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Special Colloquium/Candidate Presentation: Regularity lemma: discrete and continuous perspectives

Event Type
Seminar/Symposium
Sponsor
n/a
Location
245 Altgeld + Zoom
Date
Jan 21, 2022   4:00 pm  
Speaker
Fan Wei (Princeton)
Contact
Jozsef Balogh
E-Mail
jobal@illinois.edu
Views
184

Presentation will be in-person and via Zoom (Zoom link to come)

Abstract: Szemerédi's regularity lemma is a game-changer in extremal combinatorics and provides a global perspective to study large combinatorial objects. It has connections to number theory, discrete geometry, and theoretical computer science. One of its classical applications, the removal lemma, is the essence for many property testing problems, an active field in theoretical computer science. Unfortunately, the bound on the sample size from the regularity method typically is either not explicit or is enormous. For testing natural permutation properties, we show one can avoid the regularity proof and yield a tester with polynomial sample size. For graphs, we prove a stronger, "L_\infty'' version of the graph removal lemma, where we conjecture that the essence of this new removal lemma for cliques is indeed the regularity-type proof. The analytic interpretation of the regularity lemma also plays an important role in graph limits, a recently developed powerful theory in studying graphs from a continuous perspective. Based on graph limits, we developed a method combining with both analytic and spectral methods, to answer and make advances towards some famous conjectures on a common theme in extremal combinatorics: when does randomness give nearly optimal bounds?


These works are based on joint works with Jacob Fox, Dan Kral',  Jonathan Noel, Sergey Norin, and Jan Volec.

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