The Engaging Symmetry of Riemann Surfaces: a Historical Perspective
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Combination of Cubic and Quartic Plane Curve
IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-5728,p-ISSN: 2319-765X, Volume 6, Issue 2 (Mar. - Apr. 2013), PP 43-53 www.iosrjournals.org Combination of Cubic and Quartic Plane Curve C.Dayanithi Research Scholar, Cmj University, Megalaya Abstract The set of complex eigenvalues of unistochastic matrices of order three forms a deltoid. A cross-section of the set of unistochastic matrices of order three forms a deltoid. The set of possible traces of unitary matrices belonging to the group SU(3) forms a deltoid. The intersection of two deltoids parametrizes a family of Complex Hadamard matrices of order six. The set of all Simson lines of given triangle, form an envelope in the shape of a deltoid. This is known as the Steiner deltoid or Steiner's hypocycloid after Jakob Steiner who described the shape and symmetry of the curve in 1856. The envelope of the area bisectors of a triangle is a deltoid (in the broader sense defined above) with vertices at the midpoints of the medians. The sides of the deltoid are arcs of hyperbolas that are asymptotic to the triangle's sides. I. Introduction Various combinations of coefficients in the above equation give rise to various important families of curves as listed below. 1. Bicorn curve 2. Klein quartic 3. Bullet-nose curve 4. Lemniscate of Bernoulli 5. Cartesian oval 6. Lemniscate of Gerono 7. Cassini oval 8. Lüroth quartic 9. Deltoid curve 10. Spiric section 11. Hippopede 12. Toric section 13. Kampyle of Eudoxus 14. Trott curve II. Bicorn curve In geometry, the bicorn, also known as a cocked hat curve due to its resemblance to a bicorne, is a rational quartic curve defined by the equation It has two cusps and is symmetric about the y-axis. -
Arxiv:1012.2020V1 [Math.CV]
TRANSITIVITY ON WEIERSTRASS POINTS ZOË LAING AND DAVID SINGERMAN 1. Introduction An automorphism of a Riemann surface will preserve its set of Weier- strass points. In this paper, we search for Riemann surfaces whose automorphism groups act transitively on the Weierstrass points. One well-known example is Klein’s quartic, which is known to have 24 Weierstrass points permuted transitively by it’s automorphism group, PSL(2, 7) of order 168. An investigation of when Hurwitz groups act transitively has been made by Magaard and Völklein [19]. After a section on the preliminaries, we examine the transitivity property on several classes of surfaces. The easiest case is when the surface is hy- perelliptic, and we find all hyperelliptic surfaces with the transitivity property (there are infinitely many of them). We then consider surfaces with automorphism group PSL(2, q), Weierstrass points of weight 1, and other classes of Riemann surfaces, ending with Fermat curves. Basically, we find that the transitivity property property seems quite rare and that the surfaces we have found with this property are inter- esting for other reasons too. 2. Preliminaries Weierstrass Gap Theorem ([6]). Let X be a compact Riemann sur- face of genus g. Then for each point p ∈ X there are precisely g integers 1 = γ1 < γ2 <...<γg < 2g such that there is no meromor- arXiv:1012.2020v1 [math.CV] 9 Dec 2010 phic function on X whose only pole is one of order γj at p and which is analytic elsewhere. The integers γ1,...,γg are called the gaps at p. The complement of the gaps at p in the natural numbers are called the non-gaps at p. -
Arxiv:1406.2663V2 [Hep-Th]
Multiboundary Wormholes and Holographic Entanglement Vijay Balasubramaniana;b, Patrick Haydenc, Alexander Maloneyd;e, Donald Marolff , Simon F. Rossg aDavid Rittenhouse Laboratories, University of Pennsylvania 209 S 33rd Street, Philadelphia, PA 19104, USA bCUNY Graduate Center, Initiative for the Theoretical Sciences 365 Fifth Avenue, New York, NY 10016, USA cDepartment of Physics, Stanford University Palo Alto, CA 94305, USA dDepartment of Physics, McGill University 3600 rue Universit´e,Montreal H3A2T8, Canada eCenter for the Fundamental Laws of Nature, Harvard University Cambridge, MA 02138, USA f Department of Physics, University of California, Santa Barbara, CA 