Courses
30000, 30100. Set Theory I,II
Hirschfeldt
Math 30000 is a course on axiomatic set theory. Topics include the axioms
of Zermelo-Frankel (ZF) set theory; ordinals and cardinals; infinitary
combinatorics; Von Neumann rank and reflection principles; absoluteness;
inner models; Gödel’s constructible sets (L); and the consistency
of the Axiom of Choice (AC) and the Generalized Continuum Hypothesis (GCH).
Math 30100 deals with models of set theory; Cohen’s method of forcing
and the independence of AC and CH; Martin’s axiom and the unprovability
of Souslin’s Hypothesis; Solovay’s model in which every set
of reals is Lebesgue measurable; larger cardinals (measurable cardinals,
elementary embeddings, and compactness); the axiom of determinateness;
and possibly some descriptive set theory. Prereq: Consent of instructor.
30200. Computability Theory I (Ident to CMSC 38000)
Soare
Math 30200 begins with models for defining computable functions such as
the recursive functions and those computable by a Turing machine. Topics
include the Kleene normal form theorem for representing computable functions
and computably enumerable (c.e.) sets; the enumeration and s-m-n theorem,
unsolvable problems, classification of c.e. sets, the Kleene arithmetic
hierarchy, coding of information from one set to another, various degrees
for measuring noncomputability, many-one, truth-table, and Turing degrees.
The course also includes the Kleene recursion theorem and its applications,
other fixed point theorems such as the Arslanov completeness criterion,
elementary properties of Turing degrees, generic sets, and the construction
of various non-c.e. degrees by oracle Kleene-Post constructions. Prereq:
Math 25500 or consent of instructor.
30300. Computability Theory II (Ident to CMSC 38100)
Soare
Math 30300 develops the deeper properties of computability and the classification
of relative computability on sets and (Turing) degrees. It begins with
the finite injury priority method of Friedberg and Muchnik, continues
with the infinite injury priority method of Sacks, and minimal pair of
computably enumerable (c.e) degrees method by Lachlan. It introduces the
tree method of Lachlan for classifying more difficult priority constructions,
and it works out many properties of the c.e. degrees and the algebraic
structure of the c.e. sets. It presents results on the relationship between
a c.e. set and the degree of information it encodes such as the high maximal
set theorem of Martin. Prereq: Math 30200.
30800. Intuitionistic Logic and Constructive Mathematics
Hirschfeldt
An introduction to constructivism in mathematics, with particular emphasis
on logical aspects. Topics include deduction systems for intuitionistic
logic, Kripke semantics, relationships between classical and intuitionistic
logic, intuitionistic arithmetic, principles employed in constructive
mathematics, constructive real numbers, and constructive analysis. Prereq:
Math 27700 or equivalent logic course.
30900. Model Theory I
Completeness and compactness; elimination of quantifiers; omission of
types; elementary chains; homogeneous models; two cardinal theorems by
Vaught, Chang, and Keisler; categories and functors; inverse systems of
compact Hausdorff spaces; applications of model theory to algebra. Prereq:
Math 25500 or Math 27900, or consent of instructor.
31000. Model Theory II
Saturated models; categoricity in power; Cantor-Bendixson and Morley derivatives;
Morley and Baldwin-Lachlan theorems on categoricity; rank in model theory;
uniqueness of prime models and existence of saturated models; indiscernibles;
ultraproducts; differential fields of characteristic zero. Prereq: Math
30900.
31200. Analysis I—Measure and Integration
Ryzhik
Measure theory, including Lebesgue and Hausdorff measures, integration,
convergence theorems, the Radon-Nikodym theorem, and differentiation theory.
Normed linear spaces, Lp spaces, completeness, duality, and the Riesz
representation theorem. Fourier series, the Poisson summation formula,
and the boundedness of the Hilbert transform. Prereq: Math 26200, 27000,
27200, and 27400.
