110.106 (Q)

CALCULUS I (4) Budur

For Biological and Social Sciences Majors  Limit 25 per section    Differential and integral calculus. Includes analytic geometry, functions, limits, integrals and derivatives, introduction to differential equations, functions of several variables, linear systems, applications for systems of linear differential equations, probability distributions. Many applications to the biological and social sciences will be discussed.

Lec.

Sec. 01

02

03

MTW 10

F 9

F 12

Th 10:30-11:20

110.107 (Q)

CALCULUS II (4) Song

For Biological and Social Sciences Majors    Limit 28 per section

Prereq: Calculus I    Differential and integral calculus. Includes analytic geometry, functions, limits, integrals and derivatives, introduction to differential equations, functions of several variables, linear systems, applications for systems of linear differential equations, probability distributions. Many applications to the biological and social sciences will be discussed.

Lec.

Sec. 01

02

03

04

 05

06

MTW 10

F 9

F 9

F 12

F 12

Th 10:30-11:20

Th 10:30-11:20

110.109 (Q)

CALCULUS II (4) Howald Howard

For Physical Sciences and Engineering Majors   Limit 28 per section

Prereq: Calculus I    Differential and integral calculus. Includes analytic geometry, functions, limits, integrals and derivatives, polar coordinates, parametric equations, Taylor's theorem and applications, infinite sequences and series. Some applications to the physical sciences and engineering will be discussed, since the course is designed to meet the needs of students in these disciplines.

Sec. 05 canceled 12/29/04

Lec.

Sec. 01

02

03

04

05

06

MTW 10

F 9

F 12

F 12

Th 10:30-11:20

Th 10:30-11:20

F 9

110.201 (Q)

LINEAR ALGEBRA (4) Consani

Limit 25 per section   Prereq: Calculus I Vector spaces, matrices, and linear transformations. Solutions of systems of linear equations. Eigenvalues, eigenvectors, and diagonalization of matrices. Applications to differential equations.

Lec. I

Sec. 01

02

03

04

05

06

MTW 10

F 9

Th 10:30-11:20

F 12

F 12

F 9

Th 10:30-11:20

110.202 (Q)

CALCULUS III (4) Morava/Spinu

Limit 25 per section 

Prereq: 110.107, 110.109 or 110.112. Calculus of functions of more than one variable: partial derivatives, and applications; multiple integrals, line and surface integrals; Green's Theorem, Stokes' Theorem, and Gauss' Divergence Theorem.

Lec. I

Sec. 01

02

03

04

Lec. II

05

06

07

08

MTW 10

Th   9

Th 12

Th F   9

F 12

MTW 10

Th   9

Th 12

F   9

F 12

110.204 (Q)

ELEMENTARY NUMBER THEORY (4) Shalika   Limit 30   Prereq: a good high school background including a year of Calculus. Course provides many historical examples of topics, which serves as an illustration of and provides a background for many years of current research in number theory. Also provides concrete examples of general abstract concepts studied in 110.401-402. Primes and prime factorization, congruences, Euler's function, quadratic reciprocity, primitive roots, solutions to polynomial congruences (Chevalley's theorem), Diophantine equations including the Pythagorean and Pell equations, Gaussian integers, Dirichlet's theorem on primes.

Lec.

Sec. 01

MTW 11

F 12

110.211 (Q)

HONORS CALCULUS III (4) Zucker Limit 35    Prereq: B+ or better in Calculus II or 5 in the BC AP exam   This course includes the material in Calculus III (202) with some additional applications and theory. Recommended for mathematically able students majoring in physical science, engineering, or especially mathematics. 110.211-212 used to be an integrated year-long course, but now the two are independent courses and can be taken in either order.

Lec.

Sec. 01

MTW 12

F 12 Th 10

110.212 (Q)

HONORS LINEAR ALGEBRA  (4) Wentworth   Limit 45 35    Prereq: Calculus II or III or equivalent, preferably honors. This course includes the material in Linear Algebra (201) with some additional applications and theory. Recommended for mathematically able students majoring in physical science, engineering, or mathematics. 110.211-212 used to be an integrated year-long course, but now the two are independent courses and can be taken in either order.  This course satisfies a requirement for the math major that its non-honors sibling does not.

Lec.

Sec. 01

MTW 12

F 12 Th 10

110.302 (E,Q)

DIFFERENTIAL EQUATIONS WITH APPLICATIONS (4) Mese Limit 35 30 per section   Prereq: Calculus III This is an applied course in ordinary differential equations, which is primarily for students in the biological, physical and social sciences, and engineering. The purpose of the course is to familiarize the student with the techniques of solving ordinary differential equations. The specific subjects to be covered include first order differential equations, second order linear differential equations, applications to electric circuits, oscillation of solutions, power series solutions, systems of linear differential equations, autonomous systems, Laplace transforms and linear differential equations, mathematical models (e.g., in the sciences or economics).

