Why do mathematicians make things so complicated?


 Clementine Gibbs
 2 years ago
 Views:
Transcription
1 Why do mathematicians make things so complicated? Zhiqin Lu, The Math Department March 9, 2010
2 Introduction What is Mathematics?
3 Introduction What is Mathematics? 24,100,000 answers from Google.
4 Introduction What is Mathematics? 24,100,000 answers from Google. Such a FAQ!
5 Introduction From Wikipedia Mathematics is the study of quantity, structure, space, and change. Mathematicians seek out patterns,[2][3] formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions.[4]
6 Introduction An example The Real World vs. The Math World
7 Introduction An example The Real World vs. The Math World How to become a millionaire
8 Introduction An example The Real World vs. The Math World How to become a millionaire in a month?
9 Introduction An example Fact/Secret/Assumption: Most interest checking accounts generate interest at least one cent a month.
10 Introduction An example Fact/Secret/Assumption: Most interest checking accounts generate interest at least one cent a month. Here is How: 1 Open 100,000,000 interest checking accounts and deposit one cent to each account;
11 Introduction An example Fact/Secret/Assumption: Most interest checking accounts generate interest at least one cent a month. Here is How: 1 Open 100,000,000 interest checking accounts and deposit one cent to each account; 2 Wait for a month.
12 Introduction An example Fact/Secret/Assumption: Most interest checking accounts generate interest at least one cent a month. Here is How: 1 Open 100,000,000 interest checking accounts and deposit one cent to each account; 2 Wait for a month. The profit?
13 Introduction An example Fact/Secret/Assumption: Most interest checking accounts generate interest at least one cent a month. Here is How: 1 Open 100,000,000 interest checking accounts and deposit one cent to each account; 2 Wait for a month. The profit? 100, 000, 000 $0.01 = $1, 000, 000!
14 Introduction An example...and that is not the end of the story...
15 Introduction An example...and that is not the end of the story... Mathematicians like to say
16 Introduction An example...and that is not the end of the story... Mathematicians like to say Let n (infinity)
17 Introduction An example...and that is not the end of the story... Mathematicians like to say Let n (infinity) If we let the number of checking accounts go to infinity, what will happen?
18 Introduction An example...and that is not the end of the story... Mathematicians like to say Let n (infinity) If we let the number of checking accounts go to infinity, what will happen? One can earn the whole Universe in a month!
19 Introduction An example Since that is not possible, we get the following result by Reductio ad absurdum (proof by contradiction).
20 Introduction An example Since that is not possible, we get the following result by Reductio ad absurdum (proof by contradiction). Theorem No banks can afford a free $0.01 interest.
21 Introduction An example Since that is not possible, we get the following result by Reductio ad absurdum (proof by contradiction). Theorem No banks can afford a free $0.01 interest. (in the math world)
22 Introduction Summary I am going to talk about
23 Introduction Summary I am going to talk about 1 Why everything has to be done in an indirect way?
24 Introduction Summary I am going to talk about 1 Why everything has to be done in an indirect way? 2 The power of symbols/abstractions.
25 Introduction Summary I am going to talk about 1 Why everything has to be done in an indirect way? 2 The power of symbols/abstractions. 3 How do we choose a problem/project to work on?
26 Introduction Summary I am going to talk about 1 Why everything has to be done in an indirect way? 2 The power of symbols/abstractions. 3 How do we choose a problem/project to work on? 4 Why do we care about other sciences?
27 Introduction Summary I am going to talk about 1 Why everything has to be done in an indirect way? 2 The power of symbols/abstractions. 3 How do we choose a problem/project to work on? 4 Why do we care about other sciences? 5 Use of Computer.
28 My field Mathematics
29 My field Mathematics Differential Geometry
30 My field Mathematics Differential Geometry Complex Geometry
31 My field 1 One of my projects is in the mathematical aspects of Super String Theory.
32 My field 1 One of my projects is in the mathematical aspects of Super String Theory. 2 It is related to the Mirror Symmetry.
33 My field 1 One of my projects is in the mathematical aspects of Super String Theory. 2 It is related to the Mirror Symmetry. 3 Two Universes, quite different, but have the same Quantum Field Theory.
34 My field Figure: Brain Greene, The Elegant Universe, NY Times Best Selling Book.
35 Why everything has to be done in an... indirect way? A problem in Math 2E. Triple integralsa Problem in Math 2E
36 Why everything has to be done in an... indirect way? A problem in Math 2E. Triple integralsa Problem in Math 2E Compute W xdxdydz, where W is the region bounded by the planes x = 0, y = 0, and z = 2, and the surface z = x 2 + y 2 and lying in the quadrant x 0, y 0.
37 II IV Why do mathematicians make things so complicated? Why everything has to be done in an... indirect way? A problem in Math 2E. Triple integralsa Problem in Math 2E Compute W xdxdydz, where W is the region bounded by the planes x = 0, y = 0, and z = 2, and the surface z = x 2 + y 2 and lying in the quadrant x 0, y 0. N '<: II N >< II II >< 0 N + ~ ~ '<:
38 Why everything has to be done in an... indirect way? A problem in Math 2E. How to compute integrations over an ndimensional object?
39 Why everything has to be done in an... indirect way? A problem in Math 2E. Figure: From the internet. It is the intersection of the quintic CalabiYau threefold to our three dimensional space.
