Course Information Book
as of 3-4-2015
Department of Mathematics
Tufts University
Fall 2015
This booklet contains the complete schedule of courses offered by the Math Department in the Fall
2015 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 61-01
p. 5
Math 61-02
p. 6
Math 70
p. 7
Math 87
p. 8
Math 135
p. 9
Math 145
p. 10
Math 150
p. 11
Math 151
p. 12
Math 161
p. 13
Math 167
p. 14
Math 211
p. 15
Math 215
p. 16
Math 226
p. 17
Math 250-01
p. 18
Math 250-02
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 March 4, 2015]
TUFTS UNIVERSITY
COURSE # (OLD #)
4
COURSE TITLE
(same) Fundamentals of Mathematics
BLOCK
C
ROOM
FALL 2015
INSTRUCTOR
Zachary Faubion
19-01
02
9
Mathematics of Social Choice
G+MW
K+MW
Linda Garant
Linda Garant
21-01
02
10
Introductory Statistics
C
G+MW
Staff
Patricia Garmrian
30-01
5
Introduction to Calculus
A
Kim Ruane
32-01
02
03
04
05
06
11
Calculus I
B
B
F
F
H
H
Richard Weiss *
Hao Liang
Gail Kaufmann
TBA
TBA
TBA
34-01
02
03
12
Calculus II
C
D
F
Xiaozhe Hu
Gail Kaufmann
Mary Glaser *
50-01 Applied Calculus II
B+TRF
E+ MWF
F+MWF
TBA
Zachary Faubion *
James Adler *
E+ MWF
Zbigniew Nitecki
36-01
02
03
39
17
Honors Calculus I-II
42-01
02
03
04
05
06
13
Calculus III
B
C
D
E
F
H
TBA
George McNinch *
Kye Taylor
TBA
Mauricio Gutierrez
TBA
51-01
02
04
38
Differential Equations
B
C
H
Kye Taylor
Yusuf Mustopa
Mauricio Gutierrez *
61-01
02
22
Discrete Mathematics
D
5M
Moon Duchin
Genevieve Walsh
70-01
02
03
04
46
Linear Algebra
C
D+
E
H
Hao LIang
Mary Glaser
George McNinch
TBA
* course coordinator
DEPARTMENT OF MATHEMATICS
[Schedule as of March 4, 2015]
TUFTS UNIVERSITY
COURSE # (OLD #)
87
50
COURSE TITLE
Mathematical Modeling and Computing
135-01 (same) Real Analysis I
02
145-01 (same) Abstract Algebra I
02
BLOCK
ROOM
FALL 2015
INSTRUCTOR
D
James Adler
F+TRF
G
Todd Quinto
Loring Tu
D
J
Richard Weiss
Kim Ruane
F
Christoph Börgers
H+
Christoph Börgers
E+ MWF
Patricia Garmirian
150
(same) Neuroscience
151
(same) Applied Advanced Calculus
161
(same) Probability
167
(same) Differential Geometry
G
Zbigniew Nitecki
211
(same) Analysis
E
Moon Duchin
215
(same) Algebra
I+MW
Loring Tu
226
(same) Numerical Analysis
F+TF
Xiaozhe Hu
250-01 (same) Numerical Anaylsis II
250-02 (same) Complex Algebraic Curves
G+MW
D+
Misha Kilmer
Yusuf Mustopa
* course coordinator
Math 61
Discrete Mathematics
Course Information
Fall 2015
Block: D, Mon 9:30-10:20, Tue/Thu 10:30-11:20
Instructor: Moon Duchin
Email: moon.duchin@tufts.edu
Office: Bromfield-Pearson 113
Office hours: (Spring 2015) Thu 11-1 and by cheerful appointment
Prerequisites: Calc I, Comp 11, or instructor’s consent
Text: Rosen, Discrete Mathematics and its Applications, 7th ed. (Probably!)
Course description:
This course is an introduction to discrete mathematics. In mathematics, “discrete” is
a counterpart of “continuous,” describing things that vary in jumps rather than smoothly.