31300. Analysis II—Functional Analysis
Constantin
Frechet spaces, spaces of smooth functions, weak topologies and weak convergence,
distributions and Fourier analysis, including mollifiers, convolution,
the Paley-Wiener theorem, and local solvability of constant coefficient
PDE. Sobolev spaces and the embedding theorems. Operator theory, including
compact and bounded operators, integral operators, spectral theory and
Fredholm operators. Applications to the representation theory of compact
groups (the Peter-Weyl theorem) and an introduction to the calculus of
variations. Prereq: Math 31200.
31400. Analysis III—Complex Variables
Narasimhan
A review of the basic theory of one complex variable: Cauchy’s theorem,
the Cauchy-Riemann equations, power series expansions, the maximum modulus
principle, classification of singularities, and the residue theorem. Normal
families, conformal mapping and the Riemann mapping theorem. Prescribing
zeros and poles of meromorphic functions. Harmonic functions and the Dirichlet
problem. Introduction to Riemann surfaces. Negative curvature and Picard’s
Big Theorem. According to the inclinations of the instructor, further
topics may include: holomorphic functions of several variables (e.g. Hartogs’
Theorem), a deeper study of Riemann surfaces, the uniformization theorem,
the Dirichlet problem in higher dimensions, differential equations in
a complex domain and the Riemann-Hilbert problem, Hardy spaces. Prereq:
Math 31300.
31700. Topology And Geometry I—Smooth Manifolds
Corlette
Definition of manifolds, tangent and cotangent bundles, vector bundles.
Inverse and implicit function theorems, transversality, Sard’s theorem
and the Whitney embedding theorem. Vector fields and flows, Frobenius’
theorem, differential forms and the associated formalism of pullback,
wedge product, integration, etc. Cohomology via differential forms, and
computational tools, e.g. the Poincaré lemma and the Mayer-Vietoris
sequence. The degree of a map between compact oriented manifolds. Lie
groups and Lie algebras. Prereq: Math 26100, 26200, 26300.
31800. Topology And Geometry II—Differential Geometry
Seidel
Riemannian metrics, connections and curvature on vector bundles, the Levi-Civita
connection, and the multiple interpretations of curvature. Geodesics and
the associated variational formalism (formulas for the 1st and 2nd variation
of length), the exponential map, completeness, and the influence of curvature
on the structure of a manifold (positive versus negative curvature). The
Gauss-Bonnet theorem and possibly the Hodge Theorem. Prereq: Math 31700.
31900. Topology And Geometry III—Basic Homology
Weinberger
The fundamental group, covering space theory and Van Kampen’s theorem
(with a discussion of free and amalgamated products of groups). CW complexes,
higher homotopy groups, cellular and singular cohomology, the Eilenberg-Steenrod
axioms, computational tools including Mayer-Vietoris, cup products, Poincaré
duality, and the Lefschetz fixed point theorem. Homotopy exact sequence
of a fibration and the Hurewicz isomorphism theorem. Remarks on characteristic
classes. Prereq: Math 31800.
32000, 32100, 32200. Mathematical and Statistical Methods for the Neuro-Sciences
I, II, III
Cowan
This three-quarter sequence is for students interested in computational
and theoretical neuroscience. It introduces various mathematical and statistical
ideas and techniques used in the analysis of brain mechanisms. The first
quarter introduces mathematical ideas and techniques in a neuroscience
context. Topics include some coverage of matrices and complex variables;
eigenvalue problems, spectral methods, and Green’s functions for
differential equations; and some discussion of both deterministic and
probabilistic modeling in the neurosciences. The second quarter treats
statistical methods that are important in understanding nervous system
function. It includes basic concepts of mathematical probability; and
information theory, discrete Markov processes, and time series. The third
quarter covers more advanced topics that include perturbation and bifurcation
methods for the study of dynamical systems, symmetry, methods, and some
group theory. A variety of applications to neuroscience are described.
Prereq: Students must have completed the equivalent of one year of college
calculus and a course in linear algebra such as MATH 25000 and preferably
a course in differential equations such as MATH 27300, and at least one
course in neurobiology such as BIOS 14106 or 24236, or NURB 31800.
32500. Algebra I—Group Theory
Alperin
Group theory. Linear groups, semisimple algebras and modules, and group
representations. Prereq: Math 25400, 25500, 25600.
