Graduate Study in Geometry and Topology
The Department of Mathematics offers a strong graduate program in geometry and topology. Various areas of interest and research within the field are described below, and the courses regularly offered in each area are listed. Many of the courses are given every year, while the rest are given whenever the demand is great enough. In addition, there are special topics courses each semester on subjects not covered by the regular courses.
Area Descriptions
Differential Geometry
Modern differential geometry is concerned with the spaces on which calculus of several variables applies (differentiable manifolds) and the various geometrical structures which can be defined on them. Examples of such structures are Riemannian manifolds and homogeneous spaces. These generalize the classical geometries of Euclid, Lobatchevski, and Riemann's spherical geometry. Further examples are foliations.
In a Riemannian manifold a neighborhood of each point is given a Euclidean structure to a first order approximation. Classical differential geometry considers the second order effects of such a structure locally, that is, on an arbitrarily small piece. Modern studies are more concerned with "differential geometry in the large": how do the local second order quantities affect the geometry as a whole, especially the topological structure of the underlying space? These questions and their generalizations are taken up in Math 520, 521, and 524.
In a homogeneous space there is a distinguished group of differentiable mappings of the space into itself which acts transitively on points. An abstract space of such mappings is the prototype of a Lie group, so Math 522 and 507 are basic to their study. More advanced work in homogeneous spaces usually comes in conjunction with the other geometric structures alluded to above. The ties with physics are very important and of great current interest.
Global Analysis
The local problems studied in calculus can be formulated in any space that is locally like an open set in Euclidean space; such spaces are called differentiable manifolds. More generally, the study of function spaces leads to the notion of infinite dimensional manifolds, which look locally like a topological vector space. The solutions of many problems can be understood locally in terms of classical analysis or modern functional analysis. But when one considers these problems on entire manifolds the global geometry is often restrictive and limits the class of problems that make sense. Moreover, one needs techniques for combining local solutions to obtain global ones. The study of this influence of the entire space on problems is called global analysis.
Typical subjects in this field include the study of the relations between the singularities of a differentiable function on a manifold and the topology of the underlying space (Morse Theory), ordinary differential equations on manifolds (dynamical systems), problems in solving exterior differential equations (de Rham's Theorem), potential theory on Riemannian manifolds (Hodge's Theory), and partial differential equations on manifolds. These and other topics are dealt with in Math 520, 521, and 524.
There are numerous applications of these theories to such fields as relativit hydrodynamics, and celestial mechanics. These applications are studied in topics courses and seminars.
Differential Topology
Differential topology is the study of those properties of smooth manifolds that are invariant under smooth homeomorphisms with smooth inverses (diffeomorphisms). Typical problems are: When are two differentiable manifolds diffeomorphic? When is one manifold the boundary of another? When can one manifold be embedded (immersed) in another and when are two embeddings (immersions) isotopic (regularly homotopic)? Can every mapping between two manifolds be approximated by mappings that are stable under small perturbations? When does the image of a mapping lie in general position (transversality theory)?
Such questions are studied in topics courses, seminars and reading projects. Math 520 is the basic starting point.
Algebraic Topology
Algebraic topology has been a highly active branch of mathematics during the last thirty years due to its remarkable success in solving a number of classical questions. It has been closely related to other developments in topology and geometry, and has been instrumental in the creation of homological algebra and category theory. Math 526 and 527 are the first graduate level courses in this area.
The basic method of algebraic topology consists of associating algebraic invariants, such as homology and homotopy groups, with certain classes of topological spaces. The essence of the method is a conversion of a geometric problem into an algebraic problem which is sufficiently complex to embody the essential features of the original geometric problem, yet sufficiently simple to be solvable by standard algebraic methods. Much of the work in algebraic topology has accordingly been devoted to refining the algebraic tools. Methods of algebraic topology are frequenfly applied to problems in differential topology. These methods include the introduction of cup products, cohomology operations and other cohomology theories, such as K-theory all of which are considered in Math 533.