Calculus studies (mostly) continuous phenomena; the mathematics of the discrete largely
fits into the research area called combinatorics, where the core questions have to do with
counting or enumeration. Our topics will include sets, permutations, equivalence relations,
properties of the integers, graph theory, and propositional logic.
This course is intended as a bridge from the more computationally oriented courses like
calculus to upper level courses in mathematics and computer science. We will spend a lot
of time learning to read and write correct proofs of mathematical statements and we will
practice these skills on a variety of problems.
This course, which is co-listed as Comp 61, is required for computer science majors. It
counts toward the math major and is highly recommended for math majors (or possible
math majors) who want a first glimpse of higher mathematics.
Another section of this course is being taught by Genevieve Walsh in the G block.
Math 61-02
Discrete Mathematics
Course Information
Spring 2015
Block: 5 (Mondays 1:30 - 4)
Instructor: Genevieve Walsh
Email: genevieve.walsh@tufts.edu
Office: Robinson 156
Office hours: (Spring 2015), Tuesdays 10:30 - 12, Wednesday 12:30 - 2
Phone: (617) 627-4032
Prerequisites: Willingness to solve problems on your own and at the blackboard and ask questions in class.
Text: B. Richmond and T. Richmond , A Discrete Transition to Advanced Mathematics, American
Mathematical Society (2004)
Course description:
Discrete math is important in computer science, combinatorics, biology, engineering, economics,
game theory, and virtually every aspect of life. It is also fun.
We will start off with the foundations of logic. We will discuss what it means to prove something,
and some methods for doing so. In this class you will prove things, and decide if your classmates’
proofs are valid. We will study divisibility, counting, and the Pigeonhole Principle. This will
address the question: “What happens when there are more pigeons than there are pigeonholes?”
The answer is surprisingly useful in a variety of non-pigeon situations.
We will learn about permutations and combinations and the binomial coefficients. We will visit
the vast subject of graphs and trees, and if time permits, see some of the beauty of continued
fractions.
There will be two exams and a final, weekly homework assignments, and problems to present/discuss
in class. Participation will be a large part of this class, and it is designed to get you to think more
deeply about the material.
Math 70
Linear Algebra
Course Information
Fall 2015
Block: D+
Instructor: Mary Glaser
Email: Mary.Glaser@tufts.edu
Office: Bromfield-Pearson 4
Office hours: (Spr 2015) Tues 1:30-2:30pm,
Thurs 2:30-4:30pm.
Block: E
Instructor: George McNinch
Email: George.McNinch@tufts.edu
Office: Bromfield-Pearson 112
Office hours:
(Spr 2015), Mon,Tues
11:00am-12:00, Fri 1:30-2:30pm
Block: C
Instructor: Hao Liang
Email: Hao.Liang@tufts.edu
Office: Bromfield-Pearson 109
Office hours: (Spr 2015) Mon 10:30am12:00, Thurs 3:30-5:00pm
Block: H
Instructor: TBA
Prerequisites: Math 42, or consent of instructor.
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 setting that
provides 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 upperlevel mathematics courses. The course is also useful to majors in computer science, economics,
engineering, and the natural and social sciences.
The course will have two midterms and a final, as well as daily homework assignments.
Math 87
Mathematical Modeling and Computation
Course Information
Fall 2015
Block: D+TTh, Tue Thurs. 10:30 -11:45 AM
Instructor: James Adler
Email: james.adler@tufts.edu
Office: Bromfield-Pearson 209
Office hours: (Spring 2015) TTh 3-4:00 & Th 10:30-11:30
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...). This course will
also serve as a good starting point for those interested in participating in the Mathematical
Contest in Modeling each Spring.
Math 135
Real Analysis I
Course Information
Block: F+,F (Tue Thu 12:00–1:15p.m., Friday
12:00-12:50 (see footnote * below))
Instructor: Todd Quinto
Email: Todd.Quinto@tufts.edu
Office: Bromfield-Pearson 204
Office hours: (Spring 2015), Tues 10:00-11:00,
Thursday 1:30-2:30 Fri 12:05-12:55
Phone: (617) 627-3402
Fall 2015
Block: G (Mon Wed Fri 1:30–2:20 p.m.)
