Syllabus (subjected to changes):

1. Overview of sampling & applications. Draft notes for Lecture 1

+  Approximate sampling and approximate counting

+ Total variation distance, coupling for distributions, data processing inequality.

+ Some example algorithms/stochastic processes for sampling: the Glauber dynamics, auto-regression, masked diffusion.

2. Markov chains and mixing time: an introduction Lecture 2

+ Stationary distribution, ergodicity, reversibility, mixing time

+ Example: random walk on graphs

3. Spectral gap, Poincaré inequality, (modified) Log-Sobolev inequality

4. Conductance, Cheeger's inequality Lecture 4

5. Equivalence between approximate sampling and counting

6. Equivalence between approximate sampling and counting (continue) Concentration inequalities: Lecture 6 Concentration Inequalities

7. Path coupling + Dobrushin condition Lecture 7

8. Overview of sampling from Ising models (4/22)

9. & 10. Hardness of sampling from low-temperatured Ising models (4/29-5/4) Lecture 9-10

11. Measure decomposition via stochastic localization

12. Spectral independence, entropic independence, and bounded covariance

13. Bounding covariance via trickle-down

14. Sampling from continuous domains

15. Controlling discretization error via Girsanov

16. Parallelizing the Langevin dynamics with Picard iteration

17. Parallelizing denoising diffusion with the pinning lemma

18. Quantum Markov chains

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