40 Why everything has to be done in an... indirect way? A problem in Math 2E. How to study high dimensional geometric object? Use Calculus;
41 Why everything has to be done in an... indirect way? A problem in Math 2E. How to study high dimensional geometric object? Use Calculus; PDE, functional analysis, complex analysis, etc
42 Why everything has to be done in an... indirect way? A problem in Math 2E. How to study high dimensional geometric object? Use Calculus; PDE, functional analysis, complex analysis, etc Use Linear Algebra;
43 Why everything has to be done in an... indirect way? A problem in Math 2E. How to study high dimensional geometric object? Use Calculus; PDE, functional analysis, complex analysis, etc Use Linear Algebra; Lie algebra, commutative algebra, algebraic topology, etc
44 Why everything has to be done in an... indirect way? A problem in Math 2E. How to study high dimensional geometric object? Use Calculus; PDE, functional analysis, complex analysis, etc Use Linear Algebra; Lie algebra, commutative algebra, algebraic topology, etc Use the results in all other mathematics fields.
45 Why everything has to be done in an... indirect way? A problem in Math 2E. How to study high dimensional geometric object? Use Calculus; PDE, functional analysis, complex analysis, etc Use Linear Algebra; Lie algebra, commutative algebra, algebraic topology, etc Use the results in all other mathematics fields. Euclidean Geometry methods usually do not apply.
46 Why everything has to be done in an... indirect way? A simpler example. An even simpler example
47 Why everything has to be done in an... indirect way? A simpler example. An even simpler example 1 xdy ydx = 1. 2π circle x 2 + y 2
48 Why everything has to be done in an... indirect way? A simpler example. Conclusion: Since Human Beings can t image or sense a high dimensional object, we have to study it indirectly. Mathematics is our seventh sense organ.
49 The power of symbols/abstractions. An example The mirror map (in the simplest case) is (5ψ) 5 exp 5 (5n)! (n!) 5 (5ψ) 5n n=0 where ψ 1. { 5n } (5n)! 1 1 (n!) 5 j (5ψ) 5n j=n+1, n=1
50 The power of symbols/abstractions. An example The mirror map (in the simplest case) is (5ψ) 5 exp 5 (5n)! (n!) 5 (5ψ) 5n n=0 { 5n } (5n)! 1 1 (n!) 5 j (5ψ) 5n j=n+1, n=1 where ψ 1. Although complicated, it is very concrete.
51 The power of symbols/abstractions. An example...and we denoted it as q(ψ)
52 The power of symbols/abstractions. Another Example. Newton s Law of universal gravitation F = G m 1m 2 r 2
53 The power of symbols/abstractions. Another Example. Newton s Law of universal gravitation The Coulomb s Law F = G m 1m 2 r 2 F = k e q 1 q 2 r 2
54 The power of symbols/abstractions. Another Example. Newton s Law of universal gravitation The Coulomb s Law F = G m 1m 2 r 2 F = k e q 1 q 2 r 2 In mathematics we study the function y = C 1 r 2 which applies to both laws.
55 The power of symbols/abstractions. Another Example. The evolution of mathematics largely depends on the evolution of symbols.
56 How do we choose a problem/project to work on? Mathematicians choose problems/projects in a counterproductive way.
57 How do we choose a problem/project to work on? Mathematicians choose problems/projects in a counterproductive way. 1 Choose a problem that is unlikely to be solved.
58 How do we choose a problem/project to work on? Mathematicians choose problems/projects in a counterproductive way. 1 Choose a problem that is unlikely to be solved. 2 Choose a problem whose outcome is unexpected.
59 How do we choose a problem/project to work on? 1 Andrew Wiles proved the Fermat Last Theorem, a conjecture that lasted 398 years.
60 How do we choose a problem/project to work on? 1 Andrew Wiles proved the Fermat Last Theorem, a conjecture that lasted 398 years. 2 Grigori Perelman solved Poincaré Conjecture, almost 100 years old, using the Ricci flow method.
61 How do we choose a problem/project to work on? Pros 1 Very creative and original; 2 Usually quite deep in the discovery of new phenomena. Cons papers a year means very productive? 2 collaborative work becomes difficult. 3 the work usually finishes in the last minute.
62 Why do we care about other sciences? The evolution of Mathematics. The evolution of Mathematics How to push math forward?
63 Why do we care about other sciences? The evolution of Mathematics. 1 generalization
64 Why do we care about other sciences? The evolution of Mathematics. 1 generalization (Differential Geometry=Calculus on curved space)
65 Why do we care about other sciences? The evolution of Mathematics. 1 generalization (Differential Geometry=Calculus on curved space) 2 similar to bionical creativity engineering, get hints from other sciences
66 Why do we care about other sciences? My results in the math aspect of super string theory. There are some mathematical implications from Mirror Symmetry, one of which is the socalled BCOV Conjecture.
67 Why do we care about other sciences? My results in the math aspect of super string theory. There are some mathematical implications from Mirror Symmetry, one of which is the socalled BCOV Conjecture. BershadskyCecottiOoguriVafa Conjecture 1 Let F A be an invariant obtained from symplectic geometry of one CalabiYau manifold; 2 Let F B be an invariant obtained from complex geometry of the Mirror CalabiYau manifold. Then F A = F B.
68 Why do we care about other sciences? My results in the math aspect of super string theory. Setup of Conjecture (B) Let X be a compact Kähler manifold.