Problem Session: G+ Mon 2:20–2:50 p.m.
Instructor: Loring Tu
Email: loring.tu@tufts.edu
Office: Bromfield-Pearson 206
Office hours: (Spring 2015) Tue 2:20–2:50
(136 only), Tue 4:15–5:15, Thu 4:15–5:45
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: Patrick Fitzpatrick, Advanced Calculus, 2nd edition, American Mathematical Society, 2009.
(ISBN 978-0-8218-4791-6)
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).
* Most weeks Prof. Quinto will have class for three 50 minute periods starting at noon on Tuesdays, Thursdays,
and Fridays. However, you will need to reserve 12:00-1:15 on Tues. and Thurs. since we will use at least some of this
extra time for homework and problem/proof sessions. Also, for a couple of weeks (as announced in class) we will
have class from 12:00-1:15 TR and no class on Friday.
Math 145
Abstract Algebra
Course Information
B LOCKS : D (M 9.30-10.20, TT10:30-11:20)
I NSTRUCTORS : Richard Weiss
E MAIL : rweiss@tufts.edu
O FFICE : Bromfield-Pearson 116
O FFICE HOURS : TTF 1:30-2:30
P HONE : 7-3802
Fall 2015
B LOCK : J (M 4:30-5:20 and TT 3:00-3:50)
I NSTRUCTOR : Kim Ruane
E MAIL : kim.ruane@tufts.edu
O FFICE : Robinson 155
O FFICE HOURS : M 5-6, Tu 2-3, F 9:30-10:30
P HONE : 7-2006
P REREQUISITES : Math 70 or 72 or consent.
T EXT: Abstract Algebra, 3rd edition, by John A. Beachy and William D.Blair, Waveland
Press, 2006.
C OURSE DESCRIPTION : Algebra is one of the main branches of mathematics. 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. Formulas for the roots of cubic and fourth degree polynomial equations were found in Italy during the Renaissance. About the time of Beethoven, however,
a young French mathematician named Evariste Galois made the dramatic discovery that
for polynomials of degree greater than four, no such formula exists. To do this, Galois
had to introduce a completely new branch of mathematics which he called group theory.
The concept of a group is now one of the central unifying themes in mathematics.
Roughly speaking, group theory is the study of symmetry. In mathematics, there are
deep connections between group theory, geometry and number theory. Group theory
also plays a central role in the efforts of physicists to describe the basic laws of nature.
The principle (but by no means the only) goal of Math 145-146 is to describe the connection between group theory and the roots of polynomials. Ring theory is the natural
setting for the study of polynomials. In Math 145, we will introduce groups and rings
and investigate their basic properties.
Math 150
Mathematical and Computational Neuroscience
Course Information
Fall 2015
Block: F (Tu, Th, Fr 12:00-12:50)
Instructor: Christoph Börgers
Email: cborgers@tufts.edu
Office: Bromfield-Pearson 215
Office hours: (Spring 2015) on sabbatical leave, please e-mail
Phone: (617) 627-2366
Prerequisites: Math 38, and willingness to do a considerable amount of Matlab programming. (Programming experience is desirable but not necessary. Almost all programming
will involve modification of Matlab code given to you.) No biological background will be
assumed.
Text: None. Notes will be distributed electronically.
Course description: This is a course on differential equations in neuroscience, a field
that began with a series of papers by the physiologists A. L. Hodgkin and A. F. Huxley in
the Journal of Physiology in 1952. At the time it had been well known for a long time that
the interior voltage of a nerve cell can spike, and that voltage spikes are the basis of neuronal
communication. Hodgkin and Huxley explained the mechanism underlying the spikes, and
reproduced it in a system of differential equations modeling the nerve cell. These equations
are now called the Hodgkin-Huxley equations. Eleven years after their groundbreaking series
of papers appeared, Hodgkin and Huxley were awarded the Nobel Prize.
