Course Information Book
Department of Mathematics
Tufts University
Spring 2013
This booklet contains the complete schedule of courses offered by the Math Department in the Spring
2013 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
p. 2
Course schedule
p. 3
Math 61
p. 5
Math 70
p. 6
Math 72
p. 7
Math 128
p. 8
Math 136
p. 9
Math 146
p. 10
Math 150-01
p. 11
Math 150-02
p. 12
Math 150-03
p. 13
Math 158
p. 14
Math 162
p. 15
Math 216
p. 16
Math 250-01
p. 17
Math 250-02
p. 18
Math 252
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
Fundamentals of
Mathematics
Old
Number
4
New
Number
4
Course Name
Fundamentals of
Mathematics
New
Number
4
Old
Number
4
Introduction to Calculus
Introduction to Finite
Mathematics
5
30
10
10
6
14
Introductory Special
Topics
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
TUFTS UNIVERSITY
DEPARTMENT OF MATHEMATICS
[Schedule as of November 6, 2012]
NEW COURSE #
(OLD #)
14-01
02
03
(Old 6)
Introduction to Finite Mathematics
B
C
F
Richard Weiss*
TBA
Alex Babinski
16-01
(Old 8)
Symmetry
D
Gail Kaufmann
RA
RB
16RA
Recitation Friday 10:30-11:20
Recitation Friday 1:30-2:20
EF
GF
Jonathan Ginsberg
Jonathan Ginsberg
19-01
02
(Old 9)
The Mathematics of Social Choice
E+WF
H+
Christine Offerman
Linda Garant
Introductory Statistics
G+MW
Kei Kobayashi
Introduction to Calculus
C
B
F
George McNinch*
TBA
Linda Garant
16RB
21-01
COURSE TITLE
BLOCK
SPRING 2013
ROOM
INSTRUCTOR
30-01
02
03
(Old 5)
32-01
02
03
04
(Old 11)
Calculus I
B
C
D
F
Marjorie Hahn
Stephanie Friedhoff
Kim Ruane*
Gail Kaufmann
34-01
02
03
04
05
06
(Old 12)
Calculus II
B
F
F
F
H
H
Mauricio Gutierrez
Moon Duchin*
Marjorie Hahn
Mary Glaser
Thomas Barthelme
TBA
36-01
02
(Old 50)
Applied Calculus II
EM, E+WF
G+MW,GF
Zachary Faubion
TBA
42-01
02
03
04
05
(Old 13)
Calculus III
B
C
D
F
H
TBA
Zachary Faubion
Fulton Gonzalez*
Mauricio Gutierrez
TBA
44-01
(Old 18)
Honors Calculus III
E+
Zbigniew Nitecki
51-01
02
03
04
05
(Old 38)
Differential Equations
B
C
C
D
H
Jens Christensen
Boris Hasselblatt*
Zbigniew Nitecki
Thomas Barthelme
Jens Christensen
TUFTS UNIVERSITY
DEPARTMENT OF MATHEMATICS
[Schedule as of November 6, 2012]
SPRING 2013
61-01
02
(Old 22)
Discrete Mathematics
[Math 61 is the same as Comp 61]
K+
J
Alberto Lopez Martin
Lenore Cowen
[Computer Science]
70-01
02
03
(Old 46)
Linear Algebra
D+
E+MW
G+MW
Mary Glaser
Misha Kilmer
Alberto Lopez Martin
72-01
(Old 54)
Abstract Linear Algebra
A
Richard Weiss
128-01
Numerical Linear Algebra
D+
Christoph Borgers
136-01
Real Analysis II
F
Loring Tu
146-01
Abstract Algebra II
G+MW
Montserrat Teixidor
150-01
02
03
Mathematical Neuroscience
Linear Algebra II
Point-Set Topology
L+
I+
D+
Christoph Borgers
Misha Kilmer
Genevieve Walsh
158-01
Complex Variables
C
Moon Duchin
162-01
Statistics
E+MW
Kei Kobayashi
163-01
Computational Geometry
[Math 163 is the same as Comp163]
10+M
Diane Souvaine
[Computer Science]
212-01
Topics in Analysis
L+
Fulton Gonzalez
216-01
Representations of Finite Groups
E+WF
George McNinch
250-01
02
03
04
Numerical Methods for PDE'S
Differential Forms in Algebraic Topology
Research Topics in Geometry
Seminar in Computational & Applied Mathematics
G
H+
J+
TBA
Scott MacLachlan
Loring Tu
Genevieve Walsh
Scott MacLachlan
252-01
Nonlinear PDE's
D+
James Adler
Math 61
Discrete Mathematics
Course Information
Spring 2013
Block: K+ (Mon Wed 4:30 – 5:15)
Instructor: Alberto López Martı́n
Email: alberto.lopez@tufts.edu
Office: Bromfield-Pearson 106
Office hours: (Fall 2012) Monday 3:00 – 4:00 p.m. and Tuesday 12:00 – 1:20 p.m.
