BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Fabrizio Del Monte (University of Birmingham\, UK) DTSTART:20250416T130000Z DTEND:20250416T140000Z DTSTAMP:20260423T010754Z UID:GPL/45 DESCRIPTION:Title: Mon odromies\, Clusters\, and the WKB Approximation for q-Difference Equations \nby Fabrizio Del Monte (University of Birmingham\, UK) as part of Geo metry and Physics @ Lisbon\n\nLecture held in 6.2.33 (Seminar Room\, Math\ , FCUL).\n\nAbstract\nThe study of monodromies of differential equations h as been a rich area of mathematical physics\, interconnected with various fields in mathematics and physics. Recent discoveries reveal that monodrom y varieties naturally possess the structure of cluster varieties\, signifi cantly enhancing our understanding of their connections to string theory a nd Donaldson–Thomas invariants. A key technique in these developments is the (exact) WKB approximation. In string theory\, q-difference equations (qDEs) naturally appear as an "M-theory completion" of differential equati ons\, though defining monodromy in this context remains an active research area. In this seminar\, I will discuss how the WKB approximation\, tradit ionally formulated for second-order ODEs\, can be effectively generalized to second-order q-difference equations\, providing a natural characterizat ion of their monodromies. Central to this approach is the WKB Stokes diagr am\, known in the physics literature as the exponential network\, which of fers a framework for defining cluster coordinates for monodromies of qDEs. \n\nI will illustrate this formalism through explicit examples\, including the q-difference Mathieu equation. Remarkably\, its monodromy around the origin—known in topological string theory as the quantum mirror map—ta kes the form of the Hamiltonian of a cluster integrable system in terms of these cluster coordinates.\n LOCATION:https://researchseminars.org/talk/GPL/45/ END:VEVENT END:VCALENDAR