  We study \emph{dynamic} algorithms for maintaining spectral vertex
sparsifiers of graphs with respect to a set of terminals $T$ of our choice.
Such objects preserve pairwise resistances, solutions to systems of linear
equations, and energy of electrical flows between the terminals in $T$. We give
a data structure that supports insertions and deletions of edges, and terminal
additions, all in sublinear time. Our result is then applied to the following
problems.
  (1) A data structure for maintaining solutions to Laplacian systems
$\mathbf{L} \mathbf{x} = \mathbf{b}$, where $\mathbf{L}$ is the Laplacian
matrix and $\mathbf{b}$ is a demand vector. For a bounded degree, unweighted
graph, we support modifications to both $\mathbf{L}$ and $\mathbf{b}$ while
providing access to $\epsilon$-approximations to the energy of routing an
electrical flow with demand $\mathbf{b}$, as well as query access to entries of
a vector $\tilde{\mathbf{x}}$ such that $\left\lVert
\tilde{\mathbf{x}}-\mathbf{L}^{\dagger} \mathbf{b} \right\rVert_{\mathbf{L}}
\leq \epsilon \left\lVert \mathbf{L}^{\dagger} \mathbf{b}
\right\rVert_{\mathbf{L}}$ in $\tilde{O}(n^{11/12}\epsilon^{-5})$ expected
amortized update and query time.
  (2) A data structure for maintaining All-Pairs Effective Resistance. For an
intermixed sequence of edge insertions, deletions, and resistance queries, our
data structure returns $(1 \pm \epsilon)$-approximation to all the resistance
queries against an oblivious adversary with high probability. Its expected
amortized update and query times are $\tilde{O}(\min(m^{3/4},n^{5/6}
\epsilon^{-2}) \epsilon^{-4})$ on an unweighted graph, and
$\tilde{O}(n^{5/6}\epsilon^{-6})$ on weighted graphs.
  These results represent the first data structures for maintaining key
primitives from the Laplacian paradigm for graph algorithms in sublinear time
without assumptions on the underlying graph topologies.
