The Maxwell-Dirac equations give a model of an electron in an electromagnetic (e-m) field, in which neither the Dirac or the e-m fields are quantized. The two equations are coupled via the Dirac current which acts as a source in the Maxwell equa- tion, resulting in a nonlinear system of partial differential equations (PDE's). In this way the self-field of the electron is included.

We review our results to date and give the four real consistency conditions (one of which is conservation of charge) which apply to the components of the wavefunction and its first derivatives. These must be met by any solutions to the Dirac equation. These conditions prove to be invaluable in the analysis of the nonlinear system, and generalizable to higher dimensional supersymmetric matter.

In earlier papers, we have shown analytically that in an isolated stationary system, the surrounding electon field must be equal and opposite to the central (external) field. The nonlinearity forces electric neutrality, at least in the static case. We illustrate these properties with a numerical family of orbits which occur in the (static) spherical and cylindrical ODE cases. These solutions are highly localized and die off exponentially with increasing distance from the central charge.

Intuitionistic real numbers are constructed as sheaves on the state space of the Schrodinger representation of a CCRalgebra with a finite number of degrees of freedom. These numbers are used as the values of position and momentum variables that obey Newton's equations of motion. Heisenberg's operator equations of motion are shown to give rise to numerical equations that, on a family of open subsets of state space, are local approximations to Newton's equations of motion for the intuitionistically valued variables.

We study systems of eigenspaces arising from the representation of a Vilenkin group on a semifinite von Neumann algebra. In particular, such systems form a Schauder decomposition in the reflexive non-communative $L_p$-spaces of measurable operators affiliated with the underlying von Neumann algebra. Our results extend classical results of Paley concerning the familiar Walsh-Paley system to the non-commutative setting.

We survey the theory of wrapping maps as applied to compact groups and vector times compact semidirect products, and give an explicit description of the e-function for compact sym- metric spaces. The latter is globally defined.

We announce a new result (proved in collaboration with T.A. Gillespie) on the boundedness of a class of Schur mul- tiplier projections on the von Neumann-Schatten ideals $C_p$. We also show that for $1 \leq p \leq 2$ the average Cp norm of a $0-1$ matrix grows just as quickly as the largest norm of such a matrix.

In this article, we give a survey of spectral multipliers and present (without proof) sharp Hörmander-type multiplier theorems for a self adjoint operator $A$ under the assumption that $A$ has Gaussian heat kernel bounds and satisfies appropriate estimates of the $L^2$ norm of the kernels of spectral multipliers. Our theorems imply several important, previously known results on spectral multipliers and give new results for sharp estimates for the critical exponent for the Riesz means summability.

We review the theory of subelliptic operators on Lie groups. The principal themes are subcoercivity, weighting and Gaussian bounds.

We survey the role of non-commutative operator algebras in statistical mechanics and the relation between the classification of modular invariant partition functions in conformal field theories and braided subfactors.

Contact geometry treats such questions as the existence and classification of contact structures on manifolds of odd dimension and specified topological structure. See inequality (1) below. The geometric/analytic approach treated in this report introduces parabolic systems of partial differential equations (PDEs) in a way which complements the more algebraic methods, which until now are better known in contact geometry.

The description of electron current through a splitting is a mathematical problem of electron transport in quantum networks [5, 1]. For quantum networks constructed on the interface of narrow-gap semiconductors [29, 2] the relevant scattering problem for the multi-dimensional Schrödinger equation may be substituted by the corresponding problem on a one-dimensional linear graph with proper selfadjoint boundary conditions at the nodes [11, 10, 25, 24, 16, 19, 4, 28, 20, 18, 6, 5, 1]. However, realistic boundary conditions for splittings have not yet been derived.

Here we consider some compact domain attached to a few semiinfinite lines as a model for a quantum network. An asymptotic formula for the scattering matrix for this object is derived in terms of the properties of the compact domain. This allows us to propose designs for devices for manipulating quantum current through a splitting [3, 15, 22, 9, 21].

The existence theorem for quasiconformal mappings has found a central role in a diverse variety of areas such as holomorphic dynamics, Teichmüller theory, low dimensional topology and geometry, and the planar theory of PDEs. Anticipating the needs of future researchers we give an account of the "state of the art" as it pertains to this theorem, that is to the existence and uniqueness theory of the planar Beltrami equation, and various properties of the solutions to this equation.

In this note we survey results in recent research papers on the use of Lie groups in the study of partial differential equations. The focus will be on parabolic equations, and we will show how the problems at hand have solutions that seem natural in the context of Lie groups. The research is joint with D.W. Robinson, as well as other researchers who are listed in the references.

Recently Antoine and Vandergheynst $[1, 2]$ have produced continuous wavelet transforms on the n-sphere based on a principal series representation of $SO(n; 1)$. We present some of their calculations in a more general setting, from the point of view of Fourier analysis on compact groups and spherical function expansions.

We give a characterisation of central extensions of a Lie group $G$ by $C^\times$, in terms of a differential two-form on G and a differential one-form on $G \times G$. This is applied to the case of the central extension of the loop group.

We present a survey of recent results in the Banach space classification of non-commutative $L_p$-spaces.

The notion of a quaternionic gerbe is presented as a new way of bundling algebraic structures over a four manifold. The structure groupoid of this fibration is described in some detail. The Euclidean conformal group $\mathbb{R}^+SO(4)$ appears naturally as a (non-commutative) monoidal structure on this groupoid. Using this monoidal structure we indicate the existence of a canonical quaternionic gerbe associated to a conformal structure on a four manifold.