Geometric complexity theory: Difference between revisions

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'''Geometric complexity theory (GCT)''', is a research program in [[computational complexity theory]] proposed by [[Ketan Mulmuley]] and Milind Sohoni. The goal of the program is to answer the most famous open problem in computer science – [[P 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]] (i.e., [[geometric invariant 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 [[computing the permanent]] cannot be efficiently [[Reduction (complexity)|reduced]] to computing [[determinant]]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 by some to be the only viable currently active program to separate [[P (complexity) | P]] from [[NP (complexity) | NP]]. However, [[Ketan Mulmuley]] believes the program, if viable, is likely to take about 100 years before it can settle the [[P vs. NP]] problem.<ref>{{citation

| last = Fortnow | first = Lance

| doi = 10.1145/1562164.1562186

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| title = The Status of the P Versus NP Problem

| volume = 52

| year = 2009}}| citeseerx = 10.</ref>1.1.156.767

| s2cid = 5969255

}}.</ref>

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 known barriers such as [[Oracle machine|relativization]] and [[natural proof]]s for proving general lower bounds.<ref>{{Cite journal|last=Mulmuley|first=Ketan D.|date=2011-04-01|title=On P vs. NP and geometric complexity theory: Dedicated to Sri Ramakrishna|url=http://dl.acm.org/citation.cfm?id=1944345.1944346|journal=Journal of the ACM|volume=58|issue=2|pages=5|doi=10.1145/1944345.1944346|s2cid=7703175 |issn=0004-5411|url-access=subscription}}</ref>

== References ==

{{reflist}}

== Further Readingreading ==

K. D. Mulmuley and M. Sohoni. Geometric Complexity Theory I: An Approach to the P vs. NP and Related Problems. SIAM J. Comput. 31(2), 496–526, 2001.

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K. D. Mulmuley, H. Narayanan, and M. Sohoni. Geometric complexity theory III: on deciding nonvanishing of a Littlewood-Richardson coefficient. J. Algebraic Combin. 36 (2012), no. 1, 103–110.

K. D. Mulmuley. Geometric Complexity Theory V: EquivalenceEfficient betweenalgorithms blackboxfor derandomizationNoether ofnormalization. polynomialJ. identityAmer. testingMath. andSoc. derandomization30 of(2017), Noether's Normalization Lemmano. FOCS 20121, also225-309. [[arxiv:1209.5993|arXiv:1209.5993 [cs.CC]]]

K. D. Mulmuley. Geometric Complexity Theory VI: the flip via positivity., Technical Report, Computer Science department, The University of Chicago, January 2011.

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* [http://gct.cs.uchicago.edu/ GCT page, University of Chicago]

* [http://simons.berkeley.edu/workshop_alggeometry1.html Description on the Simons Institute webpage]

* [httphttps://cstheory.stackexchange.com/questions/tagged/gct GCT questions] on [[cstheory]]

* [httphttps://cstheory.stackexchange.com/aq/17629 Wikipedia-style explanation of Geometric Complexity Theory] by Joshua Grochow

* [https://mathoverflow.net/q/277408 What are the current breakthroughs of Geometric Complexity Theory?]

* https://mathoverflow.net/questions/243011/why-should-algebraic-geometers-and-representation-theorists-care-about-geometric/

[[Category:Computational complexity theory]]