93106, USA gCentre for Particle Theory, Department of Mathematical Sciences Durham University, South Road, Durham DH1 3LE, UK Abstract The AdS/CFT correspondence relates quantum entanglement between boundary Conformal Field Theories and geometric connections in the dual asymptotically Anti- de Sitter space-time. We consider entangled states in the n−fold tensor product of a 1+1 dimensional CFT Hilbert space defined by the Euclidean path integral over a Riemann surface with n holes. In one region of moduli space, the dual bulk state is arXiv:1406.2663v2 [hep-th] 23 Jun 2014 a black hole with n asymptotically AdS3 regions connected by a common wormhole, while in other regions the bulk fragments into disconnected components. We study the entanglement structure and compute the wave function explicitly in the puncture limit of the Riemann surface in terms of CFT n-point functions. We also use AdS minimal surfaces to measure entanglement more generally. In some regions of the moduli space the entanglement is entirely multipartite, though not of the GHZ type. -
Irreducible Canonical Representations in Positive Characteristic 3
IRREDUCIBLE CANONICAL REPRESENTATIONS IN POSITIVE CHARACTERISTIC BENJAMIN GUNBY, ALEXANDER SMITH AND ALLEN YUAN ABSTRACT. For X a curve over a field of positive characteristic, we investigate 0 when the canonical representation of Aut(X) on H (X, ΩX ) is irreducible. Any curve with an irreducible canonical representation must either be superspecial or ordinary. Having a small automorphism group is an obstruction to having irre- ducible canonical representation; with this motivation, the bulk of the paper is spent bounding the size of automorphism groups of superspecial and ordinary curves. After proving that all automorphisms of an Fq2 -maximal curve are de- fined over Fq2 , we find all superspecial curves with g > 82 having an irreducible representation. In the ordinary case, we provide a bound on the size of the auto- morphism group of an ordinary curve that improves on a result of Nakajima. 1. INTRODUCTION Given a complete nonsingular curve X of genus g 2, the finite group G := ≥ 0 Aut(X) has a natural action on the g-dimensional k-vector space H (X, ΩX ), known as the canonical representation. It is natural to ask when this representation is irre- ducible. In characteristic zero, irreducibility of the canonical representation implies that g2 G , and combining this with the Hurwitz bound of G 84(g 1), one can≤ observe | | that the genus of X is bounded. In fact, Breuer| [1]| shows ≤ that− the maximal genus of a Riemann surface with irreducible canonical representation is 14. In characteristic p, the picture is more subtle when p divides G . The Hurwitz bound of 84(g 1) may no longer hold due to the possibility of wild| | ramification in arXiv:1408.3830v1 [math.AG] 17 Aug 2014 the Riemann-Hurwitz− formula. -
Hyperbolic Vortices
Hyperbolic Vortices Nick Manton DAMTP, University of Cambridge [email protected] SEMPS, University of Surrey, November 2015 Outline I 1. Abelian Higgs Vortices. I 2. Hyperbolic Vortices. I 3. 1-Vortex on the Genus-2 Bolza Surface. I 4. Baptista’s Geometric Interpretation of Vortices. I 5. Conclusions. 1. Abelian Higgs Vortices I The Abelian Higgs (Ginzburg–Landau) vortex is a two-dimensional static soliton, stabilised by its magnetic flux. Well-known is the Abrikosov vortex lattice in a superconductor. I Vortices exist on a plane or curved Riemann surface M, with metric ds2 = Ω(z; z¯) dzdz¯ : (1) z = x1 + ix2 is a (local) complex coordinate. I The fields are a complex scalar Higgs field φ and a vector potential Aj (j = 1; 2) with magnetic field F = @1A2 − @2A1. They don’t back-react on the metric. I Our solutions have N vortices and no antivortices. On a plane, N is the winding number of φ at infinity. If M is compact, φ and A are a section and connection of a U(1) bundle over M, with first Chern number N. I The field energy is 1 Z 1 1 1 = 2 + j φj2 + ( − jφj2)2 Ω 2 E 2 F Dj 1 d x (2) 2 M Ω Ω 4 where Dj φ = @j φ − iAj φ. The first Chern number is 1 Z N = F d 2x : (3) 2π M I The energy E can be re-expressed as [E.B. Bogomolny] E = πN + 1 Z 1 Ω 2 1 2 F − (1 − jφj2) + D φ + iD φ Ω d 2x 2 1 2 2 M Ω 2 Ω (4) where we have dropped a total derivative term. -
