Geometric complexity theory: Difference between revisions

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'''Geometric Complexitycomplexity Theory''', often referred to astheory (GCT)''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 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 basic 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 [[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 by some to be the only viable currently active serious program to separate [[P (complexity) | P]] from [[NP (complexity) | NP]]. However, according to[[Ketan Mulmuley]] believes the program, if viable, is likely to take hundredsabout of100 years before it can settle the [[P vs. NP]] problem. <ref>{{citation

| last = Fortnow | first = Lance

| doi = 10.1145/1562164.1562186

| issue = 9

| journal = Communications of the ACM

| pages = 78–86

| title = The Status of the P Versus NP Problem

| volume = 52

| year = 2009| citeseerx = 10.1.1.156.767

| 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 argumentexistence of arguments for the program avoiding all known [[barriers (complexity)such as [[Oracle machine|relativization]] barriersand [[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 reading ==

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.

K. D. Mulmuley and M. Sohoni. Geometric Complexity Theory II: Towards Explicit Obstructions for Embeddings among Class Varieties. SIAM J. Comput., 38(3), 1175–1206, 2008.

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: Efficient algorithms for Noether normalization. J. Amer. Math. Soc. 30 (2017), no. 1, 225-309. [[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.

== External links ==

* [http://simonsgct.berkeleycs.uchicago.edu/workshop_alggeometry1.html Description onGCT thepage, SimonsUniversity Instuteof webpageChicago]

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

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

* [https://cstheory.stackexchange.com/q/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]]