Institute for Pure & Applied Mathematics (IPAM) Courses

Move-Reduced Graphs on a Torus vs Positroid Catalan Numbers

Explore combinatorial dynamics in affine symmetric groups, connecting torus graph classification and positroid Catalan numbers. Applications to geometry, knot theory, and Grassmannian varieties discussed.

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

Sums of Weighted Lattice Points of Polytopes

Explore connections between polytopes and algebraic combinatorics, focusing on sums of weighted lattice points and their computational implications in estimating maximal weighted lattice points.

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

The Andrews-Gordon Partition Identities and Commutative Algebra

Explore partition identities, focusing on Andrews-Gordon identities and their connections to commutative algebra. Learn about new combinatorial techniques and q-series identities in number theory.

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

Rotation-Invariant Web Bases from Hourglass Plabic Graphs and Symmetrized Six-Vertex Configurations

Explore symmetrized six-vertex model, hourglass plabic graphs, and their applications in statistical mechanics and combinatorics. Discover connections to tableaux, alternating sign matrices, and plane partitions.

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

Log-Concave Polynomials, Lattice Point Counting, and Traveling Salesperson Problem

Explore log-concave polynomials and their applications in bounds, probability, and optimization, including recent advancements in lattice point counting and the Traveling Salesperson Problem.

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

Complexity of Log-Concave Poset Inequalities

Explore log-concave inequalities in mathematics, from elementary examples to complex cases like Stanley's inequality, and examine their intrinsic complexity using combinatorics, complexity theory, and geometry.

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

Correlation Inequalities for Schur Functions and Young Tableaux

Explore correlation inequalities in mathematics, focusing on Schur functions and Young tableaux. Discover connections to probability, statistical physics, and algebraic combinatorics in this advanced lecture.

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

Sums of GUE Matrices and Concentration of Hives from Correlation Decay of Eigengaps

Explore the concentration of augmented hives in sums of random Hermitian matrices, using GUE ensemble eigenvalues and analyzing asymptotic behavior as eigenvalue count increases.

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

Global Asymptotics of Jack-Deformed Random Young Diagrams via Lukasiewicz Paths

Explores discrete analogues of log-gas systems, examining universal formulas for global asymptotics in various temperature regimes, using weighted lattice paths and revealing phase transition phenomena.

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

Jack Characters as Generating Series of Bipartite Maps and Proof of Lassalle's Conjecture

Explores Jack polynomials' connection to non-orientable map enumeration, presenting a formula for power-sum expansion and proving Lassalle's conjecture on Jack characters in Stanley's coordinates.

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

Random Partitions, Hurwitz Numbers and Counting High Genus Surfaces

Explore connections between random partitions, Hurwitz numbers, and high genus surfaces. Discover asymptotic behavior of Plancherel-Hurwitz measures and their implications for counting complex mathematical structures.

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

Toric Promotion with Reflections and Refractions

Exploration of toric promotion with reflections and refractions in combinatorial billiards, generalizing dynamics and revealing orbit structures in various graph configurations.

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

Unifying Lattices Through Hourglass Plabic Graphs

Explore connections between web bases, quantum groups, and lattice models. Discover new combinatorial frameworks linking alternating sign matrices, plane partitions, and the Tamari lattice.

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

Local Weak Convergence for Random Constraint Satisfaction Problems

Explore local weak convergence in random constraint satisfaction problems with Princeton's Allan Sly, delving into advanced statistical mechanics concepts beyond two dimensions.

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

Random Aggregation on the Complex Plane and the Hastings-Levitov Models

Explore random aggregation on complex planes through ALE models, including Hastings-Levitov models. Discover scaling limits, fluctuations, and connections to fundamental discrete models in statistical mechanics.

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