Spring 2002 courses in the RutgersNew Brunswick Math Graduate Program
Information about the table below
Go to the compact version at the bottom of this page.
Warning
All of the entries are currently correct and intended to still be
correct at the start of the semester. Updating will occur when the
maintainer of this page is informed. Experience has shown that
schedules, instructors, and topics will change. The word "Probably" is
used here as a periodic reminder of this fact.
Course number
640 is the prefix for Mathematics courses and 642 is the prefix
for Applied Mathematics courses. There is now no distinction in
degree requirements for Mathematics and Applied Mathematics, and to
some extent there has been a steady decrease in any emotional or
intellectual separation which might have existed.
Course name
This is the official course name which is in the Rutgers system for
the course corresponding to the given course number. There may be
little relationship between this name and the course contents.
Instructor
Probably.
Place
Probably. HLL refers to Hill Center. The Graduate Program and the Math
Department control the use of only a few (four) real classrooms. There
are other spaces available which have sometimes been used for class
meetings.
Days, period; times
Probably. Constraints on meeting times include times which should be
left free for faculty meetings, times left free for traditional
seminar meetings, and 8 AM. Most faculty and almost all graduate
students are quite unwilling to admit that 8 AM exists as a time for
intellectual converse. Note: Monday=M, Tuesday=T, Wednesday=W,
Thursday=Th, Friday=F. The "period" refers to the Rutgers 80minute
period. Period 1 begins at 8:10. There are 20 minutes between periods.
Informal description
Faculty were asked to supply informal descriptions of their
courses. These descriptions were edited mildly. Some descriptions
which are given may change, hopefully not too much.
Seminar on college teaching
A seminar organized by the Graduate School called Introduction to
College Teaching will run during the first twelve weeks of
the spring semester. Please see the indicated link for more
information.
Course number  Course name  Instructor  Place Days, period; times 
Informal description  

640:502  Theor Func Real Vari  A. Shadi TahvildarZadeh  HLL 525 TTh 5; 2:504:10 
This is a continuation of 501. We'll discuss basic theory of Banach and Hilbert spaces, differentiation, elements of functional analysis, and applications.  
640:504  Theor Func Comp Vari  R. Nussbaum  HLL 425 MW 6; 4:305:50 
We shall begin with normal families and the Riemann mapping theorem, if those topics are not covered in 503. We shall then cover the basic theory of entire functions and results on the location of zeros of entire functions. Applications will be given to "differential delay" equations. Other topics will be selected from the GreeneKrantz book, "Function Theory of One Complex Variable" (used the first term). In particular we shall discuss the gamma and beta functions and prove the prime number theorem. We also hope to cover some topics from the theory of dynamics of iterates of analytic maps.  
640:509  Sel Topics in Analysis  A. Soffer  HLL 425 TF 3; 11:3012:50  Introduction to spectral theory and dispersive wave
equations;
functional analytic and commutator methods in the study of the
spectral
properties of self adjoint operators in PDE and scattering theory. Required: real analysis and basics of Hilbert space theory.  
640:510 CANCELLED 
Sel Topics in Analysis  M. Kruskal  HLL 124 MW 2; 9:5011:10 
Arithmetic and analysis of surreal numbers The surreal number system is a relatively new and fascinatingly rich creation by J. H. Conway. In a compellingly natural way, it simultaneously encompasses the usual real numbers, Cantor's ordinal numbers, and a slew of infinite and infinitesimal numbers of enormously varied sizes as well as all sorts of combinations thereof. (Though superficially similar to the hyperreal number systems of nonstandard analysis, it is actually quite different). The numbers, as well as the operations and relations on them, are defined very simply and explicitly, and their elementary arithmetic properties have strikingly simple proofs with virtually no special cases, so that even restricted to the real numbers the treatment is a great improvement on the somewhat intricate classical development of the real numbers. The course will be selfcontained, since the approach adopted is considerably simpler than that available in the literature. Prerequisites: None, except for a modicum of mathematical maturity and some familiarity with elementary set theory. 

