A Brief Introduction to Singularity Theory (A. Remizov, 2010)
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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. -
Ian R. Porteous 9 October 1930 - 30 January 2011
In Memoriam Ian R. Porteous 9 October 1930 - 30 January 2011 A tribute by Peter Giblin (University of Liverpool) Będlewo, Poland 16 May 2011 Caustics 1998 Bill Bruce, disguised as a Terry Wall, ever a Pro-Vice-Chancellor mathematician Watch out for Bruce@60, Wall@75, Liverpool, June 2012 with Christopher Longuet-Higgins at the Rank Prize Funds symposium on computer vision in Liverpool, summer 1987, jointly organized by Ian, Joachim Rieger and myself After a first degree at Edinburgh and National Service, Ian worked at Trinity College, Cambridge, for a BA then a PhD under first William Hodge, but he was about to become Secretary of the Royal Society and Master of Pembroke College Cambridge so when Michael Atiyah returned from Princeton in January 1957 he took on Ian and also Rolph Schwarzenberger (6 years younger than Ian) as PhD students Ian’s PhD was in algebraic geometry, the effect of blowing up on Chern Classes, published in Proceedings of the Cambridge Philosophical Society in 1960: The behaviour of the Chern classes or of the canonical classes of an algebraic variety under a dilatation has been studied by several authors (Todd, Segre, van de Ven). This problem is of interest since a dilatation is the simplest form of birational transformation which does not preserve the underlying topological structure of the algebraic variety. A relation between the Chern classes of the variety obtained by dilatation of a subvariety and the Chern classes of the original variety has been conjectured by the authors cited above but a complete proof of this relation is not in the literature. -
Evolutes of Curves in the Lorentz-Minkowski Plane
Evolutes of curves in the Lorentz-Minkowski plane 著者 Izumiya Shuichi, Fuster M. C. Romero, Takahashi Masatomo journal or Advanced Studies in Pure Mathematics publication title volume 78 page range 313-330 year 2018-10-04 URL http://hdl.handle.net/10258/00009702 doi: info:doi/10.2969/aspm/07810313 Evolutes of curves in the Lorentz-Minkowski plane S. Izumiya, M. C. Romero Fuster, M. Takahashi Abstract. We can use a moving frame, as in the case of regular plane curves in the Euclidean plane, in order to define the arc-length parameter and the Frenet formula for non-lightlike regular curves in the Lorentz- Minkowski plane. This leads naturally to a well defined evolute asso- ciated to non-lightlike regular curves without inflection points in the Lorentz-Minkowski plane. However, at a lightlike point the curve shifts between a spacelike and a timelike region and the evolute cannot be defined by using this moving frame. In this paper, we introduce an alternative frame, the lightcone frame, that will allow us to associate an evolute to regular curves without inflection points in the Lorentz- Minkowski plane. Moreover, under appropriate conditions, we shall also be able to obtain globally defined evolutes of regular curves with inflection points. We investigate here the geometric properties of the evolute at lightlike points and inflection points. x1. Introduction The evolute of a regular plane curve is a classical subject of differen- tial geometry on Euclidean plane which is defined to be the locus of the centres of the osculating circles of the curve (cf. [3, 7, 8]). -
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). -
The Illinois Mathematics Teacher
THE ILLINOIS MATHEMATICS TEACHER EDITORS Marilyn Hasty and Tammy Voepel Southern Illinois University Edwardsville Edwardsville, IL 62026 REVIEWERS Kris Adams Chip Day Karen Meyer Jean Smith Edna Bazik Lesley Ebel Jackie Murawska Clare Staudacher Susan Beal Dianna Galante Carol Nenne Joe Stickles, Jr. William Beggs Linda Gilmore Jacqueline Palmquist Mary Thomas Carol Benson Linda Hankey James Pelech Bob Urbain Patty Bruzek Pat Herrington Randy Pippen Darlene Whitkanack Dane Camp Alan Holverson Sue Pippen Sue Younker Bill Carroll John Johnson Anne Marie Sherry Mike Carton Robin Levine-Wissing Aurelia Skiba ICTM GOVERNING BOARD 2012 Don Porzio, President Rich Wyllie, Treasurer Fern Tribbey, Past President Natalie Jakucyn, Board Chair Lannette Jennings, Secretary Ann