Course introduction
Instructor Introduction
Lecture plan
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1. How did mathematicians find the epsilondelta definition of continuity? : Introduction to basic
 1) Why should we use the mysterious epsilondelta definition of coninuity?
 2) Basic introduction to logic
 3) Basic introduction to quantifiers
 4) Deducing the epsilondelta definition of continuity
 Quiz(week 1)

2. Why should we study abstract mathematics? : Construction of real numbers.
 1) Why should we study abstract mathematics? : Construction of real numbers.
 Quiz(week 2)

3. Why should we study abstract mathematics? : Finding totally new numbers.
 1) Why should we study abstract mathematics? : Finding totally new numbers.
 Quiz(week 3)

Week 04: Why are the polynomials of degree five not solvable? : Introduction to modern abstract ...
 41 Finding roots of polynomials : story of Abel and Galois
 42 Groups of permutations of the roots
 43 Solvable groups and solvability of cubic and quartic equations.
 44 Modern formulation of Galois theory and insolvability of the quintic equations.
 Quiz(week 4)

Week 05: Solvability of ordinary differential equations.
 51 Existence and uniqueness for ordinary differential equations
 52 Global existence vs finite time blowup
 Quiz(week 5)

Week 06: Isoperimetric problem
 61 Geometric approach to isoperimetric problem
 62 EulerLagrange equations for isoperimetric problem
 Quiz(week 6)

Week 07: what is infinite dimensional space, and where is it?
 71 vector spaces with inner product
 72 Convergence of Fourier series
 Quiz(week 7)

Week 08: Why does Jacobian appear in the change of variables for multiple integrals?
 81 Jacobian of a linear transformation as volume change
 82 Jacobian of a nonlinear transformation as local volume change
 83 Jacobian and multiple integrals on differentiable manifolds I
 84 Jacobian and multiple integrals on differentiable manifolds II
 Quiz(week 8)

Week 09: I have trouble with eigenvaluse and eigenvectors. Why should I care?
 91 Eigenvalues and Eigenvectors
 92 Singular values and singular vectors
 93 Application I : Differential Equation
 94 Application II : Statistics  PCA
 Quiz(week 9)

Week 10: Should I study math to become a fund manager? (Lagrange multiplier method)
 101 Idea of Modern Portfolio Theory
 102 Formulation of portfolio selection problem
 103 Lagrange Multiplier Method
 104 Solution of MeanVariance Portfolio Optimization Problem
 Quiz(week 10)

Week 11: Should I study math to become a quant? (BlackScholes equation)
 111 Concept of Derivative Pricing
 112 BlackScholes Model
 113 BlackScholes equation
 114 Solution of BlackScholes Equation
 Quiz(week 11)

Week 12: Should I study math to become an economist? (Brouwer’s fixed point theorem)
 121 Concept of Game and Nash Equilibrium
 122 Existence of Nash equilibrium
 123 Sperner’s Lemma
 124 Proof of Brouwer’s Fixed Point Theorem
 Quiz(week 12)

Week 13: MATLAB Examples for Chapters 8 and 9
 131 Singular Value Decompositions (SVD)
 132 Dimensionality Reduction via SVD
 133 Principal Component Analysis (PCA)
 134 Numerical Comparison Between the Heat flow and the BV flow

Week 14: A taste for optimization
 141 Basic Optimization Theory Part I (Method of Least Squares)
 142 Basic Optimization Theory Part II (Method of Least Squares)
 143 Applications to Image Processing : Noise and Blur removal
 Quiz(week 14)