640:518  Partial Diff Equations  Y. Li  HLL 425 MW 2; 9:5011:10 
The course will be on elliptic equations, linear and nonlinear. In the first half of the semester, I will present some basic estimates for linear elliptic equations of second order. The second half of the course will be on fully nonlinear elliptic equations of MongeAmpere type. I will perhaps also include some current research works on some conformally invariant fully nonlinear elliptic equations arising from geometry.  
640:519 CANCELLED 
Sel Topics in Diff Equ  J. Taylor  HLL 124 TF 2; 9:5011:10 
I plan to discuss the Geometric Measure Theory alternative to PDE, as in the AlmgrenTaylorWang approach to motion by weighted mean curvature and attempts to extend to the polycrystalline case. I would also talk about the crystalline curvature approach when a surface free energy function is nondifferentiable. Novaga will be visiting in the spring for a month or two, and I hope to have him talk about his approach to crystalline problems, which is a version of PDE.  
640:532  Differential Geometry  P. Feehan  HLL 525 MTh 3; 11:3012:50 
List of possible topics: Differential manifolds, connections on principal and vector bundles, Riemannian geometry, Morse theory, variational calculus, curvature, characteristic classes. Topics may vary, depending on class background and interests.  
640:534  Sel Topics in Geometry  R. Goodman  HLL 525 MW 5; 2:504:10 
Topics in Representation Theory Text: Goodman & Wallach, Representations and Invariants of the Classical Groups (Cambridge) Prerequisites: Some knowledge of Lie algebras and/or Lie groups (e.g. 640:449 and/or 640:550), and some basic commutative algebra (elementary algebraic geometry). Possible topics:


640:541  Intro Alg Topology  F. Luo  HLL 423 TF 2; 9:5011:10 
This course is a continuation of the fall semester
course on algebraic topology.
We plan to cover the following topics: 1. Cohomology theory. 2. Elementary differential topology (Lie groups, differential forms, de Rham cohomology, integration of forms, Stokes' theorem, and Morse theory). 3. Duality theorems (Poincaré, Lefschetz, and Alexander theorems). 4. Elementary homotopy theory (homotopy groups, fiber spaces, Hurewicz theorem, EilenbergMac Lane spaces, and elementary obstruction theory). 5. Applications (Lefschetz fixed point theorem, Hopf index theorem, and Bott periodicity theorem) 

640:552  Abstract Algebra  W. Vasconcelos  HLL 423 TTh 4; 1:102:30 
This is the continuation of Math 551, aimed at
a discussion of many fundamental algebraic structures. It is strongly
recommended to students with interest in algebra and algebraic
combinatorics.
Topics include: Galois Theory Finite algebraic extensions, resolutions of equations by radicals (and without radicals) Rings of Polynomials Noetherian rings, Hibert basis theorem, Noether normalization, Nullstellensatz Basic Module Theory Projective and injective modules, resolutions, baby homological algebra, Hilbert syzygy theorem Category Theory Basic categories and their functors 

640:555:01  Sel Topics in Alg  C. Sims  HLL 525 MW 4 1:102:30 
Algebraic Algorithms The idea is to present a number of fundamental algorithms from group theory, number theory, and commutative algebra. Besides providing descriptions of the algorithms and proofs of correctness, the course would introduce students to good implementations of these algorithms. Students will use various software packages, including Maple and GAP. Prequisites should be at least concurrent enrollment in 552. 

640:555:02  Sel Topics in Alg  R. Solomon  HLL 525 MW 2 9:5011:10 
The Classical Linear Groups The course will discuss the general linear groups over fields and their classical subgroups. These groups include "most" of the simple Lie groups, linear algebraic groups and finite groups. They play an important roles in areas ranging from number theory to physics. Insofar as possible we shall work over arbitrary fields (and occasionally skew fields). 1. Definitions of the general and special linear groups, symplectic, orthogonal and unitary groups. 2. Normal subgroups and quotients: simplicity of PSL(n,K), PSp(n,K), etc. 3. Semisimple and unipotent elements and the Jordan decomposition. Tori and unipotent subgroups. 4. Parabolic and reductive subgroups. Bruhat decomposition. Stabilizers of totally singular subspaces and orthogonal decompositions. 5. Centralizers of semisimple elements and some unipotent elements. Application to HallHigman Theorem B. 6. Aschbacher's classification of maximal subgroups of classical linear groups over finite fields (with some extensions to infinite fields) and comparison with the O'NanScott classification of maximal subgroups of the finite symmetric groups. Prerequisite: the basic graduate algebra course 

640:555:03 CANCELLED 
Sel Topics in Alg  L. Carbone  HLL 423 MTh 3; 11:3012:50 
Group actions on buildings We will study groups that naturally admit actions on locally finite buildings as defined by Bruhat and Tits. Examples of such groups are simple algebraic groups over nonarchimedean local fields, and groups of KacMoody algebras over finite fields. 

640:558  Theory of Algebras  E. Taft  HLL 525 TTh 4; 1:102:30 
Hopf Algebras and Quantum Groups: The structure of Hopf algebras with quantum groups as principal examples. 