69 Why do we care about other sciences? My results in the math aspect of super string theory. Setup of Conjecture (B) Let X be a compact Kähler manifold. Let = p,q be the Laplacian on (p, q) forms;
70 Why do we care about other sciences? My results in the math aspect of super string theory. Setup of Conjecture (B) Let X be a compact Kähler manifold. Let = p,q be the Laplacian on (p, q) forms; By compactness, the spectrum of are eigenvalues: 0 λ 0 λ 1 λ n +.
71 Why do we care about other sciences? My results in the math aspect of super string theory. Setup of Conjecture (B) Let X be a compact Kähler manifold. Let = p,q be the Laplacian on (p, q) forms; By compactness, the spectrum of are eigenvalues: 0 λ 0 λ 1 λ n +. Define det = λ i 0 λ i.
72 Why do we care about other sciences? My results in the math aspect of super string theory. Setup of Conjecture (B) Let X be a compact Kähler manifold. Let = p,q be the Laplacian on (p, q) forms; By compactness, the spectrum of are eigenvalues: 0 λ 0 λ 1 λ n +. Define det = λ i 0 λ i. ζ function regularization (for example: Riemann ζfunction)
73 Why do we care about other sciences? My results in the math aspect of super string theory. Setup of Conjecture B BershadskyCeccottiOoguriVafa defined T def = (det p,q ) ( 1)p+qpq. p,q
74 Why do we care about other sciences? My results in the math aspect of super string theory. Setup of Conjecture B BershadskyCeccottiOoguriVafa defined T def = (det p,q ) ( 1)p+qpq. p,q Why define such a strange quantity?
75 Why do we care about other sciences? My results in the math aspect of super string theory. Conjecture (B) Let be the Hermitian metric on the line bundle (π K W/CP 1) 62 (T (CP 1 )) 3 CP 1 \D induced from the L 2 metric on π K W/CP 1 and from the WeilPetersson metric on T (CP 1 ). Then the following identity holds: ( ) 62 ( 1 Ωψ τ BCOV (W ψ ) = Const. F top q d ) 2 3 3, 1,B (ψ)3 y 0 (ψ) dq where Ω is the local holomorphic section of the (3, 0) forms.
76 Why do we care about other sciences? My results in the math aspect of super string theory. Conjecture (B) was proved by FangLYoshikawa. FangLYoshikawa Asymptotic behavior of the BCOV torsion of CalabiYau moduli ArXiv: JDG (80), 2008, ,
77 Why do we care about other sciences? My results in the math aspect of super string theory. Conjecture (B) was proved by FangLYoshikawa. FangLYoshikawa Asymptotic behavior of the BCOV torsion of CalabiYau moduli ArXiv: JDG (80), 2008, , Aleksey Zinger proved Conjecture (A). Combining the two results, we proved the BCOV Conjecture, which is an evidence that Super String Theory may be true.
78 Why do we care about other sciences? My results in the math aspect of super string theory. String theorists believe that there are parallel universes to our Universe. AshokDouglas developed a method to count the number of those parallel universes.
79 Why do we care about other sciences? My results in the math aspect of super string theory. String theorists believe that there are parallel universes to our Universe. AshokDouglas developed a method to count the number of those parallel universes. Joint with Michael R. Douglas, we proved that, if string theory is true, the the number of parallel universes is finite.
80 The use of computer Computer usage is absolutely important in pure math.
81 The use of computer Two kinds of math theorems Theorem π 2 > 9.8
82 The use of computer Two kinds of math theorems Theorem π 2 > 9.8 Theorem x 2 + y 2 2xy
83 The use of computer From Wikipedia A computerassisted proof is a mathematical proof that has been at least partially generated by computer.
84 The use of computer The AntunesFreitas Conjecture. AntunesFreitas Conjecture A triangle drum with its longest side equal to 1. Let λ 1, λ 2 be the two lowest frequencies. Then λ 2 λ 1 64π2 9
85 The use of computer The AntunesFreitas Conjecture. The conjecture was recently solved by BetckeLRowlett, with an extensive use of computer. It is a computer assisted proof!
86 The use of computer The AntunesFreitas Conjecture. The key part is, although there are infinitely many different triangles, we proved that by checking the conjecture for finitely many of them (In fact, we checked 10,000 triangles), the conjecture must be true for any triangles.
87 The use of computer The AntunesFreitas Conjecture. We proved that 1 for triangles with hight < 0.04, the conjecture is true;
88 The use of computer The AntunesFreitas Conjecture. We proved that 1 for triangles with hight < 0.04, the conjecture is true; 2 for triangles closed enough to the equilateral triangle, the conjecture is true;
89 The use of computer The AntunesFreitas Conjecture. We proved that 1 for triangles with hight < 0.04, the conjecture is true; 2 for triangles closed enough to the equilateral triangle, the conjecture is true; 3 If for any triangle the gap is more than 64π 2 /9, there is a neighborhood such that for any triangle in that neighborhood, the AntunesFreitas Conjecture is true.