Lec.

Sec. 01

02

03

04

MTW 1

Th 10:30-11:20

Th 12

F  9

F 12

110.402 (Q)

ADVANCED ALGEBRA II (4.5) Ono Limit 30   Prereq: 110.401 This is a continuation of 110.401. Theory of Fields Splitting field of a polynomial, algebraic closure of a field. Galois theory: correspondence between subgroups and subfields. Solvability of polynomial equations by radicals. Modules over a ring. Principal ideal domains, structure of finitely generated modules over them.

Sec. 01

ThF 12-1:15
Th 9

110.405 (Q)

ANALYSIS I (4.5) Xu     Limit 35  Prereq: Calculus III, Linear Algebra Course is designed to give a firm grounding in the basic tools of analysis. It is recommended as preparation (but may not be a prerequisite) for other advanced analysis courses. Real and complex number systems, topology of metric spaces, limits, continuity, infinite sequences and series, differentiation, Riemann-Stieltjes integration.

Sec. 01

MTW 1
F 9

110.406 (Q)

ANALYSIS II (4.5) Minicozzi   Limit 35 Prereq: 110.405     This course continues 110.405 notions of modern analysis. Sequences and series of functions, Fourier series, equicontinuity and the Arzela-Ascoli theorem, the Stone-Weierstrass theorem. Functions of several variables, the inverse and implicit function theorems, introduction to the Lebesgue integral.

Sec. 01

MTW 1
F 9

110.411 (Q)

HONORS COMPLEX ANALYSIS (4.5) Zhang Yang  Prereq: 110.201, 110.202, or Perm Req’d.     Study of functions of a complex variable, emphasis on interrelations with other parts of mathematics. Topics include Cauchy’s theorems, singularities, gamma and zeta functions, elliptic functions, theta functions, Jacobi’s triple product.

Lec.  Sec. 01

MTW 2              F 12

110.413 (Q)

INTRODUCTION TO TOPOLOGY (4.5) Boardman The basic concepts of point-set topology: topological spaces, connectedness, compactness, quotient spaces, metric spaces, and function spaces. An introduction to algebraic topology: covering spaces, the fundamental group, and other topics as time permits.

Sec. 01

MTW 2

110.417 (E,Q)

PARTIAL DIFFERENTIAL EQUATIONS FOR APPLICATIONS (4.5) Tinaglia Yang   Prereq: Calculus III, Linear Algebra    Recommended: 110.405 Characteristics: classification of second order equations, well-posed problems. Separation of variables and expansions of solutions. The wave equation: Cauchy problem, Poisson's solution, energy inequalities, domains of influence and dependence. Laplace's equation: Poisson's formula, maximum principles, Green's functions, potential theory Dirichlet and Neumann problems, eigenvalue problems. The heat equation: fundamental solutions, maximum principles.

Sec. 01

MTW 12

110.431 (Q)

KNOT THEORY (4) Morava    
The theory of knots and links is a royal road to modern topology. The prerequisite for this course is a good grade in Calc III, but the material will be mathematically sophisticated, some familiarity with the notion of groups would be helpful. We will start with braids and work up to knots and links. The fundamental group of a knot or a link complement will be the central algebraic focus, and spanning surfaces will be the main geometric tool. Together these lead very intuitively to homology groups (in low dimensions).

Sec. 01

MTW 11

110.443 (E,Q)

FOURIER ANALYSIS AND GENERALIZED FUCTIONS (4.5) Song   Prereq: 110.201, 110.202 An introduction to the Fourier transform and the construction of fundamental solutions of linear partial differential equations. Homogeneous distributions on the real line: the Dirac delta function, the Heaviside step function. Operations with distributions: convolution, differentiation, Fourier transform. Construction of fundamental solutions of the wave, heat, Laplace and Schrödinger equations. Singularities of fundamental solutions and their physical interpretations (e.g., wave fronts). Fourier analysis of singularities, oscillatory integrals, method of stationary phase.

Sec. 01

MTW 1

110.462 (Q)

PRIME NUMBERS AND RIEMANN'S ZETA FUNCTION (4) Zhang
This course is devoted to such questions as: How many prime numbers are there less than N? How are they spaced apart? Although prime numbers at first sight have nothing to do with complex numbers, the answers to these questions due to Gauss, Riemann, Hadamard) involve complex analysis and in particular the Riemann zeta function, which controls the distribution of primes. This course builds on 110.311 and is an introduction to Analytic Number Theory for undergraduates.

Course canceled 01/18/05

Sec. 01

MTW 10

110.472 (Q)

DIFFERENTIAL TOPOLOGY (4.5) Wilson   Prereq: Calculus III, and 110.405, 110.201, or 110.413 Topics include manifolds, tangent spaces, immersions, submersions, transversality, intersection theory modulo 2, intersection numbers in the integers and Lefshetz fixed point theorem, and integration of differential forms on manifolds.

