Slicing Bagels: Plane Sections of Real and Complex Tori Asilomar - December 2004 Bruce Cohen Lowell High School, SFUSD

Slides:



Advertisements
Similar presentations
Notes 6.6 Fundamental Theorem of Algebra
Advertisements

Complex Numbers Adding in the Imaginary i By Lucas Wagner.
Chapter 11-Functions of Several Variables
Proving Fermat’s last theorem FLT and lessons on the nature and practice of mathematics.
Introduction to Elliptic Curves. What is an Elliptic Curve? An Elliptic Curve is a curve given by an equation E : y 2 = f(x) Where f(x) is a square-free.
Shuijing Crystal Li Rice University Mathematics Department 1 Rational Points on del Pezzo Surfaces of degree 1 and 2.
Stick Tossing and Confidence Intervals Asilomar - December 2006 Bruce Cohen Lowell High School, SFUSD
Chapter 11 Polynomial Functions
A Brief Introduction to Real Projective Geometry Asilomar - December 2010 Bruce Cohen Lowell High School, SFUSD
The Ubiquity of Elliptic Curves Joseph Silverman (Brown University) Public Lecture – Dublin Tuesday, 4 September 2007, 7:30 PM.
Polya’s Orchard Visibility Problem and Related Questions in Geometry and Number Theory Asilomar - December 2008 Bruce Cohen Lowell High School, SFUSD
The Ubiquity of Elliptic Curves Joseph Silverman (Brown University) MAA Invited Address Baltimore – January 18, 2003.
Introduction to Gröbner Bases for Geometric Modeling Geometric & Solid Modeling 1989 Christoph M. Hoffmann.
Plane Sections of Real and Complex Tori Sonoma State - February 2006 or Why the Graph of is a Torus Based on a presentation by David Sklar and Bruce Cohen.
1 Equivalence of Real Elliptic Curves Equivalence of Real Elliptic Curves Allen Broughton Rose-Hulman Institute of Technology.
Hilbert’s Problems By Sharjeel Khan.
Precalculus CST Released Items Aligned to Stewart Text
Intro to Conic Sections. It all depends on how you slice it! Start with a cone:
Fermat’s Last Theorem Samina Saleem Math5400. Introduction The Problem The seventeenth century mathematician Pierre de Fermat created the Last Theorem.
An Introduction to Conics
By: Leonardo Ramirez Pre Calculus Per.6 Mr. Caballero.
Introduction to Conic Sections
Introduction to Conic Sections
1 1.0 Students solve equations and inequalities involving absolute value. Algebra 2 Standards.
Chapter 3 Greek Number Theory The Role of Number Theory Polygonal, Prime and Perfect Numbers The Euclidean Algorithm Pell’s Equation The Chord and Tangent.
The Ubiquity of Elliptic Curves Joseph Silverman (Brown University) MAA Invited Address – Expanded Version Baltimore – January 18, 2003.
Preperiodic Points and Unlikely Intersections joint work with Laura DeMarco Matthew Baker Georgia Institute of Technology AMS Southeastern Section Meeting.
Geometry, Trigonometry, Algebra, and Complex Numbers Palm Springs - November 2004 Dedicated to David Cohen (1942 – 2002) Bruce Cohen Lowell High School,
NONLINEAR SYSTEMS, CONIC SECTIONS, SEQUENCES, AND SERIES College Algebra.
Chapter 2 Greek Geometry The Deductive Method The Regular Polyhedra Ruler and Compass Construction Conic Sections Higher-degree curves Biographical Notes:
Lesson 4-1 Polynomial Functions.
The Ubiquity of Elliptic Curves Joseph Silverman (Brown University) SUMS – Providence – February 22, 2003.
Fundamental Theorem of Algebra and Brouwer’s Fixed Point Theorem (talk at Mahidol University) Wayne Lawton Department of Mathematics National University.
Review Day! Hyperbolas, Parabolas, and Conics. What conic is represented by this definition: The set of all points in a plane such that the difference.
Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley.
Functions of Several Variables
Circles – An Introduction SPI Graph conic sections (circles, parabolas, ellipses and hyperbolas) and understand the relationship between the.
Section 9.6: Functions and Surfaces Practice HW from Stewart Textbook (not to hand in) p. 683 # 9-13, 19, 20, 23, 24, 25 Handout Sheet 1-6, 7-27 odd.
Conic Sections By: Danielle Hayman Mrs. Guest Per. 4.
Does tropical geometry look like…. No, it looks like …
Conic Sections An Introduction. Conic Sections - Introduction Similar images are located on page 604 of your book. You do not need to try and recreate.
An Introduction to Elliptic Curve Cryptography
Section 8.5. In fact, all of the equations can be converted into one standard equation.
An Introduction to Conics
MATH 1330 Section 8.2A. Circles & Conic Sections To form a conic section, we’ll take this double cone and slice it with a plane. When we do this, we’ll.
Chapter Nonlinear Systems.
Chapter 10 – Conic Sections 1) Circles 2) Parabolas 3) Ellipses 4) Hyperbolas.
Do Now: Graph and state the type of graph and the domain and range if it is a function. Introduction to Analytic Geometry.
10.0 Conic Sections. Conic Section – a curve formed by the intersection of a plane and a double cone. By changing the plane, you can create a circle,
Chapter 15: Functions of Several Variables
MATHEMATICS B.A./B.Sc. (GENERAL) FIRST YEAR EXAMINATIONS,2012.
Solving Linear Systems
Chapter 10 Conic Sections.
33. Conic Sections in General Form
Systems: Identifying Equations, Points of Intersections of Equations
Chapter 10: Conic Sections; Polar Coordinates; Parametric Equations
Chapter 4 (Part 1): Induction & Recursion
Introduction to Functions of Several Variables
Introduction to Conic Sections
6-2 Conic Sections: Circles
Systems: Identifying Equations, Points of Intersections of Equations
Conic Sections - Circles
Introduction to Conic Sections
Systems: Identifying Equations, Points of Intersections of Equations
3.4 Zeros of Polynomial Functions: Real, Rational, and Complex
Chapter 2 Analytic Function
Systems: Identifying Equations, Points of Intersections of Equations
Systems: Identifying Equations, Points of Intersections of Equations
Chapter 2 Greek Geometry
Presentation transcript:

Slicing Bagels: Plane Sections of Real and Complex Tori Asilomar - December 2004 Bruce Cohen Lowell High School, SFUSD David Sklar San Francisco State University

Part I - Slicing a Real Circular Torus Equations for the torus in R 3 The Spiric Sections of Perseus Ovals of Cassini and The Lemniscate of Bernoulli Other Slices The Villarceau Circles A Characterization of the torus

Bibliography Part II - Slicing a Complex Torus Elliptic curves and number theory Some graphs of Hints of toric sections Two closures: Algebraic and Geometric Algebraic closure, C 2, R 4, and the graph of Geometric closure, Projective spaces P 1 (R), P 2 (R), P 1 (C), and P 2 (C) The graphs of

Elliptic curves and number theory Roughly, an elliptic curve over a field F is the graph of an equation of the form where p(x) is a cubic polynomial with three distinct roots and coefficients in F. The fields of most interest are the rational numbers, finite fields, the real numbers, and the complex numbers. Within a year it was shown that Fermats last theorem would follow from a widely believed conjecture in the arithmetic theory of elliptic curves. In 1985, after mathematicians had been working on Fermats Last Theorem for about 350 years, Gerhard Frey suggested that if we assumed Fermats Last Theorem was false, the existence of an elliptic curve where a, b and c are distinct integers such that with integer exponent n > 2, might lead to a contradiction. Less than 10 years later Andrew Wiles proved a form of the Taniyama conjecture sufficient to prove Fermats Last Theorem.

Elliptic curves and number theory The strategy of placing a centuries old number theory problem in the context of the arithmetic theory of elliptic curves has led to the complete or partial solution of at least three major problems in the last thirty years. The Congruent Number Problem – Tunnell 1983 The Gauss Class Number Problem – Goldfeld 1976, Gross & Zagier 1986 Fermats Last Theorem – Frey 1985, Ribet 1986, Wiles 1995, Taylor 1995 Although a significant discussion of the theory of elliptic curves and why they are so nice is beyond the scope of this talk I would like to try to show you that, when looked at in the right way, the graph of an elliptic curve is a beautiful and familiar geometric object. Well do this by studying the graph of the equation

Graphs of: Hints of Toric Sections If we close up the algebra to include the complex numbers and the geometry to include points at infinity, we can argue that the graph of is a torus.