90 Thank you!
How to minimize without knowing how to differentiate (in Riemannian geometry)
How to minimize without knowing how to differentiate (in Riemannian geometry) Spiro Karigiannis karigiannis@maths.ox.ac.uk Mathematical Institute, University of Oxford Calibrated Geometries and Special
More informationLecture 3. Mathematical Induction
Lecture 3 Mathematical Induction Induction is a fundamental reasoning process in which general conclusion is based on particular cases It contrasts with deduction, the reasoning process in which conclusion
More informationFrom Random Matrices to Geometry: the "topological recursion"
From Random Matrices to Geometry: the "topological recursion" Bertrand Eynard, IPHT CEA Saclay, CERN Brunel workshop on random matri theory december 17th 1 Contents 1. Random Matrices, statistical properties
More informationWHAT ARE MATHEMATICAL PROOFS AND WHY THEY ARE IMPORTANT?
WHAT ARE MATHEMATICAL PROOFS AND WHY THEY ARE IMPORTANT? introduction Many students seem to have trouble with the notion of a mathematical proof. People that come to a course like Math 216, who certainly
More informationUnderstanding Basic Calculus
Understanding Basic Calculus S.K. Chung Dedicated to all the people who have helped me in my life. i Preface This book is a revised and expanded version of the lecture notes for Basic Calculus and other
More informationHow many numbers there are?
How many numbers there are? RADEK HONZIK Radek Honzik: Charles University, Department of Logic, Celetná 20, Praha 1, 116 42, Czech Republic radek.honzik@ff.cuni.cz Contents 1 What are numbers 2 1.1 Natural
More informationHandout #1: Mathematical Reasoning
Math 101 Rumbos Spring 2010 1 Handout #1: Mathematical Reasoning 1 Propositional Logic A proposition is a mathematical statement that it is either true or false; that is, a statement whose certainty or
More informationMetric Spaces. Chapter 7. 7.1. Metrics
Chapter 7 Metric Spaces A metric space is a set X that has a notion of the distance d(x, y) between every pair of points x, y X. The purpose of this chapter is to introduce metric spaces and give some
More information6 EXTENDING ALGEBRA. 6.0 Introduction. 6.1 The cubic equation. Objectives
6 EXTENDING ALGEBRA Chapter 6 Extending Algebra Objectives After studying this chapter you should understand techniques whereby equations of cubic degree and higher can be solved; be able to factorise
More informationKnots as mathematical objects
Knots as mathematical objects Ulrich Krähmer = Uli U Glasgow Ulrich Krähmer = Uli (U Glasgow) Knots as mathematical objects 1 / 26 Why I decided to speak about knots They are a good example to show that
More informationMirror symmetry: A brief history. Superstring theory replaces particles moving through spacetime with loops moving through spacetime.
Mirror symmetry: A brief history. Superstring theory replaces particles moving through spacetime with loops moving through spacetime. A key prediction of superstring theory is: The universe is 10 dimensional.
More informationAlex, I will take congruent numbers for one million dollars please
Alex, I will take congruent numbers for one million dollars please Jim L. Brown The Ohio State University Columbus, OH 4310 jimlb@math.ohiostate.edu One of the most alluring aspectives of number theory
More informationLesson 13: Proofs in Geometry
211 Lesson 13: Proofs in Geometry Beginning with this lesson and continuing for the next few lessons, we will explore the role of proofs and counterexamples in geometry. To begin, recall the Pythagorean
More informationSouth Carolina College and CareerReady (SCCCR) PreCalculus
South Carolina College and CareerReady (SCCCR) PreCalculus Key Concepts Arithmetic with Polynomials and Rational Expressions PC.AAPR.2 PC.AAPR.3 PC.AAPR.4 PC.AAPR.5 PC.AAPR.6 PC.AAPR.7 Standards Know
More informationDELAWARE MATHEMATICS CONTENT STANDARDS GRADES 910. PAGE(S) WHERE TAUGHT (If submission is not a book, cite appropriate location(s))
Prentice Hall University of Chicago School Mathematics Project: Advanced Algebra 2002 Delaware Mathematics Content Standards (Grades 910) STANDARD #1 Students will develop their ability to SOLVE PROBLEMS
More informationEuclidean Geometry. We start with the idea of an axiomatic system. An axiomatic system has four parts:
Euclidean Geometry Students are often so challenged by the details of Euclidean geometry that they miss the rich structure of the subject. We give an overview of a piece of this structure below. We start
More informationGeometry. Unit 1. Transforming and Congruence. Suggested Time Frame 1 st Six Weeks 22 Days
Geometry Unit 1 Transforming and Congruence Title Suggested Time Frame 1 st Six Weeks 22 Days Big Ideas/Enduring Understandings Module 1 Tools of geometry can be used to solve realworld problems. Variety
More informationAn introduction to number theory and Diophantine equations
An introduction to number theory and Diophantine equations Lillian Pierce April 20, 2010 Lattice points and circles What is the area of a circle of radius r? You may have just thought without hesitation
More information1. A student followed the given steps below to complete a construction. Which type of construction is best represented by the steps given above?
