**Abstract**: Bernoulli percolation is a toy model for studying large-scale connectivity properties of networks emerging from microscale behavior. Already in the 1957 paper by Broadbent and Hammersley introducing the process, it was shown to exhibit a phase transition, which was shown to be sharp in the seminal work of Harry Kesten in 1980. In the past decades, new techniques to attack this problem in greater generality were developed, a crucial one being the study of low-revealment algorithms and the O’Donnell, Saks, Schramm, and Servedio (known as the OSSS inequality). In this talk, we will sketch the proof of sharp thresholds for Bernoulli percolation using these modern tools and show how to apply them in a dependent percolation process arising from the Majority dynamics.

**Speaker’s Bio: **Dr. Caio Alves is a mathematician specializing in discrete probability, focusing on random graph processes arising in percolation theory and preferential attachment rules. He obtained his Ph.D. in 2014 from the Federal University of Minas Gerais. He has since worked as a postdoctoral researcher at the University of Campinas, the Max Planck Institute for Mathematics in the Sciences, the University of Leipzig, and the Álfred Rényi Institute of Mathematics in Budapest. He is currently a Research Scientist in the Discrete Algorithms group at the Oak Ridge National Laboratory.

Last Updated: June 1, 2023 - 11:44 am