Geometric Closure: an Introduction to Projective Geometry Part I – Real Projective Geometry One-Dimension - the Real Projective Line P 1 (R) The real (affine) line R is the ordinary real number line The real projective line P 1 (R) is the set It is topologically equivalent to the open interval (-1, 1) by the map and topologically equivalent to a punctured circle by stereographic projection It is topologically equivalent to a closed interval with the endpoints identified and topologically equivalent to a circle by stereographic projection

Geometric Closure: an Introduction to Projective Geometry Part I – Real Projective Geometry Two-Dimensions - the Real Projective Plane P 2 (R) The real (affine) plane R 2 is the ordinary x, y -plane It is topologically equivalent to a closed disk with antipodal points on the boundary circle identified. It is topologically equivalent to the open unit disk by the map ( ) The real projective plane P 2 (R) is the set. It is R 2 together with a line at infinity,. Every line in R 2 intersects, parallel lines meet at the same point on, and nonparallel lines intersect at distinct points. Every line in P 2 (R) is a P 1 (R). Two distinct lines intersect at one and only one point.

A Projective View of the Conics Ellipse Parabola Hyperbola

A Projective View of the Conics Ellipse Parabola Hyperbola

Graphs of: Hints of Toric Sections

Graph of with x and y complex Algebraic closure

Graph of with x and y complex Algebraic closure Some comments on why the graph of the system is a surface.

Graph of with x and y complex Algebraic closure

Graph of with x and y complex

Geometric Closure: an Introduction to Projective Geometry Part II – Complex Projective Geometry One-Dimension - the Complex Projective Line or Riemann Sphere P 1 (C) The complex (affine) line C is the ordinary complex plane where (x, y) corresponds to the number z = x + iy. It is topologically a punctured sphere by stereographic projection The complex projective line P 1 (C) is the set the complex plane with one number adjoined. It is topologically a sphere by stereographic projection with the north pole corresponding to. It is often called the Riemann Sphere. (Note: 1-D over the complex numbers, but, 2-D over the real numbers)

Geometric Closure: an Introduction to Projective Geometry Part II – Complex Projective Geometry Two-Dimensions - the Complex Projective Plane P 2 (C) The complex (affine) plane C 2 or better complex 2-space is a lot like R 4. A line in C 2 is the graph of an equation of the form, where a, b and c are complex constants and x and y are complex variables. (Note: not every plane in R 4 corresponds to a complex line) (Note: 2-D over the complex numbers, but, 4-D over the real numbers) Complex projective 2-space P 2 (C) is the set. It is C 2 together with a complex line at infinity,. Every line in R 2 intersects, parallel lines meet at the same point on, and nonparallel lines intersect at distinct points. Every line in P 2 (C) is a P 1 (C), a Riemann sphere, including the line at infinity. Basically P 2 (C) is C 2 closed up nicely by a Riemann Sphere at infinity. Two distinct lines intersect at one and only one point.

Graph of with x and y complex

A Generalization: the Graph of

Bibliography 8. T. Needham, Visual Complex Analysis, Oxford University Press, Oxford E. Brieskorn & H. Knorrer, Plane Algebraic Curves, Birkhauser Verlag, Basel, K. Kendig, Elementary Algebraic Geometry, Springer-Verlag, New York Z. A. Melzak, Companion to Concrete Mathematics, John Wiley & Sons, New York, J. Stillwell, Mathematics and Its History, Springer-Verlag, New York Z. A. Melzak, Invitation to Geometry, John Wiley & Sons, New York, D. Hilbert & H. Cohn-Vossen, Geometry and the Imagination, Chelsea Publishing Company, New York, N. Koblitz, Introduction to Elliptic Curves and Modular Forms, Springer-Verlag, New York M. Villarceau, "Théorème sur le tore." Nouv. Ann. Math. 7, , M. Berger, Geometry I and Geometry II, Springer-Verlag, New York 1987