Theoretical neuroscience can be viewed as the study of coupled systems of HodgkinHuxley equations. The goal is to understand the human brain. Not surprisingly, the field
is very far from reaching this goal, but a lot has been understood about the transitions
from rest to spiking in various different types of neurons, the propagation of signals through
neuronal networks, the frequently observed synchronization of activity in large groups of
neurons, the origins and functions of oscillatory activity in neuronal networks (as observed
for instance in the EEG), and other issues related to the physics of the brain. We will study
some of the known mathematical results, and think about what they might suggest about
brain function and disease.
The two main sets of tools used for studying differential equations modeling individual
neurons and neuronal networks are qualitative methods for the study of differential equations,
and computational simulation. You will see examples of both in this course.
Math 151/ME 150
Applications of Advanced Calculus
(Linear Partial Differential Equations)
Course Information
Fall 2015
Block: H+ (Tu, Th 1:30–2:45)
Instructor: Christoph Börgers
Email: christoph.borgers@tufts.edu
Office: Bromfield-Pearson 215
Office hours: (Spring 2015) on sabbatical leave, please e-mail
Phone: (617) 627-2366
Prerequisites: Calculus III (Math 42 or 18), Ordinary Differential Equations (Math 51)
Text: None. Notes will be distributed electronically.
Course description: Most deterministic models in the sciences are partial differential
equations. The weather, the global climate, air flow around airplanes, blood flow in the heart,
the propagation of electro-magnetic waves, the forces in bridges and buildings, the growth of
tumors, the propagation of signals in nerve cells in the brain, the spread of epidemics, and
countless other phenomena are governed by partial differential equations. Quite arguably
there is no branch of mathematics with greater practical impact on the world.
The emphasis in Math 151 is on linear partial differential equations. Three fundamental,
prototypical examples are (1) the diffusion equation (describing the diffusion of a pollutant in
still air, for instance), (2) the Poisson equation (the fundamental equation of electrostatics),
and (3) the wave equation (describing the propagation of sound for instance). You will learn
about mathematical properties of the solutions of these and similar equations. In very special
cases, it is possible and useful to write down explicit, analytic expressions for the solutions,
and you will learn examples of that. Fourier analysis is a central tool in this subject, and
therefore an introduction to Fourier analysis is a part of the course.
By far the most important solution techniques for partial differential equations are numerical methods. The study of such methods is a large and rapidly growing branch of mathematics. Although this is not a course on numerical analysis, we will study the simplest,
most fundamental numerical methods whenever it is conceptually clarifying to do so.
Math 151 is to be followed by Math 152 in the spring of 2016. Math 152 will be an
introduction to nonlinear partial differential equations.
Math 161
Probability
Course Information
Fall 2015
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 167
Differential Geometry
Course Information
Fall 2015
Block: G+MW, Mon, Wed 1:30-2:45
PM
Instructor: Zbigniew Nitecki
Email: znitecki@tufts.edu
Office: Bromfield-Pearson 214
Office hours: (Spring 2015) M 9:3010:20, WF 12:30-1:30
Phone: 7-3843
Prerequisites: Math 42 or 44, and 70 or 72, or consent.
Text: Differential Geometry of Curves and Surfaces by Thomas Banchoff and Stephen
Lovett, AK Peters, Ltd, 2010, Natick, MA.
Course description:
In this course you will discover how to use the material you have learned in calculus and
linear algebra to study beautiful and important properties of curves and surfaces in threedimensional Euclidean space. These ideas generalize to the higher-dimension analogues of
surfaces, called manifolds. Differential geometry, formulated in terms of manifolds, is a large
and important branch of mathematics, interacting with and contributing to many fields in
science and technology, including engineering, economics, and certainly much of physics.
Indeed, the language of modern theoretical physics is essentially differential-geometric.
We will start by studying the local and global properties of curves in space, and then move
on to the richer and more complex arena of surfaces. We will see how to properly define such
concepts as curvature, parallelism, geodesic, vector field, and surface area intrinsically. We
will examine important topological concepts such as orientability and Euler characteristic.
