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Explore additive structures in subset sums, non-averaging sets, and convex geometry in this 19-minute conference talk by Huy Tuan Pham, a postdoctoral member at the Institute for Advanced Study. Delve into the intricate connections between these mathematical concepts as Pham presents his research findings and insights. Gain a deeper understanding of how these areas intersect and influence each other in the realm of advanced mathematics. This talk, part of the "Short Talks by Postdoctoral Members" series, offers a concise yet comprehensive overview of Pham's work, making it valuable for mathematicians, researchers, and students interested in additive combinatorics and geometric analysis.