During Summer 2024, James participated in SUMaC Program II, an intensive three-week dive into algebraic topology following Berkeley's curriculum, compressing one semester of graduate-level material into an accelerated format.
The course culminated in student presentations exploring applications of topological theorems to fields ranging from game theory to data science.
The coursework traced a deliberate path through algebraic topology's core ideas, progressing from foundations such as topological spaces and fundamental groups to sophisticated topics including manifold theory and knot theory.
Problem Set 7 features 4 problems exploring manifold theory.
James's solutions from week 3 are available on the right.
For his final presentation, Yecheng explored Kakutani's Fixed Point Theorem: a powerful result from topology that became a cornerstone of modern economics, establishing the Nash Equilibrium and the Minimax Theorem. John Nash famously used the theorem to prove that every finite game has at least one equilibrium, which later earned him the Nobel Prize in Economics.
Tracing the original proof techniques by Nash, James revealed how seemingly esoteric mathematics provides the theoretical foundation for understanding cooperation, competition, and strategic stability in economics and political science.