An Optimal Systolic Inequality for Cat(0) Metrics in Genus Two
AN OPTIMAL SYSTOLIC INEQUALITY FOR CAT(0) METRICS IN GENUS TWO MIKHAIL G. KATZ∗ AND STEPHANE´ SABOURAU Abstract. We prove an optimal systolic inequality for CAT(0) metrics on a genus 2 surface. We use a Voronoi cell technique, introduced by C. Bavard in the hyperbolic context. The equality is saturated by a flat singular metric in the conformal class defined by the smooth completion of the curve y2 = x5 x. Thus, among all CAT(0) metrics, the one with the best systolic− ratio is composed of six flat regular octagons centered at the Weierstrass points of the Bolza surface. Contents 1. Hyperelliptic surfaces of nonpositive curvature 1 2. Distinguishing 16 points on the Bolza surface 3 3. A flat singular metric in genus two 4 4. Voronoi cells and Euler characteristic 8 5. Arbitrary metrics on the Bolza surface 10 References 12 1. Hyperelliptic surfaces of nonpositive curvature Over half a century ago, a student of C. Loewner’s named P. Pu presented, in the pages of the Pacific Journal of Mathematics [Pu52], the first two optimal systolic inequalities, which came to be known as the Loewner inequality for the torus, and Pu’s inequality (5.4) for the real projective plane. The recent months have seen the discovery of a number of new sys- tolic inequalities [Am04, BK03, Sa04, BK04, IK04, BCIK05, BCIK06, KL05, Ka06, KS06, KRS07], as well as near-optimal asymptotic bounds [Ka03, KS05, Sa06a, KSV06, Sa06b, RS07]. A number of questions 1991 Mathematics Subject Classification. Primary 53C20, 53C23 . Key words and phrases. Bolza surface, CAT(0) space, hyperelliptic surface, Voronoi cell, Weierstrass point, systole. -
New Trends in Teichmüller Theory and Mapping Class Groups
Mathematisches Forschungsinstitut Oberwolfach Report No. 40/2018 DOI: 10.4171/OWR/2018/40 New Trends in Teichm¨uller Theory and Mapping Class Groups Organised by Ken’ichi Ohshika, Osaka Athanase Papadopoulos, Strasbourg Robert C. Penner, Bures-sur-Yvette Anna Wienhard, Heidelberg 2 September – 8 September 2018 Abstract. In this workshop, various topics in Teichm¨uller theory and map- ping class groups were discussed. Twenty-three talks dealing with classical topics and new directions in this field were given. A problem session was organised on Thursday, and we compiled in this report the problems posed there. Mathematics Subject Classification (2010): Primary: 32G15, 30F60, 30F20, 30F45; Secondary: 57N16, 30C62, 20G05, 53A35, 30F45, 14H45, 20F65 IMU Classification: 4 (Geometry); 5 (Topology). Introduction by the Organisers The workshop New Trends in Teichm¨uller Theory and Mapping Class Groups, organised by Ken’ichi Ohshika (Osaka), Athanase Papadopoulos (Strasbourg), Robert Penner (Bures-sur-Yvette) and Anna Wienhard (Heidelberg) was attended by 50 participants, including a number of young researchers, with broad geo- graphic representation from Europe, Asia and the USA. During the five days of the workshop, 23 talks were given, and on Thursday evening, a problem session was organised. Teichm¨uller theory originates in the work of Teichm¨uller on quasi-conformal maps in the 1930s, and the study of mapping class groups was started by Dehn and Nielsen in the 1920s. The subjects are closely interrelated, since the mapping class group is the -
Maximal Harmonic Group Actions on Finite Graphs
Maximal harmonic group actions on finite graphs Scott Corry∗ Department of Mathematics, Lawrence University, 711 E. Boldt Way – SPC 24, Appleton, WI 54911, USA Abstract This paper studies groups of maximal size acting harmonically on a finite graph. Our main result states that these maximal graph groups are exactly the finite quotients of the modular group Γ = x, y | x2 = y3 =1 of size at least 6. This characterization may be viewed as a discrete analogue of the description of Hurwitz groups as finite quotients of the (2, 3, 7)-triangle group in the context of holomorphic group actions on Riemann surfaces. In fact, as an immediate consequence of our result, every Hurwitz group is a maximal graph group, and the final section of the paper establishes a direct connection between maximal graphs and Hurwitz surfaces via the theory of