their classification is an open problem. We derive the relations describing the infinitesimal deformations of V-systems and use them to study the classification problem for V-systems in dimension 3. In particular, we prove that the isolated cases in Feigin-Veselov list admit only trivial deformations. We present the catalogue of all known 3D V-systems including graphical representations of the corresponding matroids and values of v-functions. In the second part we study the vector configurations, which form vertex sets for a new class of polyhedra called affine B-regular. They are defined by a 3-dimensional analogue of the Buffon procedure proposed by Veselov and Ward. The main result is the proof of existence of star-shaped affine B-regular polyhedron with prescribed combinatorial structure, under partial symmetry and simpliciality assumptions. The proof is based on deep results from spectral graph theory due to Colin de Verdière and Lovász.

No description available.

We consider a formally self-adjoint elliptic differential operator in IRⁿ, denoted by τ. T₀ and T are operators given by τ with specific domains. We determine conditions under which T₀ is essentially self-adjoint, introducing the topic by means of a brief historical survey of some results in this field. In Part I, we consider an operator of order 4, and in Part II, we generalise the results obtained there to ones for an operator of order 2m. Thus, the two parts run parallel. In Chapter 1, we determine the domain of T₀*, denoted by D(T₀*), where T₀* denotes the adjoint of T₀, and introduce operators <u>T</u>₀ and <u>T</u> which are modifications of T₀ and T. In Chapter 2, we use a theorem of Schechter to give conditions under which <u>T</u>₀ is essentially self-adjoint. Working with the operator <u>T</u>, in Chapter 3 ve show that we can approximate functions u in D(T₀*) by a particular sequence of test-functions, which enables us to derive an identity involving u, Tu and the coefficient functions of the operator concerned. In Chapter 4, we determine an upper bound for the integral of a function involving a derivative of u in D(T₀*) whose order is half the order of the operator concerned, and we use the identity from the previous chapter to reformulate this upper bound. In Chapter 5, we give conditions which are sufficient for the essential self-adjointness of T₀. In the main theorem itself, the major step is the derivation of the integral of the function involving the particular derivative of u in D(T₀*) whose order is half the order of the operator concerned, referred to above, itself as a term of an upper bound of an integral we wish to estimate. Hence, we can employ the upper bound from Chapter 4. This "sandwiching" technique is basic to the approach we have adopted. We conclude with a brief discussion of the operators we considered, and restate the examples of operators which we showed to be essentially self-adjoint.

Michael Anthony Isnardi. / Also issued as Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 1986. / Includes bibliographical references. / Supported in part by members of the Center for Advanced Television Studies.