32600. Algebra II—Commutative Rings and Homology
Kottwitz
Noetherian rings and modules, the Hilbert basis theorem. Integral extensions,
the going-up theorem. Localisation, exactness of localisation. Finitely
generated algebras over a field, varieties, the Noether normalisation
lemma, Hilbert’s Nullstellensatz, dimension. Discussion of the dictionary
between commutative algebra and algebraic geometry. Other possible topics
include: Kähler differentials, smoothness, completions, power series
rings, the p-adic numbers. Ext and Tor. Dedekind domains. The spectrum
of a commutative ring and the sheaf associated to a module. Prereq: Math
32500.
32700. Algebra III—Topics in Algebra
May
According to the inclinations of the instructor, this course may cover:
Galois theory, algebraic number theory, algebraic curves, multilinear
algebra (tensor, symmetric and exterior algebras), Lie algebras, homological
algebra and/or the cohomology of groups. Prereq: Math 32600.
32803,32904. Several Complex Variables I,II
Narasimhan
Intended for second and third year students. The course will treat: 1)
Domains of holomorphy, Stein Manifolds, pseudoconvexity and the Levi problem.
2) Coherent sheaves on Stein manifolds, applications. Analytic spaces,
basic local theory of such spaces (dimension, irreducibility, normality,
….) to Coherent Analytic Sheaves leading to Grauert’s theorem
on direct images and applications to deformation theory. Prereq: First
year analysis and topology
33103. Quantitative Topology
Weinberger
The aim of this course is to introduce second and third year graduate
students to some analytic methods which have proved to be very useful
in algebraic geometry. The first part of the course will be an introduction
to Kähler manifolds, curvature forms on vector bundles, Hodge theory
and the basic L? estimates (which, in some sense, generalize one aspect
of Hodge theory to non-compact manifolds). The rest of the course will
concentrate on applications (e.g. the Napier-Ramachandran use in proving
Lefschetz type theorems, deformation invariance of plurigenera of varieties
of general type, etc.).
33203, 35204. Introduction to Periods I,II
Bloch
33303,33404,33504. Applied Analysis I,II,III
Constantin and Ryzhik
The Applied Analysis sequence (I, II, III) is aimed at graduate students
who have completed the Analysis sequence. The course will cover topics
in mathematical physics (nonlinear and statistical). The course will be
self contained and taught in modules, by Constantin, Ryzhik and others.
The fall quarter will start with basic results in homogenization theory.
Prereq: Analysis sequence
33604. Elliptic curves and elliptic modules
Gaitsgory
The aim of the course is to explain how Langlands correspondence for GL(2)
is realized via the moduli space of elliptic curves over number fields
and the moduli space of elliptic modules over functions fields. The emphasis
will be on comparison between the two theories. Prereq: Basic course on
algebraic geometry; number theory (some knowledge of CFT).
33704. Topics in Algebraic Geometry
Nori
33803. Fully nonlinear equations
Nadirashvili
We will study boundary value problems and qualitative properties for solutions
of the fully nonlinear equations of elliptic type.
33904. Complex Analysis and Geometry
Webster
Geometric function theory of domains and complex manifolds; PDE methods;
Kaehler and other special metrics; CR manifolds as time permits. Prereq:
First year analysis and geometry sequences.
34004. Topics in Riemannian Geometry
Farb
This course will be an advanced course in Riemannian geometry. Topics
might include: spectral theory, harmonic maps, nonpositively curved manifolds,
Ricci flow, automorphism groups of geometric structures, affine geometry.
34100, 34200,34300. Geometric Literacy
Farb and Weinberger
This ongoing course might be subtitled: “what every good geometer
should know’’. The topics will intersperse more elementary background
with topics close to current research, and should be understandable to
second year students. The individual modules (2–5 weeks each) might
be logically interrelated, but we will try to maintain a “modular
structure” so that people who are willing to assume certain results
as “black boxes” will be able to follow more advanced modules
before formally learning all the prerequisites. This years topics might
include: basics of symplectic geometry, harmonic maps in geometry, pseudo-Anosov
homeomorphisms and Thurston’s compactification of Teichmuller space,
algebraic geometry for non-algebraic geometers. Prereq: First year graduate
sequence.
