Course Information Book
Department of Mathematics
Tufts University
Fall 2014
This booklet contains the complete schedule of courses offered by the Math Department in the Fall
2014 semester, as well as descriptions of our upper-level courses (from Math 61 [formally Math 22]
and up). For descriptions of lower-level courses, see the University catalog.
If you have any questions about the courses, please feel free to contact one of the instructors.
Course renumbering
Course schedule
p. 2
p. 3
Math 50
p. 5
Math 51-01
p. 6
Math 61
p. 7
Math 70
p. 8
Math 87
p. 9
Math 126
p. 10
Math 135
p. 11
Math 145
p. 12
Math 150
p. 13
Math 161
p. 14
Math 211
p. 15
Math 213
p. 16
Math 215
p. 17
Math 217
p. 18
Math 251
p. 19
Mathematics Major Concentration Checklist
p. 20
Applied Mathematics Major Concentration Checklist
p. 21
Mathematics Minor Checklist
p. 22
Jobs and Careers, Math Society, and SIAM
p. 23
Block Schedule
p. 24
Course Renumbering
Starting Fall, 2012, the lower level math courses will have new numbers. The
matrix below gives the map between the current numbers and the new numbers.
The course numbers on courses you take before Fall 2012 will not change, and the
content of these courses will not change.
Course Name
Old
Number
New
Number
Course Name
New
Number
Old
Number
Fundamentals of
Mathematics
4
4
Fundamentals of
Mathematics
4
4
Introduction to
Calculus
5
30
Introductory Special
Topics
10
10
Introduction to Finite
Mathematics
6
14
Introduction to Finite
Mathematics
14
6
Mathematics in
Antiquity
7
15
Mathematics in
Antiquity
15
7
Symmetry
8
16
Symmetry
16
8
The Mathematics of
Social Choice
9
19
The Mathematics of
Social Choice
19
9
Introductory Special
Topics
10
10
Introduction to
Calculus
30
5
Calculus I
11
32
Calculus I
32
11
Calculus II
12
34
Calculus II
34
12
Applied Calculus II
50-01
36
Applied Calculus II
36
50-01
Calculus III
13
42
Honors Calculus I-II
39
17
Honors Calculus I-II
17
39
Calculus III
42
13
Honors Calculus III
18
44
Honors Calculus III
44
18
Discrete Mathematics
22
61
Special Topics
50
50
Differential Equations 38
51
Differential Equations 51
38
Number Theory
41
63
Discrete Mathematics
61
22
Linear Algebra
46
70
Number Theory
63
41
Special Topics
50
50
Linear Algebra
70
46
Abstract Linear
Algebra
54
72
Abstract Linear
Algebra
72
54
DEPARTMENT OF MATHEMATICS
[Schedule as of February 20, 2014]
TUFTS UNIVERSITY
COURSE # (OLD #)
4
COURSE TITLE
(same) Fundamentals of Mathematics
19-01
02
03
9
Mathematics of Social Choice
21-01
02
10
Introductory Statistics
30-01
02
5
32-01
02
03
04
05
06
34-01
02
03
04
36-01
02
03
BLOCK
G
C
G+TR
J+TR
FALL 2014
ROOM
INSTRUCTOR
Zachary Faubion
TBA
Linda Garant
TBA
B+TR
G+ MW
TBA
Patricia Garmirian
Introduction to Calculus
C
D
Genevieve Walsh*
Gail Kaufmann
11
Calculus I
B
B
F
F
F
H
Christoph Börgers
Hao Liang
Linda Garant
Gail Kaufmann*
Kim Ruane
TBA
12
Calculus II
B
C
F
H
Marjorie Hahn*
TBA
Mary Glaser
TBA
50-01 Applied Calculus II
B+ TRF
E+ MWF
F+ TRF
TBA
Zachary Faubion
TBA
E+ MWF
Zbigniew Nitecki
39-01
17
Honors Calculus I-II
42-01
02
03
04
05
06
07
13
Calculus III
B
C
D
E
F
F
H
TBA
James Adler
Fulton Gonzalez*
TBA
TBA
TBA
TBA
(same) Graphs and Surfaces
E+WF
Genevieve Walsh
D
F
Christoph Börgers
Misha Kilmer*
50-01
51-01
02
38
Differential Equations
DEPARTMENT OF MATHEMATICS
[Schedule as of February 20, 2014]
TUFTS UNIVERSITY
COURSE # (OLD #)
COURSE TITLE
61-01
02
03
04
22
Discrete Mathematics
70-01
02
03
04
46
Linear Algebra
87-01
50
Mathematical Modeling and Computing
FALL 2014
BLOCK
INSTRUCTOR
D+TR
E+WF
G+MW
J+TR
Mary Glaser
Bruce Boghosian
TBA
TBA
H+
C
D
E
Kim Ruane
George McNinch
Hao Liang
TBA
E+WF
James Adler
(same) Numerical Analysis
H+TR
TBA
(same) Real Analysis I
F+TR
I
Todd Quinto
Loring Tu
(same) Abstract Algebra I
D+TR
Montserrat Teixidor
150-01
Random Walks, Brownian Motion, the
(same) Heat Equation and Beyond
D+TR
Marjorie Hahn
161-01
(same) Probability
E+ MWF
Patricia Garmirian
211
(same) Analysis
G+MW
Zbigniew Nitecki
213
(same) Complex Analysis
J+TR
Fulton Gonzalez
215
(same) Algebra
E
George McNinch
217
(same) Geometry and Topology
251
(same) Linear Partial Differential Equations
126
135-01
02
145
K+MW
C
Loring Tu
Bruce Boghosian
Math 50-01
Graphs and Surfaces
Course Information
Fall 2014
Block: E+WF, Wed Fri 10:30 - 11:45
Instructor: Genevieve Walsh
Email: genevieve.walsh@tufts.edu
Office: Robinson 156
Office hours: (Spring 2014) Mon Wed 11:45 - 12:30 and 2:45-3:30
Prerequisites: Geometric intuition, willingness to present mathematical ideas in class.
