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The Geometry of Resonance Tongues: a Singularity Theory Approach
INSTITUTE OF PHYSICS PUBLISHING NONLINEARITY Nonlinearity 16 (2003) 1511–1538 PII: S0951-7715(03)55769-3 The geometry of resonance tongues: a singularity theory approach Henk W Broer1, Martin Golubitsky2 and Gert Vegter3 1 Department of Mathematics, University of Groningen, PO Box 800, 9700 AV Groningen, The Netherlands 2 Department of Mathematics, University of Houston, Houston, TX 77204-3476, USA 3 Department of Computing Science, University of Groningen, PO Box 800, 9700 AV Groningen, The Netherlands Received 7 November 2002, in final form 20 May 2003 Published 6 June 2003 Online at stacks.iop.org/Non/16/1511 Recommended by A Chenciner Abstract Resonance tongues and their boundaries are studied for nondegenerate and (certain) degenerate Hopf bifurcations of maps using singularity theory methods of equivariant contact equivalence and universal unfoldings. We recover the standard theory of tongues (the nondegenerate case) in a straightforward way and we find certain surprises in the tongue boundary structure when degeneracies are present. For example, the tongue boundaries at degenerate singularities in weak resonance are much blunter than expected from the nondegenerate theory. Also at a semi-global level we find ‘pockets’ or ‘flames’ that can be understood in terms of the swallowtail catastrophe. Mathematics Subject Classification: 37G15, 37G40, 34C25 1. Introduction This paper focuses on resonance tongues obtained by Hopf bifurcation from a fixed point of a map. More precisely, Hopf bifurcations of maps occur at parameter values where the Jacobian of the map has a critical eigenvalue that is a root of unity e2πpi/q , where p and q are coprime integers with q 3 and |p| <q.Resonance tongues themselves are regions in parameter space near the point of Hopf bifurcation where periodic points of period q exist and tongue boundaries consist of critical points in parameter space where the q-periodic points disappear, typically in a saddle-node bifurcation. -
The Witten Equation, Mirror Symmetry, and Quantum Singularity Theory
Annals of Mathematics 178 (2013), 1{106 http://dx.doi.org/10.4007/annals.2013.178.1.1 The Witten equation, mirror symmetry, and quantum singularity theory By Huijun Fan, Tyler Jarvis, and Yongbin Ruan Abstract For any nondegenerate, quasi-homogeneous hypersurface singularity, we describe a family of moduli spaces, a virtual cycle, and a corresponding cohomological field theory associated to the singularity. This theory is analogous to Gromov-Witten theory and generalizes the theory of r-spin curves, which corresponds to the simple singularity Ar−1. We also resolve two outstanding conjectures of Witten. The first con- jecture is that ADE-singularities are self-dual, and the second conjecture is that the total potential functions of ADE-singularities satisfy correspond- ing ADE-integrable hierarchies. Other cases of integrable hierarchies are also discussed. Contents 1. Introduction2 1.1. Organization of the paper9 1.2. Acknowledgments9 2. W -curves and their moduli 10 2.1. W -structures on orbicurves 10 2.2. Moduli of stable W -orbicurves 20 2.3. Admissible groups G and W g;k;G 30 2.4. The tautological ring of W g;k 32 3. The state space associated to a singularity 35 3.1. Lefschetz thimble 35 3.2. Orbifolding and state space 37 H. F. was partially Supported by NSFC 10401001, NSFC 10321001, and NSFC 10631050. T. J. was partially supported by National Science Foundation grants DMS-0605155 and DMS-0105788 and the Institut Mittag-Leffler (Djursholm, Sweden). Y. R. was partially supported by the National Science Foundation and the Yangtze Center of Mathematics at Sichuan University. -
Algebraic Geometry Has Several Aspects to It