640:566  Axiomatic Set Theory  A. Hajnal  HLL 425 TTH 6; 4:305:50 
The only prerequisite for this course is a good understanding of naive
set theory. We will give an axiomatic development of set theory, we
will discuss some independence results obtained by forcing, but the
main focus will be modern results on cardinal exponentiation. Classical results on the subject were obtained in the years between 1870 and 1930. There was a general consensus that every question beyond these results would be independent from the axioms. No further progress was made until Cohen's discovery of forcing. After that the expectation was partly fulfilled. It was soon proved to be consistent that no new rules are valid for the cardinality of the powersets of regular cardinals. After a decade of futile attempts to generalize these consistency results for singular cardinals, in 1974, it turned out that for singular cardinals the situation is different. J. Silver proved that if GCH holds under a singular strong limit cardinal of uncountable cofinality, then it is true for this cardinal as well. In the wake of this discovery more general results were soon proved. Finally, the subject was developed by S.Shelah to a beautiful theory with some fantastic results. Here is one of them: (aleph_{omega})^{aleph0}< 2^{(aleph0+) }+aleph_{(omega4)}. We will discuss this theory and its main applications. Reference: Introduction to Cardinal Arithmetic by M. Holz, K. Steffens, and E. Weitz, published by Birkhauser (1999). 

640:573  Spec Top Number Theory  H. Iwaniec  HLL 124 TF 3; 11:3012:50 
This will be a
continuation of the subject I am teaching this fall.
The title will be
Applications of Spectral Theory of Automorphic Forms.
Here is a brief description of the topics to be presented.


640:574  Spec Top Number Theory  S. Miller  HLL 425 MTh 3; 11:3012:50 
Automorphic forms for SL(n,Z) and constructions
of their Lfunctions. Recommended prerequisite: Iwaniec's fall course. 

642:528  Methods of Appl Math  G. Goldin  HLL 124 TTh 6; 4:305:50 
The course focuses on complex variable methods in applied mathematics, suitable especially for students of engineering or physics or related disciplines. It is a "how to" course, with a minimum of theoretical mathematics and a lot of problem solving and applications.  
642:562  Intro Math Physics  E. Speer  HLL 425 TTh 5; 2:504:10 
We will study classical mechanics from a rigorous mathematical viewpoint.  
642:581  Applied Graph Theory  J. Kahn  HLL 425 TF 2; 9:5011:10 
We'll try to cover central topics in graph theory at introductory graduate level, e.g. matching theory; Menger's Theorem, connectivity and related; minors; coloring problems; extremal problems; probably some random and/or algebraic material.  
642:583  Combinatorics  J. Beck  HLL 525 TF 2; 9:5011:10 
This Part II of the standard Combinatorics course is
rather independent of Part I (Fall 2001). We are going to discuss
further applications of Combinatorics in other fields like Number
Theory, Geometry, Fourier Analysis, and so on.
The main source of our course is
L. Lovász:Combinatorial problems and exercises, North
Holland. A brief list of what we are going to cover is the following.
Prerequisites: Knowledge of Part I (Fall'01) doesn't hurt, but this is a moreorless independent selfcontained course. I plan to write notes. 

642:662:01  Sel Topics in Math Physics  S. Goldstein  HLL 423 MW 5; 2:504:10 
Introduction to Bohmian Mechanics and the
Foundations of Quantum Mechanics Quantum theory is the most successful physical theory yet devised. It is also bizarre, bordering on incoherent. It is widely believed that this quantum weirdness is inevitable. This course will be an introduction to the foundations of quantum mechanics (i.e., quantum, weirdness) and to Bohmian mechanics, a simple formulation of nonrelativistic quantum mechanics that is a counterexample to the claim that quantum weirdness is unavoidable. 

642:662:02  Sel Topics in Math Physics  Y. Huang  HLL 423 MW 4; 1:102:30 
We shall develop mathematical conformal field theory through the most wellunderstood examples: the WZNW models and the minimal models. Different aspects of the theory, including representation theory, differential equations, braided tensor categories and modular invariance, will be discussed in a unified framework. 
Beginning the spring semester
 The first day of class of the spring 2002 semester is Tuesday, January 22, 2001.
 Written qualifying exams will be given during the week before the semester begins, Wednesday and Thursday morning (January 16 and 17).
More complete descriptions of courses including information about texts are usually posted outside the 3^{rd} floor mailroom. Almost all introductory graduate courses in mathematics are given as a series of lectures. Most such courses have written homework, and one or more oral or written examinations. Many basic courses have assigned texts. More advanced courses depart from these rules. Students are sometimes asked to lecture, and there are rarely assigned texts.
The compact schedule
Links here are to further information above about each course.
Go to the top of this page. Last Modified 12/7/2001.