Texts: Mostly Surfaces by R. E. Schwartz, AMS (2010).
Course description: This course is designed for early math majors who want to learn
about some interesting topics in combinatorics, geometry and topology. We will begin with
graphs, discussing and exploring Eulerian graphs, Hamiltonian graphs, the Euler characteristic, and the genus of a graph. We’ll also examine some infinite graphs and discuss their
properties. We will then move to the definition of surfaces, the classification of surfaces, and
some of their properties. Finally, we will begin to explore the geometry of surfaces. Time
and interest permitting, we will discuss some topics in surface theory.
Your grade will be based on bi-weekly homework sets, class participation, and two
projects/presentations. It is understood that there may be students at vastly different levels
in this class, and you will be able to choose your projects accordingly.
Math 51, Sec. 1
Differential Equations
Course Information
Fall 2014
Please note: This page is only about Section 1, an experimental section of Math 51. It has
more stringent prerequisites than the regular Math 51, which will be offered as Math 51,
Section 2 in the fall. It also uses a different book. The two sections have somewhat different
emphasis. We offer Section 1 with the Applied Mathematics majors in mind, but all who
have fulfilled the prerequisites are welcome.
Block: D, Mon 9:30–10:20, Tue/Thu 10:30–11:20
Instructor: Christoph Börgers
Email: cborgers@tufts.edu
Office: Bromfield-Pearson 215
Office hours: (Spring 2014) Mon 12–1:30, Wed 12–1:30
Pre- and Corequisites: Math 42 or 44 (Multivariable Calculus) is a prerequisite. Math
70 or 72 (Linear Algebra) is a pre- or corequisite. That is, you should have taken it, or you
should take it in parallel with this course.
Text: Differential Equations, Fourth Edition, by Paul Blanchard, Robert Devaney, and
Glenn Hall. Brooks/Cole, 2011, ISBN 978-1133109037. This book is not the one used for
Section 2.
Course description: Differential equations relate the rates of change of quantities to
their values. Most mathematical models in the natural sciences are differential equations.
This is why we teach calculus, which underlies the study of differential equations, to so many
students.
This course will put strong emphasis on modeling, that is, on understanding how the
differential equations that we study arise in the natural sciences. You will learn about the
most elementary ways of modeling the growth of populations of people, as well as populations
of tumor cells, and about simple models of epidemics and of populations competing for
resources, resonance and the Tacoma Narrows Bridge (a suspension bridge that collapsed
into Puget Sound in 1940 — you can find video footage on youtube), and much more.
Explicit solutions of differential equations can be computed in very simple cases only, but
the methods used for doing so are useful and important. Linear algebra plays a central role
here. There will also be strong emphasis on qualitative analysis, that is, on understanding
properties of the solutions without computing them explicitly; this, too, involves linear
algebra, in particular eigenvalues and eigenvectors, in a central way.
We will also discuss numerical methods for solving differential equations, unquestionably the most important approach to solving differential equations. The study of numerical
methods for differential equations is a large and sophisticated branch of contemporary mathematics. In this course, you will learn the beginnings of that field.
Math 61
Discrete Mathematics
Course Information
Fall 2014
Block: D+TR
Instructor: Mary Glaser
Email: mary.glaser@tuft.edu
Office: Bromfield-Pearson 004
Block: E+WF
Instructor: Bruce Boghosian
Email: bruce.boghosian@tufts.edu
Block: G+MW
Instructor: TBA
Block: J+TR
Instructor: TBA
Math 70
Linear Algebra
Course Information
Fall 2014
Block: D (M 9:30-10:20 TTh 10:30-11:20)
Instructor: Hao Liang
Email: hao.liang@tufts.edu
Office: Bromfield-Pearson 109
Office hours: (Spr ’14) W 2:30–4:30pm
Fri 2:00 3:00pm
Block: C (TWF 9:30-10:20)
Instructor: George McNinch
Email: george.mcninch@tufts.edu
Office: Bromfield-Pearson 112
Office hours: (Spr ’14) T 10:30–11:45am
and W 13:30–15:00pm
Block: I+ (WM 3:00-4:20)
Instructor: Yusuf Mustopa
Block: A (WM 8:30-9:20 Th 9:30 10-10:20)
Instructor: Kim Ruane
Email: kim.ruane@tufts.edu
Office: Robinson 155
Office hours: (Spr ’14) M 1:30–2:30pm,
W 3:30–4:30pm, Th 10:30–11:30am
Prerequisites: Math 34 or 39 or consent.
