Composing and Factoring Generalized Green's Operators and Ordinary Boundary Problems

We consider solution operators of linear ordinary boundary problems with "too many" boundary conditions, which are not always solvable. These generalized Green's operators are a certain kind of generalized inverses of differential operators. We answer the question when the product of two generalized Green's operators is again a generalized Green's operator for the product of the corresponding differential operators and which boundary problem it solves. Moreover, we show that---provided a factorization of the underlying differential operator---a generalized boundary problem can be factored into lower order problems corresponding to a factorization of the respective Green's operators. We illustrate our results by examples using the Maple package IntDiffOp, where the presented algorithms are implemented.Comment: 19 page

Normal Form of Equivariant Maps and Singular Symplectic Reduction in Infinite Dimensions with Applications to Gauge Field Theory

Inspired by problems in gauge field theory, this thesis is concerned with various aspects of infinite-dimensional differential geometry. In the first part, a local normal form theorem for smooth equivariant maps between tame Fréchet manifolds is established. Moreover, an elliptic version of this theorem is obtained. The proof these normal form results is inspired by the Lyapunov–Schmidt reduction for dynamical systems and by the Kuranishi method for moduli spaces, and uses a slice theorem for Fréchet manifolds as the main technical tool. As a consequence of this equivariant normal form theorem, the abstract moduli space obtained by factorizing a level set of the equivariant map with respect to the group action carries the structure of a Kuranishi space, i.e., such moduli spaces are locally modeled on the quotient by a compact group of the zero set of a smooth map. In the second part of the thesis, the theory of singular symplectic reduction is developed in the infinite-dimensional Fréchet setting. By refining the above construction, a normal form for momentum maps similar to the classical Marle–Guillemin–Sternberg normal form is established. Analogous to the reasoning in finite dimensions, this normal form result is then used to show that the reduced phase space decomposes into smooth manifolds each carrying a natural symplectic structure. Finally,the singular symplectic reduction scheme is further investigated in the situation where the original phase space is an infinite-dimensional cotangent bundle. The fibered structure of the cotangent bundle yields a refinement of the usual orbit-momentum type strata into so-called seams. Using a suitable normal form theorem, it is shown that these seams are manifolds. Taking the harmonic oscillator as an example, the influence of the singular seams on dynamics is illustrated. The general results stated above are applied to various gauge theory models. The moduli spaces of anti-self-dual connections in four dimensions and of Yang–Mills connections in two dimensions is studied. Moreover, the stratified structure of the reduced phase space of the Yang–Mills–Higgs theory is investigated in a Hamiltonian formulation after a (3 + 1)-splitting

Normal Form of Equivariant Maps and Singular Symplectic Reduction in Infinite Dimensions with Applications to Gauge Field Theory

Inspired by problems in gauge field theory, this thesis is concerned with various aspects of infinite-dimensional differential geometry. In the first part, a local normal form theorem for smooth equivariant maps between tame Fréchet manifolds is established. Moreover, an elliptic version of this theorem is obtained. The proof these normal form results is inspired by the Lyapunov–Schmidt reduction for dynamical systems and by the Kuranishi method for moduli spaces, and uses a slice theorem for Fréchet manifolds as the main technical tool. As a consequence of this equivariant normal form theorem, the abstract moduli space obtained by factorizing a level set of the equivariant map with respect to the group action carries the structure of a Kuranishi space, i.e., such moduli spaces are locally modeled on the quotient by a compact group of the zero set of a smooth map. In the second part of the thesis, the theory of singular symplectic reduction is developed in the infinite-dimensional Fréchet setting. By refining the above construction, a normal form for momentum maps similar to the classical Marle–Guillemin–Sternberg normal form is established. Analogous to the reasoning in finite dimensions, this normal form result is then used to show that the reduced phase space decomposes into smooth manifolds each carrying a natural symplectic structure. Finally,the singular symplectic reduction scheme is further investigated in the situation where the original phase space is an infinite-dimensional cotangent bundle. The fibered structure of the cotangent bundle yields a refinement of the usual orbit-momentum type strata into so-called seams. Using a suitable normal form theorem, it is shown that these seams are manifolds. Taking the harmonic oscillator as an example, the influence of the singular seams on dynamics is illustrated. The general results stated above are applied to various gauge theory models. The moduli spaces of anti-self-dual connections in four dimensions and of Yang–Mills connections in two dimensions is studied. Moreover, the stratified structure of the reduced phase space of the Yang–Mills–Higgs theory is investigated in a Hamiltonian formulation after a (3 + 1)-splitting

Regular and singular boundary problems in Maple

We describe a new Maple package for treating boundary problems for linear ordinary differential equations, allowing two-/multipoint as well as Stieltjes boundary conditions. For expressing differential operators, boundary conditions, and Green's operators, we employ the algebra of integro-differential operators. The operations implemented for regular boundary problems include computing Green's operators as well as composing and factoring boundary problems. Our symbolic approach to singular boundary problems is new; it provides algorithms for computing compatibility conditions and generalized Green's operators