Stony Brook Mathematics Courses

Polynomially Convex Embeddings of Real Manifolds in C^n

Explore geometric embeddings of real manifolds in complex spaces, focusing on polynomial convexity and its connections to approximation theory and diverse mathematical areas.

• YouTube
• 1 hour 1 minute
• Self-Paced
• Free Video

Deterministic Localization for the Discrete Schrodinger Operator

Delve into deterministic localization for discrete Schrodinger operators, exploring how small perturbations can delocalize eigenfunctions through optimization techniques and the Hellman-Feynman formula.

• YouTube
• 1 hour 12 minutes
• Self-Paced
• Free Video

Recent Progress in Mean Curvature Flow

Delve into advanced mathematical concepts of mean curvature flow, exploring recent breakthroughs in singularity formation and the resolution of the Multiplicity One Conjecture.

• YouTube
• 1 hour 7 minutes
• Self-Paced
• Free Video

Monge-Ampere Equations: Beyond the Classical Cases

Delve into advanced mathematical theory exploring octonionic Monge-Ampère equations, their applications in geometry, and groundbreaking developments in 16-manifolds and Calabi-Yau theorem extensions.

• YouTube
• 1 hour 3 minutes
• Self-Paced
• Free Video

The Stable Bernstein Theorem for Minimal Hypersurfaces

Delve into advanced geometric analysis exploring stable minimal hypersurfaces, focusing on recent developments in the Bernstein problem and its connections to Riemannian manifold geometry.

• YouTube
• 1 hour 7 minutes
• Self-Paced
• Free Video

Cohomological Splittings in Algebraic and Symplectic Geometry

Explore fundamental results in rational cohomology of compact symplectic manifolds with Hamiltonian group actions, plus recent integral and stable homotopical refinements using chromatic homotopy theory.

• YouTube
• 1 hour 5 minutes
• Self-Paced
• Free Video

On the Applications of Topology

Explore topological methods for analyzing complex data sets, focusing on persistent homology and mapper, with applications in computer vision, biology, and medicine.

• YouTube
• 1 hour 6 minutes
• Self-Paced
• Free Video

Symplectic Capacities, Viterbo Isoperimetric Conjecture, and Contact Manifolds with Closed Reeb Orbits

Explore symplectic capacities, Viterbo's conjecture, and their impact on symplectic geometry. Gain insights into convex geometry applications and closed Reeb orbits in contact manifolds.

• YouTube
• 1 hour 2 minutes
• Self-Paced
• Free Video

Stable Degenerations of Singularities

Explore stable degenerations of singularities in algebraic geometry, focusing on Li-Xu's Stable Degeneration Conjecture and its implications for metric tangent cones in Kähler-Einstein manifolds.

• YouTube
• 1 hour 7 minutes
• Self-Paced
• Free Video

Real Loci of Non-Real Varieties - Mathematics Department Colloquium

Explore cooriented submanifolds in real algebraic geometry, focusing on their properties, intersections, and linking numbers. Gain insights into the geometry of pencils and real projective spaces.

• YouTube
• 1 hour 3 minutes
• Self-Paced
• Free Video

Homotopy Groups of Spheres

Explore continuous maps between spheres, their abelian group structures, and connections to key problems in topology with Zhouli Xu from UC San Diego.

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

Mathematical Structures in Generative Linguistics

Explore mathematical structures in generative linguistics, from formal languages to Hopf algebras and geometric spaces for modeling semantics and syntactic variation across languages.

• YouTube
• 1 hour 15 minutes
• Self-Paced
• Free Video

Geometric Variational Problems: Regularity vs Singularity Formation

Explore techniques for existence, regularity, and singularity formation in geometric problems, focusing on harmonic maps, Yang-Mills connections, and nematic liquid crystals. Learn about new gluing methods for parabolic flows.

• YouTube
• 1 hour 2 minutes
• Self-Paced
• Free Video

Vortex Sheets and Groupoids in Ideal Hydrodynamics

Explore Arnold's group-theoretic approach to ideal hydrodynamics and its applications to vortex sheets, fluids with moving boundaries, and multiphase fluids using Lie groupoid symmetries.

• YouTube
• 1 hour 2 minutes
• Self-Paced
• Free Video

Uniform Volume Doubling and Functional Inequalities on Lie Groups

Explore functional inequalities on Lie groups, focusing on uniform volume doubling and its implications for SU(2) and related structures. Gain insights into heat kernel estimates and measure contracting properties.

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