Institute for Pure & Applied Mathematics (IPAM) Courses

The Multinomial Dimer Model - Statistical Mechanics and Discrete Geometry

Explore the multinomial dimer model's limit shape theory, bistochastic gauge, and 3D Aztec diamond applications in statistical mechanics and discrete geometry.

• YouTube
• 48 minutes
• Self-Paced
• Free Video

Incidences and Tilings in Linear Geometry

Explore how classical incidence geometry theorems connect to surface tilings, unveiling a unified framework for generating new theorems and expanding existing ones in this insightful lecture.

• YouTube
• 57 minutes
• Self-Paced
• Free Video

Random Matrix Statistics in the Uniform Spanning Tree

Explore random matrix statistics in uniform spanning trees, focusing on branch behavior, scaling limits, and connections to Dyson Brownian motion and Loewner evolution.

• YouTube
• 51 minutes
• Self-Paced
• Free Video

Superport Networks: Generalizing Kirchhoff's Matrix-Tree Theorem

Exploration of multiport networks, generalizing Kirchhoff's theorem with novel mathematical approaches. Insights into electrical engineering applications and superport network theory.

• YouTube
• 49 minutes
• Self-Paced
• Free Video

Geometric Objects Associated to Planar Bipartite Graphs

Explore topological and geometric data associated with planar bipartite graphs, including knots, cluster varieties, and critical varieties, and their interconnections.

• YouTube
• 50 minutes
• Self-Paced
• Free Video

On Pfaffians in Spin Models

Explore Pfaffian relations in spin models, focusing on Ising models on planar graphs and their correlation functions. Discover insights into ferromagnetic spin models and graph simplifications.

• YouTube
• 49 minutes
• Self-Paced
• Free Video

Bipartite Graphs in Tn and Toric Mirror Symmetry

Explore bipartite graphs in Tn and their role in toric mirror symmetry, focusing on combinatorial descriptions of homological mirror symmetry and coherent sheaves on toric varieties.

• YouTube
• 44 minutes
• Self-Paced
• Free Video

Geometric Bounds for Spanning Tree Entropy of Planar Lattices

Explores geometric bounds for spanning tree entropy in planar lattices, connecting hyperbolic geometry, number theory, probability, and graph theory. Discusses conjectures and recent progress.

• YouTube
• 43 minutes
• Self-Paced
• Free Video

Constructing Knots with Specified Geometric Limits

Explore constructive proofs for geometric limits of knot complements using circle packings on conformal boundaries of tame hyperbolic 3-manifolds, with insights on related open questions.

• YouTube
• 52 minutes
• Self-Paced
• Free Video

Incidences and Dimers in Projective Geometry - Statistical Mechanics and Discrete Geometry

Explore the connection between linear incidence theorems in projective geometry and bipartite graphs on surfaces, incorporating dimer model dynamics to derive dynamic linear incidence theorems.

• YouTube
• 48 minutes
• Self-Paced
• Free Video

On Variants of Specht Polynomials and Random Geometry

Explore connections between Specht polynomials, conformal field theory, and random geometry. Discover algebraic structures in statistical mechanics models and their geometric interpretations.

• YouTube
• 52 minutes
• Self-Paced
• Free Video

Spectrum of the Ising Model

Explore the correspondence between Ising models and algebro-geometric data, extending Kenyon and Okounkov's work on dimer models and Harnack curves in this advanced mathematical presentation.

• YouTube
• 46 minutes
• Self-Paced
• Free Video

Minimal Bipartite Dimers and Maximal Riemann Surfaces

Exploration of critical weights for dimer models on isoradial graphs, extending Kenyon's results using theta functions on maximal Riemann surfaces, with applications to spectral theorems and Laplacians.

• YouTube
• 52 minutes
• Self-Paced
• Free Video

Dimers and M-curves: Limit Shapes from Riemann Surfaces

Explore dimer models, M-curves, and limit shapes through algebraic geometry, integrating Riemann surfaces and Schottky uniformization. Gain insights into statistical mechanics and discrete geometry.

• YouTube
• 53 minutes
• Self-Paced
• Free Video

Construction of Interpolation Symmetric Functions from q-Hahn Integrable Models

Explore q-Hahn integrable models and their connection to quantum loop sl2 algebra, leading to a new family of symmetric functions generalizing Macdonald functions with unique properties.

• YouTube
• 50 minutes
• Self-Paced
• Free Video