Lectures on Geometry of Plane Curves An Introduction to Algegraic Geometry ANANT R. SHASTRI Department of Mathematics Indian Institute of Technology, Mumbai Spring 1999 Contents 1 Introductory Remarks 3 2 Affine Spaces and Projective Spaces 6 3 Homogenization and De-homogenization 8 4 Defining Equation of a Curve 10 5 Relation Between Affine and Projective Curves 14 6 Resultant 17 7 Linear Transformations 21 8 Simple and Singular Points 24 9 Bezout’s Theorem 28 10 Basic Inequalities 32 11 Rational Curve 35 12 Co-ordinate Ring and the Quotient Field 37 13 Zariski Topology 39 14 Regular and Rational Maps 43 15 Closed Subspaces of Projective Spaces 48 16 Quasi Projective Varieties 50 17 Regular Functions on Quasi Projective Varieties 51 1 18 Rational Functions 55 19 Product of Quasi Projective Varieties 58 20 A Reduction Process 62 21 Study of Cubics 65 22 Inflection Points 69 23 Linear Systems 75 24 The Dual Curve 80 25 Power Series 85 26 Analytic Branches 90 27 Quadratic Transforms: 92 28 Intersection Multiplicity 94 2 Chapter 1 Introductory Remarks Lecture No. 1 31st Dec. 98 The present day algebraic geometry has several aspects to it. Let us illustrate two of the main aspects by some examples: (i) Solve geometric problems using algebraic techniques: Here is an example. To find the possible number of points of intersection of a circle and a straight line, we take the general equation of a circle and substitute for X (or Y ) using the general equation of a straight line and get a quadratic equation in one variable. -
Vanishing Cycles, Plane Curve Singularities, and Framed Mapping Class Groups
VANISHING CYCLES, PLANE CURVE SINGULARITIES, AND FRAMED MAPPING CLASS GROUPS PABLO PORTILLA CUADRADO AND NICK SALTER Abstract. Let f be an isolated plane curve singularity with Milnor fiber of genus at least 5. For all such f, we give (a) an intrinsic description of the geometric monodromy group that does not invoke the notion of the versal deformation space, and (b) an easy criterion to decide if a given simple closed curve in the Milnor fiber is a vanishing cycle or not. With the lone exception of singularities of type An and Dn, we find that both are determined completely by a canonical framing of the Milnor fiber induced by the Hamiltonian vector field associated to f. As a corollary we answer a question of Sullivan concerning the injectivity of monodromy groups for all singularities having Milnor fiber of genus at least 7. 1. Introduction Let f : C2 ! C denote an isolated plane curve singularity and Σ(f) the Milnor fiber over some point. A basic principle in singularity theory is to study f by way of its versal deformation space ∼ µ Vf = C , the parameter space of all deformations of f up to topological equivalence (see Section 2.2). From this point of view, two of the most basic invariants of f are the set of vanishing cycles and the geometric monodromy group. A simple closed curve c ⊂ Σ(f) is a vanishing cycle if there is some deformation fe of f with fe−1(0) a nodal curve such that c is contracted to a point when transported to −1 fe (0). -
Singularities Bifurcations and Catastrophes
Singularities Bifurcations and Catastrophes James Montaldi University of Manchester ©James Montaldi, 2020 © James Montaldi, 2020 Contents Preface xi 1 What’sitallabout? 1 1.1 The fold or saddle-nodebifurcation 2 1.2 Bifurcationsof contours 4 1.3 Zeeman catastrophemachine 5 1.4 Theevolute 6 1.5 Pitchfork bifurcation 11 1.6 Conclusions 13 Problems 14 I Catastrophe theory 17 2 Familiesof functions 19 2.1 Criticalpoints 19 2.2 Degeneracyin onevariable 22 2.3 Familiesoffunctions 23 2.4 Cuspcatastrophe 26 2.5 Why‘catastrophes’ 29 Problems 30 3 The ring of germs of smooth functions 33 3.1 Germs: making everythinglocal 33 3.2 Theringofgerms 35 3.3 Newtondiagram 39 3.4 Nakayama’s lemma 41 3.5 Idealsof finite codimension 43 3.6 Geometric criterion for finite codimension 45 Problems 45 4 Rightequivalence 49 4.1 Rightequivalence 49 4.2 Jacobianideal 51 4.3 Codimension 52 4.4 Nondegeneratecritical points 52 4.5 SplittingLemma 55 Problems 59 vi Contents 5 Finitedeterminacy 63 5.1 Trivial familiesofgerms 64 5.2 Finitedeterminacy 67 5.3 Apartialconverse 70 5.4 Arefinementofthefinitedeterminacytheorem 71 5.5 Thehomotopymethod 72 5.6 Proof of finite determinacy theorems 74 5.7 Geometriccriterion 77 Problems 78 6 Classificationoftheelementarycatastrophes 81 6.1 Classification of corank 1 singularities 82 6.2 Classification of corank 2 critical points 84 6.3 Thom’s 7 elementarysingularities 86 6.4 Furtherclassification 87 Problems 89 7 Unfoldingsand catastrophes 91 7.1 Geometry of families of functions 92 7.2 Changeofparameterandinducedunfoldings 94 7.3 Equivalenceof -
