110.105 Introduction to Calculus
110.106-107 Calculus I and II (Biological and Social Sciences)
110.108-109 Calculus I and II (Physical Sciences and Engineering)
110.113 Honors Calculus 2
110.201 Linear Algebra
110.202 Calculus III---Calculus of Several Variables
110.211 Honors Calculus III
110.212 Honors Linear Algebra
110.225 Putnam problem Solving
110.302 Differential Equations with Applications
110.304 Elementary Number Theory
110.311 Methods of Complex Analysis
110.328 Non - Euclidean Geometry
110.345 Basic Notions in Mathematics
110.401 Advanced Algebra I
110.402 Advanced Algebra II
110.405 Analysis I
110.406 Analysis II
110.407-408 Geometry and Relativity
110.409 Introduction to Algebraic Number Theory
110.411 Honors Complex Analysis
110.413 Introduction to Topology
110.415 Honors Analysis One
110.417 Partial Differential Equations for Applications
110.423 Lie Groups for Undergraduates
110.427 Introduction of the Calculus of Variations
110.429 Mathematics of Quantum Mechanics
110.431 Introduction to Knot Theory
110.439 Introduction to Differential Geometry
110.443 Fourier Analysis and Generalized Functions
110.462 Prime Numbers and Riemann's Zeta Function
110.472 Differential Topology
110.599 Independent Study, Undergraduate
110.105 (Q)
Introduction to Calculus
This course starts from scratch and provides students with all the
background necessary for the study of calculus. It includes a review of
algebra, trigonometry, exponential and logarithmic functions,
coordinates and graphs. Each of these tools will be introduced in its
cultural and historical context. The concept of the rate of change of a
function will be introduced. Not open to students who have studied
calculus in high school.
4 credits
110.106-107 (Q)
Calculus I, II (Biological and Social Sciences)
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.
4 credits
110.108-109 (Q)
Calculus I, II (Physical Sciences and Engineering)
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, and the courses are designed to meet the
needs of students in these disciplines.
4 credits
110.113 (Q) Honors
Calculus 2
This is an honors alternative to 107 or 109 and meets general
requirement for Calculus 2. It is a more theoretical treatment of one
variable integral calculus and is based on our modern understanding of
the real number system as explained by Cantor, Dedekind, and
Weierstrass. Students who want to know the "why's and how's" of Calculus
will find this course rewarding. Students will be expected to already
understand differential calculus (derivatives, differentiation, chain
rule, optimization, related rates, etc), and will learn about the theory
of integration, the fundamental theorem(s) of Calculus, applications of
integration,and Taylor series.
Prerequisite: A strong background in Calculus I, such as a 5 on the
AP/AB Calculus exam, an "A" in 110.106 or 110.108. 4 credits
110.201 (Q) Linear Algebra
Vector spaces, matrices, and linear transformations. Solutions of
systems of linear equations. Eigenvalues, eigenvectors, and
diagonalization of matrices. Applications to differential equations.
Prerequisite: Calculus I. Recommended: Calculus II.
4 credits
110.202 (Q) Calculus
III---Calculus of Several Variables
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.
Prerequisite: 110.107, 110.109 or 110.112.
4 credits
110.211 (Q) Honors Calculus III
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. 211-212 used to be an integrated year-long course, but now
the two are independent courses and can be taken in either order.
Prerequisite: Calculus II, or 5 on the Calculus BC AP Exam. 4 credits
110.212 (Q) Honors Linear Algebra
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.
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. Prerequisite: Calculus II or III or equivalent, preferably
honors. 4 credits
110.225 (Q) Putnam Problem Solving
Problem solving course to prepare students for the Putnam exam.
3credits
110.302 (Q,E)
Differential Equations with Applications
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). Prerequisite: Calculus II.
4 credits
110.304 (Q) Elementary
Number Theory
The student is provided with many historical examples of topics each of
which serves as an illustration of and provides a background for many
years of current research in number theory. This course also provides
the student with 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. Prerequisite: Calculus I.
4 credits
110.311 (Q) Methods of
Complex Analysis
This course is an introduction to the theory of functions of one complex
variable. Its emphasis is on techniques and applications, and it serves
as a basis for more advanced courses. Functions of a complex variable
and their derivatives; power series and Laurent expansions; Cauchy
integral theorem and formula; calculus of residues and contour
integrals; harmonic functions, Prerequisite: Calculus III.
4.5 credits
110.328 (Q)
Non-Euclidean Geometry
For 2,000 years, Euclidean geometry was the geometry. In the
19th century, new, equally consistent but very different geometries were
discovered. This course will delve into these geometries on an
elementary but mathematically rigorous level. Prerequisite: high school
geometry.
3 credits
110.345 Basics Notions in Mathematics
This seminar course is intended to introduce majors and those interested in mathematics to a large collection of topics that they may not have seen before. It meets weekly with a different speaker each week. Prerequisites: Significant experience (at least two courses) with mathematics at the 200-level or above. 1 credit.
110.401 (Q) Advanced
Algebra I
An introduction to the basic notions of modern algebra. Elements of
group theory: groups, subgroups, normal subgroups, quotients,
homomorphisms. Generators and relations, free groups, products,
commutative (Abelian) groups, finite groups. Groups acting on sets, the
Sylow theorems. Definition and examples of rings and ideals.
