Abstract: Stochastic localization is a way of decomposing a probability measure on a high-dimensional space into a mixture of simpler measures in a “Markovian” way. These simpler measures are “localized” in the sense that they are concentrated on smaller (random) subsets of the original space. The decomposition turns out to be useful in understanding the original measure. This technique has found applications in high-dimensional geometry and probability, notably in problems related to the Kannan-Lovasz-Simonovits (KLS) conjecture, and in problems of statistical physics and combinatorics of random graphs. In this reading group we plan to read the foundational papers and understand how the technique applies to various problems. The focus will be on the method, in the hope of understanding it and exploiting its full power.
Ronen’s lecture notes will be a useful guide throughout.
Meeting times
Every Monday 10-11am PST and Wednesday 11am-12pm PST except HD workshop weeks.
Each lecture below shall span between 50min and 3x50min, usually.
Zoom link is posted on discord and has been sent via e-mail.
Plan
Lecture 1 : The localization lemma and its applications.
Dates: Mon 09/07 & Wed 09/09
Speaker: Galyna Livshyts
Abstract: We shall discuss the Lovasz-Simonovitz localization lemma, its classical corollary by Kannan, Lovasz and Simonovits, the applications of these techniques to the study of the KLS conjecture and a theorem of Carbery and Wright.
Lovasz-Simonovits 1993: http://matmod.elte.hu/~lovasz/vol7.pdf
Kannan-Lovasz-Simonoits 1995: https://link.springer.com/article/10.1007/BF02574061
Carbery-Wright: https://www.maths.ed.ac.uk/~carbery/analysis/papers/cw-dim_final.pdf
Survey by Lee & Vempala: https://www.cc.gatech.edu/~vempala/papers/kls_survey.pdf
Whiteboard Monday+Wednesday: https://www.dropbox.com/s/tinqp8ysqw8nxyc/Whiteboard%20Localization.pdf?dl=0
Video Monday: https://www.dropbox.com/s/cpg2jgo8q0eyr9c/zoom_0.mp4?dl=0
Video Wednesday: https://berkeley.zoom.us/rec/share/xxW30z5QRBYj9em4tsmgCp8l_Rn69lttEfOkZHur96eM3vT-43tTQgAKjTIEai2I.NEq-QDmu1RHV1nPz
Lecture 2 : Informal preliminaries on stochastic analysis.
Date: Mon 09/14
Speaker: Ahmed El Alaoui
Brownian motion. Ito processes. Ito’s formula. SDEs.
Ronen’s lecture notes http://www.wisdom.weizmann.ac.il/~ronene/GFANotes.pdf
Oksendal- Stochastic differential equations
(See chapters 1-5 for a rigorous treatment.)
Lecture 3: Poincaré inequality for SK at high temperature
Date: Wed 09/16& Mon 09/28
Speaker: Frederic Koehler
https://arxiv.org/abs/2007.08200v1 (Eldan Koehler Zeitouni 2020)
Lecture Wednesday: https://jamboard.google.com/d/1nIziEc8ZSKE5mePH018umwEqT-D1s5_WzM4wSKIFsGM/edit?usp=sharing
Lecture Monday: https://cdn.discordapp.com/attachments/751116214918578225/760224488159313940/Note_Sep_28_2020.pdf
Video Wednesday: https://berkeley.zoom.us/rec/share/IpQnkBLKckEL2fJoXnWBVh6QR8zTBPewZnwo-k8mKR9Bst6J9IeIPU_Omu3U1mvP.X-xZ8Zj0imslfqKU
Mon 09/21– Fri 09/25 —> Workshop week
Lecture 4: Taming correlations of high-dimensional measures
Date: Wed 09/30 & Mon 10/05
Speaker: Ronen Eldan
https://arxiv.org/abs/1811.11530v2 (Eldan 2019)