Arxiv:Math/0507171V1 [Math.AG] 8 Jul 2005 Monodromy
Monodromy Wolfgang Ebeling Dedicated to Gert-Martin Greuel on the occasion of his 60th birthday. Abstract Let (X,x) be an isolated complete intersection singularity and let f : (X,x) → (C, 0) be the germ of an analytic function with an isolated singularity at x. An important topological invariant in this situation is the Picard-Lefschetz monodromy operator associated to f. We give a survey on what is known about this operator. In particular, we re- view methods of computation of the monodromy and its eigenvalues (zeta function), results on the Jordan normal form of it, definition and properties of the spectrum, and the relation between the monodromy and the topology of the singularity. Introduction The word ’monodromy’ comes from the greek word µoνo − δρoµψ and means something like ’uniformly running’ or ’uniquely running’. According to [99, 3.4.4], it was first used by B. Riemann [135]. It arose in keeping track of the solutions of the hypergeometric differential equation going once around arXiv:math/0507171v1 [math.AG] 8 Jul 2005 a singular point on a closed path (cf. [30]). The group of linear substitutions which the solutions are subject to after this process is called the monodromy group. Since then, monodromy groups have played a substantial rˆole in many areas of mathematics. As is indicated on the webside ’www.monodromy.com’ of N. M. Katz, there are several incarnations, classical and l-adic, local and global, arithmetic and geometric. Here we concentrate on the classical lo- cal geometric monodromy in singularity theory. More precisely we focus on the monodromy operator of an isolated hypersurface or complete intersection singularity. -
Geometries, the Principle of Duality, and Algebraic Groups
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Expo. Math. 24 (2006) 195–234 www.elsevier.de/exmath Geometries, the principle of duality, and algebraic groups Skip Garibaldi∗, Michael Carr Department of Mathematics & Computer Science, Emory University, Atlanta, GA 30322, USA Received 3 June 2005; received in revised form 8 September 2005 Abstract J. Tits gave a general recipe for producing an abstract geometry from a semisimple algebraic group. This expository paper describes a uniform method for giving a concrete realization of Tits’s geometry and works through several examples. We also give a criterion for recognizing the auto- morphism of the geometry induced by an automorphism of the group. The E6 geometry is studied in depth. ᭧ 2005 Elsevier GmbH. All rights reserved. MSC 2000: Primary 22E47; secondary 20E42, 20G15 Contents 1. Tits’s geometry P ........................................................................197 2. A concrete geometry V , part I..............................................................198 3. A concrete geometry V , part II .............................................................199 4. Example: type A (projective geometry) .......................................................203 5. Strategy .................................................................................203 6. Example: type D (orthogonal geometry) ......................................................205 7. Example: type E6 .........................................................................208 -
On Dynkin Diagrams, Cartan Matrices