combinatorial maps. Keywords: harmonic group action, Hurwitz group, combinatorial map 2010 MSC: 14H37, 05C99 1. Introduction Many recent papers have explored analogies between Riemann surfaces and finite graphs (e.g. [2],[3],[4],[6],[7],[8],[14],[15],[17]). Inspired by the Accola- Maclachlan [1], [21] and Hurwitz [18] genus bounds for holomorphic group ac- tions on compact Riemann surfaces, we introduced harmonic group actions on finite graphs in [14], and established sharp linear genus bounds for the maximal size of such actions. As noted in the introduction to [14], it is an interesting problem to characterize the groups and graphs that achieve the upper bound 6(g − 1). Such maximal groups and graphs may be viewed as graph-theoretic arXiv:1301.3411v2 [math.CO] 27 Mar 2015 analogues of Hurwitz groups and surfaces—those compact Riemann surfaces S of genus g ≥ 2 such that Aut(S) has maximal size 84(g−1). -
Period Relations for Riemann Surfaces with Many Automorphisms
PERIOD RELATIONS FOR RIEMANN SURFACES WITH MANY AUTOMORPHISMS LUCA CANDELORI, JACK FOGLIASSO, CHRISTOPHER MARKS, AND SKIP MOSES Abstract. By employing the theory of vector-valued automorphic forms for non- unitarizable representations, we provide a new bound for the number of linear rela- tions with algebraic coefficients between the periods of an algebraic Riemann surface with many automorphisms. The previous best-known general bound for this num- ber was the genus of the Riemann surface, a result due to Wolfart. Our new bound significantly improves on this estimate, and it can be computed explicitly from the canonical representation of the Riemann surface. As observed by Shiga and Wolfart, this bound may then be used to estimate the dimension of the endomorphism algebra of the Jacobian of the Riemann surface. We demonstrate with a few examples how this improved bound allows one, in some instances, to actually compute the dimension of this endomorphism algebra, and to determine whether the Jacobian has complex multiplication. 1. Introduction The theory of vector-valued modular forms, though nascent in the work of various nineteenth century authors, is a relatively recent development in mathematics. Perhaps the leading motivation for working out a general theory in this area comes from two- dimensional conformal field theory, or more precisely from the theory of vertex operator algebras (VOAs). Indeed, in some sense the article [Zhu96] – in which Zhu proved that the graded dimensions of the simple modules for a rational VOA constitute a weakly holomorphic vector-valued modular function – created a demand for understanding how such objects work in general, and what may be learned about them by studying the representations according to which they transform. -
The Fifteenth Annual Meeting of the American Mathematical Society
THE ANNUAL MEETING OF THE SOCIETY. 275 THE FIFTEENTH ANNUAL MEETING OF THE AMERICAN MATHEMATICAL SOCIETY. SINCE the founding of the Society in 1888, the regular, including the annual, meetings have been held almost Without exception in New York City, as the most convenient center for the members living in the eastern states and others who might from time to time attend an eastern meeting. The summer meeting, migratory between limits as far apart as Boston and St. Louis, has afforded an annual opportunity for a fully repre sentative gathering, and provision has been made for the con venience of the central and western members by the founding of the Chicago Section in 1897, the San Francisco Section in 1902, and the Southwestern Section in 1906. The desire has, however, often been expressed that the annual meeting of the Society might, when geographic and other conditions were exceptionally favorable, be occasionally held like that of many other scientific bodies in connection with the meeting of the American association for the advancement of science, a gather ing which naturally affords many conveniences of travel and scientific advantages. It was therefore decided to hold the annual meeting of 1908 at Baltimore in affiliation with the Association, the days chosen being Wednesday and Thursday, December 30-31. Two sessions were held on each day in the Biological Laboratory of Johns Hopkins University. The total atten dance numbered about seventy-five, including the following fifty-seven members of the Society : Miss C. C. Barnum, Dr. E. G. Bill, Professor G. A. Bliss, Professor E. -
