Lehigh University logo
Lehigh University logo
Mechanical Engineering and Mechanics
Subhrajit Bhattacharya

Teaching >>
Ideas in Mathematics (Math 170-001, Fall 2015, UPenn)
Aug 16, 2015

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

Hours and Location

Lectures: Tuesdays and Thursdays, 10:30am - noon (1.5 hours each).
Lecture Location: DRL room A2.
Recitations: Mondays & Wednesdays, 8am-9am & 9am-10am (please check your respective schedules for the exact hours and locations).

Important Dates

  • Lectures start on Aug 27. Last lecture on Dec 8. (Total number of lectures: approx. 23)
  • List of days without lectures: Oct 8 (Fall Break), Nov 26 (Thanksgiving)
  • Drop Period ends: Oct 9. Last day to withdraw from a course: Nov 6.
  • First mid-term: Oct 1 (Thursday), 10:30am. (It will be a ~45min exam starting at 10:30am during the usual lecture time and location. The exam will be followed by usual lecture for the remaining time of the class.)
  • Second mid-term: Oct 29 (Thursday), 10:30am. (It will be a ~45min exam starting at 10:30am during the usual lecture time and location. The exam will be followed by usual lecture for the remaining time of the class.)
  • Final exam: Dec 15, 9-11am (See http://www.upenn.edu/registrar/finals/index.html for details.)


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

Amelie Dougherty
E-mail: See canvas page
Office location: See canvas page
Office hours: Monday 5:30-6:30, Wednesday 11am-12pm


Attendance, Homeworks and Exams

  • Attendance and participation in class is expected. You are expected to attend every lecture and recitation. In case you miss a lecture, you will need to figure out what was covered in that class either by talking to your fellow students, or by stopping by my office hours. It is your responsibility that you do these to catch up with the class.
  • There will be weekly homeworks assigned on Wednesdays, and due on the following week's Thursdays 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/1293744
  • Late homework submissions will NOT be accepted. This is so that we can ensure a smooth flow in the grading of the homeworks (avoid backlogs) and to ensure uniformity and fairness. Likewise, if you miss a surprise quiz, NO make-up quiz will be allowed. Exams dates will be announced well in advance. So, if you miss an exam, NO make-up exam will be allowed either.
  • Conditions for exceptions on late submissions and make-up quiz/exams: Late submission of homeworks will be allowed only if you obtain and send an official course absence report (on medical grounds or otherwise) from the university in advance, reporting your absence during the period in which the homework was assigned. For make-up quiz or make-up exam (on serious medical grounds or emergencies), in addition to sending an official course absence report, you'll also need to have an email sent to me by your academic advisor confirming your valid/serious reason of absence. No other form of letter/certificate will be accepted.