Physics 220, Lecture 16 ? Reference: Georgi chapters 8-9, a bit of 20. • Continue with Dynkin diagrams and the Cartan matrix, αi · αj Aji ≡ 2 2 : αi The j-th row give the qi − pi = −pi values of the simple root αi's SU(2)i generators acting on the root αj. Again, we always have αi · µ 2 2 = qi − pi; (1) αi where pi and qi are the number of times that the weight µ can be raised by Eαi , or lowered by E−αi , respectively, before getting zero. Applied to µ = αj, we know that qi = 0, since E−αj jαii = 0, since αi − αj is not a root for i 6= j. 0 2 Again, we then have AjiAij = pp = 4 cos θij, which must equal 0,1,2, or 3; these correspond to θij = π=2, 2π=3, 3π=4, and 5π=6, respectively. The Dynkin diagram has a node for each simple root (so the number of nodes is 2 2 r =rank(G)), and nodes i and j are connected by AjiAij lines. When αi 6= αj , sometimes it's useful to darken the node for the smaller root. 2 2 • Another example: constructing the roots for C3, starting from α1 = α2 = 1, and 2 α3 = 2, i.e. the Cartan matrix 0 2 −1 0 1 @ −1 2 −1 A : 0 −2 2 Find 9 positive roots. • Classify all simple, compact Lie algebras from their Aji. Require 3 properties: (1) det A 6= 0 (since the simple roots are linearly independent); (2) Aji < 0 for i 6= j; (3) AijAji = 0,1, 2, 3. -
Semi-Simple Lie Algebras and Their Representations
i Semi-Simple Lie Algebras and Their Representations Robert N. Cahn Lawrence Berkeley Laboratory University of California Berkeley, California 1984 THE BENJAMIN/CUMMINGS PUBLISHING COMPANY Advanced Book Program Menlo Park, California Reading, Massachusetts ·London Amsterdam Don Mills, Ontario Sydney · · · · ii Preface iii Preface Particle physics has been revolutionized by the development of a new “paradigm”, that of gauge theories. The SU(2) x U(1) theory of electroweak in- teractions and the color SU(3) theory of strong interactions provide the present explanation of three of the four previously distinct forces. For nearly ten years physicists have sought to unify the SU(3) x SU(2) x U(1) theory into a single group. This has led to studies of the representations of SU(5), O(10), and E6. Efforts to understand the replication of fermions in generations have prompted discussions of even larger groups. The present volume is intended to meet the need of particle physicists for a book which is accessible to non-mathematicians. The focus is on the semi-simple Lie algebras, and especially on their representations since it is they, and not just the algebras themselves, which are of greatest interest to the physicist. If the gauge theory paradigm is eventually successful in describing the fundamental particles, then some representation will encompass all those particles. The sources of this book are the classical exposition of Jacobson in his Lie Algebras and three great papers of E.B. Dynkin. A listing of the references is given in the Bibliography. In addition, at the end of each chapter, references iv Preface are given, with the authors’ names in capital letters corresponding to the listing in the bibliography. -
Algebraic Groups of Type D4, Triality and Composition Algebras
1 Algebraic groups of type D4, triality and composition algebras V. Chernousov1, A. Elduque2, M.-A. Knus, J.-P. Tignol3 Abstract. Conjugacy classes of outer automorphisms of order 3 of simple algebraic groups of classical type D4 are classified over arbitrary fields. There are two main types of conjugacy classes. For one type the fixed algebraic groups are simple of type G2; for the other type they are simple of type A2 when the characteristic is different from 3 and are not smooth when the characteristic is 3. A large part of the paper is dedicated to the exceptional case of characteristic 3. A key ingredient of the classification of conjugacy classes of trialitarian automorphisms is the fact that the fixed groups are automorphism groups of certain composition algebras. 2010 Mathematics Subject Classification: 20G15, 11E57, 17A75, 14L10. Keywords and Phrases: Algebraic group of type D4, triality, outer au- tomorphism of order 3, composition algebra, symmetric composition, octonions, Okubo algebra. 