Text: Linear Algebra and Its Applications , 4th edition, by David Lay, Addison-Wesley
(Pearson), 2011.
Course description:
Linear algebra begins the study of systems of linear equations in several unknowns. The
study of linear equations quickly leads to the important concepts of vector spaces, dimension,
linear transformations, eigenvectors and eigenvalues. These abstract ideas enable efficient
study of linear equations, and more importantly, they fit together in a beautiful whole that
will give you a deeper understanding of these ideas.
Linear algebra arises everywhere in mathematics – it plays an important role in almost
every upper level math course –; it is also crucial in physics, chemistry, economics, biology,
and a range of other fields. Even when a problem involves nonlinear equations, as is often the
case in applications, linear systems still play a central role, since “linearizing” is a common
approach to non-linear problems.
This course introduces students to axiomatic mathematics and proofs as well as fundamental mathematical ideas. Mathematics majors and minors are required to take linear
algebra (Math 70 or Math 72) and are urged to take it as early as possible, as it is a prerequisite for many upper-level mathematics courses. The course is also useful to majors in
computer science, engineering, and the natural and social sciences.
The course will have two midterms and a final, as well as daily assignments.
Math 87
Mathematical Modeling and Computation
Course Information
Fall 2014
Block: E+WF, Wed Fri 10:30 -11:45 AM
Instructor: James Adler
Email: james.adler@tufts.edu
Office: Bromfield-Pearson 209
Office hours: (Spring 2014) on leave
Prerequisites: Math 34 or 39, or consent.
Text: none
Course description:
This course is about using elementary mathematics and computing to solve practical problems. Single-variable calculus is a prerequisite; other mathematical and computational tools,
such as elementary probability, matrix algebra, elementary combinatorics, and computing in
MATLAB, will be introduced as they come up.
Mathematical modeling is an important area of study, where we consider how mathematics can be used to model and solve problems in the real world. This class will be driven
by studying real-world questions where mathematical models can be used as part of the
decision-making process. Along the way, we’ll discover that many of these questions are best
answered by combining mathematical intuition with some computational experiments.
Some problems that we will study in this class include:
1. The managed use of natural resources. Consider a population of fish that has a natural growth rate, which is decreased by a certain amount of harvesting. How much
harvesting should be allowed each year in order to maintain a sustainable population?
2. The optimal use of labor. Suppose you run a construction company that has fixed
numbers of tradespeople, such as carpenters and plumbers. How should you decide
what to build to maximize your annual profits? What should you be willing to pay to
increase your labor force?
3. Project scheduling. Think about scheduling a complex project, consisting of a large
number of tasks, some of which cannot be started until others have finished. What
is the shortest total amount of time needed to finish the project? How do delays in
completion of some activities affect the total completion time?
Using basic mathematics and calculus, we will address some of these issues and others, such
as dealing with the MBTA T system and penguins (separately of course...).
Math/CS 126
Numerical Analysis
Course Information
Fall 2014
B LOCK : H+T/Th: Tuesday, Thursday 1:30-2:45pm
I NSTRUCTOR : TBD (new tenure-track hire)
P REREQUISITES : For Math 126/CS 126: Math 51 (old number: 38) and experience programming in a language such as C, C++, Fortran, Matlab, etc.
T EXT: TBD
C OURSE DESCRIPTION :
“Numerical analysis is the study of algorithms for the problems of continuous mathematics.” (L. N. Trefethen, “The definition of numerical analysis”, SIAM News, November
1992.) Continuous mathematics means mathematics involving the continuum of real or
complex numbers.
There are many striking examples illustrating the importance of the neccessity of accurate numerical solution methods. Here are just three of them. (1) In 1991, a Norwegian
offshore oil platform called the Sleipner A sank. The resulting economic loss was estimated at $700 million. The post-accident investigation traced the problem to a faulty numerical method for predicting shear stresses. (2) Ordinary differential equations are used
to model the spread of an infection through a population. Numerical solution techniques
are usually the only practical option for solving these problems, but without an underlying understanding of the properties of both the model and the algorithm, numerical
results are not to be trusted. (3) CT scanners are used throughout the world for medical
diagnostics. This technology is based on numerical algorithms for the solution of certain
integral equations. Then, of course, there are the more simple questions, “How does my
calculator compute the square root of 2??”
This course is an introduction to the field, treating linear algebra fairly lightly, instead
emphasizing numerical solution of nonlinear equations and differential equations, numerical integration techniques, and unconstrained optimization, often in the context of
applications. (For a thorough treatment of numerical linear algebra, take Math 128/CS
128 or its 200-level branch.) Computer programming will be a substantial component of
the homework, with Matlab as the suggested tool.