Hanson, Executive Director Directors: Mary Modene, Catherine Moushon, Early Childhood (2009-2012) Com. College/Univ. (2010-2013) Polly Hill, Marshall Lassak, Early Childhood (2011-2014) Com. College/Univ. (2011-2014) Cathy Kaduk, George Reese, 5-8 (2010-2013) Director-At-Large (2009-2012) Anita Reid, Natalie Jakucyn, 5-8 (2011-2014) Director-At-Large (2010-2013) John Benson, Gwen Zimmermann, 9-12 (2009-2012) Director-At-Large (2011-2014) Jerrine Roderique, 9-12 (2010-2013) The Illinois Mathematics Teacher is devoted to teachers of mathematics at ALL levels. The contents of The Illinois Mathematics Teacher may be quoted or reprinted with formal acknowledgement of the source and a letter to the editor that a quote has been used. The activity pages in this issue may be reprinted for classroom use without written permission from The Illinois Mathematics Teacher. (Note: Occasionally copyrighted material is used with permission. Such material is clearly marked and may not be reprinted.) THE ILLINOIS MATHEMATICS TEACHER Volume 61, No. -
Some Remarks on Duality in S3
GEOMETRY AND TOPOLOGY OF CAUSTICS — CAUSTICS ’98 BANACH CENTER PUBLICATIONS, VOLUME 50 INSTITUTE OF MATHEMATICS POLISH ACADEMY OF SCIENCES WARSZAWA 1999 SOME REMARKS ON DUALITY IN S 3 IAN R. PORTEOUS Department of Mathematical Sciences University of Liverpool Liverpool, L69 3BX, UK e-mail: [email protected] Abstract. In this paper we review some of the concepts and results of V. I. Arnol0d [1] for curves in S2 and extend them to curves and surfaces in S3. 1. Introduction. In [1] Arnol0d introduces the concepts of the dual curve and the derivative curve of a smooth (= C1) embedded curve in S2. In particular he shows that the evolute or caustic of such a curve is the dual of the derivative. The dual curve is just the unit normal curve, while the derivative is the unit tangent curve. The definitions extend in an obvious way to curves on S2 with ordinary cusps. Then one proves easily that where a curve has an ordinary geodesic inflection the dual has an ordinary cusp and vice versa. Since the derivative of a curve without cusps clearly has no cusps it follows that the caustic of such a curve has no geodesic inflections. This prompts investigating what kind of singularity a curve must have for its caustic to have a geodesic inflection, by analogy with the classical construction of de l'H^opital,see [2], pages 24{26. In the second and third parts of the paper the definitions of Arnol0d are extended to surfaces in S3 and to curves in S3. The notations are those of Porteous [2], differentiation being indicated by subscripts. -
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. -
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]. -
Life and Work of Egbert Brieskorn (1936-2013)
Life and work of Egbert Brieskorn (1936 – 2013)1 Gert-Martin Greuel, Walter Purkert Brieskorn 2007 Egbert Brieskorn died on July 11, 2013, a few days after his 77th birthday. He was an impressive personality who left a lasting impression on anyone who knew him, be it in or out of mathematics. Brieskorn was a great mathematician, but his interests, knowledge, and activities went far beyond mathematics. In the following article, which is strongly influenced by the authors’ many years of personal ties with Brieskorn, we try to give a deeper insight into the life and work of Brieskorn. In doing so, we highlight both his personal commitment to peace and the environment as well as his long–standing exploration of the life and work of Felix Hausdorff and the publication of Hausdorff ’s Collected Works. The focus of the article, however, is on the presentation of his remarkable and influential mathematical work. The first author (GMG) has spent significant parts of his scientific career as a arXiv:1711.09600v1 [math.AG] 27 Nov 2017 graduate and doctoral student with Brieskorn in Göttingen and later as his assistant in Bonn. He describes in the first two parts, partly from the memory of personal cooperation, aspects of Brieskorn’s life and of his political and social commitment. In addition, in the section on Brieskorn’s mathematical work, he explains in detail 1Translation of the German article ”Leben und Werk von Egbert Brieskorn (1936 – 2013)”, Jahresber. Dtsch. Math.–Ver. 118, No. 3, 143-178 (2016). 1 the main scientific results of his publications.