Finally, we will unite the geometry and the topology of a compact surface by means of the
all-important Gauss-Bonnet theorem.
This course is intended especially, but not exclusively, for students in mathematics,
physics, economics, and engineering, who have an interest in higher-level geometry.
Math 211
Analysis
Course Information
Fall 2015
Block: E+MW Mon/Wed 10:30-11:45
Instructor: Moon Duchin
Email: moon.duchin@tufts.edu
Office: Bromfield-Pearson 113
Office hours: (Spring 2015) Thu 11-1 and by cheerful appointment
Prerequisites: graduate student standing or instructor’s consent.
Course description:
This is an introductory graduate-level course covering central topics in real and functional
analysis. Topics will include measure theory (general measures, Lebesgue integration, convergence theorems, product measures), metrics and topology, and Banach spaces (norms, Lp
spaces, Hilbert spaces, orthonormal sets and bases, linear forms, duality, Riesz representation
theorems, signed measures).
The course provides preparation for the Ph.D. oral examinations in analysis. Grades will
be based on weekly homework as well as exams.
Undergraduates who have done well in Math 135-136 are strongly encouraged to consider
taking the course, especially if they intend to apply to graduate school in pure or applied
mathematics.
Math 215
Algebra
Course Information
Fall 2015
Block: I+MW, Mon Wed 3:00–4:15 PM
Instructor: Loring Tu
Email: loring.tu@tufts.edu
Office: Bromfield-Pearson 206
Office hours: (Spring 2015) Tue 2:20–2:50 (136 only), Tue 4:15–5:15, Thu 4:15–5:45
Phone: (617) 627-3262
Prerequisites: Math 146 or the equivalent.
Text: Abstract Algebra, 3rd ed., by David S. Dummit and Richard M. Foote, John Wiley & Sons,
2004, New York. (ISBN 0-471-43334-9)
Course description:
Algebra is the study of patterns and structures. For this reason it is the underpinning of many
areas of mathematics. This course concentrates on four basic algebraic structures: groups, rings,
modules, and fields. Since one of the purposes of the course is to prepare the students for the
algebra core exam, we will follow closely its syllabus, reproduced below.
1. Generalities:
Quotients and isomorphism theorems for groups, rings, and modules.
2. Groups:
The action of a group on a set; applications to conjugacy classes and the class equation.
The Sylow theorems; simple groups.
Simplicity of the alternating group for n ≥ 5.
3. Rings and Modules:
Polynomial rings, Euclidean domains, principal ideal domains.
Unique factorization; the Gauss lemma and Eisenstein’s criteria for irreducibility.
Free modules; the tensor product.
Structure of finitely generated modules over a PID; applications (finitely generated
abelian groups, canonical forms of linear transformations).
4. Fields:
Algebraic, transcendental, separable, and Galois extensions, splitting fields.
Finite fields, algebraic closures.
The fundamental theorem of Galois theory for a finite extension of a field of arbitrary
characteristic.
Many of these topics are covered in Math 145 and 146, but we will study them in greater depth.
It is unlikely that we can cover all of these topics in the first semester. Students wishing to take
the algebra core exam in January will need to do some additional study on their own.
This course is suitable for undergraduates who have done very well in Math 145 and/or 146,
especially those wishing to attend graduate school.
Math 226
Numerical Analysis
Course Information
Fall 2015
Block: F+ (Tuesday, Friday 12:00 pm -1:15 pm)
Instructor: Xiaozhe Hu
Email: Xiaozhe.Hu@tufts.edu
Office: Bromfield-Pearson 212
Office hours: (Spring 2015), Tuesday, Thursday 9:30 am- 11:30 am, or by appointment
Prerequisites: Math 126 and some programming ability in a language such as C, C++,
Fortran, Matlab, etc.
Text: TBD
Course description:
“Numerical analysis is the study of algorithms that uses numerical approximations for
the problems of mathematical analysis” – Wikipedia
Numerical analysis creates, analyzes, and implements algorithms for numerical solving
real-world problems, such as fluid dynamics, image processing, social sciences, petroleum
engineering, and many other areas. The overall goal is the design and analysis of numerical
tools to give efficient and reliable approximations to hard problems. For example, computing
the trajectory of a rocket, predicting the weather, and deciding the price of airplane tickets.
