Motivated by spectral gluing patterns in the Betti Langlands program, we show that for any reductive group G, a parabolic subgroup P, and a topological surface M, the (enhanced) spectral Eisenstein series category of M is the factorization homology over M of the E2-Hecke category H_{G,P} =IndCoh(LS_{G,P}(D^2, S^1)), where LS_{G,P}(D^2, S^1) denotes the moduli stack of G-local systems on a disk together with a P-reduction on the boundary circle. More generally, for any pair of stacks Y → Z satisfying some mild conditions and any map between topological spaces N → M, we define (Y, Z)^{N,M} = Y^N ×_{Z^N} {Z^M} to be the space of maps from M to Z along with a lift to Y of its restriction to N. Using the pair of pants construction, we define an E_n-category H_n(Y, Z) = IndCoh_0((Y, Z)^{S^n−1,D^n})_{Y}), and compute its factorization homology on any d-dimensional manifold M with d ≤ n, \int_M H_n(Y, Z) = IndCoh_0(Y, Z)^{∂(M×D^n−d),M}_{Y^M}, where IndCoh_0 is the sheaf theory introduced by ArinkinGaitsgory and Beraldo. Our result naturally extends previous known computations of Ben-Zvi–Francis–Nadler and Beraldo.

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We prove the complete monotonicity on $${(0, \infty)^{n}}$$ for suitable inverse powers of the spanning-tree polynomials of graphs and, more generally, of the basis generating polynomials of certain classes of matroids. This generalizes a result of Szegő and answers, among other things, a long-standing question of Lewy and Askey concerning the positivity of Taylor coefficients for certain rational functions. Our proofs are based on two ab-initio methods for proving that $${P^{-\beta}}$$ is completely monotone on a convex cone$C$: the determinantal method and the quadratic-form method. These methods are closely connected with harmonic analysis on Euclidean Jordan algebras (or equivalently on symmetric cones). We furthermore have a variety of constructions that, given such polynomials, can create other ones with the same property: among these are algebraic analogues of the matroid operations of deletion, contraction, direct sum, parallel connection, series connection and 2-sum. The complete monotonicity of $${P^{-\beta}}$$ for some $${\beta > 0}$$ can be viewed as a strong quantitative version of the half-plane property (Hurwitz stability) for$P$, and is also related to the Rayleigh property for matroids.

In this paper we prove the invariance of quantum rings for general ordinary flops, whose local models are certain non-split toric bundles over arbitrary smooth bases. An essential ingredient in the proof is a quantum splitting principle which reduces a statement in Gromov--Witten theory on non-split bundles to the case of split bundles.