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.