University of Chicago Department of Mathematics Courses

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

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

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

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

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

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

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

Toward Physical Generative Models

Explore random dynamical systems in generative models, examining when algorithms produce "physical" samples and how to recover target distributions from noisy processes.

• YouTube
• 1 hour 4 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

Measure and Topological Rigidity beyond Homogeneous Dynamics

Explore rigidity theorems for smooth dynamical systems with hyperbolicity, extending beyond homogeneous structures to classify ergodic measures and orbit closures.

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

Limit Measures for Topologically and Geometrically Random Surfaces

Explore limit measures for topologically and geometrically random surfaces in hyperbolic 3-manifolds, examining how surface sequences with vanishing curvatures induce probability measures.

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

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