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Mechanical Engineering and Mechanics
Subhrajit Bhattacharya

Teaching >>
Ideas in Mathematics (Math 170-002, Fall 2014, UPenn)
Aug 20, 2014


Topics from among the following: logic, sets, topology, graph theory, geometry, and their relevance to contemporary science and society.

Hours and Location

Lectures: Monday, Wednesday and Friday, 9-10am. Location: DRL room A5. (Total number of lectures: approx. 35)
Recitations: Tuesday and Thursday (please check your respective schedules for the exact hours and locations).

Important Dates

Lectures start on Aug 27 (Wednesday). Last lecture on Dec 8 (Monday). (Review session on Dec 10)
List of days without lectures: Sept 1 (M), Oct 10 (F), Oct 24 (F), Nov 28 (F)
First mid-term: September 29 (Monday), during the usual lecture hour.
Second mid-term: November 3 (Monday), during the usual lecture hour.
Final exam: December 15 (Monday), 9-11am (http://www.upenn.edu/registrar/finals/index.html)

Instructor

Subhrajit Bhattacharya
E-mail: subhrabh@math.upenn.edu
Web-site: http://www.math.upenn.edu/~subhrabh
Office location: DRL 3C7
Office hours: Fridays 1-2pm (or by appointment)

Teaching Assistant

Matthew Tai
Web-site: http://www.math.upenn.edu/~mtai/
Office hours: Mondays and Tuesdays 2-3pm

Textbooks

  • "Book of Proofs" (2nd Edition) by Richard Hammack.
  • "The Heart of Mathematics: An invitation to effective thinking" (4th Edition) by Edward B. Burger and Michael Starbird. (Chapters 2-6)

Attendance, Homeworks and Exams

  • Attendance and participation in class is expected.
  • There will be weekly homeworks assigned on Wednesdays, and due on the following Wednesday at the beginning of class. You can discuss among yourselves on the homework problems. But each of your homework submission must be your original work.
  • There will be a few surprise quizzes, either during the lecture hours or during the recitation hours.
  • There will be three exams: two mid-terms and one final. The first mid-term will tentatively take place around end of September, and the second mid-term around the end of October.
  • The homeworks and announcements will be posted on the canvas site. So make sure you are signed up there and regularly check the announcements: https://canvas.upenn.edu/courses/1255759

Grading Scheme

Class participation: 5%
Homeworks and Quizzes: 25%
Midterm 1: 20%
Midterm 2: 20%
Finals: 30%


