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Register a new userGeometric Complexity Theory: Introduction
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Ketan Mulmuley D
Publication date
2007
fields
Informatics Engineering
and research's language is
English
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These are lectures notes for the introductory graduate courses on geometric complexity theory (GCT) in the computer science department, the university of Chicago. Part I consists of the lecture notes for the course given by the first author in the spring quarter, 2007. It gives introduction to the basic structure of GCT. Part II consists of the lecture notes for the course given by the second author in the spring quarter, 2003. It gives introduction to invariant theory with a view towards GCT. No background in algebraic geometry or representation theory is assumed. These lecture notes in conjunction with the article cite{GCTflip1}, which describes in detail the basic plan of GCT based on the principle called the flip, should provide a high level picture of GCT assuming familiarity with only basic notions of algebra, such as groups, rings, fields etc.
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We study a basic algorithmic problem in algebraic geometry, which we call NNL, of constructing a normalizing map as per Noethers Normalization Lemma. For general explicit varieties, as formally defined in this paper, we give a randomized polynomial-time Monte Carlo algorithm for this problem. For some interesting cases of explicit varieties, we give deterministic quasi-polynomial time algorithms. These may be contrasted with the standard EXPSPACE-algorithms for these problems in computational algebraic geometry. In particular, we show that: (1) The categorical quotient for any finite dimensional representation $V$ of $SL_m$, with constant $m$, is explicit in characteristic zero. (2) NNL for this categorical quotient can be solved deterministically in time quasi-polynomial in the dimension of $V$. (3) The categorical quotient of the space of $r$-tuples of $m times m$ matrices by the simultaneous conjugation action of $SL_m$ is explicit in any characteristic. (4) NNL for this categorical quotient can be solved deterministically in time quasi-polynomial in $m$ and $r$ in any characteristic $p$ not in $[2, m/2]$. (5) NNL for every explicit variety in zero or large enough characteristic can be solved deterministically in quasi-polynomial time, assuming the hardness hypothesis for the permanent in geometric complexity theory. The last result leads to a geometric complexity theory approach to put NNL for every explicit variety in P.
Geometric complexity theory (GCT) is an approach to the P vs. NP and related problems. This article gives its complexity theoretic overview without assuming any background in algebraic geometry or representation theory.
We show that most arithmetic circuit lower bounds and relations between lower bounds naturally fit into the representation-theoretic framework suggested by geometric complexity theory (GCT), including: the partial derivatives technique (Nisan-Wigderson), the results of Razborov and Smolensky on $AC^0[p]$, multilinear formula and circuit size lower bounds (Raz et al.), the degree bound (Strassen, Baur-Strassen), the connected components technique (Ben-Or), depth 3 arithmetic circuit lower bounds over finite fields (Grigoriev-Karpinski), lower bounds on permanent versus determinant (Mignon-Ressayre, Landsberg-Manivel-Ressayre), lower bounds on matrix multiplication (B{u}rgisser-Ikenmeyer) (these last two were already known to fit into GCT), the chasms at depth 3 and 4 (Gupta-Kayal-Kamath-Saptharishi; Agrawal-Vinay; Koiran), matrix rigidity (Valiant) and others. That is, the original proofs, with what is often just a little extra work, already provide representation-theoretic obstructions in the sense of GCT for their respective lower bounds. This enables us to expose a new viewpoint on GCT, whereby it is a natural unification and broad generalization of known results. It also shows that the framework of GCT is at least as powerful as known methods, and gives many new proofs-of-concept that GCT can indeed provide significant asymptotic lower bounds. This new viewpoint also opens up the possibility of fruitful two-way interactions between previous results and the new methods of GCT; we provide several concrete suggestions of such interactions. For example, the representation-theoretic viewpoint of GCT naturally provides new properties to consider in the search for new lower bounds.
Geometric complexity theory (GCT) is an approach to the $P$ vs. $NP$ and related problems. A high level overview of this research plan and the results obtained so far was presented in a series of three lectures in the Institute of Advanced study, Princeton, Feb 9-11, 2009. This article contains the material covered in those lectures after some revision, and gives a mathematical overview of GCT. No background in algebraic geometry, representation theory or quantum groups is assumed.
This article describes a {em nonstandard} quantum group that may be used to derive a positive formula for the plethysm problem, just as the standard (Drinfeld-Jimbo) quantum group can be used to derive the positive Littlewood-Richardson rule for arbitrary complex semisimple Lie groups. The sequel cite{GCT8} gives conjecturally correct algorithms to construct canonical bases of the coordinate rings of these nonstandard quantum groups and canonical bases of the dually paired nonstandard deformations of the symmetric group algebra. A positive $#P$-formula for the plethysm constant follows from the conjectural properties of these canonical bases and the duality and reciprocity conjectures herein.
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