1. A student followed the given steps below to complete a construction. Step 1: Place the compass on one endpoint of the line segment. Step 2: Extend the compass from the chosen endpoint so that the width
More informationNumerical Analysis Lecture Notes
Numerical Analysis Lecture Notes Peter J. Olver 5. Inner Products and Norms The norm of a vector is a measure of its size. Besides the familiar Euclidean norm based on the dot product, there are a number
More informationProblem of the Month The Shape of Things
Problem of the Month The Problems of the Month (POM) are used in a variety of ways to promote problem solving and to foster the first standard of mathematical practice from the Common Core State Standards:
More informationalgebra, geometry, and Schroedinger atoms
algebra, geometry, and Schroedinger atoms PILJIN YI Korea Institute for Advanced Study QUC Inauguration Conference February 2014 AtiyahSinger Index Theorem ~ 1963 CalabiYau ~ 1978 Calibrated Geometry
More informationQuantum Mechanics and Representation Theory
Quantum Mechanics and Representation Theory Peter Woit Columbia University Texas Tech, November 21 2013 Peter Woit (Columbia University) Quantum Mechanics and Representation Theory November 2013 1 / 30
More informationMA257: INTRODUCTION TO NUMBER THEORY LECTURE NOTES
MA257: INTRODUCTION TO NUMBER THEORY LECTURE NOTES 2016 47 4. Diophantine Equations A Diophantine Equation is simply an equation in one or more variables for which integer (or sometimes rational) solutions
More informationTHREE DIMENSIONAL GEOMETRY
Chapter 8 THREE DIMENSIONAL GEOMETRY 8.1 Introduction In this chapter we present a vector algebra approach to three dimensional geometry. The aim is to present standard properties of lines and planes,
More informationUltraproducts and Applications I
Ultraproducts and Applications I Brent Cody Virginia Commonwealth University September 2, 2013 Outline Background of the Hyperreals Filters and Ultrafilters Construction of the Hyperreals The Transfer
More information8.1 Examples, definitions, and basic properties
8 De Rham cohomology Last updated: May 21, 211. 8.1 Examples, definitions, and basic properties A kform ω Ω k (M) is closed if dω =. It is exact if there is a (k 1)form σ Ω k 1 (M) such that dσ = ω.
More informationStudent Name: Teacher: Date: District: MiamiDade County Public Schools. Assessment: 9_12 Mathematics Geometry Exam 1
Student Name: Teacher: Date: District: MiamiDade County Public Schools Assessment: 9_12 Mathematics Geometry Exam 1 Description: GEO Topic 1 Test: Tools of Geometry Form: 201 1. A student followed the
More informationINTRODUCTION TO EUCLID S GEOMETRY
CHAPTER 5 INTRODUCTION TO EUCLID S GEOMETRY (A) Main Concepts and Results Points, Line, Plane or surface, Axiom, Postulate and Theorem, The Elements, Shapes of altars or vedis in ancient India, Equivalent
More informationMaster of Arts in Mathematics
Master of Arts in Mathematics Administrative Unit The program is administered by the Office of Graduate Studies and Research through the Faculty of Mathematics and Mathematics Education, Department of
More informationModule 3 Congruency can be used to solve realworld problems. What happens when you apply more than one transformation to
Transforming and Congruence *CISD Safety Net Standards: G.3C, G.4C Title Big Ideas/Enduring Understandings Module 1 Tools of geometry can be used to solve realworld problems. Variety of representations
More informationPrentice Hall Algebra 2 2011 Correlated to: Colorado P12 Academic Standards for High School Mathematics, Adopted 12/2009
Content Area: Mathematics Grade Level Expectations: High School Standard: Number Sense, Properties, and Operations Understand the structure and properties of our number system. At their most basic level
More informationSECTION 102 Mathematical Induction
73 0 Sequences and Series 6. Approximate e 0. using the first five terms of the series. Compare this approximation with your calculator evaluation of e 0.. 6. Approximate e 0.5 using the first five terms
More informationLAKE ELSINORE UNIFIED SCHOOL DISTRICT
LAKE ELSINORE UNIFIED SCHOOL DISTRICT Title: PLATO Algebra 1Semester 2 Grade Level: 1012 Department: Mathematics Credit: 5 Prerequisite: Letter grade of F and/or N/C in Algebra 1, Semester 2 Course Description:
More informationNonlinear Fourier series and applications to PDE
CNRS/Cergy CIRM Sept 26, 2013 I. Introduction In this talk, we shall describe a new method to analyze Fourier series. The motivation comes from solving nonlinear PDE s. These PDE s are evolution equations,
More informationalgebra, geometry, and Schroedinger atoms
algebra, geometry, and Schroedinger atoms PILJIN YI Korea Institute for Advanced Study USTC, Hefei, May 26 th 2016 what can string theory do, besides its yettoberealized promise of a unified fundamental
More informationSection 13.5 Equations of Lines and Planes
Section 13.5 Equations of Lines and Planes Generalizing Linear Equations One of the main aspects of single variable calculus was approximating graphs of functions by lines  specifically, tangent lines.
More informationMathematics Undergraduate Student Learning Objectives
Mathematics Undergraduate Student Learning Objectives The Mathematics program promotes mathematical skills and knowledge for their intrinsic beauty, effectiveness in developing proficiency in analytical
More informationBetweenness of Points
Math 444/445 Geometry for Teachers Summer 2008 Supplement : Rays, ngles, and etweenness This handout is meant to be read in place of Sections 5.6 5.7 in Venema s text [V]. You should read these pages after
More informationold supersymmetry as new mathematics
old supersymmetry as new mathematics PILJIN YI Korea Institute for Advanced Study with help from Sungjay Lee AtiyahSinger Index Theorem ~ 1963 CalabiYau ~ 1978 Calibrated Geometry ~ 1982 (Harvey & Lawson)
More informationCourse 421: Algebraic Topology Section 1: Topological Spaces
Course 421: Algebraic Topology Section 1: Topological Spaces David R. Wilkins Copyright c David R. Wilkins 1988 2008 Contents 1 Topological Spaces 1 1.1 Continuity and Topological Spaces...............
