1. How did mathematicians find the epsilon-delta definition of continuity? : Introduction to basic
- 1) Why should we use the mysterious epsilon-delta definition of coninuity?
- 2) Basic introduction to logic
- 3) Basic introduction to quantifiers
- 4) Deducing the epsilon-delta 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 ...
- 4-1 Finding roots of polynomials : story of Abel and Galois
- 4-2 Groups of permutations of the roots
- 4-3 Solvable groups and solvability of cubic and quartic equations.
- 4-4 Modern formulation of Galois theory and insolvability of the quintic equations.
- Quiz(week 4)
Week 05: Solvability of ordinary differential equations.
- 5-1 Existence and uniqueness for ordinary differential equations
- 5-2 Global existence vs finite time blow-up
- Quiz(week 5)
Week 06: Isoperimetric problem
- 6-1 Geometric approach to isoperimetric problem
- 6-2 Euler-Lagrange equations for isoperimetric problem
- Quiz(week 6)
Week 07: what is infinite dimensional space, and where is it?
- 7-1 vector spaces with inner product
- 7-2 Convergence of Fourier series
- Quiz(week 7)
Week 08: Why does Jacobian appear in the change of variables for multiple integrals?
- 8-1 Jacobian of a linear transformation as volume change
- 8-2 Jacobian of a nonlinear transformation as local volume change
- 8-3 Jacobian and multiple integrals on differentiable manifolds I
- 8-4 Jacobian and multiple integrals on differentiable manifolds II
- Quiz(week 8)
Week 09: I have trouble with eigenvaluse and eigenvectors. Why should I care?
- 9-1 Eigenvalues and Eigenvectors
- 9-2 Singular values and singular vectors
- 9-3 Application I : Differential Equation
- 9-4 Application II : Statistics - PCA
- Quiz(week 9)
Week 10: Should I study math to become a fund manager? (Lagrange multiplier method)
- 10-1 Idea of Modern Portfolio Theory
- 10-2 Formulation of portfolio selection problem
- 10-3 Lagrange Multiplier Method
- 10-4 Solution of Mean-Variance Portfolio Optimization Problem
- Quiz(week 10)
Week 11: Should I study math to become a quant? (Black-Scholes equation)
- 11-1 Concept of Derivative Pricing
- 11-2 Black-Scholes Model
- 11-3 Black-Scholes equation
- 11-4 Solution of Black-Scholes Equation
- Quiz(week 11)
Week 12: Should I study math to become an economist? (Brouwer’s fixed point theorem)
- 12-1 Concept of Game and Nash Equilibrium
- 12-2 Existence of Nash equilibrium
- 12-3 Sperner’s Lemma
- 12-4 Proof of Brouwer’s Fixed Point Theorem
- Quiz(week 12)
Week 13: MATLAB Examples for Chapters 8 and 9
- 13-1 Singular Value Decompositions (SVD)
- 13-2 Dimensionality Reduction via SVD
- 13-3 Principal Component Analysis (PCA)
- 13-4 Numerical Comparison Between the Heat flow and the BV flow
Week 14: A taste for optimization
- 14-1 Basic Optimization Theory Part I (Method of Least Squares)
- 14-2 Basic Optimization Theory Part II (Method of Least Squares)
- 14-3 Applications to Image Processing : Noise and Blur removal
- Quiz(week 14)