Math 135
Real Analysis I
Course Information
Block: F+ (Tue Thu 12:00–1:15 p.m.)
Instructor: Todd Quinto
Email: Todd.Quinto@tufts.edu
Office: Bromfield-Pearson 204
Office hours: (Spring 2014), Tues 10:15-11:15,
Wed Fri 1:30-2:30
Phone: (617) 627-3402
Fall 2014
Block: I (Mon Wed 3:00–3:50 p.m., Fri
3:30–4:20 p.m.)
Instructor: Loring Tu
Email: loring.tu@tufts.edu
Office: Bromfield-Pearson 206
Office hours: (Spring 2014) on sabbatical,
please email professor.
Phone: (617) 627-3262
Prerequisites: Math 42 or 44, and 70 or 72, or consent. (In the old numbering scheme, Math 13
or 18, and 46 or 54, or consent.)
Text:
• Prof. Quinto’s section: TBA (Either Fitzpatrick or Marsden and Hoffman)
• Prof. Tu’s section: Patrick Fitzpatrick, Advanced Calculus, 2nd edition, American Mathematical Society, 2009. (ISBN-10: 0821847910)
Course description:
Real analysis is the rigorous study of real functions, their derivatives and integrals. It provides the
theoretical underpinning of calculus and lays the foundation for higher mathematics, both pure
and applied. Unlike Calculus I, II, and III, where the emphasis is on intuition and computation,
the emphasis in real analysis is on justification and proofs.
Is this rigor really necessary? This course will convince you that the answer is an unequivocal
yes, because intuition not grounded in rigor often fails us or leads us astray. This is especially
true when one deals with the infinitely large or the infinitesimally small. For example, it is not
intuitively obvious that, although the set of rational numbers contains the set of integers as a proper
subset, there is a one-to-one correspondence between them. These two sets, in this sense, are the
same size! On the other hand, there is no such correspondence between the real numbers and the
rational numbers, and therefore the set of real numbers is uncountably infinite.
A metric space is a set on which there is a distance function. In this course, we will study
the topology of the real line and metric spaces, compactness, connectedness, continuous mappings,
and uniform convergence. The topics constitute essentially the first five chapters of the textbook.
Along the way, we will encounter theorems of calculus, such as the intermediate-value theorem
and the maximum-minimum theorem, but in a more general setting that enlarges their range of
applicability. This is one of the exciting aspects of real analysis and of upper-level math in general;
by studying fundamental ideas in a general setting, we understand them more deeply and gain a
broader perspective on and appreciation of the ideas. We anticipate ending with the contraction
mapping principle, a fundamental theorem using which one can prove the existence and uniqueness
of solutions of differential equations.
In addition to introducing a core of basic concepts in analysis, a companion goal of the course
is to hone your skills in distinguishing the true from the false and in reading and writing proofs.
Math 135 is required of all pure and applied math majors. A math minor must take Math 135
or 145 (or both).
Math 145
Abstract Algebra
Course Information
Fall 2014
Block: D+ (Tuesday, Thursday 10:30-11:45)
Instructor: montserat teixidor
Email: mteixido@tufts.edu
Office: Bromfield-Pearson 115
Office hours: (Spring 2014) T/Th 12-1.30, Fr. 8.30-9.20 and by appointment
Phone: (617) 627-2358
Prerequisites: Linear Algebra (Math 70 or 72).
Text: A first course in Abstract Algebra seventh edition by John B. Fraleigh , Addison
Wesley 2003.
Course description: Algebra, along with Analysis and Geometry is one of the main
pillars of mathematics. It has ancient roots, especially in Europe, India and China. Historically, algebra was concerned with the manipulation of equations and, in particular, with the
problem of finding the roots of polynomials. This is the algebra that you know from high
school. There are clay tablets from 1700B.C. that show that the Babylonians knew how to
solve quadratic equations. The solutions to cubic and fourth degree polynomial equations
were solved in Italy during the Renaissance. About the time of Beethoven, a young French
mathematician Evariste Galois made the dramatic discovery that for polynomials of degree
greater than four, no similar solution exists. To do this, Galois introduced the branch of
mathematics known as group theory. This was not, of course, the end of the story. Algebra
has continued its development to the present day most notably with the classification of
finite simple groups and with Andrew Wiles proof of Fermat’s Last Theorem.
The concept of a group is now one of the most important in mathematics. Roughly
speaking, group theory is the study of symmetry. There are deep connections between
group theory, geometry and number theory. Groups pop up in every area of mathematics:
symmetry groups can be used to find solutions to differential equations, associating groups
(and rings) to topological spaces allows to distinguish among them. Groups also appear in
the attempts of physicists to describe the basic laws of nature.
In Math 145, we introduce the concepts of group and ring. Their properties mimic the
arithmetic properties of numbers and polynomials. In Math 146, we describe the connection
Galois discovered between group theory and the roots of polynomials.