More informationG C.3 Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.
Performance Assessment Task Circle and Squares Grade 10 This task challenges a student to analyze characteristics of 2 dimensional shapes to develop mathematical arguments about geometric relationships.
More informationAdvanced Higher Mathematics Course Assessment Specification (C747 77)
Advanced Higher Mathematics Course Assessment Specification (C747 77) Valid from August 2015 This edition: April 2016, version 2.4 This specification may be reproduced in whole or in part for educational
More informationWhen Is the Sum of the Measures of the Angles of a Triangle Equal to 180º?
Sum of the Measures of Angles Unit 9 NonEuclidean Geometries When Is the Sum of the Measures of the Angles of a Triangle Equal to 180º? Overview: Objective: This activity illustrates the need for Euclid
More informationTrigonometric Functions and Equations
Contents Trigonometric Functions and Equations Lesson 1 Reasoning with Trigonometric Functions Investigations 1 Proving Trigonometric Identities... 271 2 Sum and Difference Identities... 276 3 Extending
More informationRICCI SUMMER SCHOOL COURSE PLANS AND BACKGROUND READING LISTS
RICCI SUMMER SCHOOL COURSE PLANS AND BACKGROUND READING LISTS The Summer School consists of four courses. Each course is made up of four 1hour lectures. The titles and provisional outlines are provided
More informationClassical theorems on hyperbolic triangles from a projective point of view
tmcsszilasi 2012/3/1 0:14 page 175 #1 10/1 (2012), 175 181 Classical theorems on hyperbolic triangles from a projective point of view Zoltán Szilasi Abstract. Using the CayleyKlein model of hyperbolic
More informationMath Real Analysis I
Math 431  Real Analysis I Solutions to Homework due October 1 In class, we learned of the concept of an open cover of a set S R n as a collection F of open sets such that S A. A F We used this concept
More informationIntroduction to Topology
Introduction to Topology Tomoo Matsumura November 30, 2010 Contents 1 Topological spaces 3 1.1 Basis of a Topology......................................... 3 1.2 Comparing Topologies.......................................
More informationPrerequsites: Math 1A1B, 53 (lower division calculus courses)
Math 151 Prerequsites: Math 1A1B, 53 (lower division calculus courses) Development of the rational number system. Use the number line (real line), starting with the concept of parts of a whole : fractions,
More informationThe following is Dirac's talk on projective geometry in physics at Boston University on October 30, 1972.
Audio recording made by John B. Hart, Boston University, October 30, 1972 Comments by Roger Penrose [JOHN HART] The following is Dirac's talk on projective geometry in physics at Boston University on October
More informationRIGIDITY OF HOLOMORPHIC MAPS BETWEEN FIBER SPACES
RIGIDITY OF HOLOMORPHIC MAPS BETWEEN FIBER SPACES GAUTAM BHARALI AND INDRANIL BISWAS Abstract. In the study of holomorphic maps, the term rigidity refers to certain types of results that give us very specific
More informationThe Mathematics of Origami
The Mathematics of Origami Sheri Yin June 3, 2009 1 Contents 1 Introduction 3 2 Some Basics in Abstract Algebra 4 2.1 Groups................................. 4 2.2 Ring..................................
More informationDecember 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B. KITCHENS
December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B KITCHENS The equation 1 Lines in twodimensional space (1) 2x y = 3 describes a line in twodimensional space The coefficients of x and y in the equation
More informationANALYTICITY OF SETS ASSOCIATED TO LELONG NUMBERS AND THE EXTENSION OF MEROMORPHIC MAPS 1
BULLETIN OF THE AMERICAN MATHEMATICAL SOCIETY Volume 79, Number 6, November 1973 ANALYTICITY OF SETS ASSOCIATED TO LELONG NUMBERS AND THE EXTENSION OF MEROMORPHIC MAPS 1 BY YUMTONG SIU 2 Communicated
More informationSection 1.1. Introduction to R n
The Calculus of Functions of Several Variables Section. Introduction to R n Calculus is the study of functional relationships and how related quantities change with each other. In your first exposure to
More informationNot Even Wrong, ten years later: a view from mathematics February on prospects 3, 2016for fundamenta 1 / 32. physics without experiment
Not Even Wrong, ten years later: a view from mathematics on prospects for fundamental physics without experiment Peter Woit Columbia University Rutgers Physics Colloquium, February 3, 2016 Not Even Wrong,
More informationA Foundation for Geometry
MODULE 4 A Foundation for Geometry There is no royal road to geometry Euclid 1. Points and Lines We are ready (finally!) to talk about geometry. Our first task is to say something about points and lines.
More informationLinear Systems and Gaussian Elimination
Eivind Eriksen Linear Systems and Gaussian Elimination September 2, 2011 BI Norwegian Business School Contents 1 Linear Systems................................................ 1 1.1 Linear Equations...........................................
