everything.study
18.994

Seminar in Geometry

3-0-9 units · Undergraduate

Students present and discuss subject matter taken from current journals or books. Topics vary from year to year. Instruction and practice in written and oral communication provided. Enrollment limited.

Fall 2004

Topics

  • Introduction
  • A Review on Differentiation
  • Inverse Function Theorem
  • Implicit Function Theorem
  • First Fundamental Form
  • Curves
  • Gauss Map I: Background and Definition
  • Gauss Map II: Geometric Interpretation
  • Gauss Map III: Local Coordinates
  • Introduction to Minimal Surfaces I
  • Introduction to Minimal Surfaces II
  • Review on Complex Analysis I
  • Review on Complex Analysis II
  • Isothermal Parameters and Harmonic Functions
  • Bernstein’s Theorem
  • Manifolds and Geodesics I
  • Manifolds and Geodesics II
  • Complete Minimal Surfaces I
  • Complete Minimal Surfaces II
  • Weierstrass-Enneper Representations
  • Gauss Maps and Minimal Surfaces
  • Project Talk

Readings

  • Osserman, Robert. A Survey of Minimal Surfaces. Dover Phoenix Editions. New York: Dover Publications, May 1, 2002. ISBN: 0486495140.
    Listed reading
  • Do Carmo, Manfredo. Differential Geometry of Curves and Surfaces. Englewood Cliffs, N. J.: Prentice Hall, February 1, 1976. ISBN: 0132125897.
    Listed reading
  • Spivak, Michael. Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus. New York: Westview Press, June 1, 1965. ISBN: 0805390219.
    Listed reading
  • Spivak, Michael. A Comprehensive Introduction to Differential Geometry. Vol 3. 3rd ed. Boston: Publish or Perish, January 1, 1999. ISBN: 0914098721.
    Listed reading
  • Spivak, Michael. A Comprehensive Introduction to Differential Geometry. Vol 4. 3rd ed. Berkeley: Publish or Perish, January 1, 1999. ISBN: 091409873X.
    Listed reading
  • Ahlfors, Lars. Complex Analysis. 3rd ed. New York: McGraw-Hill Science/Engineering/Math, January 1, 1979. ISBN: 0070006571.
    Listed reading
  • Ahlfors, L. V., and L. Sario. Riemann Surfaces . Princeton, N.J.: Princeton University Press, 1960. ISBN: 0691080275.
    Listed reading
  • lec #: 2 · TOPICS: A Review on Differentiation · READINGS: Spivak ISBN: 0914098721 pp. 15-31 or 33
    Reading assignment
  • lec #: 3 · TOPICS: Inverse Function Theorem · READINGS: Spivak ISBN: 0805390219 pp. 31-33
    Reading assignment
  • lec #: 4 · TOPICS: Implicit Function Theorem · READINGS: Spivak ISBN: 0805390219 pp. 40-43
    Reading assignment
  • See examples of parametrised surfaces: Do Carmo, Spivak ISBN: 0914098721
    Listed reading
  • lec #: 5 · TOPICS: First Fundamental Form · READINGS: Osserman p. 10
    Reading assignment
  • lec #: 6 · TOPICS: Curves · READINGS: Osserman pp. 8-11
    Reading assignment
  • lec #: 7 · TOPICS: Gauss Map I: Background and Definition · READINGS: Do Carmo 2.4, props 2-3, 3.2: definition of Gauss map, p. 135, appendix to chapter 3, p. 214
    Reading assignment
  • lec #: 8 · TOPICS: Gauss Map II: Geometric Interpretation · READINGS: Osserman p. 12
    Reading assignment
  • lec #: 9 · TOPICS: Gauss Map III: Local Coordinates · READINGS: Do Carmo 3.2, definitions 5, 7, 9
    Reading assignment
  • lec #: 10 · TOPICS: Introduction to Minimal Surfaces I · READINGS: Do Carmo 3.3 up to p. 156
    Reading assignment
  • Refer to Do Carmo 3.5B and Spivak ISBN: 0914098721 chapter 3 for more details
    Listed reading
  • lec #: 11 · TOPICS: Introduction to Minimal Surfaces II · READINGS: Osserman p. 18, 20-23
    Reading assignment
  • Ahlfors ISBN: 0070006571 chapter 2, 1.1: just definition 4, example of real function of a complex variable, 1.2
    Listed reading
  • lec #: 12 · TOPICS: Review on Complex Analysis I · READINGS: Ahlfors ISBN: 0070006571 chapter 2, 2.4, chapter 5, 1.2, p. 179, chapter 2, 3.1-3.4, chapter 3, 2.2, 2.3, chapter 4, 3.2 pp. 126-127
    Reading assignment
  • lec #: 13 · TOPICS: Review on Complex Analysis II · READINGS: Ahlfors ISBN: 0070006571 chapter 4, pp.127-129
    Reading assignment
  • lec #: 14 · TOPICS: Isothermal Parameters · READINGS: Osserman pp. 15-19, pp. 27-30
    Reading assignment
  • lec #: 15 · TOPICS: Bernstein’s Theorem · READINGS: Osserman pp. 23-26, 31-32
    Reading assignment
  • lec #: 16 · TOPICS: Manifolds and Geodesics I · READINGS: Osserman pp. 40-44
    Reading assignment
  • lec #: 17 · TOPICS: Manifolds and Geodesics II · READINGS: Osserman pp. 44-47 Observe (Do Carmo 5.10), p. 49
    Reading assignment
  • lec #: 19 · TOPICS: Complete Minimal Surfaces II · READINGS: Do Carmo 4.4, 5.3 pp. 238-248, pp. 325-328
    Reading assignment
  • lec #: 21 · TOPICS: Gauss Maps and Minimal Surfaces · READINGS: Osserman pp. 70-78, 81-83
    Reading assignment

Prerequisites & course information

Prereq: ( 18.06 , 18.700 , or 18.701 ) and ( 18.100A , 18.100B , 18.100P , or 18.100Q ) U (Spring) 3-0-9 units

Sources