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.
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 [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. Many of the theorems in these lecture notes are stated without proofs, but after giving enough motivation so that they can be taken on faith. For the readers interested in further study, Figure 1 shows logical dependence among the various papers of GCT and a suggested reading sequence.
The first author is grateful to Paolo Codenotti, Joshua Grochow, Sourav Chakraborty and Hari Narayanan for taking notes for his lectures. The basic structure of GCT By Ketan D. Mulmuley
Chapter 1
Scribe: Joshua A. Grochow
Goal: An overview of GCT.
The purpose of this course is to give an introduction to Geometric Complexity Theory (GCT), which is an approach to proving P = NP via algebraic geometry and representation theory. A basic plan of this approach is described in [GCTflip1,GCTflip2]. It is partially implemented in a series of articles [GCT1]- [GCT11]. The paper [GCTconf] is a conference announcement of GCT. The paper [Ml] gives an unconditional lower bound in a PRAM model without bit operations based on elementary algebraic geometry, and was a starting point for the GCT investigation via algebraic geometry.
The only mathematical prerequisites for this course are a basic knowledge of abstract algebra (groups, ring, fields, etc.) and a knowledge of computational complexity. In the first month we plan to cover the representation theory of finite groups, the symmetric group S n , and GL n (C), and enough algebraic geometry so that in the remaining lectures we can cover basic GCT. Most of the background results will only be sketched or omitted.
This lecture uses slightly more algebraic geometry and representation theory than the reader is assumed to know in order to give a more complete picture of GCT. As the course continues, we will cover this material.
Here is an outline of the GCT approach. Consider the P vs. NP question in characteristic 0; i.e., over integers. So bit operations are not allowed, and basic operations on integers are considered to take constant time. For a similar approach in nonzero characteristic (characteristic 2 being the classical case from a computational complexity point of view), see GCT 11.
The basic principle of GCT is the called the flip [GCTflip1]. It “reduces” (in essence, not formally) the lower bound problems such as P vs. NP in characteristic 0 to upper bound problems: showing that certain decision problems in algebraic geometry and representation theory belong to P . Each of these decision problems is of the form: is a given (nonnegative) structural constant associated to some algebro-geometric or representation theoretic object nonzero? This is akin to the decision problem: given a matrix, is its permanent nonzero? (We know how to solve this particular problem in polynomial time via reduction to the perfect matching problem.)
Next, the preceding upper bound problems are reduced to purely mathematical positivity hypotheses [GCT6]. The goal is to show that these and other auxilliary structural constants have positive formulae. By a positive formula we mean a formula that does not involve any alternating signs like the usual positive formula for the permanent; in contrast the usual formula for the determinant involves alternating signs.
Finally, these positivity hypotheses are “reduced” to conjectures in the theory of quantum groups [GCT6,GCT7,GCT8,GCT10] intimately related to the Riemann hypothesis over finite fields proved in [Dl2], and the related works [BBD,KL2,Lu2]. A pictorial summary of the GCT approach is shown in Figure 1.1, where the arrows represent reductions, rather than implications.
To recap: we move from a negative hypothesis in complexity theory (that there does not exist a polynomial time algorithm for an NP-complete problem) to a positive hypotheses in complexity theory (that there exist polynomial-time algorithms for certain decision problems) to positive hypotheses in mathematics (that certain structural constants have positive formulae) to conjectures on quantum groups related to the Riemann hypothesis over finite fields, the related works and their possible extensions. The first reduction here is the flip: we red
This content is AI-processed based on open access ArXiv data.