Math 150
Random Walks, Brownian Motion, the Heat Equation
and Beyond
Course Information
Fall 2014
Block: D+TR, Tue Thurs 10:30-11:45 PM
Instructor: Marjorie Hahn
Email: marjorie.hahn@tufts.edu
Office: Bromfield-Pearson 202
Office hours: T, Th, F 9:30-10
Prerequisites: Math 161 preferred or another calculus based probability course plus Math
42 or consent of instructor.
Text: Most likely a small inexpensive book by Greg Lawler plus handouts.
Course description:
Stochastic processes are random processes that evolve in discrete or continuous time and
provide useful models for many phenomena. This is an interactive course focused on two
of the most important kinds of stochastic processes–random walks and Brownian motion.
The course is meant to be accessible and of interest to both undergraduate and graduate
students who have had both multivariate calculus and a suitable calculus based probability
course. No previous exposure to stochastic processes is needed. The intent is to develop
theses processes and use them to explore current areas of active research.
This course is based on materials prepared and used by Greg Lawler for several active
summer student research experiences in probability. These materials lend themselves to a
series of accessible interconnected lectures and investigations that give rise to many ideas
in probability and analysis via random walks on the line, in the plane, on groups and on
graphs as well as their simulations and generalizations to Brownian motion, Markov chains,
and beyond. In the process students will not only get a taste of modern day probability and
current areas of research, but see the development of beautiful interconnections with topics
they have already learned, as well as many new thought provoking ideas and directions. For
instance, are there curves that are nowhere differentiable? If so, do they arise naturally? Are
they important and abundant? Are there important uncountable subsets of the line with
zero length? If so, where do they arise and how are their relative sizes compared?
The following should give a sense of tentative topics: simple random walk and Stirling’s
formula, Simple random walk in many dimensions, Self-avoiding walk, Brownian motion,
Shuffling and random permutations, Seven shuffles are enough (sort of), Markov chains on
finite sets, Markov chain Monte Carlo, Random walks and electrical networks, Uniform spanning trees, Random walk and other simulations including applications to finance, Random
walk and the discrete heat equation, Brownian motion and the heat equation, Self-similarity
and fractals; Random fractals.
The course is meant to be interactive. Working individually and in groups, students will
have opportunities to experience a sense of the discovery that propels research.
Math 161
Probability
Course Information
Fall 2014
Block: E+ MWF 10:30-11:45 AM
Instructor: Patricia Garmirian
Email: patricia.garmirian@tufts.edu
Office: Bromfield-Pearson 201
Office hours: MW 3:00-4:30 PM
Phone: (617) 627-2682
Prerequisites: Math 42 or consent.
Text: Probability and Stochastic Processes by Frederick Solomon, Prentice-Hall, Inc., 1987,
Upper Saddle River, NJ.
Course description:
Many things we experience in the real world are unpredictable. Consider flipping a coin. Can
we predict if it will land heads or tails? If the exact position of your finger, the composition
of the table, and the air currents were known, then we could predict the outcome of the
coin flip. However, due to the lack of information, we cannot predict the result. This lack
of information is explained as being due to “randomness.” In this course, we will study the
probability distributions of outcomes of random experiments.
In this course, we will cover sample spaces associated with a random experiment, the axioms of probability, combinatorics, conditional probability, independence of events, discrete
and continuous random variables, joint probability distributions, the central limit theorem,
and the law of large numbers. The probability distributions that we will cover include
geometric, binomial, uniform, exponential, poisson, gamma, and the normal distribution.
Knowledge of single and multivariable calculus is required for this course. There will be
weekly problem sets, two midterms, and a final exam.
Math 211
Analysis
Course Information
Fall 2014
Block: G+ MW: Mon, Wed 1:30-2:45
Instructor: Zbigniew Nitecki
Email: znitecki@tufts.edu
Office: Bromfield-Pearson 214
Office hours: (Spring 2014) MWF 1:30-2:30
Prerequisites: Math 135, or consent.
Course description:
The focus of this course is graduate level real analysis, which is a beautiful, coherent subject.
It is comprised of three main topics that provide essential foundations for all other areas of
analysis, as well as for geometry, topology, and applied mathematics.1 The main topics
are measure theory (general measures, Lebesgue integration, convergence theorems, product
measures), point set topology (topologies, metrics, compactness, completeness), and Banach
spaces (norms, Lp spaces, Hilbert spaces, orthonormal sets and bases, linear forms, duality,
Riesz representation theorems, signed measures). The course provides excellent preparation
for the Ph.D. oral examinations in analysis.
Lectures will be blackboard style, providing motivation for the theory, emphasizing basic
ideas of the proofs, and providing examples. Grades will be based on weekly homework as
well as exams.
Strong undergraduates who have done well in Math 135 and/or Math 136 are encouraged
to consider taking the course, especially if they intend to apply to graduate school in pure
or applied mathematics.
1
As Dennis Sullivan once famously remarked, “The subject is topology, the object is analysis,” referring to
the fact that much of Poincaré’s work inventing algebraic topology was motivated by the study of differential
equations.
