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This lecture from the ZhengTong Chern-Weil Symposium Spring 2025 features Huy Tuan Pham (IAS & Caltech) exploring additive combinatorics through probabilistic and combinatorial perspectives. Delve into the fundamental concept of sumsets in additive combinatorics, beginning with Freiman's celebrated theorem that characterizes the structure of integer sets with small sumsets. Discover new combinatorial and probabilistic approaches to sumsets that emerged from studies of random Cayley graphs, providing robust, flexible, and quantitative descriptions of sets with small sumsets. Explore the fascinating intersection between additive combinatorics and probabilistic combinatorics, which has led to new understandings of Cayley graphs and random Cayley graphs in general groups, as well as solutions to longstanding questions in the field of additive combinatorics.
