University of Chicago Department of Mathematics Courses

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

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

What's Special About Special? - Understanding Nilpotent Matrices and Weyl Group Representations

Explore the relationship between nilpotent matrices, Weyl group representations, and special orbits as David Vogan explains Springer and Lusztig's theories in representation theory of reductive groups.

• YouTube
• 1 hour 3 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

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

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

Diophantine Results for Shimura Varieties

Explore Diophantine results for Shimura varieties, focusing on exceptional settings without moduli interpretation, including finiteness of S-rational points, Tate conjecture, and construction of canonical integral models.

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

Additive Combinatorics from Probabilistic and Combinatorial Perspectives

Explore additive combinatorics through new probabilistic and combinatorial perspectives on sumsets, revealing connections to Cayley graphs and resolving longstanding questions in the field.

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

Bounds on the Density of Zeroes of the Riemann Zeta Function in Strips

Explore Larry Guth's research on bounds of Riemann zeta function zeroes, their relation to prime distribution, and recent findings with Maynard in this mathematical symposium talk.

• 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

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

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