Math 213
Complex Analysis
Course Information
Fall 2014
Block: F+TR, Tue Thu 12:00-1:15 PM
Instructor: Fulton Gonzalez
Email: fulton.gonzalez@tufts.edu
Office: Bromfield-Pearson 203
Office hours: (Spring 2014) Mondays 10:30 a.m. – 12:00 p.m.,
Tuesdays 3:30 – 5:00 p.m.
Prerequisites: Math 211, or consent.
Possible Texts: Real and Complex Analysis, Third Edition by Walter Rudin, McGrawHill, 1987.
Functions of a Complex Variable I, Second Edition, by John Conway, Springer, New York,
1978.
A Guide to Complex Variables, by Steven Krantz, Mathematical Association of America,
2008. (Available online for free.)
Course description: This is a course on complex function theory which will assume a
bit of graduate real analysis.
Complex function theory is a beautiful subject with many surprising and deep results, and
with connections to many other areas of mathematics, including number theory, representation theory, and harmonic analysis.
We’ll start at the beginning: Cauchy’s theorem and integral formula, the residue theorem,
and the classification of singularities. Then we’ll look at various properties of harmonic
functions and the Poisson kernel. We’ll examine consequences of the maximum modulus
principle, including Phragmen-Lindelof methods and interpolation theorems.
We will then explore two sister theorems: the Mittag-Leffler theorem, which posits the
existence of meromorphic function with preassigned poles, and the Weierstrass interpolation
theorem (along with infinite products), which posits the existence of holomorphic functions
with preassigned zeros.
Further topics will include normal families, conformality and the Riemann mapping theorem, subharmonicity and Hardy spaces, analytic continuation and the monodromy theorem,
Riemann surfaces and modular functions, value distribution theory and Picard’s little and
big theorems.
Math 215
Graduate Algebra I
Course Information
Fall 2014
Block: E block MWF 10:30–11:20
Instructor: George McNinch
Email: george.mcninch@tufts.edu
Office: Bromfield-Pearson 112
Office hours: (Spring 2014) Tues 10:30–11:45 and Wed 13:30–15:00
Prerequisites: Math 145 or consent.
Text: D.Dummit, S. Foote, Abstract Algebra, Prentice Hall.
Course description:
Algebra is an important part of modern mathematics, both as a subject in itself, and as a
ubiquitous tool. Historically, algebra was concerned with the manipulation of equations and,
in particular, with the problem of finding the roots of polynomials; this is the algebra you
know from high school. Modern algebra systematically studies groups, rings, fields and
modules. Roughly speaking, group theory is the study of symmetry; of course, symmetries
are important in mathematics but also in physics, chemistry and other physical sciences.
The study of rings and fields is part of arithmetic – i.e. of number theory – and of
geometry – since the geometry of a space may be investigated by study of (suitable) rings
of functions on the space. The study of modules over rings is in some sense a generalization
of linear algebra.
The course is suitable for students who have successfully taken at least one semester of
undergraduate abstract algebra and wish to acquire a solid foundation in algebra.
There will be weekly problem sets, a midterm, and a final exam.
Math 217
Geometry and Topology
Course Information
Fall 2014
Block: K+, Mon Wed 4:30–5:45 p.m.
Instructor: Loring Tu
Email: loring.tu@tufts.edu
Office: Bromfield-Pearson 206
Office hours: (Spring 2014) on sabbatical, please email professor.
Phone: (617) 627-3262
Prerequisites: Undergraduate real analysis (Math 135, 136) and abstract algebra (Math
145), some point-set topology.
Text: Loring Tu, An Introduction to Manifolds, 2nd edition, Springer, 2011.
Course description:
Undergraduate calculus progresses from differentiation and integration of functions on
the real line to functions on the plane and in 3-space. Then one encounters vector-valued
functions and learns about integrals on curves and surfaces. Real analysis extends differential
and integral calculus from R3 to Rn . This course is about the extension of calculus from
curves and surfaces to higher dimensions.
The higher-dimensional analogues of smooth curves and surfaces are called manifolds.
The constructions and theorems of vector calculus become simpler in the more general setting
of manifolds; gradient, curl, and divergence are all special cases of the exterior derivative,
and the fundamental theorem for line integrals, Green’s theorem, Stokes’ theorem, and
the divergence theorem are different manifestations of a single general Stokes’ theorem for
manifolds.
Higher-dimensional manifolds arise even if one is interested only in the three-dimensional
space which we inhabit. For example, if we call a rotation followed by a translation an affine
motion, then the set of all affine motions in R3 is a six-dimensional manifold. As another
example, the zero set of a system of equations is often, though not always, a manifold.
We will study conditions under which a topological space becomes a manifold. Combining
aspects of algebra, topology, and analysis, the theory of manifolds has found applications to
many areas of mathematics and even classical mechanics, general relativity, and quantum
field theory.
Topics to be covered included manifolds and submanifolds, smooth maps, tangent spaces,
vector bundles, vector fields, Lie groups and their Lie algebras, differential forms, exterior
differentiation, orientations, and integration. If time permits, there may be a short introduction to fundamental groups and covering spaces.