More informationGeometry 2: Remedial topology
Geometry 2: Remedial topology Rules: You may choose to solve only hard exercises (marked with!, * and **) or ordinary ones (marked with! or unmarked), or both, if you want to have extra problems. To have
More informationDiablo Valley College Catalog 20142015
Mathematics MATH Michael Norris, Interim Dean Math and Computer Science Division Math Building, Room 267 Possible career opportunities Mathematicians work in a variety of fields, among them statistics,
More informationClassical Physics Prof. V. Balakrishnan Department of Physics Indian Institution of Technology, Madras. Lecture No. # 13
Classical Physics Prof. V. Balakrishnan Department of Physics Indian Institution of Technology, Madras Lecture No. # 13 Now, let me formalize the idea of symmetry, what I mean by symmetry, what we mean
More informationMathematics Program Description Associate in Arts Degree Program Outcomes Required Courses............................. Units
Program Description Successful completion of this maj will assure competence in mathematics through differential and integral calculus, providing an adequate background f employment in many technological
More information7. The GaussBonnet theorem
7. The GaussBonnet theorem 7. Hyperbolic polygons In Euclidean geometry, an nsided polygon is a subset of the Euclidean plane bounded by n straight lines. Thus the edges of a Euclidean polygon are formed
More informationAlgebraic Geometry. Andreas Gathmann. Notes for a class. taught at the University of Kaiserslautern 2002/2003
Algebraic Geometry Andreas Gathmann Notes for a class taught at the University of Kaiserslautern 2002/2003 CONTENTS 0. Introduction 1 0.1. What is algebraic geometry? 1 0.2. Exercises 6 1. Affine varieties
More informationMetric Spaces. Chapter 1
Chapter 1 Metric Spaces Many of the arguments you have seen in several variable calculus are almost identical to the corresponding arguments in one variable calculus, especially arguments concerning convergence
More informationStudievereniging A Eskwadraat. Druk. Jaargang 09/10 Nummer 1
Studievereniging A Eskwadraat Druk Jaargang 09/10 Nummer 1 VAKidioot Wiskunde VAK Spaces becoming noncommutative Door: Jorge Plazas Noncommutative geometry has developed over the last three decades building
More informationMathematics (MAT) MAT 061 Basic Euclidean Geometry 3 Hours. MAT 051 PreAlgebra 4 Hours
MAT 051 PreAlgebra Mathematics (MAT) MAT 051 is designed as a review of the basic operations of arithmetic and an introduction to algebra. The student must earn a grade of C or in order to enroll in MAT
More informationMATH HISTORY: POSSIBLE TOPICS FOR TERM PAPERS
MATH HISTORY: POSSIBLE TOPICS FOR TERM PAPERS Some possible seeds for historical developmental topics are: The Platonic Solids Solution of nth degree polynomial equation (especially quadratics) Difference
More informationMA651 Topology. Lecture 6. Separation Axioms.
MA651 Topology. Lecture 6. Separation Axioms. This text is based on the following books: Fundamental concepts of topology by Peter O Neil Elements of Mathematics: General Topology by Nicolas Bourbaki Counterexamples
More informationVenn Symmetry and Prime Numbers: A Seductive Proof Revisited
Venn Symmetry and Prime Numbers: A Seductive Proof Revisited Stan Wagon and Peter Webb When he was an undergraduate at Swarthmore College in 1960, David W. Henderson [4] discovered an important and surprising
More informationalternate interior angles
alternate interior angles two nonadjacent angles that lie on the opposite sides of a transversal between two lines that the transversal intersects (a description of the location of the angles); alternate
More informationMATH. ALGEBRA I HONORS 9 th Grade 12003200 ALGEBRA I HONORS
* Students who scored a Level 3 or above on the Florida Assessment Test Math Florida Standards (FSAMAFS) are strongly encouraged to make Advanced Placement and/or dual enrollment courses their first choices
More informationMathematics Course 111: Algebra I Part IV: Vector Spaces
Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 19967 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are
More informationMAT2400 Analysis I. A brief introduction to proofs, sets, and functions
MAT2400 Analysis I A brief introduction to proofs, sets, and functions In Analysis I there is a lot of manipulations with sets and functions. It is probably also the first course where you have to take
More informationQuantum manifolds. Manuel Hohmann
Quantum manifolds Manuel Hohmann Zentrum für Mathematische Physik und II. Institut für Theoretische Physik, Universität Hamburg, Luruper Chaussee 149, 22761 Hamburg, Germany Abstract. We propose a mathematical
More informationDiscrete Mathematics
Slides for Part IA CST 2014/15 Discrete Mathematics Prof Marcelo Fiore Marcelo.Fiore@cl.cam.ac.uk What are we up to? Learn to read and write, and also work with, mathematical
More informationTriangle Congruence and Similarity: A CommonCoreCompatible Approach
Triangle Congruence and Similarity: A CommonCoreCompatible Approach The Common Core State Standards for Mathematics (CCSSM) include a fundamental change in the geometry curriculum in grades 8 to 10:
More informationTExES Mathematics 7 12 (235) Test at a Glance
TExES Mathematics 7 12 (235) Test at a Glance See the test preparation manual for complete information about the test along with sample questions, study tips and preparation resources. Test Name Mathematics
More informationChapter 1  Matrices & Determinants
Chapter 1  Matrices & Determinants Arthur Cayley (August 16, 1821  January 26, 1895) was a British Mathematician and Founder of the Modern British School of Pure Mathematics. As a child, Cayley enjoyed