1Partially supported by the Canada Research Chairs Program and an NSERC research grant. 2Supported by the Spanish Ministerio de Econom´ıa y Competitividad and FEDER (MTM2010-18370-C04-02) and by the Diputaci´onGeneral de Arag´on|Fondo Social Europeo (Grupo de Investigaci´on de Algebra).´ 3Supported by the F.R.S.{FNRS (Belgium). J.-P. Tignol gratefully acknowledges the hospitality of the Zukunftskolleg of the Universit¨atKonstanz, where he was in residence as a Senior Fellow while this work was developing. 2 Chernousov, Elduque, Knus, Tignol 1. Introduction The projective linear algebraic group PGLn admits two types of conjugacy classes of outer automorphisms of order two. -
From Singularities to Graphs
FROM SINGULARITIES TO GRAPHS PATRICK POPESCU-PAMPU Abstract. In this paper I analyze the problems which led to the introduction of graphs as tools for studying surface singularities. I explain how such graphs were initially only described using words, but that several questions made it necessary to draw them, leading to the elaboration of a special calculus with graphs. This is a non-technical paper intended to be readable both by mathematicians and philosophers or historians of mathematics. Contents 1. Introduction 1 2. What is the meaning of those graphs?3 3. What does it mean to resolve a surface singularity?4 4. Representations of surface singularities around 19008 5. Du Val's singularities, Coxeter's diagrams and the birth of dual graphs 10 6. Mumford's paper on the links of surface singularities 14 7. Waldhausen's graph manifolds and Neumann's calculus on graphs 17 8. Conclusion 19 References 20 1. Introduction Nowadays, graphs are common tools in singularity theory. They mainly serve to represent morpholog- ical aspects of surface singularities. Three examples of such graphs may be seen in Figures1,2,3. They are extracted from the papers [57], [61] and [14], respectively. Comparing those figures we see that the vertices are diversely depicted by small stars or by little circles, which are either full or empty. These drawing conventions are not important. What matters is that all vertices are decorated with numbers. We will explain their meaning later. My aim in this paper is to understand which kinds of problems forced mathematicians to associate graphs to surface singularities. -
Fundamental Theorems in Mathematics
SOME FUNDAMENTAL THEOREMS IN MATHEMATICS OLIVER KNILL Abstract. An expository hitchhikers guide to some theorems in mathematics. Criteria for the current list of 243 theorems are whether the result can be formulated elegantly, whether it is beautiful or useful and whether it could serve as a guide [6] without leading to panic. The order is not a ranking but ordered along a time-line when things were writ- ten down. Since [556] stated “a mathematical theorem only becomes beautiful if presented as a crown jewel within a context" we try sometimes to give some context. Of course, any such list of theorems is a matter of personal preferences, taste and limitations. The num- ber of theorems is arbitrary, the initial obvious goal was 42 but that number got eventually surpassed as it is hard to stop, once started. As a compensation, there are 42 “tweetable" theorems with included proofs. More comments on the choice of the theorems is included in an epilogue. For literature on general mathematics, see [193, 189, 29, 235, 254, 619, 412, 138], for history [217, 625, 376, 73, 46, 208, 379, 365, 690, 113, 618, 79, 259, 341], for popular, beautiful or elegant things [12, 529, 201, 182, 17, 672, 673, 44, 204, 190, 245, 446, 616, 303, 201, 2, 127, 146, 128, 502, 261, 172]. For comprehensive overviews in large parts of math- ematics, [74, 165, 166, 51, 593] or predictions on developments [47]. For reflections about mathematics in general [145, 455, 45, 306, 439, 99, 561]. Encyclopedic source examples are [188, 705, 670, 102, 192, 152, 221, 191, 111, 635].