Geometric and Polyhedral Topology
This is the study of spaces defined by fitting together standard blocks that are usually cells or simplexes. When simplexes are used, the study is sometimes called piecewise linear topology. Sometimes the fitting of blocks is done with smooth cells and the study extends heavily into differential topology. There are many problems in this area, for example the Poincare Conjecture, knot problems, and a surprizing number of problems from group theory. The problems and techniques seem to appeal to people with a strongly geometrical turn of mind. Math 534 and many of the topics courses offered as Math 595 center around geometric and polyhedral topology.
Algebraic Geometry
Classically, algebraic geometry is the study of the zeroes of a collection of polynomials in a finite number of variables over a field K. By Hilbert's Nullstellensatz this is equivalent to studying the set of maximal ideals in the ring of all polynomials in the same number of indeterminates with coefficients in the algebraic closure of K, modulo the ideal generated by the original collection of polynomials.
There are two main objectives in this type of geometry. First, classify by means of algebraic invariants (e.g., rational functions, numerical invariants, homology) the geometric objects which arise in this way. Second, describe topologically the geometric objects attached to such algebraic structures (Riemann surfaces, compact complex manifolds, zeta functions).
Today the study of these problems has been generalized to the study of the geometric object which one can attach to any commutative ring - the set of all primes of the ring. This is the theory of schemes developed by Grothendieck and others.
Some of the outstanding problems are:
- given a scheme X find a scheme Y which has no singularities and is birationally equivalent to X,
- describe the algebraic invariants which classify a scheme up to birational equivalence,
- classify those schemes which occur as homogeneous spaces of group schemes (the analogue of Lie Groups).
The subject has many applications to (and draws inspiration from) the fields of complex manifolds, number theory, and commutative algebra. Math 511 is the first graduate level course in algebraic geometry.
General Topology
General topology has been an active research area for many years, and is broadly the study of topological spaces and their associated continuous functions. Sometimes called point set topology, the field has many applications in other branches of mathematics. Since the definitions are less restrictive than in differential or polyhedral topology, a much wider variety of situations can arise in this category. The ideas tend to be more abstract and less geometrical. Problems range from those with a strong algebraic content to others which are close to logic and set theory. Math 535 presents the basic graduate level material.
Problems in Classical Geometry
There are many easily understood, unsolved problems concerning convex sets, geometric inequalities, packings and coverings, distance geometry, combinatorial geometry, the geometry of numbers, and other like branches of classical geometry. Their solution often depends more on insight, ingenuity and originality than on the development and application of abstract theories. Students interested in problems of this kind should prepare by developing a strong background in the fundamentals of analysis and algebra. Advanced topics courses in these areas are occasionally offered to lead students into interesting problem areas.
Courses in Geometry and Topology
The following courses are listed in the Course Catalog and are offered regularly or upon demand.
- Math 423. Differential Geometry
- Math 432. Set Theory and Topology
- Math 507. Lie Algebras
- Math 511. Algebraic Geometry
- Math 520. Differentiable Manifolds
- Math 521. Riemannian Geometry
- Math 522. Lie Groups
- Math 524. Linear Analysis on Manifolds
- Math 595. Topics in Geometry. The following sections of this topics
course have been offered in recent years:
- Computer Geometry, George Francis
- Riemannian Foliations, Philippe Tondeur
- Representations of Compact Lie Groups, Ranga Rao
- Elementary Geometry from a Modern Viewpoint, John Wetzel
- Riemann Surfaces, Robert Carroll
- Geometry of Gauge Fields, Philippe Tondeur
- Metric Spaces of Bounded Curvature, Igor Nikolaev
- Complex Differential Geometry, Steven Bradlow
- Math 526. Algebraic Topology
- Math 527. Homotopy Theory
- Math 533. Fiber Spaces and Characteristic Classes
- Math 535. General Topology
- Math 595. Topics in Topology.