Honor Seminar: Can One hear the Shape of a Drum? (Instructor: Siqi Fu)
- Syllabus
- Topics and Notes.
- Introduction to the course: A PowerPoint presentation that (somewhat) explains
the title of the course.
- Introduction to Euclidean and non-Euclidean geometry.
- Historical background on Euclidean geometry.
- Overview of Euclid's 13 books of Elements.
- Euclid's Elements, book I in detail.
- From Euclidean to non-Euclidean: A Powerpoint presentation.
- A game of sprouts: PDF file.
- Homework assignment #1: PDF file.
- Number systems.
- Conversion between different bases.
- Principle of mathematical induction.
- Prime numbers: Euclid's proof of infinity of prime numbers. Fundamental theorem of arithmetic.
Goldbach conjecture. Twin Prime conjecture.
- Congruence: Rules for integers divisible by 3, 5, 4, 7, 11.
- Commensurable segments. Infinite decimal fractions. Pythogoras' proof of irrationality of square
root of 2.
- The set of real numbers. Continuity principles.
- Infinite sets: Introduction to Cantor's theory of sets. Continuum hypothesis.
- Complex numbers.
- Homework assignment #2: Do the following problems in the textbook What is Mathematics: P17 #2; P35;
P37 #3; P61 #1, #2; P66 #6; P67 #1; P80 #1; P91 #3; P97 #1, #2, #8; P100 #1, #4.
- Functions and limits.
- Function concepts.
- Sequential limits. Definitions of e and pi. Proof of the irrationality of e.
- Derivatives. Chain Rule. Implicit Differentiation. Optimization problems.
- Integrals. Fundamental theorem of calculus.
- Partial derivatives. Differentials. Gradients. Directional derivatives. Laplace operator.
- Homework assignment #3.
- Introduction to Newton's Principia.
- An overview of Newton's Principia.
- Viewing a biographical movie about Newton.
- One cannot hear the shape of a drum.
- Mathematical model for vibrating strings.
- Mathematical model for vibrating membranes.
- Explicit calculation of eigenvalues of the Laplacian for rectangles, right isosceles triangles.
- Gordon, Webb, and Wolpert's construction of isospectral nonisometric drums.