| 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.
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| 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$). Numbers: Natural numbers; completion of the addition operation giving integers; additive inverse, additive identity; Modular/clock arithmetic: notations, basic properties/formulae, example problems. (Ref: Chapter 3 of Hammack, Notes on Numbers Δ.) |
| Oct 1 |
Midterm 1; Prime numbers, prime counting function, prime number theorem, Fermat's little theorem, Chinese remainder theorem -- Problems involving these; (Ref: Class notes, Notes on Numbers Δ.) |
| Oct 6 |
Review of theorems discussed in previous class, problems involving those; Coprime numbers; General discussion on prime numbers -- open problems (twin prime conjecture, Goldbach's conjecture), application of prime numbers to cryptography (SSL/RSA); Numbers: Rational numbers, irrational numbers (including proof that square root of 2 is irrational), algebraic numbers, transcendental numbers. (Ref: Class notes, Notes on Numbers Δ.) |
| Oct 13 |
Quick review of rational, irrational, algebraic, transcendental numbers; Rational numbers: Remarks on multiplicative identity and multiplicative inverse in context of integers and rational numbers, decimal representation of rational numbers -- conversion to and from; Introduction to complex numbers (requirement of ($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$) (Ref: Class notes, Notes on Numbers Δ, Video notes on cardinality.) |
| Oct 15 |
Reviewed discussion on cardinality from previous class, cardinality of ($\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}$), and higher order infinities; Introduction to topology (rubber-sheet geometry) -- idea of topological equivalence (homeomorphism), examples (1-dimensional: circle, line segment; 2-dimensional: sphere, torus, sphere with a puncture, torus with a puncture). (Ref: Class notes, Notes on Numbers Δ, Video notes on cardinality.) |
| Oct 20 |
Topology / "rubber-sheet geometry": Topological equivalences, examples; Manifolds; Cutting (torus example); First Betti number (as the maximum number of cuts); Gluing (cylinder, torus, Mobius band). (Ref: Class notes, Notes on Topology Δ) |
| Oct 22 |
Topology: Manifolds and their dimension (with examples of manifolds and examples of spaces that are not manifolds); Cuts -- cutting a torus into a flat square; Developed representation; Glueing: glueing opposite edges of a square -- construction of a cylinder, a torus and a Mobius band, construction of Klein bottle; Construction of spheres by gluing boundary of disks; First Betti number of 2-dimensional manifolds. Embedding -- a topological space "sitting inside" another; Topological equivalence between two different embeddings of figure 8 in ($\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 Δ) |
| 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 Δ) |
| 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 Δ) |
| 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 Δ) |
| 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 Δ) |
| 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 Δ) |
| 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 Δ) |
| 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 Δ) |
| 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) |
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