Introduction to field theory. Linear algebra over a field. Field
extensions, constructible polygons, non-trisectability. Prerequisite:
Linear Algebra
4.5 credits
110.402 (Q) Advanced
Algebra II
This is a continuation of 110.401. Theory of fields (continued).
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.
4.5 credits
110.405 (Q) Analysis I
This 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.
Prerequisites: Calculus III, Linear Algebra.
4.5 credits
110.406 (Q) Analysis
II
This course continues
110.405, with an emphasis on the fundamental 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. Prerequisite:
110.405.
4.5 credits
110.407-408 (Q,N)
Geometry and Relativity
Special relativity: Lorentz transformation, Minkowski spacetime, mass,
energy-momentum, stress-energy tensor, electrodynamics. Introduction to
differential geometry: theory of surfaces, first and second fundamental
forms, curvature. Gauss's theorema egregium, differentiable
manifolds, connections and covariant differentiation, geodesics,
differential forms, Stokes theorem. Gravitation as a geometric theory:
Lorentz metrics, Riemann curvature tensor, tidal forces and geodesic
deviation, gravitational redshift, Einstein field equation, the
Schwarzschild solution, perihelion precession, the deflection of light,
black holes, cosmology. Prerequisites: Calculus II, Linear Algebra,
General Physics II.
4.5 credits
110.409 (Q) Intro to
Algebraic Number Theory
This is an introduction to the arithmetic of rings of algebraic integers
and more general Dedekind domains. It covers topics such as: the unique
factorization theorem for ideals in rings of algebraic integers,
integral bases, the discriminant, the different, ramification, the
finiteness theorem for ideal-class groups, Dirichlet's theorem on groups
of units of rings of algebraic integers etc. The prerequisites are
Algebra 401-402 or an equivalent one year course in algebra.
4 credits
110.411 (Q) Honors Complex Analysis
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. Prerequisite: 110.201, 110.202 or permission from instructor.
4.5 credits
110.413 (Q)
Introduction to Topology
The basic concepts of point-set topology: topological spaces,
connectedness, compactness, quotient spaces, metric spaces, function
spaces. An introduction to algebraic topology: covering spaces, the
fundamental group, and other topics as time permits. Prerequisite:
Calculus III.
4.5 credits
110.415 (Q) Honors Analysis I
This highly theoretical sequence in analysis is reserved for the most able students. The sequence covers the real number system, metric spaces, basic functional analysis, the Lebesgue integral, and other topics. 4.5 credits
110.417 (Q,E) Partial
Differential Equations for Applications
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. Prerequisites: Calculus III.
Recommended:
110.405.
4.5 credits
110.423 (Q) Lie Groups
for Undergraduates
This course is an introduction to Lie Groups and their representations
at the upper undergraduate level. It will cover basic Lie Groups such as
SU (2), U (n) , the Euclidean Motion Group and Lorentz Group. This
course is useful for students who want a working knowledge of group
representations. We will also discuss some aspects of the role of
symmetry groups in particle physics such as some of the formal aspects
of the electroweak and the strong interactions. A good reference is the
book Lie Algebras in Particle Physics by Howard Georgi. Prior knowledge
of group theory would be helpful.
4 credits
110.427 (Q) Introduction to the Calculus of
Variations
The calculus of variations is concerned with finding optimal solutions
(shapes, functions, etc.) where optimality is measured by minimizing a
functional (usually an integral involving the unknown functions)
possibly with constraints. In this introductory (self-contained)
course, we will concern ourselves with one dimensional (often geometric)
problems: brachistochrone, geodesics, minimum surface area of
revolution, isoperimetric problem, curvature flows. We will run the
course in a seminar style with active participation required. I will
teach additional material as required (some differential geometry of
curves and surfaces) to hold prerequisites to a minimum. Prerequisite:
Calculus I, II, III.
4 credits
110.429 (Q) Mathematics of Quantum Mechanics
The basis of quantum mechanics is the Schrodinger equation. The focus of
this course will be on one dimensional Schrodinger equations. Topics
include eigenvalue problems, bound states, scattering states, tunneling,
uncertainty principle, dynmaics, semi-classical limit. The ideas will be
illustrated through many examples. Pre-requisite: 110.302
4 credits
110.431 (Q) Introduction to Knot Theory
The theory of knots and links is a royal road to modern topology. The
pre-requisite for this course is a good grade in Calc 3, 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).
4 credits
110.439 (Q)
Introduction to Differential Geometry
Theory of curves and surfaces in Euclidean space: Frenet equations,
fundamental forms, curvatures of a surface, theorems of Gauss and
Mainardi-Codazzi, curves on a surface; introduction to tensor analysis
and Riemannian geometry; theorema egregium; elementary global theorems.
Prerequisites: Calculus III, Linear Algebra.
4.5 credits
110.443 (Q) Fourier
Analysis and Generalized Functions
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. Prerequisites:
Calculus III, Linear Algebra. Recommended:
110.405.
4.5 credits
110.462 (Q) Prime
Numbers and Riemann's Zeta Function
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
the distribution of primes. This course builds on 110.311 and is an
introduction to Analytic Number Theory for undergraduates.
Pre-requisite: 110.311
4 credits
110.472 (Q) Differential Topology
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. Prerequisite: 201, 202 and either
110.405 or 110.413.
4.5 credits
110.599 Independent Study, Undergraduate
Cross-Listed
171-204 Theoretical Mechanics