Geometric complexity theory

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 – whether P = NP – by showing that the complexity class P is not equal to the complexity class 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 algebraic complexity classes. Proving that computing the permanent cannot be efficiently reduced to computing determinants is considered to be a major milestone for the program. These computational problems can be characterized by their symmetries. The program aims at utilizing these symmetries for proving lower bounds.

The approach is considered by some to be the only viable currently active program to separate P from 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.[1]

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 relativization and natural proofs for proving general lower bounds.[2]

## References

1. ↑ Fortnow, Lance (2009), "The Status of the P Versus NP Problem", Communications of the ACM 52 (9): 78–86, doi:10.1145/1562164.1562186 .
2. ↑ Mulmuley, Ketan D. (2011-04-01). "On P vs. NP and geometric complexity theory: Dedicated to Sri Ramakrishna". Journal of the ACM 58 (2): 5. doi:10.1145/1944345.1944346. ISSN 0004-5411. http://dl.acm.org/citation.cfm?id=1944345.1944346.

## 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 [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

* GCT page, University of Chicago
* Description on the Simons Institute webpage
* GCT questions on cstheory
* Wikipedia-style explanation of Geometric Complexity Theory by Joshua Grochow
* What are the current breakthroughs of Geometric Complexity Theory?
* https://mathoverflow.net/questions/243011/why-should-algebraic-geometers-and-representation-theorists-care-about-geometric/

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