35104. Topics in Calculus of Variations
Venkataramani
The course will focus on the interplay between weak converence methods
for PDEs and convex analysis. We will discuss the analysis of variational
problems, the role of convexity, and some applications to nonlinear elasticity
and material science. Depending on the interests of the class, we might
also discuss some geometric measure theory. Prereq: A knowledge of measure
theory and Functional analysis, at the level of the first year analysis
grad classes.
35304. Group actions and homotopy theory
Grodal
The aim of this course to give an modern introduction to group actions,
seen from the viewpoint of homotopy theory. After building up some useful
tools, we will look at applications. One application is the theory of
p-compact groups, which are homotopy theoretic versions of compact Lie
groups and p-local finite groups, which are homotopy theoretic versions
of finite groups. Another application is to use the general tools to classify
group actions on spaces such as to use the general tools to classify group
actions on spaces such as spheres. The exact contents will depend on the
participants. The course should be accessible to graduate students who
have completed the first year program, and might be of interest to students
focusing on algebra or topology in its various guises.
36000, 36100, 36200. Topology Proseminar
May
As a regular feature of the graduate mathematics program, there is an
informal topology ``proseminar” that is devoted primarily to algebraic
topology, but is often concerned with topics of interest to people in
such neighboring fields as algebraic geometry, geometric topology, and
group theory. The proseminar is run by Professor May, who often talks
on requested topics or on topics relevant to upcoming topology seminars.
Talks are also given by junior faculty, by graduate students working in
algebraic topology, and by visitors to the department. Sometimes there
is no prearranged topic, but rather informal discussion initiated by questions
asked by participants.
36303. Analytic functions of several complex variables
Baily
This will include Cauchy's theorem and other standard facts for several
complex variables, complex analytic spaces, and a proof of Chow's theorem
that a complex analytic subspace of complex projective space is a projective
algebraic variety, as well as Fourier expansions of complex analytic functions
periodic in their real parts with respect to some lattice in R^n. Prereq:
Functions of a complex variable and the necessary basic material in algebra.
36404. Siegel's modular forms, functions, and results on algebraic dependence
for such modular forms and functions
Baily
I shall consider conections with moduli of Abelian varieties and compactifications
of the Siegel modular space. This will naturally depend on the results
of the autumn course, and will include the rationality of the Fourier
expansions of Eisenstein series under certain hypotheses.
37500. Algorithms In Finite Groups (Ident to: CMSC 36500)
Babai
We shall consider the asymptotic complexity of some of the basic problems
of computational group theory. The two classes of groups highlighted will
be permutation groups and matrix groups. The course will demonstrate the
relevance of a delightful mix of mathematical techniques, ranging from
combinatorial ideas, the elements of probability theory, and elementary
group theory, to the theories of rapidly mixing Markov chains, applications
of simply stated consequences of the Classification of Finite Simple Groups
(CFSG), and occasionally, detailed information about finite simple groups.
We shall go in some depth into the theory of permutation groups, combining
19th century style combinatorial approaches with techniques relying on
CFSG. Prereq: Linear algebra, finite fields, a first course in group theory
(Jordan-Holder and Sylow theorems), elements of probability theory (Chebyshev's
inequality). All other requisite subjects will be reviewed in class before
used. No prior knowledge of the theory of algorithms is required.
38004. Applications of harmonic analysis to PDE
Kenig
The course will consist of three modules in which the applications of
harmonic analysis to 3 areas of pde will be discussed.
•Module 1:Applications to free boundary problems.
•Module 2:Applications to unique continuation problems.
•Module 3:Applications to non-linear evolution equations.
Prereq: First year analysis sequence.
38300. Numerical Solutions to PDEs (Ident to: CMSC 38300)
Dupont
This course covers the basic mathematical theory behind numerical solution
of partial differential equations. The course will investigate the convergence
properties of finite element, finite difference and other discretization
methods for solving partial differential equations. A brief introduction
to Sobolev spaces and polynomial approximation theory will be given. Special
emphasis on error estimators, adaptivity and optimal-order solvers for
linear systems arising from PDEs. Special topics include (from time to
time) PDEs of fluid mechanics, max-norm error estimates, and Bananch-space
operator-interpolation techniques. Prereq: Consent of instructor.