Arxiv:2105.03871V2 [Math.GT] 21 May 2021
THE EXTREMAL LENGTH SYSTOLE OF THE BOLZA SURFACE MAXIME FORTIER BOURQUE, D´IDAC MART´INEZ-GRANADO, AND FRANCO VARGAS PALLETE Abstract. We prove that the extremal length systole of genusp two surfaces attains a strict local maximum at the Bolza surface, where it takes the value 2. 1. Introduction Extremal length is a conformal invariant that plays an important role in complex analysis, complex dynamics, and Teichm¨ullertheory [Ahl06, Ahl10, Jen58]. It can be used to define the notion of quasiconformality, upon which the Teichm¨ullerdistance between Riemann surfaces is based. In turn, a formula of Kerckhoff [Ker80, Theorem 4] shows that Teichm¨ullerdistance is determined by extremal lengths of (homotopy classes of) essential simple closed curves, as opposed to all families of curves. The extremal length systole of a Riemann surface X is defined as the infimum of the extremal lengths of all essential closed curves in X. This function fits in the framework of generalized systoles (infima of collections of \length" functions) developed by Bavard in [Bav97] and [Bav05]. In contrast with the hyperbolic systole, the extremal length systole has not been studied much so far. For flat tori, we will see that the extremal length systole agrees with the systolic ratio, from which it follows that the regular hexagonal torus uniquely maximizes the extremal length systole in genus one (c.f. Loewner's torus inequality [Pu52]). In [MGP19], the last two authors of the present paper conjectured that the Bolza surface maximizes the extremal length systole in genus two. This surface, which can be obtained as a double branched cover of the regular octahedron branched over the vertices, is the most natural candidate since it maximizes several other invariants in genus two such as the hyperbolic systole [Jen84], the kissing number [Sch94b], and the number of automorphisms [KW99, Section 3.2]. -
Computational Algebraic and Analytic Geometry
572 Computational Algebraic and Analytic Geometry AMS Special Sessions on Computational Algebraic and Analytic Geometry for Low-Dimensional Varieties January 8, 2007: New Orleans, LA January 6, 2009: Washington, DC January 6, 2011: New Orleans, LA Mika Seppälä Emil Volcheck Editors American Mathematical Society Computational Algebraic and Analytic Geometry AMS Special Sessions on Computational Algebraic and Analytic Geometry for Low-Dimensional Varieties January 8, 2007: New Orleans, LA January 6, 2009: Washington, DC January 6, 2011: New Orleans, LA Mika Seppälä Emil Volcheck Editors 572 Computational Algebraic and Analytic Geometry AMS Special Sessions on Computational Algebraic and Analytic Geometry for Low-Dimensional Varieties January 8, 2007: New Orleans, LA January 6, 2009: Washington, DC January 6, 2011: New Orleans, LA Mika Seppälä Emil Volcheck Editors American Mathematical Society Providence, Rhode Island EDITORIAL COMMITTEE Dennis DeTurck, Managing Editor Michael Loss Kailash Misra Martin J. Strauss 2010 Mathematics Subject Classification. Primary 14HXX, 30FXX, and 68WXX. Library of Congress Cataloging-in-Publication Data Computational algebraic and analytic geometry : AMS special sessions on computational algebraic and analytic geometry for low-dimensional varieties, January 8, 2007, New Orleans, LA, January 6, 2009, Washington, DC, [and] January 6, 2011, New Orleans, LA / Mika Sepp¨al¨a, Emil Volcheck, editors. p. cm. — (Contemporary mathematics ; v. 572) Includes bibliographical references. ISBN 978-0-8218-6869-0 (alk. paper) 1. Curves, Algebraic–Data processing–Congresses. 2. Riemann surfaces–Congresses. I. Sepp¨al¨a, Mika. II. Volcheck, Emil, 1966– QA565.C658 2012 512’.5–dc23 2012009188 Copying and reprinting. Material in this book may be reproduced by any means for edu- cational and scientific purposes without fee or permission with the exception of reproduction by services that collect fees for delivery of documents and provided that the customary acknowledg- ment of the source is given.