There will be weekly problem sets, a midterm, and a final. Undergraduates who have
done well in the prerequisite courses should find this course within their competence.
Math 251
Linear Partial Differential Equations
Course Information
Block: C
Instructor: Bruce Boghosian
Email: bruce.boghosian@tufts.edu
Fall 2014
MATHEMATICS MAJOR CONCENTRATION CHECKLIST
For students matriculating Fall 2012 and after (and optionally for others)
(To be submitted with University Degree Sheet)
Name:
I.D.#:
E-Mail Address:
College and Class:
Other Major(s):
(Note: Submit a signed checklist with your degree sheet for each major.)
Please list courses by number. For transfer courses, list by title, and add “T”. Indicate
which courses are incomplete, in progress, or to be taken.
Note: If substitutions are made, it is the student’s responsibility to make sure the substitutions are
acceptable to the Mathematics Department.
Ten courses distributed as follows:
I. Five courses required of all majors. (Check appropriate boxes.)
1.
2.
3.
Math 42: Calculus III or
4.
Math 44: Honors Calculus
5.
Math 70: Linear Algebra or
Math 72: Abstract Linear Algebra
Math 135: Real Analysis I
Math 145: Abstract Algebra I
Math 136: Real Analysis II or
Math 146: Abstract Algebra II
We encourage all students to take Math 70 or 72 before their junior year. To prepare for the
proofs required in Math 135 and 145, we recommend that students who take Math 70 instead of 72
also take another course above 50 (in the new numbering scheme) before taking these upper level
courses.
II. Two additional 100-level math courses.
III. Three additional mathematics courses numbered 50 or higher (in the new numbering
scheme); up to two of these courses may be replaced by courses in related fields
including:
Chemistry 133, 134; Computer Science 15, 126, 160, 170; Economics 107, 108, 154, 201,
202; Electrical Engineering 18, 107, 108, 125; Engineering Science 151, 152; Mechanical
Engineering 137, 138, 150, 165, 166; Philosophy 33, 103, 114, 170; Physics 12, 13 any
course numbered above 30; Psychology 107, 108, 140.
1. ______________
2. ______________
3. ______________
Advisor’s signature:
Date:
Chair’s signature:
Date:
Note: It is the student’s responsibility to return completed, signed degree sheets
to the Office of Student Services, Dowling Hall.
(form revised September 23, 2013)
APPLIED MATHEMATICS MAJOR CONCENTRATION CHECKLIST
(To Be Submitted with University Degree Sheet)
Name:
I.D.#:
E-Mail Address:
College and Class:
Other Major(s):
(Note: Submit a signed checklist with your degree sheet for each major.)
Please list courses by number. For transfer courses, list by title, and add “T”. Indicate
which courses are incomplete, in progress, or to be taken.
Note: If substitutions are made for courses listed as “to be taken”, it is the student’s responsibility
to make sure the substitutions are acceptable.
Thirteen courses beyond Calculus II. These courses must include:
I. Seven courses required of all majors. (Check appropriate boxes.)
1.
2.
3.
Math 42: Calculus III or
4.
Math 44: Honors Calculus III
5.
Math 70: Linear Algebra or
6.
Math 72: Abstract Linear Algebra 7.
Math 51: Differential Equations
II. One of the following:
1. ______________
Math 145: Abstract Algebra I
Comp 15: Data Structures
Math 87: Mathematical Modeling
Math 158: Complex Variables
Math 135: Real Analysis I
Math 136: Real Analysis II
Math 61/Comp 61: Discrete Mathematics
Math/Comp 163: Computational Geometry
III. One of the following three sequences:
1. ______________
Math 126/128: Numerical Analysis/Numerical Algebra
Math 151/152: Applications of Advanced Calculus/Nonlinear Partial Differential
Equations
Math 161/162: Probability/Statistics
IV. An additional course from the list below but not one of the courses chosen in section III:
1. ______________
Math 126
Math 128
Math 151
Math 152
Math 161
Math 162
V. Two electives (math courses numbered 61 or above are acceptable electives. With the
approval of the Mathematics Department, students may also choose as electives courses
with strong mathematical content that are not listed as Math courses.)
1. ______________
2. _____________
Advisor’s signature:
Date:
Chair’s signature:
Date:
Note: It is the student’s responsibility to return completed, signed degree sheets
to the Office of Student Services, Dowling Hall.
(form revised December 23, 2013)
DECLARATION OF MATHEMATICS
MINOR
Name:
I.D.#:
E-mail address:
College and Class:
Major(s)
Faculty Advisor for Minor (please print)
MATHEMATICS MINOR CHECKLIST
Please list courses by number. For transfer courses, list by title and add “T”. Indicate which
courses are incomplete, in progress, or to be taken.
Courses numbered under 100 will be renumbered starting in the Fall 2012 semester. Courses are
listed here by their new number, with the old number in parentheses.
Note: If substitutions are made for courses listed as “to be taken”, it is the student’s
responsibility to make sure that the substitutions are acceptable.
