Topics in advanced mathematics (Advanced Geometry), MATH 789.1001, Fall 2011

This is the home page and first day handout for MATH 789, meeting MW from 2:30 to 3:45 in CBC-C316.  My name is Dr. Arthur Baragar, my office is CDC 1016, and I can be reached at 895-0378. Important information announced in class will also be posted on WebCampus.


Preferred prerequisites: (If you do not have all of these prerequisites and would like to take the course, please contact me.)

  • Geometry, MATH 480/680:
  • Abstract Algebra: I will assume knowledge of groups and fields.
  • Linear Algebra: I will assume working knowledge of matrices, matrix groups, and diagonalization.
  • Calculus III: I will assume working knowledge of differentials, dot product, and cross product.



  • A Survey of Classical and Modern Geometries, Baragar, Prentice Hall, 2001. Students who took MATH 480/680 will already have this text. Other students should first contact me, as mentioned above.
  • Foundations of Hyperbolic Manifolds, Ratcliffe. Potential reference? Available electronically, through our library.
  • Foundations of geometry, David Hilbert. Suggested reference? Kindle copy is $5?



  • Hyperbolic geometry (Reference: Baragar. Four or five weeks?)
    • Poincare upper half plane model, using arclength differentials to define length
    • Fractional linear transformations and the cross ratio
    • The Poincare disc model and pseudosphere model
    • Three dimensional hyperbolic geometry -- the Poincare upper half space and ball models.
    • Isometries and lattices
  • Projective geometry (Reference? Two weeks?)
    • Coordinates, homogeneous equations
    • Desargues theorem
    • Conic sections, Pascal's theorem, duality, Briachon's theorem
    • Basic varieties, smooth and singular points
    • Bezout's theorem (?)
    • Elliptic geometry and metrics
  • Hilbert's axioms of incidence geometry (Reference: Baragar, Hilbert. Three weeks?)
    • Weak and strong incidence geometry
    • Non-Desarguian geometry (Reference?)
    • Finite hyperbolic geometry
  • Other topics (References? Time permitting?)
    • Modular surfaces and automorphic forms?
    • Introduction to elliptic curves
    • Introduction to algebraic geometry?
    • The Banach-Tarski paradox?

Grading Scheme:

I have the philosophy that, as graduate students, it is up to you to decide what you want to get out of this course. The minimum requirement for a B is regular class attendance and completed homework. I will tolerate some missed classes and a couple late homeworks. A higher grade will depend on the student's performance on the final exam. Some homework will be assigned individually, and some will be assigned in groups.

Office Hours: If you need to see me, please look for me during the office hours posted below, or please make an appointment. Office hours begin the second week of classes.

  • Monday and Wednesday, 10am -- 11am,
  • Monday, 4pm -- 5pm.
  • ... or by appointment.

Please note that on Monday afternoon, if no one is in my office at 4:30pm, I will feel free to leave, so please arrive early.


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One-on-one or small group assistance with writing is available free of charge to UNLV students at the Writing Center, located in CDC-3-301. Although walk-in consultations are sometimes available, students with appointments will receive priority assistance.

Appointments may be made in person or by calling 895-3908. The student’s Rebel ID Card, a copy of the assignment (if possible), and two copies of any writing to be reviewed are requested for the consultation. More information can be found at:

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