Geometric Complexity Theory
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Geometric Complexity Theory seeks to address fundamental complexity lower bound questions such as P versus NP by means of algebraic geometry and representation theory. There has recently been a burst of activity in these areas that has revealed connections between the original program and other questions in complexity theory, as well as several longstanding open questions in representation theory and algebraic geometry. Orbit closure problems are of particular importance here. The analysis of the symmetries via representions leads to a bunch of challenging mathematical questions.
