Volume 2017, Issue 728

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. Consequences include closed-form descriptions of the intersection numbers (for example, in terms of gamma functions). We use these closed-form descriptions to prove Harer–Zagier’s formula for the Euler characteristic of ℳg,1{\mathcal{M}_{g,1}}. Finally, 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 Brézin and Hikami.

We prove generating series formulae for suitable twisted characteristic classes of symmetric products of a singular complex quasi-projective variety. More concretely, we study homology Hirzebruch classes for motivic coefficients, as well as for complexes of mixed Hodge modules. As a special case, we obtain a generating series formula for the (intersection) homology Hirzebruch classes of symmetric products. In some cases, the latter yields a similar formula for twisted homology L -classes generalizing results of Hirzebruch–Zagier and Moonen. Our methods also apply to the study of Todd classes of (complexes of) coherent sheaves, as well as Chern classes of (complexes of) constructible sheaves, generalizing to arbitrary coefficients results of Moonen and respectively Ohmoto.

We show that every isoperimetric set in ℝN{\mathbb{R}^{N}} with density is bounded if the density is continuous and bounded above and below. This improves the previously known boundedness results, which basically needed a Lipschitz assumption; on the other hand, the present assumption is sharp, as we show with an explicit example. To obtain our result, we observe that the main tool which is often used, namely a classical “ε -- ε” property already discussed by Allard, Almgren and Bombieri, admits a weaker counterpart which is still sufficient for the boundedness, namely, an “ε --εβ\varepsilon^{\beta}” version of the property. And in turn, while for the validity of the first property the Lipschitz assumption is essential, for the latter the continuity alone is enough. We conclude by deriving some consequences of our result for the existence and almost-everywhere regularity of isoperimetric sets.

Let 𝒥{{\mathcal{J}}} be an ideal sheaf on a reduced analytic space X with zero set Z . We show that the Lelong numbers of the restrictions to Z of certain generalized Monge–Ampère products ( d dcd^{c}log|f|2|f|^{2})k)^{k}, where f is a tuple of generators of 𝒥{{\mathcal{J}}}, coincide with the so-called Segre numbers of 𝒥{{\mathcal{J}}}, introduced independently by Tworzewski, Achilles–Manaresi, and Gaffney–Gassler. More generally we show that these currents satisfy a generalization of the classical King formula that takes into account fixed and moving components of Vogel cycles associated with 𝒥{{\mathcal{J}}}. A basic tool is a new calculus for products of positive currents of Bochner–Martinelli type. We also discuss connections to intersection theory.

We construct a generalization of the Hasse invariant for any Shimura variety of PEL-type A over a prime of good reduction, whose non-vanishing locus is the open and dense μ-ordinary locus.

As a step to establish the McKay conjecture on character degrees of finite groups, we verify the inductive McKay condition introduced by Isaacs–Malle–Navarro for simple groups of Lie type 𝖠n-1{\mathsf{A}_{n-1}}, split or twisted. Key to the proofs is the study of certain characters of SLn⁢(q){{\mathrm{SL}}_{n}(q)} and SUn⁢(q){{\mathrm{SU}}_{n}(q)} related to generalized Gelfand–Graev representations. As a by-product we can show that a Jordan decomposition for the characters of the latter groups is equivariant under outer automorphisms. Many ideas seem applicable to other Lie types.

In this paper we introduce and study the Euler characteristic (denoted by χ) associated with algebraic modules generated by arbitrary elements of certain noncommutative polyballs. We provide several asymptotic formulas for χ and prove some of its basic properties. We show that the Euler characteristic is a complete unitary invariant for the finite rank Beurling type invariant subspaces of the tensor product of full Fock spaces F2F^{2}(Hn1H_{n_{1}})⊗\otimes⋯\cdots⊗\otimesF2F^{2}(HnkH_{n_{k}}), and prove that its range coincides with the interval [0,∞\infty). We obtain an analogue of Arveson’s version of the Gauss–Bonnet–Chern Theorem from Riemannian geometry, which connects the curvature to the Euler characteristic. In particular, we prove that if ℳ{{\mathcal{M}}} is an invariant subspace of F2F^{2}(Hn1H_{n_{1}})⊗\otimes⋯\cdots⊗\otimesF2F^{2}(HnkH_{n_{k}}), nin_{i}≥\geq 2, which is graded (generated by multi-homogeneous polynomials), then the curvature and the Euler characteristic of the orthocomplement of ℳ{{\mathcal{M}}} coincide.

We consider the evolution of hypersurfaces with boundary under inverse mean curvature flow. The boundary condition is of Neumann type, i.e. the evolving hypersurface moves along, but stays perpendicular to, a fixed supporting hypersurface. In this setup, we prove existence and uniqueness of weak solutions. Furthermore, we indicate the existence of a monotone quantity which is the analog of the Hawking mass for closed hypersurfaces.

Let p be a prime. We prove that there is an anti-equivalence between the category of unipotent strongly divisible lattices of weight p-1{p-1} and the category of Galois stable ℤp{\mathbb{Z}_{p}}-lattices in unipotent semi-stable representations with Hodge–Tate weights in {0,…,p-1}\{0,\ldots,p-1\}. This completes the last remaining piece of Breuil’s conjecture ([6, Conjecture 2.2.6]).