Spring 2002 courses in the Rutgers-New Brunswick Math Graduate Program

Spring 2002 courses in the Rutgers-New 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 80-minute 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 Tahvildar-Zadeh HLL 525
TTh 5; 2:50-4:10
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; 4:30-5:50
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 Greene-Krantz 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; 11:30-12:50
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; 9:50-11:10
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 self-contained, 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; 9:50-11:10
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 Monge-Ampere 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; 9:50-11:10
I plan to discuss the Geometric Measure Theory alternative to PDE, as in the Almgren-Taylor-Wang 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 non-differentiable. 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; 11:30-12:50
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; 2:50-4:10
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:
  • Classical Groups and their irreducible finite-dimensional representations: Theorem of highest weight, Weyl models on tensor spaces, Weyl character formula
  • Duality Theory: Double Commutant Theorem, Schur Duality, Howe Duality
  • Classical Invariant Theory: First and Second Fundamental Theorems for GLn and other classical groups
  • Spectral Analysis: Multiplicity-free spaces, Spherical Harmonics
640:541 Intro Alg Topology F. Luo HLL 423
TF 2; 9:50-11:10
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, Eilenberg-Mac 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; 1:10-2:30
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 1:10-2:30
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 9:50-11:10
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 Hall-Higman 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'Nan-Scott classification of maximal subgroups of the finite symmetric groups.
Pre-requisite: the basic graduate algebra course
640:555:03
CANCELLED
Sel Topics in Alg L. Carbone HLL 423
MTh 3; 11:30-12:50
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 non-archimedean local fields, and groups of Kac-Moody algebras over finite fields.
640:558 Theory of Algebras E. Taft HLL 525
TTh 4; 1:10-2:30
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; 4:30-5:50
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: (alephomega)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; 11:30-12:50
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.
  1. Spectrum distribution
  2. The theory of Kloosterman sums
  3. The closed geodesic theorem
  4. Equidistribution problems
    1. for points on the sphere
    2. for the CM points
    3. for roots of quadratic congruences
  5. The automorphic L-functions (if time permits?)
640:574 Spec Top Number Theory S. Miller HLL 425
MTh 3; 11:30-12:50
Automorphic forms for SL(n,Z) and constructions of their L-functions.
Recommended prerequisite: Iwaniec's fall course.
642:528 Methods of Appl Math G. Goldin HLL 124
TTh 6; 4:30-5:50
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; 2:50-4:10
We will study classical mechanics from a rigorous mathematical viewpoint.
642:581 Applied Graph Theory J. Kahn HLL 425
TF 2; 9:50-11:10
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; 9:50-11:10
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.
  1. Combinatorial geometry
  2. Analytic methods like Roth's theorem on 3-term arithmetic progressions
  3. Fourier analysis and combinatorics
  4. Erdös type combinatorial number theory
  5. More of the Probabilistic Method
Students are expected to solve homework.
Prerequisites: Knowledge of Part I (Fall'01) doesn't hurt, but this is a more-or-less independent self-contained course.
I plan to write notes.
642:662:01 Sel Topics in Math Physics S. Goldstein HLL 423
MW 5; 2:50-4:10
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; 1:10-2:30
We shall develop mathematical conformal field theory through the most well-understood 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).
"Tea" (coffee/tea/etc.) and "cookies" (variable in type and quantity) are available most afternoons (3:30-4:30 PM) during the semester, usually in the lounge on the 7th floor. On rare occasions even more food appears. There are many interesting seminars almost every week.

More complete descriptions of courses including information about texts are usually posted outside the 3rd 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.

Period & times Monday Tuesday Wednesday Thursday Friday
2  9:50-11:10 AM 640:518 HLL 425
640:555:02 HLL 525
640:510 HLL 124
640:541 HLL 423
642:581 HLL 425
642:583 HLL 525
640:518 HLL 425
640:555:02 HLL 525
640:510 HLL 124
  640:541 HLL 423
642:581 HLL 425
642:583 HLL 525
3  11:30 AM-12:50 PM 640:532 HLL 525
640:555:03 HLL 423
640:574 HLL 425
640:509 HLL 425
640:573 HLL 124
Faculty should leave this time open. 640:532 HLL 525
640:555:03 HLL 423
640:574 HLL 425
640:509 HLL 425
640:573 HLL 124
4  1:10-2:30 PM 640:555:01 HLL 525
642:662:02 HLL 423
640:552 HLL 423
640:558 HLL 525
640:555:01 HLL 525
642:662:02 HLL 423
640:552 HLL 423
640:558 HLL 525
Faculty should leave this time open.
5  2:50-4:10 PM 640:534 HLL 525
642:662:01 HLL 423
640:502 HLL 525
642:562 HLL 425
640:534 HLL 525
642:662:01 HLL 423
640:502 HLL 525
642:562 HLL 425
 
6  4:30-5:50 PM 640:504 HLL 425 640:566 HLL 425
642:528 HLL 124
640:504 HLL 425
640:566 HLL 425
642:528 HLL 124
Colloquium (4:30-5:30)
Almost always in HLL 705.
7  6:10-7:30 PM          

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