University of Chicago Department of Mathematics Courses

Higher Expansion, Representation Stability and Minimal Submanifolds

Explore higher expansion, representation stability, and minimal submanifolds through geometric approaches to k-expander families and cosystolic expansion in locally symmetric spaces.

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

The Square Root Method in the Algebraization of Dirichlet Series

Explore advanced algebraization methods for Dirichlet series through functional equations and automorphic forms, connecting arithmetic theory to class number problems.

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

On the Multiplicity One Conjecture for Mean Curvature Flows of Surfaces

Explore the Multiplicity One Conjecture for Mean Curvature Flows, covering singularity analysis, embedded surfaces evolution, and new regularity theory for geometric flows.

• YouTube
• 1 hour
• 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

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

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

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

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

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

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

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