Title: Multi-instantons in 2D string theory and median resummation

Abstract: The normalization of multi-instanton effects in two-dimensional string theory involves numerical factors that are hard to determine. I'll show how they can be found using a chiral representation of the dual matrix quantum mechanics.The result is confirmed by a string field theory calculation of annuli amplitudes and nicely agrees with a median resummation of the Gamma function.

Title: Divergence and resurgence

Abstract: In the mathematical phenomenon of resurgence, typically associated with divergent series, the tail of the series – the late terms - can be resummed to generate a series of corrections corresponding to the early terms of the original series or one or more closely related series. The first of two examples concerns the saddle-point approximation to integrals. The divergent tail of the series corresponding to an integration path through one saddle is represented, after resummation, by the terms in the integals through other saddles; this is

‘hyperasymptotics’[1]. The second example concerns the Riemann zeta function at height t on the critical line, in particular the real function Z(t), proportional to ζ(1/2+it). Representing Z(t) by the Dirichlet series gives a divergent series of complex terms. But if the Dirichlet series is truncated by O(t1/2) terms, the tail, approximated by the Poisson sum formula, consists of terms that are the complex conjugates of those in the original Dirichlet series [2-4]. The resulting approximation to Z(t) is real, and equal to the ‘main sum’ in the Riemann-Siegel formula.

[1] Berry, M. V., & Howls, C. J. 1991 Hyperasymptotics for integrals with saddles Proc. Roy. Soc. Lond. A434 657-675.

[2] Berry, M. V. 1986. Riemann's zeta function: a model for quantum chaos? In T. H. Seligman & H. Nishioka (Eds.), Quantum chaos and statistical nuclear physics (Vol. 263, pp. 1-17).

[3] Edwards, H. M. 2001.Riemann's Zeta Function Mineola, New York Dover Publications:

[4] Berry, M. V. 1995 The Riemann-Siegel formula for the zeta function: high orders and remainders Proc.Roy.Soc.Lond. A450 439 - 462.

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Title: Resurgence's first and second youth

Abstract: To begin with, I propose to indulge in some reminiscing, briefly revisit those aspects of resurgence theory that have best stood the test of time, and tentatively sketch a few paths ahead. Then I shall turn to specifics - to a type of resurgence of special relevance to knot theory. Associated with power series whose Taylor coefficients are syntactically sum-product combinations, it lays bare, in striking fashion, the mechanisms that call singularities into existence and cause them to self-reproduce (that latter part draws on joint work with Shweta Sharma).

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