University of Chicago Department of Mathematics Courses

How Curved are Level Sets of Solutions to Elliptic PDE? - Part 2

Explore the geometry of level sets in semilinear elliptic equations with David Jerison, examining Jacobi functions and Levi-Civita affine connections inspired by Hamilton and Perelman's work on curvature flows.

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

How Curved are Level Sets of Solutions to Elliptic PDE? - Part 2

Delve into the geometry of level sets in semilinear elliptic equations, exploring Jacobi functions and Levi-Civita's affine connection as tools for understanding curvature in solutions to elliptic PDE.

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

How Curved are Level Sets of Solutions to Elliptic PDE? - Part 3

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 1 minute
• Self-Paced
• Free Video

Kakeya Sets in R^3

Explore the Kakeya set conjecture in R^3, examining compact subsets containing unit line segments in every direction, and discover a proof through the analysis of tube unions.

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

Improved Bounds for Szemeredi's Theorem Using Higher Order Fourier Analysis

Dive into higher order Fourier analysis and explore improved bounds for Szemeredi's Theorem, including inverse theorem for Gowers norms and proof strategies in arithmetic progressions.

• YouTube
• 1 hour 2 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

Minimal Submanifolds in Riemannian Geometry

Explore the connection between stable minimal hypersurfaces and Riemannian manifolds with scalar curvature lower bounds, including recent progress in higher codimension theory for surfaces.

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

Interpolation for Brill-Noether Curves

Explore the complete solution to the interpolation problem for Brill--Noether curves, examining when curves of specific types can pass through general points.

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

How Curved are Level Sets of Solutions to Elliptic PDE? - Part 2

Explore the geometry of level sets in semilinear elliptic equations with David Jerison, examining Jacobi functions and Levi-Civita affine connections inspired by Hamilton and Perelman's work on curvature flows.

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

Hilbert's Sixth Problem: Derivation of the Boltzmann and Fluid Equations

Delve into the derivation of Boltzmann and fluid equations from hard sphere dynamics, resolving Hilbert's Sixth Problem through mathematical analysis in both Euclidean and periodic settings.

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

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

What's the Matter with Geometry? How String Theory Generalizes General Relativity

Explore how string theory blurs the lines between gauge fields, matter, and geometry, challenging the standard field theory paradigm with illustrative examples.

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