Stony Brook Mathematics Courses

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
• Self-Paced
• 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
• Self-Paced
• 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
• Self-Paced
• 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
• Self-Paced
• 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
• Self-Paced
• 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
• Self-Paced
• Free Video

Supersymmetry and Trace Formulas: From Atiyah-Singer to Jacobi-Poisson and Selberg

Explore supersymmetry's mathematical origins and applications, from Atiyah-Singer formula to Jacobi-Poisson and Selberg trace formulas, in this advanced mathematics colloquium.

• YouTube
• 1 hour 4 minutes
• Self-Paced
• Free Video

Supersymmetry and Trace Formulas: From Atiyah-Singer to Jacobi-Poisson and Selberg - Part 1

Explore supersymmetry's mathematical origins and applications, from Atiyah-Singer formula to Jacobi-Poisson and Selberg trace formulas, in this advanced mathematics colloquium.

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

The High-Degree Rational Cohomology of the Special Linear Group

Explore high-degree rational cohomology of SL_n(Z) and SL_n(R) for number rings. Examine Steinberg modules and associated simplicial complexes to understand special linear group cohomology.

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

Positivity, Representations, and Non-Commutative Hyperbolic Geometry

Explore total positivity in matrices, its generalization to split real Lie groups, and a new notion of positivity in flag varieties with applications to higher Teichmüller spaces.

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

Local-Global Principles for Homogeneous Spaces Over Arithmetic Function Fields

Explore local-global principles for homogeneous spaces over arithmetic function fields, focusing on function fields of curves over complete discretely valued fields and related patching methods.

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

Mirror Structure Constants via Non-Archimedean Analytic Disks

Explore mirror structure constants for smooth affine log Calabi-Yau varieties using non-archimedean analytic disks, generalizing previous constructions and discussing positivity, integrality, and potential extensions.

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

Nonabelian Mixed Hodge Structures

Explore nonabelian mixed Hodge structures, their relation to rational homotopy, and applications to character varieties and schematic homotopy types. Examine weights from noncompact bases and parabolic structures.

• YouTube
• 1 hour 38 minutes
• Self-Paced
• Free Video

Quantitative Ell-Adic Sheaf Theory

Explore quantitative bounds for étale cohomology groups and their applications in algebraic geometry. Learn about complexity measures for sheaves and how they interact with various functors.

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

Counting Curves in Calabi-Yau Threefolds

Explore enumerative invariants and a universal approach to counting curves in Calabi-Yau threefolds. Discover how cluster formalism aids computations and potentially simplifies conjectural identities between invariants.

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