Geometric complexity theory: Difference between revisions

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'''Geometric Complexity Theory''', often shortened to '''GCT''', is a research program in [[computational complexity theory]] proposed by [[Ketan Mulmuley]]. The goal of the program is to answer the most famous open problem in computer science – [[P vs.versus NP problem|whether P = NP]] – by showing that the complexity class [[P (complexity) | P]] is not equal to the complexity class [[NP (complexity) | NP]].

The idea behind the approach is to adopt and develop advanced tools in [[algebraic geometry]] and [[representation theory]] to prove lower-bounds for problems. Currently the main focus of the program is on [[Arithmetic circuit complexity#Algebraic P and NP | algebraic complexity]] classes. Proving that [[Permanantcomputing the permanent]] cannot be efficiently [[Reduction (complexity)|reduced]] to computing [[Determinantdeterminant]]s is considered to be a major milestone for the program. These computational problems can be characterized by their [[symmetry (mathematics) | symmetries]]. The program aims at utilizing these symmetries for proving lower-bounds.

The approach is often considered the only viable currently active program to separate [[P (complexity) | P]] from [[NP (complexity) | NP]]. However, according to Mulmuley the program is likely to take hundreds of years before it can settle the [[P vs. NP]] problem.

The program is pursued by several researchers in mathematics and theoretical computer science. Part of the reason for the interest in the program is the existence of arguments for the program avoiding all known [[barriers (complexity)such as [[Oracle machine|relativization]] barriersand [[natural proof]]s for proving general lower-bounds.

== External links ==