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, rational numbers, real numbers); Cardinality of a set; Infinite sets (countably infinite, uncountably infinite); The concept of infinity; Subsets; Set operations (union, difference); (Ref: sections 1.1, 1.3, 1.5 of Hammack)
Sept 1 Set operations (union, intersection, difference); Indexed sets (and their union/intersection); Venn diagram; Universal set and complement; Power set; Cartesian product. (Ref: Chapter 1 of Hammack)
Sept 3 More on Cartesian product of sets; Euclidean spaces of dimension 1, 2, 3, 4 and higher; Practice/example problems; Logic: Statements, conditional statements and mathematical implications; Converse statement; Contrapositive statement; Examples. (Ref: sections 2.1, 2.3, 2.4 of Hammack)
Sept 8 (Lecture by Dr. Emanuel Lazar) Representing numbers, different number bases, practice problems.
Sept 10 (Lecture by Dr. Emanuel Lazar) Multiplication using Roman Numerals, addition in other bases, examples in base 5. Adding two numbers in an arbitrary basis (base 2 through 10). Definitions in mathematics and how to use them in a proof. Definition of even/odd numbers as numbers that could be written as n = 2k and n = 2k+1, respectively, for some integer k. Prove that if n is odd, then so is n^2, or that for even/odd n, n^2+n is even.
Sept 15 Logic - Review of statements, converse of a statement, contrapositive of a statement; Logical operations - AND, OR, NOT, XOR; Truth tables - using truth tables to show if two statements are equivalent; Definitions (divisibility, prime numbers); Methods of proof - direct proof, contrapositive proof, examples. (Ref: Chapters 4 and 5 of Hammack)
Sept 17 Methods of proof - contrapositive proof, proof by contradiction, examples. Euclid's proof of infinitely many primes. Proof by induction (examples), introduced the Fibonacci sequence. (Ref: Chapters 5, 6 and 10 of Hammack)
Sept 22 Examples of proof by induction. If-and-only-if proofs (examples). Counting (number of ways n distinct objects can be placed in n bins -- allowing / not allowing more than one object in a bin). (Ref: Chapters 10 and 3 of Hammack)
Sept 24 Counting: Number of ways m distinct objects can be placed in n bins -- allowing / not allowing more than one object in a bin. Number of ways m identical objects can be placed in n bins (not allowing more than one object in a bin) -- "n choose m". Introduction to Binomial Coefficients. Example problems. Proof for the formula ($\left( \begin{array}{c} n \\ k \end{array}\right) = \left( \begin{array}{c} n-1 \\ k \end{array}\right) + \left( \begin{array}{c} n-1 \\ k-1 \end{array}\right) $). Introduction to Pascal's triangle.
Sept 29 Pascal's triangle: Description, properties. Terms in the n-th row of Pascal's triangle are coefficients in the expansion of ($(a+b)^n$). Numbers: Natural numbers; completion of the addition operation giving integers; additive inverse, additive identity; Modular/clock arithmetic: notations, basic properties/formulae, example problems. (Ref: Chapter 3 of Hammack, Notes on Numbers.)
Oct 1 Midterm 1; Prime numbers, prime counting function, prime number theorem, Fermat's little theorem, Chinese remainder theorem -- Problems involving these; (Ref: Class notes, Notes on Numbers.)
Oct 6 Review of theorems discussed in previous class, problems involving those; Coprime numbers; General discussion on prime numbers -- open problems (twin prime conjecture, Goldbach's conjecture), application of prime numbers to cryptography (SSL/RSA); Numbers: Rational numbers, irrational numbers (including proof that square root of 2 is irrational), algebraic numbers, transcendental numbers. (Ref: Class notes, Notes on Numbers.)
Oct 13 Quick review of rational, irrational, algebraic, transcendental numbers; Rational numbers: Remarks on multiplicative identity and multiplicative inverse in context of integers and rational numbers, decimal representation of rational numbers -- conversion to and from; Introduction to complex numbers (requirement of ($i$) in order to write solutions to some polynomial equations, addition & multiplication of complex numbers, fundamental theorem of algebra); Cardinality -- comparing cardinality of two sets by establishing one-to-one correspondence between elements in the set (due to Georg Cantor); Cardinality of natural numbers, the set of natural numbers greater than 1, the set of even natural numbers, the set of integers -- all equal to cardinality of countably infinite sets, ($\aleph_0$) (Ref: Class notes, Notes on Numbers, Video notes on cardinality.)
Oct 15 Reviewed discussion on cardinality from previous class, cardinality of ($\mathbb{Z}\times\mathbb{Z}$) and ($\mathbb{Q}$) using the "spiral argument" -- both equal to ($\aleph_0$), brief discussion on the cardinality of algebraic numbers; Cardinality of the continuum, ($\aleph_1$) -- the cardinality of real numbers; Cantor's diagonal argument to prove that the cardinality of reals is strictly greater than ($\aleph_0$); Brief discussion on the relationship ($\aleph_1 = |\mathscr{P}(\mathbb{N})| = 2^{\aleph_0}$), and higher order infinities; Introduction to topology (rubber-sheet geometry) -- idea of topological equivalence (homeomorphism), examples (1-dimensional: circle, line segment; 2-dimensional: sphere, torus, sphere with a puncture, torus with a puncture). (Ref: Class notes, Notes on Numbers, Video notes on cardinality.)
Oct 20 Topology / "rubber-sheet geometry": Topological equivalences, examples; Manifolds; Cutting (torus example); First Betti number (as the maximum number of cuts); Gluing (cylinder, torus, Mobius band). (Ref: Class notes, Notes on Topology)
