Course Descriptions, Fall 2001 - CUNY Ph.D. Program in Mathematics

- Ph.D. Program of the City University of New York -

Course Descriptions - Fall 2001

MATH 70100: Functions of a Real Variable
Prof. Edgar Feldman
T&Th, 12:45 - 2pm

[Description Not Submitted]

MATH 70300: Functions of a Complex Variable
Prof. Burton Randol
M&W, 2:30 - 3:45pm

[Description Not Submitted]

MATH 70500: Algebra I
Prof. Lucien Szpiro
M&W, 4:15 - 5:30pm

We will cover groups, vector spaces, determinants, polynomial rings and local rings in a two semester course. The recommended book is Basic Abstract Algebra by P.B.Bhattacharya, S.K.Jain and S.R.Naipaul, edited by Cambridge University Press. Later we will use Introduction to Commutative Algebra by M.F.Atiyah and I.G.Macdonald. The Monday Class will be expository; the Wednesday class will be mostly exercises and problems.

MATH 70700: Topology I
Prof. Alphonse Vasquez
M&W, 1:00 - 2:15pm

The first topic will be metric spaces. The discussion will include completeness, completions, the contraction mapping theorem and Baire’s category theorem. (See, e.g., chapters 8 and 14 of Lipschutz’s General Topology.) Topological spaces and continuity will be introduced next.
The notions of connectedness and compactness will be explored. Compactness for metric spaces will be explored in some detail. (See, e.g., chapter 4 of Simmons’ Introduction to Topology and Modern Analysis.) Subspaces, product spaces (the Tychonov theorem) and quotient spaces will be studied. The compact-open topology on function spaces will be introduced and the corresponding exponential law will be proved. (See Dugundji’s Topology.) This account of some of the high points of general topology will probably be too rapid for someone encountering these notions for the first time. The fundamental group(oid) will be introduced and computed in some "easy" instances. Covering spaces will be introduced and the basic theorems relating these two topics will be established. A version of Van-Kampen’s theorem will be proved. Applications will include the topology of compact, connected, surfaces. The classification of surfaces will be discussed. (See, e.g., Massey’s Algebraic Topology: an Introduction. This text is now incorporated in his GTM volume 127.) Many textbooks will be on reserve in the library. You should consult them regularly as a source of examples and alternative approaches.

MATH 70900: Differential Geometry I
Prof. Adam Koranyi
T&TH, 2:00 - 3:15pm

Prerequisites: Linear algebra, advanced calculus.

This course will be an introduction to Riemannian geometry starting with a concise discussion of hypersurfaces in Euclidean space, then gradually introducing the general machinery of manifolds, tensors and forms, connexions, and Lie groups, in order to start a general study of Riemannian manifolds. It will cover roughly the first half of the short book of N. J. Hicks, Notes on Differential Geometry (Van Nostrand, (1965). This book is a very good outline, but is too sketchy to be sufficient in itself. Good supplementary sources are, for foundational material, F. W. Warner, Foundations of Differentiable Manifolds and Lie Groups (Scott Foresman, 1971), and M. Spivak, A Comprehensive Introduction to Differential Geometry (Publish or Perish, Boston, 1970); for Riemannian geometry, see I. Chavel, Riemannian Geometry: a Modern Introduction (Cambridge U. Press, 1993). These books will be on reserve in the Mina Rees Library.

MATH 71100: Logic I
Prof. Roman Kossak
T&TH, 4:15 - 5:30

This is an introductory course. No prerequisites required, however some knowledge of abstract algebra and “naïve” set theory will be helpful. The emphasis will be on the basic concepts and tools of mathematical logic and their applications in algebra and analysis. Topics will include:

1. Isomorphism and elementary equivalence of first order structures.
2. Ehrenfeucht-Fraisse games.
3. Compactness theorem and ultraproduct constructions.
4. Saturation, recursive saturation, and some applications.

The text will be A Course in Model Theory by Bruno Poizat (Springer, 2000).
Additional reading: Basic Model Theory by Kees Doets (CSLI Publications,
1996).

MATH 80900: Lie Groups & Symmetric Spaces
Prof. Martin Moskowitz
M&W, 1:00 - 2:15pm

[Description Not Submitted]

MATH 81700: Topology II: Lusternik-Schnirelmann Category
Prof. Joseph Roitberg
T&Th, 11:00 - 12:15pm

Prerequisite: A first-year graduate course in topology, such as the GC's TOPOLOGY I

Consider covers of a "well-behaved" topological space by subsets each of which is contractible in X. The Lusternik-Schnirelmann category of X, cat(X), is the minimum of the cardinalities of such covers, with 1 subtracted for convenience. The numerical invariant cat(X) was introduced in the 1930's by L. Lusternik and L. Schnirelmann, who showed that for X a smooth manifold, cat(X) is a lower bound for the number of critical points of an arbitrary bounded-from-below, smooth real-valued function on X. Subsequently, topological studies of cat(X) were undertaken by several authors, notably R. H. Fox (1940's), G. W. Whitehead (1950's) and T. Ganea (1960's). Whitehead reformulated the definition of cat(X) in a useful homotopy-theoretic way and proved that for G a connected topological group, the group [X,G] of pointed homotopy classes of pointed maps from X to G has nilpotency class bounded above by cat(X), if cat(X) is finite. He deduced from this a "Jacobi identity" for (J. H. C.) Whitehead products, thus marking the first appearance of a (graded) Lie algebra structure in algebraic topology. Later, Ganea recast Whitehead's definition in a yet more useful way, developed the theory further and formulated some pertinent questions, most famously: is cat(X x n-sphere) = cat(X) + 1? The aims of this course are to survey LS theory, discussing the work described above, and to give an introduction to some of the recent hectic activity in the field, including negative examples to Ganea's question (using Hopf invariant technology) and applications to symplectic topology.

