## Rick Miranda

### M470, Euclidean and Non-Euclidean Geometry

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

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.

```Syllabus:
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.

Syllabus:

• Recognizing Linear Systems and solutions.
• Matrices of linear systems, equivalent matrices.
• Elementary Row Operations, reduced row echelon form, general solutions.
• Matrix addition, scalar multiplication, matrix-vector multiplication, matrix form of linear system.
• Basic solutions, polynomial interpolation.
• Matrix multiplication, Inverses.
• Changing plane coordinates.
• f(A)v two ways, Eigenvalues and eigenvectors.
• A-annihilator of a vector, the minimal polynomial.
• Projection matrices.
• Determinant computation.
• Determinants geometrically.
• Spectral radius. Graphing characteristic polynomials.
• Evolution of discrete dynamical systems.
• Markov Chains and Leslie population models.

### M519, Complex Analysis

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

Syllabus:

• Complex Numbers and Functions
• Power Series
• Cauchy's Theorem
• Winding Numbers
• Laurent Series and Isolated Singularities
• Residues
• Further Topics

### M666, Graduate Algebra III, Fall 1996

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

Syllabus:

• Basics of Group Theory
• The General Linear Group
• Sylow Theorems and Finite p-groups
• Composition Series and Solvability
• Semisimple Algebras and Wedderburn Theory
• Group Representations and Character Theory
• Further Topics

### 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.

Syllabus:

• Injections, Surjections, Bijections, Cardinality
• Finiteness, Countability
• Topologies: open sets, closed sets, neighborhoods, interiors, boundaries
• Basic open sets, abstract bases
• Metric spaces, the metric topology
• Sequences
• Separation axioms
• Subspaces
• Cauchy sequences, complete metric spaces
• Function spaces
• Homeomorphisms
• Open covers and compactness
• Heine-Borel Theorem
• Uniform continuity
• Normal spaces
• Connectedness, path-connectedness
• Products

### 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:

• Affine Geometry and Projective Geometry
• Definition of Projective Space
• Homogeneous Coordinates in Projective Space
• Linear Subspaces, Points, Lines, Planes, Collinearity Conditions
• Schubert Conditions, Changes of Coordinates
• Curves in the plane, degree
• Classification of conics
• Quadric surfaces and hypersurfaces, classification by rank
• Varieties in projective space
• Homogeneous polynomials, the homogeneous coordinate ring
• The Ideal of a projective variety
• Rational functions
• Cubic plane curves, classification, singular cubics
• Smooth cubics, Weierstrass form, the group law
• Cubic surfaces and the 27 lines
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.

Syllabus:

• Divisibility, Congruence
• Prime Numbers
• Numerical Functions, Mobius Inversion
• Algebra of Congruence Classes, Chinese Remainder Theorems
• Higher Degree Congruences, Quadratic Residues and Reciprocity
• Sums of Squares

### M619, Complex Analysis II

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

Syllabus:

• Complex Charts and Complex Structures
• First Examples of Riemann Surfaces
• Projective Curves
• Functions and Maps
• Examples of Meromorphic Functions
• Holomorphic Maps between Riemann Surfaces
• Global Properties of Holomorphic Maps
• More Elementary Examples of Riemann Surfaces
• Less Elementary Examples of Riemann Surfaces
• Group Actions on Riemann Surfaces
• Monodromy
• Basic Projective Geometry
• Differential Forms
• Operations on Differential Forms
• Integration on a Riemann Surface
• Divisors
• Linear Equivalence of Divisors
• Spaces of Functions and Forms Associated to a Divisor
• Divisors and Maps to Projective Space
• Algebraic Curves
• Laurent Tail Divisors
• The Riemann-Roch Theorem and Serre Duality
• First Applications of Riemann-Roch
• The Canonical Map
• The Degree of Projective Curves
• Inflection Points and Weierstrass Points
• Abel's Theorem
• Sheaves and Cohomology
• Algebraic Sheaves
• Invertible Sheaves, Line Bundles, and H^1

### M667, Graduate Algebra IV, Spring 1995

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

Syllabus:

• Representations of Finite Groups
• Characters
• Examples; Induced Representations, Groups Algebras, Real Representations
• Representations of the Symmetric Group: Young Diagrams and Frobenius' Formula
• Representations of the Alternating Group and GL(2,q)
• Lie Groups and Lie Algebras

### 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.

Syllabus:

• Basic Constructions
• Localization
• Associated Primes and Primary Decomposition
• Integral Dependence and the Nullstellensatz
• Filtrations and the Artin-Rees Lemma
• Flatness
• Completions and Hensel's Lemma
• Basic Definitions of Dimension Theory
• The Principal Ideal Theorem and Systems of Parameters
• Dimension and Codimension One
• Hilbert-Samuel Polynomials

### 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.

Syllabus:

• Multilinear Algebra
• Homological Algebra
• Grobner Bases
• Modules of Differentials
• Regular Sequences and the Koszul Complex
• Depth, Codimension, and Cohen-Macaulay Rings
• Homological Theory of Regular Local Rings
• Free Resolutions and Fitting Invariants
• Duality, Canonical Modules, and Gorenstein Rings

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.

Syllabus:

• Matrix Operations
• Groups
• Vector Spaces and Linear Transformations
• Symmetry
• More Group Theory
• Rings
• Factorization
• Modules
• Fields
• Galois Theory