Journal of Symbolic Computation
Arithmetic Circuits: A survey of recent results and open questions
Foundations and Trends® in Theoretical Computer Science
On P vs. NP and geometric complexity theory: Dedicated to Sri Ramakrishna
Journal of the ACM (JACM)
Geometric complexity theory and tensor rank
Proceedings of the forty-third annual ACM symposium on Theory of computing
The GCT program toward the P vs. NP problem
Communications of the ACM
On the geometry of tensor network states
Quantum Information & Computation
Geometric complexity theory III: on deciding nonvanishing of a Littlewood---Richardson coefficient
Journal of Algebraic Combinatorics: An International Journal
Explicit lower bounds via geometric complexity theory
Proceedings of the forty-fifth annual ACM symposium on Theory of computing
The Graph Isomorphism Problem and approximate categories
Journal of Symbolic Computation
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In [K. D. Mulmuley and M. Sohoni, SIAM J. Comput., 31 (2001), pp. 496-526], henceforth referred to as Part I, we suggested an approach to the $P$ vs. $NP$ and related lower bound problems in complexity theory through geometric invariant theory. In particular, it reduces the arithmetic (characteristic zero) version of the $NP \not \subseteq P$ conjecture to the problem of showing that a variety associated with the complexity class $NP$ cannot be embedded in a variety associated with the complexity class $P$. We shall call these class varieties associated with the complexity classes $P$ and $NP$. This paper develops this approach further, reducing these lower bound problems—which are all nonexistence problems—to some existence problems: specifically to proving the existence of obstructions to such embeddings among class varieties. It gives two results towards explicit construction of such obstructions. The first result is a generalization of the Borel-Weil theorem to a class of orbit closures, which include class varieties. The second result is a weaker form of a conjectured analogue of the second fundamental theorem of invariant theory for the class variety associated with the complexity class $NC$. These results indicate that the fundamental lower bound problems in complexity theory are, in turn, intimately linked with explicit construction problems in algebraic geometry and representation theory. The results here were announced in [K. D. Mulmuley and M. Sohoni, in Advances in Algebra and Geometry (Hyderabad, $2001$), Hindustan Book Agency, New Delhi, India, 2003, pp. 239-261].