We review the general theory of duality rotations which, in four dimensions, exchange electric with magnetic fields. Necessary and sufficient conditions in order for a theory to have duality symmetry are established. A nontrivial example is Born-Infeld theory with n abelian gauge fields and with Sp(2n,R) self-duality. We then review duality symmetry in supergravity theories. In the case of N=2 supergravity duality rotations are in general not a symmetry of the theory but a key ingredient in order to formulate the theory itself. This is due to the beautiful relation between the geometry of special Kaehler manifolds and duality rotations.

I shall discuss some basic facts about supersymmetry, in a way that I hope will be understandable to non experts. A (very little) knowledge of relativistic quantum mechanics will be assumed. I shall restrict myself almost entirely to rigid (global) supersymmetry in four space-tine dimensions, although the super-Higgs effect will be discussed briefly. Supergravity and supersymmetry in diverse dimesnions form a sizable body of work that has become an integral part of superstring theory, and that can be described clearly only in lengthy monographs written for readers with considerable technical background knowledge. Even within the restricted scope have imposed on myself, I was constrained to select a few topics that I consider important and which do not require a highly technical descriprtion. No attempt to completeness is made in the text: or in the references. Topics which could not be covered include holomorphic techniques, duality and "BPS saturation" and various aspects of dynamical supersymmetry breaking.

A brief description of some salient aspects of four-dimensional supersymmetry: early history, supermanifolds, the MSSM, cold dark matter, the cosmological constant and the string landscape.

First, we review a result in our previous paper, of how a ten-dimensional superparticle, taken off-shell, has a hidden eleven-dimensional super Poincare symmetry. Then, we show that the physical sector is defined by three first-class constraints which preserve the full eleven-dimensional symmetry. Applying the same concepts to the eleven-dimensional superparticle, taken off-shell, we discover a hidden twelve-dimensional super Poincare symmetry that governs the theory.

The Seiberg-Witten map for noncommutative Yang-Mills theories is considered from the point of view of deformation quantization. Methods for the explicit construction of the map are described which are valid for any gauge group. In particular the use of the evolution equation is described in some detail and its relation to he cohomological approach is elucidated. Cohomological methods which are applicable to gauge theories requiring the Batalin-Vilkoviskii antifield formalism are briefly mentioned. The generalization to a position dependent Poisson tensor is considered.

A covariant quantization method is developed for the off-shell superparticle in 10 dimensions. On shell it is consistent with light cone quantization, while off shell it gives a noncommutative superspace that realizes nonlinearly a hidden 11-dimensional super Poincare symmetry. The nonlinear commutation rules are then used to construct the supersymmetric generalization of the covariant Moyal star product in noncommutative superspace. As one of the possible applications, we propose this new product as the star product in supersymmetric string field theory. Furthermore, the formalism introduces new techniques and concepts in noncommutative (super)geometry.

The Seiberg-Witten map for noncommutative Yang-Mills theories is studied and methods for its explicit construction are discussed which are valid for any gauge group. In particular the use of the evolution equation is described in some detail and its relation to the cohomological approach is elucidated. Cohomological methods which are applicable to gauge theories requiring the Batalin-Vilkoviskii antifield formalism are briefly mentioned. Also, the analogy of the Weyl-Moyal star product with the star product of open bosonic string field theory and possible ramifications of this analogy are briefly mentioned.

In this paper, we describe a method for obtaining the nonabelian Seiberg-Witten map for any gauge group and to any order in θ. The equations defining the Seiberg-Witten map are expressed using a coboundary operator, so that they can be solved by constructing a corresponding homotopy operator. The ambiguities, of both the gauge and covariant type, which arise in this map are manifest in our formalism.

We briefly review the results of our paper(4): We study certain perturbative solutions of left-unilateral matrix equations. These are algebraic equations where the coefficients and the unknown are square matrices of the same order, or, more abstractly, elements of an associative, but possibly noncommutative algebra, and all coefficients are on the left. Recently such equations have appeared in a discussion of generalized Born-Infeld theories. In particular, two equations, their perturbative solutions and the relation between them are studied, applying a unified approach based on the generalized Bezout theorem for matrix polynomials.

We present a cohomological method for obtaining the non-abelian Seiberg-Witten map for any gauge group and to any order in theta. By introducing a ghost field, we are able to express the equations defining the Seiberg-Witten map through a coboundary operator, so that they can be solved by constructing a corresponding homotopy operator.

We consider the problem of constructing the fully gauged effective action in 2n-dimensional space-time for Nambu-Goldstone bosons valued in a homogeneous space G/H, with the requirement that the action is a solution of the anomalous Ward identity and that it is invariant under the gauge transformations of H. We show that this can be done whenever the homotopy group , π2n(G/H) is trivial, G/H is reductive and H is embedded in G so as to be anomaly free, in particular when H is an anomaly safe group. We construct the necessary generalization of the Bardeen counterterm and give explicit forms for the anomaly and the effective action. When G/H is a symmetric space, the counterterm and the anomaly decompose into a parity-even and a parity-odd part. In this case, for the parity-even part of the action, one does not need the anomaly free embedding of H.

In a recent paper the quantum 2-sphere $S^2_q$ was described as a quantum complex manifold. Here we consider several copies of $S^2_q$ and derive their braiding commutation relations. The braiding is extended to the differential and to the integral calculus on the spheres. A quantum analogue of the classical anharmonic ratio of four points on the sphere is given, which is invariant under the coaction of $SU_q(2)$.

A generalization of the differential geometry of forms and vector fields to the case of quantum Lie algebras is given. In an abstract formulation that incorporates many existing examples of differential geometry on quantum spaces we combine an exterior derivative, inner derivations, Lie derivatives, forms and functions all into one big algebra, the ``Cartan Calculus''. (This is an extended version of a talk presented by P. Schupp at the XXII$^{th}$ International Conference on Differential Geometric Methods in Theoretical Physics, Ixtapa, Mexico, September 1993)

Theq-differential calculus for theq-Minkowski space is developed. The algebra of theq-derivatives with theq-Lorentz generators is found giving theq-deformation of the Poincaré algebra. The reality structure of theq-Poincaré algebra is given. The reality structure of theq-differentials is also found. The real Laplacian is constructed. Finally the comultiplication, counit and antipode for theq-Poincaré algebra are obtained making it a Hopf algebra.

The nonlinear reality structure of the derivatives and the differentials for the Euclidean q-spaces are found. A real Laplacian is constructed and reality properties of the exterior derivative are given.

The six generator deformation of the Lorentz algebra is presented. The Hopf algebra structure and the reality conditions are found. The chiral decomposition of SL(2, C) is generalized to theq-case. Casimir operators for theq-Lorentz algebra are given.

We derive a q-deformed version of the Lorentz algebra by deforming the algebraSL(2,C). The method is based on linear representations of the algebra on the complex quantum spinor space. We find that the generators usually identified withSLq(2,C) generateSU q (2) only. Four additional generators are added which generate Lorentz boosts. The full algebra of all seven generators and their coproduct is presented. We show that in the limitq→1 the generators are those of the classical Lorentz algebra plus an additionalU(1). Thus we have a deformation ofSL(2,C)×U(1).

We construct a right-invariant differential calculus on the quantum supergroupGL q (1/1) and we show that the quantum Lie algebra generators satisfy the undeformed Lie superalgebra. The deformation becomes apparent when one studies the comultiplication for these generators. We bring the algebra into the standard Drinfeld-Jimbo form by performing a suitable change of variables, and we check the consistency of the map with the induced comultiplication.