Phone: (617) 627-2357
Prerequisites: Math 32, Comp 11, or consent.
Required for: Computer Science majors. Recommended for Math majors.
Text: Discrete Mathematics and its Applications, 7th edition, by K. Rosen (McGraw Hill)
Course description: Discrete mathematics is not the name of a branch of mathematics; it
is the description of several branches that all have a common feature: they deal with objects
that can assume only distinct, separated values. The term “discrete mathematics” is therefore
used in opposition to “continuous mathematics,” which are the branches in math dealing with
objects that can vary smoothly (and which includes, for example, Calculus). While discrete
objects can often be characterized by integers, continuous objects require real numbers.
We will start by studying propositional logic and learning how to read, understand, and
write proofs. We will then use our newly-acquired skills to study other fields that are considered
parts of discrete mathematics, such as Combinatorics (the study of how discrete objects combine
with one another and the probabilities of various outcomes) and Graph Theory. We will also
examine sets, permutations, equivalence relations and properties of the integers.
Your grade will be based on two midterm exams and a final exam, as well as homework.
An old riddle. . .
The Prussian city of Königsberg spanned both
banks of the river Pregel, and two islands in it.
Seven bridges connected both banks and both islands with each other. A popular pastime among
the citizens of Königsberg was to attempt a solution to the following problem: How to walk across
both banks and both islands by crossing each of the
seven bridges only once?
. . . and the modern use of discrete math
Google Maps gives driving directions between (almost) any two points in the US quite
accurately. For that, Google uses many basic algorithms from Graph Theory - for example,
Dijkstra’s algorithm - to find the shortest path between two nodes in a graph.
Google’s search engine ranks pages based on how important they are to the world. This is
done with a complex math algorithm that uses Linear Algebra and Graph Theory!
Math 70
Block: E+ (MW 10:30-11:45)
Instructor: Misha Kilmer
misha.kilmer@tufts.edu
Office: Bromfield-Pearson 114
Office hours: (Fall 2012)
On leave - by appointment only
Phone: (617) 627-2005
Linear Algebra
Course Information
Block: G+ (MW 1:30-2:45)
Instructor: Alberto López
alberto.lopez@tufts.edu
Office: Bromfield-Pearson 106
Office hours: (Fall 2012)
M 3:00-4:00, T 12:00-1:30,
or by appointment
Phone: (617) 627-2357
Spring 2013
Block: D+ (TR 10:30-11:45)
Instructor: Mary Glaser
mary.glaser@tufts.edu
Office: Bromfield-Pearson 4
Office hours: (Fall 2012)
T 1:00-2:00,
R 2:30-4:30,
or by appointment
Phone: (617) 627-5045
Prerequisites: Math 34 (or 39) or consent.
Text: Linear Algebra and its applications, 4th edition, by David Lay. Pearson-Addison-Wesley,
2011.
Course description: Linear algebra is the study of matrices, vector spaces, and linear transformations. In Math 70 we start by studying systems of linear equations. For example, 2x+3y = 5 is a
linear equation in the unknowns x and y, whereas ex = y is nonlinear. The study of linear equations
quickly leads to useful concepts such as vector spaces, dimension (you will learn about four-, five-,
and infinite-dimensional spaces), linear transformations, and eigenvalues. These concepts will help
you solve linear equations efficiently, and more importantly, they fit together in a beautiful whole
that will give you a deeper understanding of mathematics.
Linear algebra arises everywhere in mathematics (you will use it in almost every one of our upper
level courses) as well as 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 the most common methods for studying nonlinear systems approximate
them by linear systems.
