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.

For a smooth projective toric surface we determine the Donaldson invariants and their wallcrossing in terms of the Nekrasov partition function. Using the solution of the Nekrasov conjecture [33, 38, 3] and its refinement [34], we apply this result to give a generating function for the wallcrossing of Donaldson invariants of good walls of simply connected projective surfaces with b+ = 1 in terms of modular forms. This formula was proved earlier in [19] more generally for simply connected 4-manifolds with b+ = 1, as- suming the Kotschick-Morgan conjecture, and it was also derived by physical arguments in [31].

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We define a partition of Z into intervals {$I$_{j}} and prove the Littlewood-Paley inequality ‖$f$‖_{$p$}≦$C$_{p}‖$Sf$‖_{$p$}, 2≦$p$<∞. Here$f$is a function on [o, 2π) and $$Sf = (\sum |\Delta _j |^2 )^{1/2} , \hat \Delta j = \hat f\chi _{Ij} $$ . This is a new example of a partition having the Littlewood-Paley property since the {$I$_{j}} are not of the type obtained by iterating lacunary partitions finitely many times.

A local-global principle is shown to hold for all conjugacy classes of any inner form of GL($n$), SL($n$), U($n$), SU($n$), and for all semisimple conjugacy classes in any inner form of Sp($n$), over fields$k$with vcd($k$)≤1. Over number fields such a principle is known to hold for any inner form of GL($n$) and U($n$), and for the split forms of Sp($n$), O($n$), as well as for SL($p$) but not for SL($n$),$n$non-prime. The principle holds for all conjugacy classes in any inner form of GL($n$), but not even for semisimple conjugacy classes in Sp($n$), over fields$k$with vcd($k$)≤2. The principle for conjugacy classes is related to that for centralizers.