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The University of Chicago Symmetry and Equivalence

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The University of Chicago Symmetry and Equivalence THE UNIVERSITY OF CHICAGO SYMMETRY AND EQUIVALENCE RELATIONS IN CLASSICAL AND GEOMETRIC COMPLEXITY THEORY A DISSERTATION SUBMITTED TO THE FACULTY OF THE DIVISION OF THE PHYSICAL SCIENCES IN CANDIDACY FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF COMPUTER SCIENCE BY JOSHUA ABRAHAM GROCHOW CHICAGO, ILLINOIS JUNE 2012 To my parents, Jerrold Marvin Grochow and Louise Barnett Grochow ABSTRACT This thesis studies some of the ways in which symmetries and equivalence relations arise in classical and geometric complexity theory. The Geometric Complexity Theory program is aimed at resolving central questions in complexity such as P versus NP using techniques from algebraic geometry and representation theory. The equivalence relations we study are mostly algebraic in nature and we heavily use algebraic techniques to reason about the computational properties of these problems. We first provide a tutorial and survey on Geometric Complexity Theory to provide perspective and motivate the other problems we study. One equivalence relation we study is matrix isomorphism of matrix Lie algebras, which is a problem that arises naturally in Geometric Complexity Theory. For certain cases of matrix isomorphism of Lie algebras we provide polynomial-time algorithms, and for other cases we show that the problem is as hard as graph isomorphism. To our knowledge, this is the first time graph isomorphism has appeared in connection with any lower bounds program. Finally, we study algorithms for equivalence relations more generally (joint work with Lance Fortnow). Two techniques are often employed for algorithmically deciding equivalence relations: 1) finding a complete set of easily computable invariants, or 2) finding an algorithm which will compute a canonical form for each equivalence class. Some equivalence relations in the literature have been solved efficiently by other means as well. We ask whether these three conditions—having an efficient solution, having an efficiently computable complete invariant, and having an efficiently computable canonical form—are equivalent. We show that this question requires non-relativizing techniques to resolve, and provide new connections between this question and factoring integers, probabilistic algorithms, and quantum computation. iii ACKNOWLEDGMENTS First I thank my advisors. I thank Lance Fortnow for his advice, support, guidance, and collaboration on Chapter 5 and several other projects which have yet to bear fruit. I thank Ketan Mulmuley for his advice and support, as well as countless hours of discussion through- out the course of my graduate career. Without these discussions, it would not have been possible to seriously work on problems related to Geometric Complexity Theory—such as matrix isomorphism of Lie algebras (Chapter 4)—let alone to write a survey on it (Chapter 3). I thank Benson Farb for his advice, guidance, and many fruitful discussions, even when he was not officially my advisor. I thank my thesis committee for their continued advice, prodding, and editorial support. I thank Anne Rogers for her support and for the countless decisions regarding my career trajectory, both large and small, she helped me understand how to make. I thank Sasha Razborov for many interesting discussions, and for enforcing a much-needed kick-in-the-pants in the middle of my graduate career that ensured I graduated in a timely fashion. I cannot imagine this duty was much fun for him, but it was a tremendous help to me. I thank anonymous reviewers for feedback that improved the quality, clarity, and presen- tation of the works on which Chapters 4 and 5 are based. In particular, one of the reviewers pointed out the importance of the complexity of factoring polynomials for Chapter 4. One of the reviewers suggested that we define some sort of hybrid notion of Cohen and transi- tive genericity, as well as suggested the notion of UP-transitive genericity that are used in Chapter 5. I also thank Lane Hemaspaandra—who was our editor for the corresponding paper—and Paolo Codenotti for useful comments on a draft of Chapter 5. I thank Laci Babai for useful comments on a draft of Chapter 4, as well as pointing me to several results [20, 62] related to that chapter, and suggesting that I consider the corresponding questions for associative algebras. I thank Stuart Kurtz and Laci Babai for several useful discussions regarding Chapter 5. In particular, Stuart suggested the use of the equivalence relation RL, which led to Theorem iv 5.3.3, and Laci pointed out the canonical form for subgroup equality of permutation groups [23]. I thank Scott Aaronson for the observations leading to Section 5.3.1. I thank Andreas Blass for pointing me to the original