The course will focus on the development of numerical algorithms and the primary objective of the course is to understand the construction of numerical algorithms and, more
importantly, the applicability and limits of their usages. We will discuss the mathematical
foundations of numerical algorithms and also pay attention to their computational complexity, computer storage, and finite precision arithmetic. The emphasis will be the accuracy,
numerical stability, efficiency, robustness, and scalability. The topics include: approximation
theory, numerical integration, solving linear and nonlinear equations, numerical optimization, and numerical solution to differential equations. In-class examples, homework, and
projects will feature some of the practical applications of these algorithms to solve realworld problems. Homework problems will consist of written problems as well as computer
programming assignments in Matlab (or your favorite programming language).
This course is intended as a rigorous first graduate level course in Numerical Analysis.
We will cover a large portion of the Numerical Analysis qualifying exam core topics, which
include numerical integration and orthogonal polynomials, interpolation, numerical solution
of differential equations, unconstrained optimization, and solution of nonlinear equations.
Math 250-NAII
Numerical Analysis II
Course Information
Fall 2015
Block: G+MW, 1:30-2:45
Instructor: Misha Kilmer
Email: misha.kilmer@tufts.edu
Office: Bromfield-Pearson 103
Office hours: (Spring 2015) By Appointment
Prerequisites: 226 and 228, or permission
Text: TBD
Course description:
In this course, we will cover, in depth, a selection of important topics from Numerical Analysis
and Numerical Linear Algebra. It is assumed that students have already had our 226/228
sequence (or equivalent). Homework will consist of proofs as well as coding assignments, so
the ability and willingness to program are musts. It is likely there will be multiple sources,
and not a single textbook for the course, given the variety of topics.
An list of likely topics includes: Matrix polynomials and invariant subspaces; eigenvalue
and singular value perturbation bounds; perturbation of invariant subspaces. Detailed investigation of short-term recurrence Krylov methods for symmetric and unsymmetric systems,
and the connection to the topics previously listed as well as preconditioning. Quasi-Newton
methods. Trust-region methods for non-linear least squares problems. Designing timestepping methods for realistic problems. Introduction to Lattice-Boltzmann method. Time
permitting, intro to parallel computing.
MATH 250 (COMPLEX ALGEBRAIC CURVES) FALL 2015
Instructor: Yusuf Mustopa
E-mail: Yusuf.Mustopa@tufts.edu
Office Hours: Bromfield-Pearson 106, TBD
Course Meeting Time: Block D+ (Tuesday and Thursday, 10:30AM-11:45AM).
Prerequisites: A course in complex variable theory (e.g. Math 158), or instructor’s consent.
Text: Miranda’s book Algebraic Curves and Riemann Surfaces (Graduate Studies in Mathematics vol. 5,
AMS) will be our primary reference.
Description: The theory of complex algebraic curves—which originated with the study of elliptic integrals—
ranks among the greatest successes of late-19th/early-20th-century mathematics. It has substantial connections with areas ranging from number theory to partial differential equations, and is a pillar of modern
algebraic geometry as well as a subject of research interest in its own right.
This course introduces complex algebraic curves as compact oriented surfaces with complex structure,
and provides a tour of their geometry whose viewpoint shifts from complex-analytic to algebro-geometric
as the semester progresses. The following is a list of topics we plan to cover; more advanced topics will be
included if time and interest allow.
•
•
•
•
•
•
•
•
•
•
•
•
Complex structures on 2-dimensional manifolds
Examples of compact Riemann surfaces
Holomorphic and meromorphic functions
Group actions and monodromy
Elementary complex projective geometry
Integration and the residue theorem
Divisors and maps to projective space
Riemann-Roch Theorem
The canonical embedding
Castelnuovo’s bound for curves in projective 3-space
Inflectionary behavior and Weierstrass points
Abel’s Theorem
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 https://sites.google.com/site/tuftssiam/ 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.