Mathematical Sciences Research Institute

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Holomorphic Differentials in Mathematics and Physics August 12, 2019 to December 13, 2019
Organizers LEAD Jayadev Athreya (University of Washington), Steven Bradlow (University of Illinois at Urbana-Champaign), Sergei Gukov (California Institute of Technology), Andrew Neitzke (Yale University), Anna Wienhard (Max Planck Institute for Mathematics in the Sciences), Anton Zorich (Institut de Mathematiques de Jussieu)
Some holomorphic differentials on a genus 2 surface, with close up views of singular points, image courtesy Jian Jiang.
Holomorphic differentials on Riemann surfaces have long held a distinguished place in low dimensional geometry, dynamics and representation theory. Recently it has become apparent that they constitute a common feature of several other highly active areas of current research in mathematics and also at the interface with physics. In some cases the areas themselves (such as stability conditions on Fukaya-type categories, links to quantum integrable systems, or the physically derived construction of so-called spectral networks) are new, while in others the novelty lies more in the role of the holomorphic differentials (for example in the study of billiards in polygons, special - Hitchin or higher Teichmuller - components of representation varieties, asymptotic properties of Higgs bundle moduli spaces, or in new interactions with algebraic geometry). It is remarkable how widely scattered are the motivating questions in these areas, and how diverse are the backgrounds of the researchers pursuing them. Bringing together experts in this wide variety of fields to explore common interests and discover unexpected connections is the main goal of our program. Our program will be of interest to those working in many different elds, including low-dimensional dynamical systems (via the connection to billiards); differential geometry (Higgs bundles and related moduli spaces); and different types of theoretical physics (electron transport and supersymmetric quantum field theory). Bibliography Preprints - Masur-Veech volumes, frequencies of simple closed geodesics and intersection numbers of moduli spaces of curves - The vanishing rate of Weil-Petersson sectional curvatures - Higgs bundles -- Recent applications - Orthogonal Higgs bundles with singular spectral curves - What if Planet 9 is a Primordial Black Hole? - Ultraviolet Freeze-in and Non-Standard Cosmologies - Exponential BPS graphs and D brane counting on toric Calabi-Yau threefolds: Part I - Multi-cover skeins, quivers, and 3d \mathcal{N}=2 dualities - Higgs bundles without geometry - Minimal percolating sets for mutating infectious diseases - Reviving Z and Higgs Mediated Dark Matter Models in Matter Dominated Freeze-out - Generalized B-Opers - Stability conditions, cluster varieties, and Riemann-Hilbert problems from surfaces
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Programmatic Workshops
August 15, 2019 - August 16, 2019 Connections for Women: Holomorphic Differentials in Mathematics and Physics
August 19, 2019 - August 23, 2019 Introductory Workshop: Holomorphic Differentials in Mathematics and Physics
November 18, 2019 - November 22, 2019 Holomorphic Differentials in Mathematics and Physics