[ + ]   Detailed List of Topics Covered by Date.
Day Topics covered
Aug 27 "Structures" - The primary subject of interest in mathematics; Sets: Definitions, list notation, set-builder notation, examples, some standard sets (natural numbers, integers, real numbers); Cardinality of a set; The concept of infinity; Subsets. (Ref: sections 1.1, 1.3 of Hammack)
Aug 29 Set operations (union, intersection, difference); Indexed sets (and their union/intersection); Venn diagram; Universal set and complement. (Ref: sections 1.5, 1.6, 1.7, 1.8 of Hammack)
Sept 3 Cartesian product of sets; Practice/example problems; Introduction to maps between sets. (Ref: section 1.2 of Hammack)
Sept 5 Logic: Statements, conditional statements and mathematical implications; Converse statement; Contrapositive statement; Examples. (Ref: sections 2.1, 2.3, 2.4 of Hammack)
Sept 8 Logic: AND, OR, XOR, NOT and Truth tables; Different types of statements (definition, theorem, proposition); Introduction to direct proof with an example. (Ref: sections 2.2, 2.5, 2.6, 2.8, 2.10, 4.2, 4.3 of Hammack)
Sept 10 Methods of proof I -- Direct proof, contrapositive proof; Examples of proofs; Introduction to prime numbers. (Ref: the examples from chapter 4 and section 5.1 of Hammack)
Sept 12 Methods of proof II -- More contrapositive proofs, proof by contradiction, proof for "if and only if" statements, proof by induction. (Ref: the examples from sections 5.1, 6.1, 6.2, 6.4, 7.1, 10.0, 10.3 of Hammack)
Sept 15 Fibonacci sequence; More proof by induction; Russell's paradox. (Ref: Sections 1.10, 10.0, 10.3 of Hammack)
Sept 17 Euclid's proof for infinitely many primes; Introduction to numbers: From natural numbers to integers -- completion of a structure. (Ref: Section 6.1 of Hammack, class lecture)
Sept 19 Natural numbers, Integers, Modular arithmetic. (Ref: Section 2.4 of Burger and Starbird)
Sept 22 Modular arithmetic; Fundamental Theorem of Arithmetic; Prime numbers (Coprime numbers, Prime number theorem, Fermat's Little theorem, general discussion on large prime numbers and their application in cryptography). (Ref: Sections 2.3, 2.4, 2.5 of Burger and Starbird)
Sept 24 Rational numbers, irrational numbers, proof of irrationality of , algebraic numbers, transcendental numbers. (Ref: Sections 2.6, 2.7 of Burger & Starbird, and class lecture)
Sept 26 Review of natural numbers, integers, rational numbers, algebraic numbers, real numbers, transcendental and irrational numbers; Decimal representation of rational and irrational numbers, and how to distinguish between them; Introduction to cardinality of infinite sets -- establishing one-to-one correspondences; proof that natural numbers and integers have the same cardinality, . (Ref: Sections 2.6, 2.7, 3.1 of Burger & Starbird, and class lecture)
Sept 29 Midterm 1
Oct 1 Cardinality of and that of the rational numbers, using the "spiral" argument. Brief discussion on the cardinality of algebraic numbers. Cantor's diagonal argument as proof that cardinality of the set of real numbers is strictly greater than . Defined . Power set of a set. Mention that is the cardinality of the power set of , thus relating and . (Ref: Whole of chapter 3 of Burger & Starbird, and class lecture)
Oct 3 Cardinality of closed and open line segments; Introduction to complex numbers - definition, addition, multiplication and division of complex numbers; The complex plane, ; Mentioned the Euler's formula. (Ref: class lecture, supplementary note on complex numbers: complex_numbers.pdf)
Oct 6 Complex numbers (Fundamental Theorem of Algebra, a few additional remarks on Euler's formula); Topology -- stretching and squeezing of spaces, "topological equivalence" or "homeomorphism", examples of topological equivalence (sphere, torus, punctured sphere, punctured torus). (Ref: Class notes, supplementary notes on Cmplex numbers: complex_numbers.pdf, Section 5.1 of Burger & Starbird)
Oct 8 Topology: Topological equivalents of punctured torus; torus of genus 2 and higher; Manifolds and their dimension (with examples of manifolds and examples of spaces that are not manifolds); Cuts -- cutting a torus into a flat square; Glueing: glueing opposite edges of a square -- construction of a cylinder, a torus and a Mobius band. (Ref: Class notes, supplementary notes on Complex numbers: complex_numbers.pdf, Sections 5.1, 5.2 of Burger & Starbird)
Oct 13 Mobius band and some of its properties; Cutting the Mobius band along its central line; Klein bottle, its construction and some properties; Construction of spheres by gluing boundary of disks; First Betti number of 2-dimensional manifolds. (Ref: Class lectures, supplementary notes on topology: topology.pdf, and Sections 5.1, 5.2 of Burger & Starbird)
Oct 15 Introduction to three and higher dimensional spaces: Euclidean spaces of dimensions 2, 3, 4; Spheres of different dimensions as subsets of Euclidean spaces; Simply-connected spaces (with examples); Statement of Poincare conjecture and some historic notes. (Ref: Class lectures, supplementary notes on topology: topology.pdf, and Sections 5.1, 5.2 of Burger & Starbird)
Oct 17 Review of the definition of spheres of different dimension; Simply-connected spaces (with examples); Statement of Poincare conjecture and some historic notes; Embedding -- a topological space "sitting inside" another; The intuition behind topological equivalence between two different embeddings of figure 8 in , different embeddings of the cylinder in , and different embeddings of cylinder in (a.k.a., knots); Introduction to different types of knots -- unknot, trefoil knot, figure-8 knot, square knot, granny knot. (Ref: Class lectures, supplementary notes on topology: topology.pdf, and Sections 5.1, 5.2, 5.5 of Burger & Starbird)
Oct 20 Review of different types of knots; Knot equivalence; Reidemeister moves; in-class examples of knot equivalences; knot sum; Introduction to graphs (definition). (Ref: Class lectures, supplementary notes on topology: topology.pdf, and Sections 5.3, 5.5 of Burger & Starbird)
Oct 22 Review of the definition of a graph (vertex and edge sets), with examples of application; Definitions: Degree, Neighbors, Path, Circuit, Euler circuit; Euler circuit theorem; Representation of a "map" using a graph, the map/graph coloring problem, the Four color theorem; Counting the number of "faces" created by a graph embedded on a plane, Euler characteristic, Euler Characteristic Theorem for a graph on a pane (with the exterior counted as a face). (Ref: Class lectures, supplementary notes on topology: topology.pdf, and Sections 5.3, 5.4 of Burger & Starbird)
