Explore cutting-edge research at the intersection of statistical mechanics, discrete geometry, and cluster algebras through this comprehensive workshop from the Institute for Pure & Applied Mathematics. Delve into the geometric structures associated with bipartite graphs on surfaces and their crucial role in solving complex mathematical problems across multiple disciplines. Discover how probabilists in statistical mechanics seek appropriate embeddings of planar graphs to develop discrete complex analysis theories suitable for observing conformally invariant objects in scaling limits. Examine the deep connections between specific graph immersions, model integrability, and Harnack curves that have emerged from this research. Learn about spaces of geometric objects parametrized by weighted bipartite graphs on surfaces and their relationships to cluster algebras and integrable systems, including discrete differential geometry objects like Q-nets and Darboux maps, positive Grassmannians, and higher Teichmüller spaces. Investigate the emerging connections between knot theory and the dimer model through presentations covering planar site percolation, tree embeddings, Ising model spectrum analysis, multinomial dimer models, Specht polynomials, Aztec diamond graphs, discrete maximal Lorentz surfaces, mirror coamoebae, M-curves, knot construction with geometric limits, free boundary Schur processes, superport networks, Pfaffians in spin models, and spanning tree entropy of planar lattices. Gain insights from leading researchers as they present their latest findings and foster interdisciplinary collaboration in these rapidly evolving mathematical fields.

Syllabus

Zhongyang Li - Planar Site Percolation via Tree Embeddings - IPAM at UCLA
Sergey Fomin - Incidences and tilings - IPAM at UCLA
Terrence George - Spectrum of the Ising model - IPAM at UCLA
Richard Kenyon - The multinomial dimer model - IPAM at UCLA
Eveliina Peltola - On variants of Specht polynomials and random geometry - IPAM at UCLA
Matthew Nicoletti - Perfect t-embeddings of uniform Aztec diamond graphs - IPAM at UCLA
Niklas Affolter - S-embeddings and discrete maximal Lorentz surfaces - IPAM at UCLA
Harold Williams - Bipartite graphs and mirror coamoebae - IPAM at UCLA
Alexander Bobenko - Dimers and M-curves: Limit shapes from Riemann surfaces - IPAM at UCLA
Jessica Purcell - Constructing knots with specified geometric limits - IPAM at UCLA
Mirjana Vuletić - Bulk scaling limits in free boundary Schur process - IPAM at UCLA
Pavel Galashin - Geometric objects associated to planar bipartite graphs - IPAM at UCLA
Pavlo Pylyavskyy - Superport networks - IPAM at UCLA
Mikhail Basok - Dimers on a Riemann surface and compactified free field - IPAM at UCLA
Marcin Lis - On Pfaffians in spin models - IPAM at UCLA
Tomas Berggren - Geometry of the doubly periodic Aztec dimer model - IPAM at UCLA
Anton Izosimov - Incidences and dimers - IPAM at UCLA
Abhijit Champanerkar - Geometric bounds for spanning tree entropy of planar lattices - IPAM at UCLA
Cédric Boutillier - Minimal bipartite dimers and maximal Riemann surfaces - IPAM at UCLA