Video Mon https://drive.google.com/file/d/1-BHuey_1fNKzRUv6spzDb-DCJjVTnykU/view?usp=sharing
Notes: https://cdn.discordapp.com/attachments/751116214918578225/762999544391925760/Note_30_Sep_2020_2.pdf
Lecture 5: Best to-date estimate in the KLS conjecture via stochastic localization
Date: Wed 10/07 & Mon 10/12
Speaker: Kevin Tian
The Lee-Vempala results:
https://arxiv.org/abs/1612.01507, https://arxiv.org/abs/1712.01791
Survey: https://www.cc.gatech.edu/~vempala/papers/kls_survey.pdf
Lecture 6: The Lovasz-Simonovits localization lemma via Stochastic Localization (an unpublished proof)
Date: Wed 10/14
Speaker: Santosh Vempala
Board: https://www.dropbox.com/s/xntado70ftcqtz2/talk.pdf?dl=0
Mon 10/19– Fri 10/23 —> Workshop week
Lecture 7: Tubular neighborhoods of complex-analytic sets and the stochastic localization
Date: Mon 10/26
Speaker: Bo’az Klartag
https://www.weizmann.ac.il/math/klartag/sites/math.klartag/files/uploads/1702.02315_1.pdf (Klartag 2017)
Notebook: https://www.dropbox.com/s/ntfe35lq34tiff7/Math%2026%20Oct%202020%2017_46_48.pdf?dl=0
Lecture 8: CLT in high dimensions
Date: Wed 10/28& Mon 11/02
Speaker: Dan Mikulincer
https://arxiv.org/abs/1806.09087 (Eldan Mikulincer Zhai 2018)
Skorokhod embedding: https://arxiv.org/abs/1303.3315 (Eldan 2013)
Notes: https://www.dropbox.com/s/p9d9t500pf3e26n/Martingale%20embeddings%20-%20notes.pdf?dl=0
Lecture 9: Nonlinear large deviations and the condition of low complexity gradient
Date: Wed 11/04
Speaker: Ronen Eldan
https://arxiv.org/abs/1612.04346 (Eldan 2016)
Lecture 10: Thin-shell implies KLS
Date: Mon 11/16 & Wed 11/18
Speaker: Daniel Dadush
https://arxiv.org/abs/1203.0893 (Eldan 2012)
Ronen’s lecture notes: http://www.wisdom.weizmann.ac.il/~ronene/GFANotes.pdf
Survey: https://www.cc.gatech.edu/~vempala/papers/kls_survey.pdf
Whiteboard: https://drive.google.com/file/d/1ro3gdjUEGsdgS5QRgHIJQAOmpqL_VhHE/view?usp=sharing Video: https://berkeley.zoom.us/rec/share/JG1q20HHnBN_rL11V92Al_cQvFWv29OJgOjZDub7MfryE4vaFevVBLETAUzD_ZSa.6PtwNV45Ul_Uk3Ub And https://berkeley.zoom.us/recording/detail?meeting_id=W7oPlYLiReO%2FzDpyCuQDtg%3D%3D
Mon 11/16– Fri 11/20 —> Workshop week
Lecture 11: An almost constant bound on the KLS conjecture by Yuansi Chen
Date: Wed 12/23
Speaker: Ronen Eldan
https://arxiv.org/abs/2011.13661
Whiteboard: https://www.dropbox.com/s/5x6mj1f714x0aes/Eldan-KLS-notebook-Dec%E2%80%9923.pdf?dl=0
Further readings:
https://arxiv.org/abs/1708.05859 (Eldan Gross 2017)
Follmer’s process. Boué-Dupuis, and Lehec’s formula for the entropy
https://arxiv.org/abs/1006.3028v1 (Lehec 2010)
https://projecteuclid.org/download/pdfview_1/euclid.aihp/1372772648
(Lehec 2013, More recent & rigorous version of the same paper)
Generalizations of the localization lemma:
https://www.sciencedirect.com/science/article/pii/S0001870805001726 (Fradelizi-Guedon 2006)
Best quantitative bounds on the mean-field approximation (optimal transport):
https://arxiv.org/pdf/1903.08021.pdf (Augeri 2019)
A notion of log-concavity on the hypercube:
https://arxiv.org/abs/2007.13108v1 (Eldan Shamir 2020)
Needle decompositions in Riemannian geometry
https://arxiv.org/abs/1408.6322 (Klartag 2014)
Organizers: Ahmed El Alaoui, Galyna Livshyts, Vishesh Jain