Stony Brook Mathematics Courses

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
• On-Demand
• 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
• On-Demand
• 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
• On-Demand
• 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
• On-Demand
• 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
• On-Demand
• 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
• On-Demand
• 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
• On-Demand
• 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
• On-Demand
• 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
• On-Demand
• Free Video

The Nonlinear Stability of Black Holes: An Overview

Explore the mathematical properties of black holes, focusing on their stability as solutions to Einstein's equation and recent proof of nonlinear stability for slowly rotating Kerr black holes.

• YouTube
• 1 hour 10 minutes
• On-Demand
• Free Video

Towards a Mathematical Theory of 3D Mirror Symmetry

Explore the mathematical implications of 3D mirror symmetry, connecting quantum field theory dualities to algebraic geometry and symplectic topology. Uncover links to geometric Langlands and symplectic duality.

• YouTube
• 59 minutes
• On-Demand
• Free Video

On Exotic Embeddings of Submanifolds in 4-Manifolds

Explore exotic embeddings of submanifolds in 4-manifolds, focusing on phenomena that survive stabilizations. Delve into advanced 4-dimensional topology concepts and recent research findings.

• YouTube
• 1 hour 5 minutes
• On-Demand
• Free Video

Chain Level String Bracket and Punctured Holomorphic Discs

Explore Audin's conjecture and its generalization to Liouville manifolds. Discover connections between holomorphic discs and string topology operations, focusing on chain level string brackets.

• YouTube
• 1 hour 2 minutes
• On-Demand
• Free Video

Billiards in Conics Revisited

Explore optical properties of conics, billiard reflections, and geometric consequences. Delve into classic results, Ivory's lemma, Poncelet Porism, and related theorems in projective geometry.

• YouTube
• 1 hour 7 minutes
• On-Demand
• Free Video

Homology Cobordism and the Geometry of Hyperbolic Three-Manifolds

Explore the interaction between hyperbolic geometry and homology cobordism in low-dimensional topology using monopole Floer homology to study subgroups of the homology cobordism group.

• YouTube
• 1 hour 5 minutes
• On-Demand
• Free Video