Abstract We provide a complete characterization of theories of tracial von Neumann algebras that admit quantifier elimination. We also show that the theory of a separable tracial von Neumann algebra script upper M $\mathcal {M}$ M is never model complete if its direct integral decomposition contains upper I upper I Subscript 1 $\mathrm {II}_1$ I I 1 factors script upper N $\mathcal {N}$ N such that upper M 2 left parenthesis script upper N right parenthesis $M_2(\mathcal {N})$ M 2 ( N ) embeds into an ultrapower of script upper N $\mathcal {N}$ N . The proof in the case of upper I upper I Subscript 1 $\mathrm {II}_1$ I I 1 factors uses an explicit construction based on random matrices and quantum expanders.

We present the first iterative spectral algorithm to find near-optimal solutions for a random quadratic objective over the discrete hypercube, resolving a conjecture of Subag [Sub21].

We propose an algebraic framework for generalized graph Laplacians which unifies the study of resistor networks, the critical group, and the eigenvalues of the Laplacian and adjacency matrices. Given a graph with boundary $G$ together with a generalized Laplacian $L$ with entries in a commutative ring $R$, we define a generalized critical group $\Upsilon_R(G,L)$. We relate $\Upsilon_R(G,L)$ to spaces of harmonic functions on the network using the Hom, Tor, and Ext functors of homological algebra. We study how these algebraic objects transform under combinatorial operations on the network $(G,L)$, including harmonic morphisms, layer-stripping, duality, and symmetry. In particular, we use layer-stripping operations from the theory of resistor networks to systematize discrete harmonic continuation. This leads to an algebraic characterization of the graphs with boundary that can be completely layer-stripped, an algorithm for simplifying computation of $\Upsilon_R(G,L)$, and upper bounds for the number of invariant factors in the critical group and the multiplicity of Laplacian eigenvalues in terms of geometric quantities.