MATH 82530: Algebraic Topology & Spaces of Riemann Surfaces
Prof. Dennis Sullivan
W, 10:00 - 12:00pm

We will study various general techniques in algebraic topology, geometry and combinatorics. These will be tested in the study of the space of all Riemann Surfaces.

MATH 84530: Introduction to Combinatorics
Prof. Janos Pach
Wed., 4:15 - 6:15pm+

We give a systematic introduction to some core areas of combinatorics with special emphasis on links between seemingly unrelated areas and on applications to problems in other parts of mathematics and computer science. Some basic knowledge of linear algebra, calculus, and familiarity with the notion of finite fields are required. We cover the following topics:

Combinatorial counting
Double-counting, parity arguments
The number of spanning trees
Dilworth' theorem and extremal set theory
Finite projective planes, latin squares
Proofs by counting: probabilistic proofs
Generating functions
Partitions
Applications of linear algebra
Combinatorial designs

Textbook: J. Matousek and J. Nesetril: Invitation to Discrete Mathematics, Oxford University Press, 1998.

MATH 85500: Probability
Prof. Michael Marcus
T&Th, 2:00 - 3:15pm

This is the first course in a four semester sequence of courses on the theory of probability. A student completing all four courses is in a very good position to write a Ph.D. thesis in probability. The first two semesters gives a good background in probability to any mathematician. Many problems in algebra, analysis, and geometry are approached using probabilistic arguments. This is a trend which is very likely to get stronger in the coming years. A basic year of probability would also be very valuable to applied statisticians. (Theoretical statisticians need more.)

In 1998 and 2000 the third semester of the probability sequence was devoted to a rigourous treatment of the basic theorems in mathematical finance. The first year of probability is essential to understand this material.

This course assumes knowledge of measure theory and the other topics covered in a first year graduate course in real analysis. A strong student can take this concurrently. Some review of these concepts will take place when they are used.

A rough outline of this first semester is:

Brief review of measure theory
Probabiity spaces
Laws of large numnbers
Characteristics functions
Central limit theorems
Infinitely divisible laws
Conditional expectations
Martingales
Convergence of laws on separable metric spaces.

The main text book is Real Analysis and Probability, by R. M. Dudley, 1989, Wadsworth & Brooks/Cole. The first half of this book is devoted to real analysis so the student can review the prerequisites.

Professor Michael Marcus
email: mbmarcus@earthlink.net

MATH 86300: Functional Analysis
Prof. Stanley Kaplan
M&W, 2:30 - 3:45pm

An introduction to such topics as: topological vector spaces, Banach Algebras and operators in Hilbert space, culminating in the spectral theorem for bounded and unbounded self-adjoint operators. If there is sufficient student interest, the hope is that a second semester follow-up course would be offered. Prerequisites: Real Analysis and a bit
of Complex Analysis.

MATH 87100: Group Theory
Prof. Gilbert Baumslag
Fri, 11:45 - 1:45pm

The purpose of this course is to discuss various embedding theorems for finitely presented groups. These embeddings theorems will include the famous theorem of Higman, characterising the finitely generated subgroups of finitely presented groups, work of Philip Hall on metabelian groups, the theorem that all finitely generated metabelian groups can be embedded into finitely presented metabelian groups and some new embedding theorems whereby a number of wreath-like products are explicitly embedded into finitely presented groups.&nbps; knowledge of amalgamated products, HNN extensions and some basic group theory will be assumed. Much use will be made of the book Combinatorial Group Theory by Lyndon and Schupp, where much of the necessary pre-requisites for this course can be found.

MATH 87400: Analytic Methods in Diophantine Geometry
Prof. Carlos Moreno
T&Th, 10 - 11:45am

Part I: Survey of the modern theory of L-functions

1. Hecke's theory of abelian L-functions(Tate's local approach)
2. Artin's L-functions on the Weil group of a number field (Proof of Weil's
modification of the Brauer induction Theorem)
3. Automorphic L-functions [Arithmetic and analytic consequences of the local
Langlands reciprocity law for GL(n)]

Part II: The explicit formulas of number theory

1. The Riemann explicit formula
2. Weil's explicit formula for Artin-Hecke L-functions
3. Explicit formulas for automorphic L-functions on GL(n)

Part III: Bounds on discriminants and conductors

1. The Stark-Odlysko bounds on discriminants and conductors
2. The Serre-Poitou method based on the explicit formulas
3. Bounds for the conductors of isogeny classes of abelian varieties (Mestre
bounds)
4. Diophantine consequences of the Multiplicity One Theorem for GL(n)

GENERAL REMARKS

The course will run in two parts: one on Tuesdays, and another on Thursdays.
I: The Tuesday class will develop the basic language:

Definitions of Zeta and L-functions
Proof of functional equations
Siegel's proof of Minkowski's theorem
Weil's proof of the explicit formulas
Poitou-Serre bounds for discriminants

This will be accomplished by working out a carefully chosen set of excercises from algebraic number theory. The prerequisites are a basic knowledge of algebra and complex variables.

II: The Thursday class will be run as a topics seminar and will attempt to survey the present state of the art concerning L-functions, Explicit formulas, and open research problems in related areas of analytic number theory.

Information: cmoreno@gc.cuny.edu