38500. Applied Mathematics Literacy
Scott
This ongoing course, analogous to Geometric Literacy, might be subtitled:
``some things every good applied mathematician should know.'' The topics
will intersperse elementary background with topics in current research,
and will be understandable by second year math grad students. The individual
modules (hopefully 3 weeks long, but maybe 2-5 weeks each) will allow
people to re-start if interest or focus diverges. Topics for fall 2003
will include:
•models for fluids from Newton to Rivlin and Eriksen (existence,
uniqueness, computational algorithms);
•models for economic equilibrium based on Monge-Ampere type equations
Guest lectures by experts on particular subjects will be featured.
Prereq: None, but Analysis I or equivalent would be useful.
40103. Topics in algebra
Kottwitz
This course will consist of three (or maybe more) separate (but related)
topics in algebra. Some likely choices for topics are 1) algebraic stacks
(with emphasis on examples rather than general theory) 2) simplicial sets
and spaces, including sheaves on them 3) closed model categories. I'll
be trying to get across some of the basic ideas rather than trying to
cover lots of material. Prereq: First year courses.
40203. Introduction to Lie Algebras
Ginzburg
This is a course on the basics of Lie algebras. We begin with nilpotent
and solvable Lie algebras, and prove Engel and Lie theorems. The rest
of the course is mostly devoted to the structure of semisimple Lie algebras,
i.e., Cartan subalgebras, the Killing form, root system, etc. The course
is completed by proving most fundamental results on finite dimensional
representations of semisimple Lie algebras: complete reducibility, Weyl
character formula, highest weight classification of irreducible finite
dimensional representations.
41504. Topics in representation theory
Kottwitz
The course will probably start with the basics of the classical representation
theory of p-adic groups but may include other topics as well, for instance
topics in geometric representation theory. Prereq: First year courses.
42004. Computational Neuroscience
Cowan
42104. Topics in Representation theory
Ginzburg
The course is intended to be an introduction to both finite and infinite
dimensional representation theory of Lie groups and Lie algebras. We begin
by reviewing classical theory of finite-dimensional representations, in
particular, Borel-Weil-Bott theorem. Then, we are going to discuss the
Bernstein-Gelfand-Gelfand theory, Harish-Chandra modules, and also applications
to the structure of unitary representations of reductive groups.
42204. Lie algebra homology, Equivariant cohomology and Koszul duality
Ginzburg
We will survey the "public domain" area between homological
algebra, Lie theory and Topology, and will try to describe as many applications
as possible of objects involved in the title beyond those areas. We begin
with basic definitions for Lie algebra cohomology, and prove classical
results on cohomology of semisimple Lie algebras. Applications to the
topology of compact groups, e.g. computation of Betti numbers of a compact
Lie group will be given. The role of Lie algebra cohomology in Deformation
theory and general Moduli problems (e.g. moduli of vector bundles) will
be discussed. We then introduce equivariant cohomology in two different
ways: one is based on the notion of the classifying space, the other is
the "infinitesimal model" of Cartan-Weil. Localization theorem
for equivariant cohomology of a torus action will be proved. We finally
discuss the general notion of a Koszul algebra, and Koszul duality, the
basic example being a duality between the Symmetric and Exterior algebra.
It turns out that Koszul duality clarifies relationships between equivariant
and ordinary cohomology (recent work by Goresky-Kottwitz-MacPherson).
47000, 47100, 47200. Geometric Langlands Seminar
Beilinson and Drinfeld
This seminar is devoted not only to the Geometric Langlands theory but
also to related subjects (including topics in algebraic geometry, algebra
and representation theory). We will try to learn some modern homological
algebra (Kontsevich’s A-infinity categories) and some “forgotten”
parts of D-module theory (e.g. the microlocal approach).
This list was last revised on 9/23/2003.
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