two papers he co-authored with Gurevich [56, 57]. I thank my collaborators, on projects both finished and in progress: Lance Fortnow, L´aszl´o Babai, Paolo Codenotti, Youming “Jimmy” Qiao, Jonah Blasiak, and Thomas Church. It was and continues to be a pleasure to work with them. In particular, Chap- ter 5 is based on joint work with Lance, and Jonah helped me clarify my thoughts on matrix isomorphism of Lie algebras and together realize the equivalence with graph isomorphism in Chapter 4. I find it incredibly useful, rewarding, and fun to talk through mathematics with others, and it is my great pleasure and honor to thank Lance Fortnow, Ketan Mulmuley, Ben- son Farb, Thomas Church, Ian Shipman, Spencer Dowdall, Anna Marie Bohmann, Daniel Studenmund, Vipul Naik, Paolo Codenotti, Youming “Jimmy” Qiao, Chris Umans, J. M. Landsberg, Jerzy Weyman, Shrawan Kumar, Neeraj Kayal, Arkadev Chatthopadhyay, Pascal Koiran, Gerald J. Sussman, and Jonah Blasiak for not only useful and interesting discus- sions, but also for their infectious enthusiasm. Many discussions regarding GCT and matrix isomorphism of Lie algebras took place at the Brown-ICERM Workshop on Mathe- matical Aspects of P vs. NP and its Variants in August 2011, for which I would like to thank ICERM and the organizers of the workshop—J. M. Landsberg, Saugata Basu, and J. Maurice Rojas—for the invitation and support to attend the workshop. I would especially like to thank Stuart Kurtz and Gerald J. Sussman for sharing with me some small portion of their incredible breadth of knowledge and depth of philosophy. They have both made my research career and my life more interesting. This thesis was partially supported by K. Mulmuley’s NSF Grant CCF-1017760, L. Fort- now et al.’s NSF Grant DMS-0652521 and fellowships from the University Chicago Depart- ment of Computer Science. I would like to thank the members of the University of Chicago Department of Computer Science Techstaff. They’ve setup such a great system and were so helpful that I barely noticed all the technology I was using: I could do what I wanted, how I wanted, when I wanted. I think this is the mark of a truly great technical staff. I would also like to thank v the staff of the University of Chicago Library, especially those in Eckhart Library: I am likely one of their most frequent patrons. Finally, I thank my family and extended family. My extended family, who were also my roommates at various points throughout my graduate career: Spencer Dowdall, Ian Shipman, Ann Herbert, Rebecca Lordan, and (honorary roommate) Thomas Church; it’s not so much that they made graduate school worth the time and effort, but that they made it worthwhile at least ten times over. I especially thank my grandparents Samuel and Frances Grochow, and Marvin and Hazel Barnett, my parents Jerrold and Louise Grochow, and my sister, Rebecca Grochow, for all their love and support in so many ways over the years. Last but by no means least, I thank my fianc´eNikki Pfarr. I thank her for her patience, support, and partnership; for her humor; for her love, romance, and companionship; for her wisdom, wit, humor, and intelligence; and for her smile. vi TABLE OF CONTENTS ABSTRACT ........................................ iii ACKNOWLEDGMENTS ................................. iv LISTOFFIGURES .................................... x LISTOFTABLES ..................................... xi Chapter 1 INTRODUCTION ................................... 1 1.1 Computationalcomplexity ............................ 2 1.1.1 Computational problems and complexity measures . 2 1.1.2 Degreesofcomplexity .......................... 3 1.2 Equivalencerelations ............................... 5 1.3 Symmetry..................................... 7 1.3.1 Continuous symmetries and Lie algebras . 9 1.3.2 Symmetry-based equivalence relations . 9 1.4 Symmetry and equivalence relations in complexity . .. 12 1.5 Organization ................................... 15 2 BACKGROUND .................................... 16 2.1 ComplexityTheory................................ 16 2.1.1 Computationalproblems . 17 2.1.2 Reductions ................................ 18 2.1.3 Complexityclasses ............................ 18 2.1.4 Circuit complexity . 26 2.1.5 Algebraic complexity . 28 2.1.6 Barriers: relativization, algebrization, and natural proofs . ..... 31 2.2 Algebra ...................................... 33 2.2.1 Equivalence relations . 33 2.2.2 Groups................................... 33 2.2.3 Rings,fields,andmodules ........................ 38 2.2.4 Liealgebras ................................ 40 vii 3 A TUTORIAL AND SURVEY OF GEOMETRIC COMPLEXITY THEORY . 49 3.1 Introduction.................................... 49 3.1.1 Outline .................................. 50 3.2 The1,000-footview................................ 51 3.2.1 Theplanofattack ...........................
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