Direct boundary integral equation method for electromagnetic scattering by partly coated dielectric objects

We present a new variational direct boundary integral equation approach for solving the scattering and transmission problem for dielectric objects partially coated with a PEC layer. The main idea is to use the electromagnetic Calderón projector along with transmission conditions for the electromagnetic fields. This leads to a symmetric variational formulation which lends itself to Galerkin discretization by means of divergence-conforming discrete surface currents. A wide array of numerical experiments confirms the efficacy of the new metho

Classical field theory on Lie algebroids: Variational aspects

The variational formalism for classical field theories is extended to the setting of Lie algebroids. Given a Lagrangian function we study the problem of finding critical points of the action functional when we restrict the fields to be morphisms of Lie algebroids. In addition to the standard case, our formalism includes as particular examples the case of systems with symmetry (covariant Euler-Poincare and Lagrange Poincare cases), Sigma models or Chern-Simons theories.Comment: Talk deliverd at the 9th International Conference on Differential Geometry and its Applications, Prague, September 2004. References adde

Semi-linear parabolic equations on homogeneous Lie groups arising from mean field games

The existence and the uniqueness of solutions to some semilinear parabolic equations on homogeneous Lie groups, namely, the Fokker-Planck equation and the Hamilton-Jacobi equation, are addressed. The anisotropic geometry of the state space plays a crucial role in our analysis and creates several issues that need to be overcome. Indeed, the ellipticity directions span, at any point, subspaces of dimension strictly less than the dimension of the state space. Finally, the above results are used to obtain the short-time existence of classical solutions to the mean field games system defined on an homogenous Lie group

Conformal symmetries of the super Dirac operator

In this paper, the Dirac operator, acting on super functions with values in super spinor space, is defined along the lines of the construction of generalized Cauchy-Riemann operators by Stein and Weiss. The introduction of the superalgebra of symmetries osp(m|2n) is a new and essential feature in this approach. This algebra of symmetries is extended to the algebra of conformal symmetries osp(m + 1, 1|2n). The kernel of the Dirac operator is studied as a representation of both algebras. The construction also gives an explicit realization of the Howe dual pair osp(1|2) x osp(m|2n) \subset osp(m + 4n|2m + 2n). Finally, the super Dirac operator gives insight into the open problem of classifying invariant first order differential operators in super parabolic geometries

An introduction to quantized Lie groups and algebras

We give a selfcontained introduction to the theory of quantum groups according to Drinfeld highlighting the formal aspects as well as the applications to the Yang-Baxter equation and representation theory. Introductions to Hopf algebras, Poisson structures and deformation quantization are also provided. After having defined Poisson-Lie groups we study their relation to Lie-bi algebras and the classical Yang-Baxter equation. Then we explain in detail the concept of quantization for them. As an example the quantization of $sl_2$ is explicitly carried out. Next we show how quantum groups are related to the Yang-Baxter equation and how they can be used to solve it. Using the quantum double construction we explicitly construct the universal $R$-matrix for the quantum $sl_2$ algebra. In the last section we deduce all finite dimensional irreducible representations for $q$ a root of unity. We also give their tensor product decomposition (fusion rules) which is relevant to conformal field theory.Comment: 38 page

On the Geometry of Inhomogeneous Quantum Groups

We first give a pedagogical introduction to the differential calculus on q-groups and analize the relation between differential calculus and q-Lie algebra. Equivalent definitions of bicovariant differential calculus are studied and their geometrical interpretation is explained. Vectorfields, contraction operator and Lie derivative are defined and their properties discussed. After a review of the geometry of the (multiparametric) linear q-group $GL_{q,r}(N)$ we construct the inhomogeneous q-group $IGL_{q,r}(N)$ as a projection from $GL_{q,r}(N+1)$, i.e. as a quotient of $GL_{q,r}(N+1)$ with respect to a suitable Hopf algebra ideal. The semidirect product structure of $IGL_{q,r}(N)$ given by the $GL_{q,r}(N)$ q-subgroup times translations is easily analized. A bicovariant calculus on $IGL_{q,r}(N)$ is explicitly obtained as a projection from the one on $GL_{q,r}(N+1)$. The universal enveloping algebra of $IGL_{q,r}(N)$ and its $R$-matrix formulation are constructed along the same lines. We proceed similarly in the orthogonal and symplectic case. We find the inhomogeneous multiparametric q-groups of the $B_n,C_n,D_n$ series via a projection from $B_{n+1}, C_{n+1},D_{n+1}$. We give an $R$-matrix formulation and discuss real forms. We study their universal enveloping algebras and differential calculi. In particular we obtain the bicovariant calculus on a dilatation-free minimal deformation of the Poincar\'e group $ISO_q(3,1)$. The projection procedure is also used to construct differential calculi on multiparametric q-orthogonal planes in any dimension $N$. Real forms are studied and in particular we obtain a q-Minkowski space and its q-deformed phase-space with hermitian operators $x^a$ and $p_a$.Comment: Ph.D. thesis, 181 pages. Added ref. [72] and [73]. Added ref. in [66] and [76

Flat vacua of maximal supergravity

The main topic of this thesis is the study of the maximal supergravity theory in 4 spacetime dimensions, in order to find, with algebraic techniques, its Minkowski vacua