Oct 22 Topology: Manifolds and their dimension (with examples of manifolds and examples of spaces that are not manifolds); Cuts -- cutting a torus into a flat square; Developed representation; Glueing: glueing opposite edges of a square -- construction of a cylinder, a torus and a Mobius band, construction of Klein bottle; Construction of spheres by gluing boundary of disks; First Betti number of 2-dimensional manifolds. Embedding -- a topological space "sitting inside" another; Topological equivalence between two different embeddings of figure 8 in ($\mathbb{R}^2$) -- extend to higher dimensions: The definition of knots as different embeddings of ($\mathbb{S}^1$) in ($\mathbb{R}^3$). Introduction to different types of knots -- unknot, trefoil knot, figure-8 knot, square knot, granny knot. (Ref: Class notes, Notes on Topology)
Oct 27 Knot theory: Review of different types of knots; Knot equivalence; Reidemeister moves; multiple in-class examples of knot equivalences; knot sum; (Ref: Class notes, Notes on Topology)
Oct 29 Midterm 2; Introduction to Platonic solids; The 5 different Platonic solids, their properties (number of faces, edges, vertices); (Ref: Class notes, Notes on Platonic Solids and Distances)
Nov 3 Review of the 5 platonic solids; Euler Characteristic theorem; Proof of the fact that there are only 5 platonic solids (introduced definition of degree of a vertex, the number of sides of each face, and how they relate to number of edges, vertices and faces); Dual Platonic solids; Developed representation of Platonic solids; (Ref: Class notes, Notes on Platonic Solids and Distances)
Nov 5 Review of developed representation of Platonic solids (and how they are not unique); Distance between points on the surface of a platonic solid; Euclidean distance computation using Pythagoras' theorem; A proof of Pythagoras' theorem (introduced relationship between ratio of area of two similar planar objects and the ratio of two of their corresponding line elements); Computation of Euclidean distance between points on the surface of a cube by setting up coordinate axes; Computation of geodesic distance between points on surface of a cube using developed representations; (Ref: Class notes, Notes on Platonic Solids and Distances)
Nov 10 Review of distance computation between points on a Platonic solids (Euclidean distance, geodesic distance); Basic coordinate geometry for computing Euclidean distances; Examples involving cube and tetrahedron; Geodesic and Euclidean distances between points on the surface of a sphere; Few remarks on other types of distances (e.g., Manhattan/taxicab distance); Axioms/properties of metric; (Ref: Class notes)
Nov 12 ''(Lecture by Amelie)''. Definition of a graph (vertex and edge sets); 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 plane (with the exterior counted as a face); (Ref: Class notes)
Nov 17 A few remarks on graphs an counting "faces" for the Euler characteristic theorem; Golden rectangle, golden ration and Fibonacci sequence; Symmetry: Definition, illustration of the symmetries of the equilateral triangle, reflection and rotation symmetries; (Ref: Class notes, Notes on Symmetry)
Nov 19 Review of the symmetries of the equilateral triangle -- listing all the reflection and rotation symmetries, Identifying an independent/generating set, Writing the other symmetries as a composition of the independent/generating symmetries, general procedure for identifying that two symmetries are the same (using labels); Point symmetry; Symmetries of the square and other polygons; More examples; (Ref: Class notes, Notes on Symmetry)
Nov 24 Quick review of symmetries discussed so far; Translation symmetry -- tiling of the plane. Examples: Square tiling, equilateral triangle tiling, hexagonal tiling; Symmetries of tilings and examples of choosing generating/independent symmetries; Other examples of tilings that have symmetries (L-shaped tiles); Introduction to pinwheel tiling. (Ref: Class notes, Notes on Symmetry)
Dec 1 Pinwheel tiling and its properties (no symmetry); Penrose tiling (using two rhombus tiles); Introduction to scale invariance/similarity -- with integer scaling (symmetric tilings); Motivation for fractals -- scale invariance/similarity with fractional scaling; Construction of Koch curve, Sierpinski triangle -- explicit description of all transformations involved; Self-similarity (scale) property of fractals. (Ref: Class notes)
Dec 3 Fractals: Revisit construction and description of the Koch curve; Pythagoras' tree; Re-visit dimension of a figure (manifold); An alternative way of computing 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 square) and a solid object (a cube); (Ref: Class notes)
Dec 8 Dimension: Review of definition of dimension of a manifold, alternative way of computing dimension (an informal computation of "box dimension"), Computation of dimension of fractals; A brief discussion on fractals that can be created using transformations other than affine transformations -- description of Julia sets. The Mandelbrot set. (Ref: Class notes)

Course Outline (Subject to Change)

Elementary Set theory (1-2 lectures)
- Set theory.
- Venn diagram.

Logic and Methods of proof (2-3 lectures)
- Logic.
- Statements in mathematics.
- Methods of proof.

Numbers (3-4 lectures)
- Counting (elementary combinatorics).
- Types of numbers, number theory.
- Cardinality.
- Introduction to complex numbers.

Elementary Topology (3-4 lectures)
- Concepts and examples from topology.
- Topological invariants (Euler's characteristics).
- Three and higher dimensional spaces.

Elementary graph theory (2-3 lectures)
- Definition and description of graphs.
- Elementary properties and computations.

Geometry - Distances (3-4 lectures)
- Introduction to Metric and their various types.
- Introduction to higher dimensional measures (area, volume).
- Measuring distances on different surfaces.

Geometry - Symmetry and Measures (2-3 lectures)
- Symmetry, Platonic solids.
- Fractals.

Additional Resources


short URL: