Rick Miranda

Courses Taught Recently:

M470, Euclidean and Non-Euclidean Geometry

Text: Modern Geometries, Fifth Edition, by James R. Smart
Brooks/Cole Publishing (1998); ISBN 0-534-35188-3
Course Home Page Here.

This course will discuss the basic notions of Euclidean and Non-Euclidean geometry, in two dimensions. We will focus on the three main examples: the Euclidean Plane, the Projective Plane, and the Hyperbolic Plane. Each will take up approximately one-third of the course material. In the discussion of each of these three geometries, we will begin with the synthetic approach, via an axiomatic treatment. Then we will turn to the introduction of coordinates in each geometry. Finally we will discuss changes of coordinates in each geometry, leading to the groups of automorphisms of each geometry. Tesselations or tilings in each geometry will cap each unit.

1. Introduction to Geometry. 
Synthetic and Analytic Geometry.
Easy Examples of Small Geometries. Models for a Geometry. 
2. Euclidean Geometry 
a. Synthetic Euclidean Geometry: Euclid's Axioms.
Axioms incorporating length and angle. 
b. Euclidean Coordinates.  The Cartesian Coordinate System. 
c. Changes of Coordinates: the general linear group,
the euclidean affine group.  Orthogonal transformations. Matrices. 
d. Tesselations in the plane.  Ribbon Patterns.  Wallpaper Groups. 
3. Projective and Spherical Geometry. 
a. Synthetic Projective Geometry.  Axioms for a Projective Plane. 
b. Models for the Projective Plane. 
c. Homogeneous Coordinates. 
d. Changes of Homogeneous Coordinates: the Projective General Linear Group. 
e. Tesselations in the Projective Plane.  Regular Solids. 
4. Hyperbolic Geometry. 
a. Synthetic Hyperbolic Geometry.  Axioms for the Hyperbolic Plane. 
b. Models for the Hyperbolic Plane:
the complex upper-half plane and the Poincare Disc. 
c. Complex Coordinates for the Hyperbolic Plane. 
d. Automorphisms of the Hyperbolic Plane: PSL(2) 

M229, Matrices and Linear equations

Text: Matrices, Linear Equations, and Applications: An Algorithmic Approach, by R. Liebler.


M519, Complex Analysis

Text: Complex Analysis, Third Edition, by Serge Lang. Graduate Text in Mathematics No. 103, Springer-Verlag.


M666, Graduate Algebra III, Fall 1996

Text: Groups and Representations, by J. Alperin and R. Bell. Graduate Text in Mathematics No. 162, Springer-Verlag.


M400D, Topology

This course was taught using a modified Moore method. Each day the course met, I handed out a list of definitions and theorems to be proved. During the next class period, students were called on in turn to present solutions to the problems and proofs of the theorems. If a student was called on to present a particular problem, he or she was responsible for writing the solution to hand in. At the end of the class period, I handed another sheet of definitions, problems, and theorems.


M675-775: Projective Geometry

I have taught this year-long course every other year since 1987. It varies slightly from one incarnation to the next.

The study of Projective Geometry has its roots in the art and science of the late middle ages. The basic objects of study in Projective Geometry is the same as in all geometries: points, lines, planes, conics, etc. However Projective Geometry systematically introduces the concept of infinity into the geometric universe, thereby ``compactifying'' the ordinary space R^n (or C^n) by adding points at infinity.

This first semester course covers the basic theory of Projective Geometry over fields, concentrating on the complex numbers (although much of the first part of the theory works over any field). The prerequisites are a good understanding of linear algebra and a little abstract algebra (notion of a ring and ideal is enough). Don't be scared by the high number: M566 should be plenty as a prerequisite.

We start by covering the linear theory of projective space (points, lines, planes, etc.) and then some of the non-linear aspects: conic curves, quadric surfaces, cubic curves, etc.

Fall Syllabus:

We have used "Undergraduate Algebraic Geometry", by Miles Reid, London Mathematical Society Student Text No. 12, Cambridge University Press as a text for the fall semester. In 1993 we used "Algebraic Geometry" by Joe Harris, Graduate Texts in Mathematics No. 133, Springer-Verlag.

Spring Syllabus: This has varied somewhat, depending on the interests and level of the students. In 1996 we read chapters of "Basic Algebraic Geometry I: Varieties in Projective Space", by I. R. Shafarevich, Springer-Verlag. In 1994 we continued in Joe Harris' book in the second semester. In 1992 I taught a course in Algebraic Surfaces, with no text.

M400C, Number Theory

Text: "Elementary Number Theory", by Charles Vanden Eynden, McGraw-Hill.


M619, Complex Analysis II

Text: Algebraic Curves and Riemann Surfaces, by Rick Miranda. Graduate Studies in Mathematics No. 5, AMS.

See a Postscript version of the Table of Contents HERE


M667, Graduate Algebra IV, Spring 1995

Text: Representations Theory, by William Fulton and Joe Harris. Graduate Text in Mathematics No. 129, Springer-Verlag.


M666, Graduate Algebra III, Fall 1994

Text: Commutative Algebra, with a view toward Algebraic Geometry, by David Eisenbud. Graduate Text in Mathematics No. 150, Springer-Verlag.


M666, Graduate Algebra IV, Spring 1997

Text: Commutative Algebra, with a view toward Algebraic Geometry, by David Eisenbud. Graduate Text in Mathematics No. 150, Springer-Verlag.


M566, Graduate Algebra I-II

This year-long sequence is the basic graduate algebra course in our department. The Algebra Qualifying Examination is based on these two semester courses. Several texts have been used in the past few years, including Herstein, Lang, and Hungerford. In 1993-4 I used:

Text: Algebra, by Michael Artin. Prentice-Hall.