More informationLecture 4 DISCRETE SUBGROUPS OF THE ISOMETRY GROUP OF THE PLANE AND TILINGS
1 Lecture 4 DISCRETE SUBGROUPS OF THE ISOMETRY GROUP OF THE PLANE AND TILINGS This lecture, just as the previous one, deals with a classification of objects, the original interest in which was perhaps
More informationarxiv: v1 [math.mg] 6 May 2014
DEFINING RELATIONS FOR REFLECTIONS. I OLEG VIRO arxiv:1405.1460v1 [math.mg] 6 May 2014 Stony Brook University, NY, USA; PDMI, St. Petersburg, Russia Abstract. An idea to present a classical Lie group of
More informationDoug Ravenel. October 15, 2008
Doug Ravenel University of Rochester October 15, 2008 s about Euclid s Some s about primes that every mathematician should know (Euclid, 300 BC) There are infinitely numbers. is very elementary, and we
More informationLiouville Quantum gravity and KPZ. Scott Sheffield
Liouville Quantum gravity and KPZ Scott Sheffield Scaling limits of random planar maps Central mathematical puzzle: Show that the scaling limit of some kind of discrete quantum gravity (perhaps decorated
More informationCHAPTER 1 BASIC TOPOLOGY
CHAPTER 1 BASIC TOPOLOGY Topology, sometimes referred to as the mathematics of continuity, or rubber sheet geometry, or the theory of abstract topological spaces, is all of these, but, above all, it is
More informationEuclidean Geometry: An Introduction to Mathematical Work
Euclidean Geometry: An Introduction to Mathematical Work Math 3600 Fall 2016 Introduction We have two goals this semester: First and foremost, we shall learn to do mathematics independently. Also, we shall
More informationTHE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS
THE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS KEITH CONRAD 1. Introduction The Fundamental Theorem of Algebra says every nonconstant polynomial with complex coefficients can be factored into linear
More informationMathematics Review for MS Finance Students
Mathematics Review for MS Finance Students Anthony M. Marino Department of Finance and Business Economics Marshall School of Business Lecture 1: Introductory Material Sets The Real Number System Functions,
More informationComplex Function Theory. Second Edition. Donald Sarason >AMS AMERICAN MATHEMATICAL SOCIETY
Complex Function Theory Second Edition Donald Sarason >AMS AMERICAN MATHEMATICAL SOCIETY Contents Preface to the Second Edition Preface to the First Edition ix xi Chapter I. Complex Numbers 1 1.1. Definition
More information3. INNER PRODUCT SPACES
. INNER PRODUCT SPACES.. Definition So far we have studied abstract vector spaces. These are a generalisation of the geometric spaces R and R. But these have more structure than just that of a vector space.
More informationAllocation of Mathematics Modules at Maynooth to Areas of Study for PME (Professional Masters in Education)
Allocation of Mathematics Modules at Maynooth to Areas of Study for PME (Professional Masters in Education) Module Code Module Title Credits Area 1 Area 2 MT101 Differential Calculus In One Real Variable
More informationBasic Concepts of Point Set Topology Notes for OU course Math 4853 Spring 2011
Basic Concepts of Point Set Topology Notes for OU course Math 4853 Spring 2011 A. Miller 1. Introduction. The definitions of metric space and topological space were developed in the early 1900 s, largely
More informationMCS 563 Spring 2014 Analytic Symbolic Computation Wednesday 15 January. Elimination Methods
Elimination Methods In this lecture we define projective coordinates, give another application, explain the Sylvester resultant method, illustrate how to cascade resultants to compute discriminant, and
More informationThere is no royal road to geometry (Euclid)
THE CRAFOORD PRIZE IN MATEMATICS 2016 POPULAR SCIENCE BACKGROUND There is no royal road to geometry (Euclid) Yakov Eliashberg is one of the leading mathematicians of our time. For more than thirty years
More informationGrade 7/8 Math Circles Greek Constructions  Solutions October 6/7, 2015
Faculty of Mathematics Waterloo, Ontario N2L 3G1 Centre for Education in Mathematics and Computing Grade 7/8 Math Circles Greek Constructions  Solutions October 6/7, 2015 Mathematics Without Numbers The
More informationCHAPTER II THE LIMIT OF A SEQUENCE OF NUMBERS DEFINITION OF THE NUMBER e.
CHAPTER II THE LIMIT OF A SEQUENCE OF NUMBERS DEFINITION OF THE NUMBER e. This chapter contains the beginnings of the most important, and probably the most subtle, notion in mathematical analysis, i.e.,
More informationDEGREE OF NEGATION OF AN AXIOM
DEGREE OF NEGATION OF AN AXIOM Florentin Smarandache, Ph D Professor of Mathematics Chair of Department of Math & Sciences University of New Mexico 200 College Road Gallup, NM 87301, USA Email: smarand@unm.edu
More informationvertex, 369 disjoint pairwise, 395 disjoint sets, 236 disjunction, 33, 36 distributive laws
Index absolute value, 135 141 additive identity, 254 additive inverse, 254 aleph, 466 algebra of sets, 245, 278 antisymmetric relation, 387 arcsine function, 349 arithmetic sequence, 208 arrow diagram,
More informationDirect Proofs. CS 19: Discrete Mathematics. Direct Proof: Example. Indirect Proof: Example. Proofs by Contradiction and by Mathematical Induction
Direct Proofs CS 19: Discrete Mathematics Amit Chakrabarti Proofs by Contradiction and by Mathematical Induction At this point, we have seen a few examples of mathematical proofs. These have the following
More informationEuclid s 5 th Axiom (on the plane):
Euclid s 5 th Axiom (on plane): That, if a straight line falling on two straight lines makes interior angles on same side less than two right angles, two straight lines, if produced indefinitely, meet
More information