We derive an effective recursion for Witten's r-spin intersection numbers, using Witten's conjecture relating r-spin numbers to the Gel'fand-Dikii hierarchy (Theorem 4.1). Consequences include closed-form descriptions of the intersection numbers (for example, in terms of gamma functions: Propositions 5.2 and 5.4, Corollary 5.5). We use these closed-form descriptions to prove Harer-Zagier's formula for the Euler characteristic of M_ {g, 1}. Finally in Section 6, we extend Witten's series expansion formula for the Landau-Ginzburg potential to study r-spin numbers in the small phase space in genus zero. Our key tool is the calculus of formal pseudodifferential operators, and is partially motivated by work of Brezin and Hikami.

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The main results of this paper involve general algebraic differentials ω on a general pencil of algebraic curves. We show how to determine whether ω is integrable in elementary terms for infinitely many members of the pencil. In particular, this corrects an assertion of James Davenport from 1981 and provides the first proof, even in rather strengthened form. We also indicate analogies with work of Andre and Hrushovski and with the Grothendieck–Katz Conjecture. To reach this goal, we first provide proofs of independent results which extend conclusions of relative Manin–Mumford type allied to the Zilber–Pink conjectures: we characterise torsion points lying on a general curve in a general abelian scheme of arbitrary relative dimension at least 2. In turn, we present yet another application of the latter results to a rather general pencil of Pell equations A^2−DB^2=1 over a polynomial ring. We determine whether the Pell equation (with squarefree D) is solvable for infinitely many members of the pencil.

Let the vector bundle E be a deformation of the tangent bundle over the Grassmannian G(k, n). We compute the ring structure of sheaf cohomology valued in exterior powers of E, also known as the polymology. This is the first part of a project studying the quantum sheaf cohomology of Grassmannians with deformations of the tangent bundle, a generalization of ordinary quantum cohomology rings of Grassmannians. A companion physics paper [6] describes physical aspects of the theory, including a conjecture for the quantum sheaf cohomology ring, and numerous examples.

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