Course Notes | 21-317 Functional Inequalities
21-317 · Fall 2026
Course notes
Notes, reading guides, exercises, and resources for mathematical exposition.
Round I Dates updated September 19, 2026. Presentations run September 28–October 19; topics are unchanged.
Class meeting notes
Student-facing notes and exercises for completed class meetings are available below. Each handout reflects the material covered in class.
Meeting 1: Why should a function be almost constant?
A course panorama through Gaussian concentration, the two-point Poincaré identity, variance relaxation, Gibbs free energy, and the logarithmic Sobolev inequality.
August 24, 2026 · PDF · 6 pagesMeeting 2: Finite probability and entropy
Finite probability, expectation and variance, the two-point model, Shannon-Boltzmann entropy, and entropy additivity for independent systems.
August 26, 2026 · PDF · 4 pagesMeeting 3: Relative entropy and Gibbs equilibrium
Jensen's inequality, relative entropy, Gibbs inequality, maximum entropy for the uniform law, and the finite Gibbs variational principle.
August 28, 2026 · PDF · 4 pagesMeeting 4: The Gibbs variational principle and discrete-time Markov chains
The finite Gibbs variational principle, sample paths, transition kernels, evolution of observables and laws, stationarity, and detailed balance.
August 31, 2026 · PDF · 5 pagesMeeting 5: Continuous-time Markov chains I: Exponential clocks and Poissonization
A discrete-time recap followed by memoryless waiting times and rates, the Poisson clock, Poissonization, the transition semigroup, and preservation of stationarity.
September 2, 2026 · PDF · 7 pagesMeeting 6: Continuous-time Markov chains II: Generators, jump rates, stationarity, and reversibility
The infinitesimal generator, jump and exit rates, holding times, uniformization and competing clocks, forward evolution of laws, stationarity, and detailed balance.
September 4, 2026 · PDF · 6 pagesMeeting 7: Dirichlet forms and the Poincaré inequality: Variance dissipation, spectral gap, and relaxation
Reversible Dirichlet forms and edge energies, the exact variance-dissipation identity, the Poincaré inequality, the Rayleigh-quotient characterization of the spectral gap, and exponential variance relaxation.
September 9, 2026 · PDF · 6 pagesMeeting 8: The Poincaré inequality in action: Spectral-gap proofs and elementary model chains
Functional-analysis and linear-algebra terminology, irreducibility and reversibility, proofs of the Rayleigh-quotient and exponential-relaxation theorems, and the symmetric two-state and reset chains.
September 11, 2026 · PDF · 6 pagesMeeting 9: Cycle, bottlenecks, and logarithmic Sobolev inequalities: Revised notes through the start of entropy dissipation
The cycle spectral gap and diffusive time scale, the bottleneck indicator test and conductance bound, the logarithmic Sobolev inequality, its linearization to Poincaré, and the opening entropy-derivative calculation.
September 14, 2026 · PDF · 4 pagesMeeting 10: Entropy dissipation and free-energy relaxation
The exact entropy-dissipation identity, comparison with square-root Dirichlet energy, exponential entropy and free-energy relaxation, and the distinction between the diffusion chain rule and the discrete inequality.
September 16, 2026 · PDF · 3 pagesMeeting 11: Measure spaces and the integral
Sigma-algebras, countably additive measures, measurable functions and almost-everywhere language, and construction of the integral from indicators and simple functions.
September 18, 2026 · PDF · 2 pagesMeeting 12: The operational integration toolkit I
Construction of the integral, Lebesgue spaces, Hölder's inequality, probability-space norm monotonicity, and the monotone convergence, Fatou, and dominated convergence theorems.
September 21, 2026 · PDF · 2 pagesMeeting 13: From integration to weak derivatives
The moving-spike counterexample, product measures and Tonelli-Fubini, absolute continuity and Radon-Nikodym densities, and the definition of weak derivative through integration by parts.
September 23, 2026 · PDF · 2 pagesMeeting 14: Weak derivatives and the Euclidean branch
Weak-derivative examples, Sobolev spaces, mollification and zero modes, scaling and the Euclidean Sobolev inequality, with Nash's inequality and heat-flow smoothing as continuation material.
September 25, 2026 · PDF · 4 pagesStudent presentation resources
The Round I materials are available below. Individual assignments and presentation dates are posted in the assignment announcement, updated September 19 and on the course schedule. Round I runs September 28–October 19, with the last presentation after Fall Break; topic assignments are unchanged. The ranked-preference period ended on August 30, 2026.
Book your 20-minute preparation consultation before your presentation. See the September 18 announcement for the meeting windows and preparation reminder.
Quick-browse booklet
The topic catalogue followed by the first-page overview of all ten packets.
PDF · 14 pagesComplete packet set
All ten packets in one file, including required proof content, examples, exercises, and reading routes.
PDF · 33 pagesRound I assignments
The complete presentation order, assigned topics, and revised Round I dates.
Published September 9 · Dates updated September 18, 2026Individual topic packets
The schedule is the canonical place for assigned preparation. Newly posted or substantially revised materials will also be identified in a dated announcement.