Title:

Algebraic topology of PDES

We consider a compact, oriented,smooth Riemannian manifold $M$ (with or without boundary) and wesuppose $G$ is a torus acting by isometries on $M$. Given $X$ in theLie algebra of $G$ and corresponding vector field $X_M$ on $M$, onedefines Witten's inhomogeneous coboundary operator $\d_{X_M} =\d+\iota_{X_M}: \Omega_G^\pm \to\Omega_G^\mp$ (even/odd invariantforms on $M$) and its adjoint $\delta_{X_M}$. First, Witten [35] showed that the resulting cohomology classeshave $X_M$harmonic representatives (forms in the null space of$\Delta_{X_M} = (\d_{X_M}+\delta_{X_M})^2$), and the cohomologygroups are isomorphic to the ordinary de Rham cohomology groups ofthe set $N(X_M)$ of zeros of $X_M$. The first principal purpose isto extend Witten's results to manifolds with boundary. Inparticular, we define relative (to the boundary) and absoluteversions of the $X_M$cohomology and show the classes haverepresentative $X_M$harmonic fields with appropriate boundaryconditions. To do this we present the relevant version of theHodgeMorreyFriedrichs decomposition theorem for invariant forms interms of the operators $\d_{X_M}$ and $\delta_{X_M}$; the proofinvolves showing that certain boundary value problems are elliptic.We also elucidate the connection between the $X_M$cohomology groupsand the relative and absolute equivariant cohomology, followingwork of Atiyah and Bott. This connection is then exploited to showthat 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 definethe \emph{$X_M$Poincar\'{e} duality angles} between the interiorsubspaces of $X_M$harmonic fields on $M$ with appropriate boundaryconditions.Second, In 2008, Belishev and Sharafutdinov [9] showed thatthe DirichlettoNeumann (DN) operator $\Lambda$ inscribes into thelist of objects of algebraic topology by proving that the de Rhamcohomology 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 $\partial M$which we call the $X_M$DN map and using this we recover the longexact $X_M$cohomology sequence of the topological pair $(M,\partialM)$ from an isomorphism with the long exact sequence formed from thegeneralized boundary data. Consequently, This shows that for aZariskiopen subset of the Lie algebra, $\Lambda_{X_M}$ determinesthe free part of the relative and absolute equivariant cohomologygroups 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 theequivariant algebraic topology of $M$ than does the operator$\Lambda$ on $\partial M$. Finally, we elucidate the connectionbetween BelishevSharafutdinov's boundary data on $\partial N(X_M)$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 ofcertain subcomplex of the complex $(\Omega^{*}_G,\d_{X_M})$ and weprove that it is isomorphic to the total absolute and relative$X_M$cohomology groups.
