IAS | PCMI Park City Mathematics Institute Courses

Probabilistic and Extremal Combinatorics

Explore advanced discrete mathematics through extremal and probabilistic combinatorics, covering graph theory, Ramsey theory, enumeration methods, and connections to analysis and computer science.

• YouTube
• 1 day 21 hours 57 minutes
• Self-Paced
• Free Video

Motivic Homotopy Theory

Explore advanced mathematical concepts connecting algebraic geometry, topology, and arithmetic through motivic homotopy theory with leading experts in this specialized field.

• YouTube
• 1 day 23 hours 16 minutes
• Self-Paced
• Free Video

Quantum Computation - The 2023 IAS-PCMI Graduate Summer School

Dive into advanced quantum computing mathematics through expert-led minicourses covering quantum learning, error correction, complexity theory, and near-term quantum device analysis.

• YouTube
• 1 day 19 hours 46 minutes
• Self-Paced
• Free Video

An Introduction to Lattices, Lattice Reduction, and Lattice-Based Cryptography

Explore lattice theory and quantum-resistant cryptography through comprehensive lectures covering lattice problems, reduction algorithms, and practical cryptosystem construction.

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

Quadratic Forms, Milnor K-Theory, and Arithmetic - 2021

Explore quadratic forms theory, Milnor K-theory, and arithmetic through the Hasse-Minkowski theorem, quadratic reciprocity law, and Siegel mass formula with experimental mathematics.

• YouTube
• 17 hours 6 minutes
• Self-Paced
• Free Video

Heegaard Floer Homology

Dive into Heegaard Floer homology through comprehensive lectures covering fundamental concepts and advanced applications in low-dimensional topology and knot theory.

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

Advanced Topics in Mathematical Gauge Theory - Higgs Bundles

Delve into advanced mathematical gauge theory and Higgs bundles through expert lectures covering sophisticated theoretical frameworks and their geometric applications.

• YouTube
• 4 hours 11 minutes
• Self-Paced
• Free Video

Introduction to Gauge Theory

Dive into gauge theory fundamentals through comprehensive mathematical lectures covering connections, curvature, and applications in modern geometry and physics.

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

Introduction to Knot Theory

Explore the mathematical foundations of knot theory through comprehensive lectures covering fundamental concepts, invariants, and applications in this advanced mathematics series.

• YouTube
• 6 hours 11 minutes
• Self-Paced
• Free Video

Harmonic Analysis - PCMI 28th Annual Summer Session 2018

Dive into advanced harmonic analysis through expert lectures on Almgren's center manifolds, elliptic homogenization, rectifiable measures, and minimal sets from leading mathematicians.

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

Lectures on Harmonic Analysis

Dive into advanced harmonic analysis through expert lectures covering Almgren's theory, elliptic homogenization, rectifiable measures, and minimal sets from leading mathematicians.

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

Almgren's Center Manifold in a Simple Setting

Explore Almgren's center manifold theory through a simplified toy case, proving C³,α regularity for minimal surfaces using elementary methods without Schauder estimates.

• YouTube
• 4 hours 16 minutes
• Self-Paced
• Free Video

Riemann-Hilbert Problems - Theory and Applications

Dive into Riemann-Hilbert problems theory, inverse scattering, nonlinear steepest-descent methods for integrable systems, random matrices, and orthogonal polynomials applications.

• YouTube
• 4 hours 14 minutes
• Self-Paced
• Free Video

Wigner Random Matrices and the Semicircle Law

Explore Wigner's semicircle law through eigenvalues, graph theory, and Catalan numbers in random matrix theory with proofs and spectral analysis techniques.

• YouTube
• 3 hours 54 minutes
• Self-Paced
• Free Video

Universality for Random Matrix Ensembles of Wigner Type

Dive into advanced random matrix theory with Terry Tao, exploring universality phenomena in Wigner-type ensembles through rigorous mathematical analysis and proofs.

• YouTube
• 4 hours 8 minutes
• Self-Paced
• Free Video