Six courses distributed as follows:
I. Two courses required of all minors. (Check appropriate boxes.)
1.
2.
Math 42 (old: 13): Calculus III
or
Math 70 (old: 46): Linear Algebra or
Math 44 (old: 18): Honors Calculus
Math 72 (old: 54): Abstract Linear
Algebra
II. Four additional math courses with course numbers Math 50 or higher (in the new
numbering scheme).
These four courses must include Math 135: Real Analysis I or 145: Abstract Algebra
(or both).
Note that Math 135 and 145 are only offered in the fall.
1. ______________
2. ______________
3. ______________
4. ______________
Advisor’s signature
Date
Note: Please return this form to the Mathematics Department Office
(Form Revised September 23, 2013)
Jobs and Careers
The Math Department encourages you go to discuss your career plans with your professors. All of
us would be happy to try and answer any questions you might have. Professor Quinto has built
up a collection of information on careers, summer opportunities, internships, and graduate
schools and his web site (http://equinto.math.tufts.edu) is a good source.
Career Services in Dowling Hall has information about writing cover letters, resumes and jobhunting in general. They also organize on-campus interviews and networking sessions with
alumni. There are job fairs from time to time at various locations. Each January, for example,
there is a fair organized by the Actuarial Society of Greater New York.
On occasion, the Math Department organizes career talks, usually by recent Tufts graduates. In
the past we had talks on the careers in insurance, teaching, and accounting. Please let us know if
you have any suggestions.
The Math Society
The Math Society is a student run organization that involves mathematics beyond the classroom.
The club seeks to present mathematics in a new and interesting light through discussions,
presentations, and videos. The club is a resource for forming study groups and looking into career
options. You do not need to be a math major to join! See any of us about the details. Check out
http://ase.tufts.edu/mathclub for more information.
The SIAM Student Chapter
Students in the Society for Industrial and Applied Mathematics (SIAM) student chapter organize
talks on applied mathematics by students, faculty and researchers in industry. It is a great way to
talk with other interested students about the range of applied math that’s going on at Tufts. You
do not need to be a math major to be involved, and undergraduates and graduate students from a
range of fields are members. Check out http://neumann.math.tufts.edu/~siam for more
information.
BLOCK SCHEDULE
50 and 75 Minute
Classes
Mon
8:05-9:20 (A+,B+)
A+
Mon
Tue
B+
0+
8:30-9:20 (A,B)
Tue
A
Wed
Wed
A+
Thu
Fri
B+
1+
B
Thu
2+
A
Fri
150/180 Minute Classes
and Seminars
B+
3+
4+
B
B
8:30-11:30 (0+,1+,2+,3+,4+)
9-11:30 (0,1,2,3,4)
9:30-10:20 (A,C,D)
10:30-11:20 (D,E)
10:30-11:45 (D+,E+)
D
0
C
1
C
2
A
3
E
D
E
D
E
E+
D+
E+
D+
E+
12:00-12:50 (F)
Open
F
12:00-1:15 (F+)
Open
F+
1:30-2:20 (G,H)
G
1:30-2:45 (G+,H+)
G+
5
H
6
H+
G
7
G+
F
F
F+
F+
H
8
G
4
9
H+
1:30-4:00 (5,6,7,8,9)
1:20-4:20 (5+,6+,7+,8+,9+)
2:30-3:20 (H on Fri)
3:00-3:50 (I,J)
3:00-4:15 (J+,I+)
3:30-4:20 (I on Fri)
C
H (2:30-3:20)
I
I+
J
J+
5+
I
I+
6+
J
J+
7+
4:30-5:20 ( K,L)
J/K
L
K
L
4:30-5:20 (J
( on Mon)
4:30-5:45 (K+,L+i)
K+
L+
K+
L+
6:00-6:50 (M,N)
N/M
6:00-7:15 (M+,N+)
M+
10+
11+
N
N+
10
12+
M
9+
13+
N
M+
11
I (3:30-4:20)
8+
N+
12
13
7:30-8:15 (P,Q)
Q/P
Q
P
Q
6:00-9:00 (10+,11+,12+,13+)
7:30-8:45 (P+,Q+)
P+
Q+
P+
Q+
6:30-9:00 (10,11,12,13)
Notes
* A plain letter (such as B) indicates a 50 minute meeting time.
* A letter augmented with a + (such as B+) indicates a 75 minute meeting time.
* A number (such as 2) indicates a 150 minute class or seminar. A number with a + (such as 2+) indicates a 180 minute meeting time.
* Lab schedules for dedicated laboratories are determined by department/program.
* Monday from 12:00-1:20 is departmental meetings/exam block.
* Wednesday from 12:00-1:20 is the AS&E-wide meeting time.
* If all days in a block are to be used, no designation is used. Otherwise, days of the week (MTWRF) are designated (for example, E+MW).
* Roughly 55% of all courses may be offered in the shaded area.
* Labs taught in seminar block 5+-9+ may run to 4:30. Students taking these courses are advised to avoid courses offered in the K or L block.