Oct 24 Review of Euler Characteristic Theorem; The Four color theorem; Putting the graph on surface of a sphere and re-interpretation of Euler Characteristic Theorem; The Platonic solids and the Euler Characteristic; Classification of Platonic solids -- there are only 5 platonic solids -- and the proof; (Ref: Class lectures, supplementary notes on topology: topology.pdf, and chapter 5 of Burger & Starbird)
Oct 27 Classification of Platonic solids -- there are only 5 platonic solids -- and the proof of that; Duals of Platonic solids; Distances -- Euclidean distance, Manhattan distance; (Ref: Class lectures, and chapters 5 & 4 of Burger & Starbird)
Oct 29 A proof of Pythagorean theorem; Euclidean distance in and in ; Distance on the surface of the cube -- development of the cube by cutting it; Geodesic distance on the surface of a sphere. (Ref: Class lectures, and chapter 4 of Burger & Starbird)
Oct 31 Geodesic distance -- distances on platonic solids (using development), and distances on the surface of a sphere (using computation of angle subtended at center). (Ref: Class lectures, and chapter 4 of Burger & Starbird)
Nov 3 Midterm 2
Nov 5 Geodesic distance computation on platonic solids and the surface of a sphere, example; "Triangles" on the surface of sphere and hyperbolic surfaces (saddles) -- their angles not summing up to 180 degrees; (Ref: Class lectures, and chapter 4 of Burger & Starbird)
Nov 7 Geodesic distance computation on the surface of a sphere -- completed an example; Introduction to Fibonacci sequence (Ref: Class lectures, and chapter 4 of Burger & Starbird)
Nov 10 Fibonacci sequence; Golden ratio; Golden rectangle; Introduction to symmetries (Ref: Class lectures, and chapter 4 of Burger & Starbird)
Nov 12 Symmetries of the equilateral triangle -- reflection and rotation -- introduction to the notations; Composition of symmetries; Identity transformation; Inverse of symmetries; Identifying equivalence of two symmetries; Examples involving the equilateral triangle and the square. (Ref: Class lectures, chapter 4 of Burger & Starbird, and supplementary notes on symmetry: symmetry.pdf)
Nov 14 More examples of rotational and reflection symmetries; Translational symmetry -- tiling of the pane. (Ref: Class lectures, chapter 4 of Burger & Starbird, and supplementary notes on symmetry: symmetry.pdf)
Nov 17 Tiling of the plane -- Tiling using square tiles and equilateral triangle tiles, and their symmetries (rotation, reflection, translation, "scale symmetry"); Pinwheel tiling and its properties ("scale symmetry", but no translation symmetry); Penrose tiling (using two rhombus tiles). (Ref: Class lectures, chapter 4 of Burger & Starbird)
Nov 19 Finished Penrose tiling; Motivation for fractals -- "scale symmetry" with fractional scaling; Construction of Koch curve, Sierpinski triangle -- explicit description of all transformations involved; Self-similarity (scale) property of fractals. (Ref: chapter 4 & 7 (4 & 6 if you are using the 3rd edition) from the Burger and Starbird book)
Nov 21 Revisit construction and description of the Koch curve & Sierpinski triangle; Independence on choice of initial figure for construction of fractals; Description of a fractal fern (Barnsley fern); Introduction to (an informal approach to) computation of Box dimension. (Ref: chapter 7 (6 if you are using the 3rd edition) from the Burger and Starbird book)
Nov 24 Dimension of a figure -- an informal way of computing box dimension: Ratio of d-dimensional "volume" of similar figures and its relation to ratio of corresponding length elements in the figures, additivity of d-dimensional volumes, equation for computing dimension and logarithm; Computation of dimension of a planar object (a triangle and a square) and a solid object (a tetrahedron); Computation of dimensions of fractals: Koch curve & Sierpinski triangle. (Ref: chapter 7 (6 if you are using the 3rd edition) from the Burger and Starbird book)
Nov 26 An alternative way of creating fractals -- Basic idea of a dynamical system, convergence & divergence; Dynamical system constituting of functions from the plane to the plane -- construction of the Julia set (as boundary between the basin of convergence and that of divergence), couple of examples of connected and disconnected Julia sets; Mandelbrot set; Introduction to counting -- number of ways objects can be placed in different bins, introduced factorial; Definition of probability. (Ref: chapter 7 & 8 (6 & 7 if you are using the 3rd edition) from the Burger and Starbird book and Chapter 3 from Hammack's book.)
Dec 1 Introduction to counting: The "multiplication principle", Number of ways different things can be placed in different bins, introduced factorial; Introduction to probability: Definition of probability of an event, Probability that a particular or a particular type of placement of different things in different bins will happen; Number of ways different things can be placed in different bins (with no bins allowing more than one object, ); Number of ways different things can be placed in different bins (with bins allowing more than one object); Number of ways identical things can be placed in different bins (with no bins allowing more than one object, ) -- introduced Binomial coefficients. (Ref: chapter 7 & 8 (6 & 7 if you are using the 3rd edition) from the Burger and Starbird book and Chapter 3 from Hammack's book.)
Dec 3 Review of counting; Pascal's triangle; Example problems on counting.
Dec 5 Example/practice problems on counting; Elementary probability theory.
Dec 8 Review for final exam.

Course Outline

Elementary Set theory (2-3 lectures)
- Set theory.
- Venn diagram.

Logic and Methods of proof (3-4 lectures)
- Logic.
- Statements in mathematics.
- Methods of proof.

Numbers (4-5 lectures)
- Types of numbers, number theory.
- Cardinality.
- Introduction to complex numbers.

Elementary Topology (5-6 lectures)
- Concepts and examples from topology.
- Topological invariants (Euler's characteristics).
- Three and higher dimensional spaces.

Elementary graph theory (4-5 lectures)
- Definition and description of graphs.
- Elementary computations.
- Elementary graph algorithms.

Geometry - Distances (4-5 lectures)
- Introduction to Metric and their various types.
- Introduction to higher dimensional measures (area, volume).

Geometry - Symmetry and Measures (4-5 lectures)
- Symmetry, Platonic solids.
- Fractals.

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