Sec. 01

MTW 10

110.512

INTERNSHIP - UNDERGRADUATE Howald  Course added 03/18/05

TBA

110.602

ALGEBRA Kong Prereq: 110.401-402 An introductory graduate course on fundamental topics in algebra to provide the student with the foundations for Number Theory, Algebraic Geometry, and other advanced courses. Topics include group theory, commutative algebra, Noetherian rings, local rings, modules, and rudiments of category theory, homological algebra, field theory, Galois theory, and non-commutative algebras.

Sec. 01

MT 11:30-12:45
12-1:15

110.607

COMPLEX VARIABLES Shiffman Prereq: 110.311, 110.405 Analytic functions of one complex variable.  Topics include Mittag-Leffler Theorem, Weierstrass factorization theorem, elliptic functions, Riemann-Roch theorem, Picard theorem, and Nevanlinna theory.

Sec. 01

MW 1-2:30
1:30-3

110.616

ALGEBRAIC TOPOLOGY Boardman Prereq: 110.401, 110.413   Polyhedra, simplicial and singular homology theory, Lefschetz fixed-point theorem, cohomology and products, homological algebra, Künneth and universal coefficient theorems, Poincaré and Alexander duality theorems.

Sec. 01

ThF 2-3:15
MTW 3

110.632

PARTIAL DIFFERENTIAL EQUATIONS Spruck  Prereq: 110.605-606 An introductory graduate course in partial differential equations. Classical topics include first order equations and characteristics, the Cauchy-Kowalevski theorem, Laplace's equation, heat equation, wave equation, fundamental solutions, weak solutions, Sobolev spaces, maximum principles. The second term focuses on special topics such as second order elliptic theory.

Sec. 01

MTW 11-12:30

110.636

MICROLOCAL ANALYSIS Sogge Prereq: 110.605-606   Recom: 110.631 Microlocal analysis is the geometric study of singularities of solutions of partial differential equations.  The course will begin by introducing the geometric theory of (Schwartz) distributions:  Fourier transform and Sobolev spaces, pseudo-differential operators, wave front set of a distribution, elliptic operators, Lagrangean distributions, oscillatory integrals, method of stationary phase, Fourier integral operators. 

Sec. 01

MTW 10

110.637

EIGENFUNCTIONS OF LAPLACIAN Zelditch  Billiards on an elliptic drum are predictable, while those on a stadium drum are chaotic. This course relates modes of vibration of a drum (eigenfunctions) to the motion of billiard balls (geodesics) on it.

Sec. 01

MTW 10

110.644

ALGEBRAIC GEOMETRY Shokurov Affine varieties and commutative algebra. Hilbert's theorems about polynomials in several variables with their connections to geometry. General varieties and projective geometry. Dimension theory and smooth varieties. Sheaf theory and cohomology. Applications of sheaves to geometry; e.g., the Riemann-Roch Theorem. Other topics may include Jacobian varieties, resolution of singularities, geometry on surfaces, schemes, connections with complex analytic geometry and topology.

Sec. 01

MTW 12

110.645

RIEMANNIAN GEOMETRY Minicozzi   Prereq: 110.405, 110.413 Differential manifolds, vector fields, Frobenius’ theorem. Differential forms, deRham’s theorem, vector bundles, connections, curvature, Chern classes, Cartan structure equations. Riemannian manifolds, Bianchi identities, geodesics, exponential maps. Geometry of submanifolds, hypersurfaces in Euclidean space. Other topics as time permits, e.g. harmonic forms and Hodge’s theorem, Jacobi equation, variation of arc length and area, Chern-Gauss-Bonnet theorems.

Sec. 01

MTW 9

110.660

QUALIFYING EXAM PROBLEMS Chen

Sec. 01

TTh 4-5:15

110.677

MODEL MODULI SPACES Faber 
Moduli spaces occupy a central space in algebraic geometry. After an introduction and some general theory, focus will be on the moduli spaces of curves and of abelian varieties.

Sec. 01

MT 11-12:15

110.727

TOPICS IN ALGEBRAIC TOPOLOGY Wilson

Sec. 01

MW 2-3:30

110.730

TOPICS IN COMPLEX GEOMETRY Shiffman Course added 11/29/04

Sec. 01

W 3:45-5

110.734

TOPICS IN ALGEBRAIC NUMBER THEORY Ono

Sec. 01

ThF 3:30-5

110.777

LOGARITHMIC STRUCTURES IN GEOMETRY AND HODGE THEORY Zucker     We give the definition of logarithmic structures, illustrated by several examples.  We then show how it is used to give a common framework for treating a variety of problems in geometry and Hodge theory.  As time permits, we will present some of these in greater detail.

Sec. 01

Th 2-3:30

110.799

THESIS RESEARCH Staff

110.800

INDEPENDENT STUDY