This joint work with Calvin Tsay, Jan Kronqvist, Alexander Thebelt is based on two papers (https://arxiv.org/abs/2102.04373, https://arxiv.org/abs/2101.12708).

]]>This talk is based on:

*On exact-WKB analysis, resurgent structure, and quantization conditions*, N.Sueishi, S.K, T.Misumi, and M.Ünsal, arXiv:2008.00379.*Exact-WKB, complete resurgent structure, and mixed anomaly in quantum mechanics on $S^1$*, N.Sueishi, S.K, T.Misumi, and M.Ünsal, arXiv.2103.06586

Our goal in this talk is to show that an asymptotic nonlinear mean value formula holds for the classical Monge-Ampère equation.

Joint work with P. Blanc (Jyvaskyla), F. Charro (Detroit), and J.J. Manfredi (Pittsburgh).

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[1] E. Fiorelli et al., Physical Review Letters 125, 070604 (2020)

[2] F. Carollo and I. Lesanovsky, arXiv:2009.13932 (2020)

[3] V.D. Vaidya et al., Physical Review X 8, 011002 (2018)

[4] P. Rotondo et al., Journal of Physics A 51, 115301 (2018)

[5] E. Fiorelli et al., Physical Review A 99, 032126 (2019)

]]>In particular we show that the P=W conjecture holds for character varieties which admit a symplectic resolution, namely in genus 1 and arbitrary rank and in genus 2 and rank 2. In the talk I will first mention basic notions of non abelian Hodge theory and introduce the P=W conjecture for smooth moduli spaces; then I will explain how to extend these phenomenas to the singular case, showing the proof our results in a specific example.

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