University of Chicago Department of Mathematics Courses

Bordered Floer Homology - Lecture 3

Delve into Bordered Floer homology, a three-manifold invariant that enables Mayer-Vietoris descriptions of Heegaard Floer homology, with recent developments in the unspecialized theory.

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

Heegaard Floer Homology: From Heegaard Diagrams to Holomorphic Disks - Lecture 2

Delve into Heegaard Floer homology, a tool for studying three- and four-dimensional manifolds using symplectic geometry-inspired methods, including applications and computational advances.

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

How Curved are Level Sets of Solutions to Elliptic PDE? - Lecture 1

Delve into the geometry of level sets in semilinear elliptic equations, exploring Jacobi functions and Levi-Civita's affine connection with applications to harmonic functions.

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

Unni Namboodiri Lectures in Geometry and Topology (2025) - Lecture 1

Explore the combinatorial formulation of Knot Floer homology, a bigraded vector space invariant for knots that categorifies the Alexander polynomial, along with its applications to knot theory.

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

Entropic Selection Principle for Monge's Optimal Transport

Delve into the convergence of entropic regularized optimal transport to classical Monge transport, exploring selection principles, large deviation theory, and unique characterization of limiting transport plans in higher dimensions.

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

Theory of Valuations on Manifolds

Explore the concept of valuations on smooth manifolds, including their algebraic structure, relationship to smooth functions and measures, and foundational principles, as introduced by Semyon Alesker.

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

Geometry and Topology at Many Scales

Explore multiscale and nonsmooth geometry concepts, from Kaufman's surjective Lipschitz map to the limits of multiscale constructions in maps and surfaces.

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

Symplectic Manifolds and Derived Orbifolds - Lecture 3

Explore symplectic manifolds and derived orbifolds in this advanced geometry lecture, examining 40 years of developments since Floer's groundbreaking work in symplectic topology.

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

Uniform Stability of High-Rank Arithmetic Groups

Explore uniform stability properties of high-rank arithmetic groups through asymptotic cohomology theory and its applications to almost-representations.

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

Symplectic Manifolds and Bordism - Lecture 2

Explore symplectic manifolds and bordism in this advanced topology lecture, building on Gromov-Floer theory to understand modern developments combining homotopy theory with Morse theory.

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

Cotangent Bundles and the Hopf Map - Symplectic Topology 40 Years After Floer - Lecture 1

Explore symplectic topology's evolution from Poincaré to modern homotopy theory methods, focusing on cotangent bundles, Hopf maps, and Lagrangian submanifolds.

• YouTube
• 1 hour 9 minutes
• Self-Paced
• Free Video

Function Spaces as Geometric Objects

Explore algebraic topology through function spaces, learning to navigate nonlinear components and discover real-world applications in this mathematical journey.

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

The Massless Sine-Gordon Model - From Solitons to Integrability and Spontaneous Symmetry Breaking

Explore the massless sine-Gordon model in 2d quantum field theory, covering solitons, integrability, symmetry breaking, and connections to dimer models and free fermion points.

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

Periodic Orbits of Vector Fields in Three Dimensions

Explore periodic orbits in 3D vector fields, their existence, quantity, and properties, culminating in recent advances on Reeb vector fields in three-dimensional manifolds.

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

Theory of Valuations on Manifolds

Explore the theory of valuations on smooth manifolds, covering finitely additive measures, topological filtered algebras, and their relationship to smooth functions and measures.

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