Algebraic Topology of PDEs

We consider a compact, oriented, smooth Riemannian manifold M (with or without boundary) and we suppose G is a torus acting by isometries on M. Given X in the Lie algebra of G and corresponding vector field X_M on M, one defines Witten’s inhomogeneous coboundary operator $d_{X_M} = d+i_{X_M} : \Omega_G^\pm \to \Omega_G^\pm$ (even/odd invariant forms on M) and its adjoint $\delta_{X_M}$. First,Witten [35] showed that the resulting cohomology classes have X_M-harmonic representatives (forms in the null space of $\Delta_{X_M} = (d_{X_M} + \delta_{X_M})^2$), and the cohomology groups are isomorphic to the ordinary de Rham cohomology groups of the set N(X_M) of zeros of X_M. The first principal purpose is to extend Witten’s results to manifolds with boundary. In particular, we define relative (to the boundary) and absolute versions of the X_M-cohomology and show the classes have representative X_M-harmonic fields with appropriate boundary conditions. To do this we present the relevant version of the Hodge-Morrey-Friedrichs decomposition theorem for invariant forms in terms of the operators d_{X_M} and \deta_{X_M}; the proof involves showing that certain boundary value problems are elliptic. We also elucidate the connection between the X_M-cohomology groups and the relative and absolute equivariant cohomology, following work of Atiyah and Bott. This connection is then exploited to show that every harmonic field with appropriate boundary conditions on N(X_M) has a unique corresponding an X_M-harmonic field on M to it, with corresponding boundary conditions. Finally, we define the interior and boundary portion of X_M-cohomology and then we define the X_M-Poincar´e duality angles between the interior subspaces of X_M-harmonic fields on M with appropriate boundary conditions. Second, in 2008, Belishev and Sharafutdinov [9] showed that the Dirichlet-to-Neumann (DN) operator \Lambda inscribes into the list of objects of algebraic topology by proving that the de Rham cohomology groups are determined by \Lambda. In the second part of this thesis, we investigate to what extent is the equivariant topology of a manifold determined by a variant of the DN map?. Based on the results in the first part above, we define an operator \Lambda_{X_M} on invariant forms on the boundary ¶M which we call the X_M-DN map and using this we recover the long exact X_M-cohomology sequence of the topological pair (M;\partial M) from an isomorphism with the long exact sequence formed from the generalized boundary data. Consequently, This shows that for a Zariski-open subset of the Lie algebra, \Lambda_{X_M} determines the free part of the relative and absolute equivariant cohomology groups of M. In addition, we partially determine the mixed cup product of X_M-cohomology groups from \Lambda_{X_M}. This shows that \Lambda_{X_M} encodes more information about the equivariant algebraic topology of M than does the operator \Lambda on the boundary. Finally, we elucidate the connection between Belishev-Sharafutdinov’s boundary data on N(X_M) and ours on \partial M. Third, based on the first part above, we present the (even/odd) X_M-harmonic cohomology which is the cohomology of certain subcomplex of the complex (\Omega_G^*, d_{X_M}) and we prove that it is isomorphic to the total absolute and relative X_M-cohomology groups