We introduce a web of strongly correlated interacting 3+1D topological superconductors/ insulators of 10 particular global symmetry groups of Cartan classes, realizable in electronic condensed matter systems, and their generalizations. The symmetries include SU(N), SU(2), U(1), fermion parity, time reversal and relate to each other through symmetry embeddings. We overview the lattice Hamiltonian formalism. We complete the list of bulk symmetry-protected topological invariants (SPT invariants/partition functions that exhibit boundary 't Hooft anomalies) via cobordism calculations, matching their full classification. We also present explicit 4-manifolds that detect these SPTs. On the other hand, once we dynamically gauge part of their global symmetries, we arrive in various new phases of $SU(N)$ Yang-Mills (YM) theories, realizable as quantum spin liquids with emergent gauge fields. We discuss how coupling YM theories to time reversal-SPTs affects the strongly coupled theories at low energy. For example, we point out a possibility of having two deconfined gapless time-reversal symmetric $SU(2)$ YM theories at $\theta=\pi$ as two distinct conformal field theories, which although are secretly indistinguishable by gapped SPT states nor by correlators of local operators on oriented spacetimes, can be distinguished on non-orientable spacetimes or potentially by correlators of extended operators.

We develop a local cohomology theory for FI^m-modules, and show that it in many ways mimics the classical theory for multi-graded modules over a polynomial ring. In particular, we define an invariant of FI^m-modules using this local cohomology theory which closely resembles an invariant of multi-graded modules over Cox rings defined by Maclagan and Smith. It is then shown that this invariant behaves almost identically to the invariant of Maclagan and Smith.

We give bounds for various homological invariants (including Castelnuovo-Mumford regularity, degrees of local cohomology, and injective dimension) of finitely generated VI-modules in the non-describing characteristic case. It turns out that the formulas of these bounds for VI-modules are the same as the formulas of corresponding bounds for FI-modules.

Using the Nakayama functor, we construct an equivalence from a Serre quotient category of a category of finitely generated modules to a category of finite-dimensional modules. We then apply this result to the categories FI_G and VI_q, and answer positively an open question of Nagpal on representation stability theory.

We introduce new boundary conditions for differential forms on symplectic manifolds with boundary. These boundary conditions, dependent on the symplectic structure, allows us to write down elliptic boundary value problems for both second-order and fourth-order symplectic Laplacians and establish Hodge theories for the cohomologies of primitive forms on manifolds with boundary. We further use these boundary conditions to define a relative version of the primitive cohomologies and to relate primitive cohomologies with Lefschetz maps on manifolds with boundary. As we show, these cohomologies of primitive forms can distinguish certain Kahler structures of Kahler manifolds with boundary.