Among the applications of linear algebra are solutions of linear systems of equations, determining
conic curves and quadric surfaces not in standard form, the second-derivative test in vector calculus,
computer graphics, linear economic models, and differential equations. Linear algebra also enters
in further study of calculus, where the derivative is viewed as a linear transformation.
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 most upper-level mathematics
courses. The course is also useful to majors in computer science and engineering, as well as those
in the natural and social sciences. In addition to computation and problem-solving, the course will
introduce the students to axiomatic mathematics and simple proofs.
Math 72
Abstract Linear Algebra
Course Information
Spring 2013
Block: A (Monday, Wednesday 8:30-9:20, Thursday 9:30-10:20)
Instructor: Richard Weiss
Email: rweiss@tufts.edu
Office: Bromfield-Pearson 116
Office hours: (Spring 2013) Tuesday, Wednesday, Friday 1:00-2:00
Phone: (617) 627-3802
Prerequisites: Math 34
Text: Linear Algebra, 4th ed., S. Friedberg, A. Insel and L. Spence, Prentice-Hall, 2003.
Course description: Linear algebra is the study of vector spaces, matrices and linear
transformations. Linear algebra provides a fundamental link between our geometric intuition
of the world around us and the powerful tools of algebra.
The results and concepts of linear algebra are essential in virtually every branch of higher
mathematics, from the most pure to the most applied. In fact, one could almost define higher
mathematics as the study of ideas that are based on linear algebra, except that this definition
would include most of physics as well!
This course is the honors version of Math 70. From the beginning, we will adopt a more
sophisticated point of view. In particular, proofs and “theory” will play a greater role than
in Math 70.
This course is recommended for all those wanting to learn linear algebra who also enjoy
a little extra challenge. Math 72 of course counts as a replacement for Math 70 wherever
Math 70 is required as a prerequisite.
There will be three exams in this course, two during the semester and one at the end,
and there will be daily assignments.
Math 128
Numerical Linear Algebra
Course Information
Spring 2013
Block: J+ (Tuesday, Thursday 3:00-4:15 )
Instructor: Christoph Börgers
Email: christoph.borgers@tufts.edu
Office: Bromfield-Pearson 215
Office hours: (Fall 2012) Monday 1:30–3:00, Thursday 1:30–3:00, and by appointment
Phone: (617) 627-2366
Prerequisites: Math 70 or 72, and willingness and ability to do significant Matlab programming with only a small amount of programming instruction.
Text: Numerical Linear Algebra, by Lloyd N. Trefethen and David Bau, SIAM, 1997
Course description: We will study the computational solution of two types of problems
in this course: (1) systems of equations: Ax = b, where b is a given vector, A is a given
rectangular matrix, and the vector x to be computed, and (2) eigenvalue problems: Ax = λx,
where A is a given square matrix, and the vector x 6= 0 and the number λ are to be
computed. The importance of these problems could hardly be over-stated. Large linear
systems of equations are central parts of almost all computational problems in the physical
sciences and in engineering. Eigenvalue problems are the key to understanding the stability
of physical systems. Even though these problems are so fundamental, their computational
solution remains the subject of a large and active current research field.
Of course, linear systems of equations can in principle be solved using the row reduction
method that you learned about in your linear algebra course, if not in high school. However,
the systems that one must solve in practice are often gigantic: A system with a million
equations in equally many unknowns would be considered of moderate size. For systems of
this size, row reduction is hopelessly inefficient, and one must invent faster methods, typically
exploiting special properties of the systems to be solved.
General eigenvalue problems cannot be solved explicitly if the matrix A is larger than
4 × 4. In fact, if one could solve all n × n eigenvalue problems, one could solve all polynomial
equations of degree n (you will learn why in the course), but it is known that there is no
explicit formula for solving all polynomial equations of degree n > 4. One must therefore
develop algorithms for computing approximations to the solutions of eigenvalue problems.
We will study the most fundamental algorithms for linear systems and eigenvalue problems and their mathematical foundations. We will implement the algorithms in Matlab.
Math 136
Real Analysis II
Course Information
Spring 2013
Block: F (Tuesday, Thursday, Friday noon–12:50 p.m.)
Instructor: Loring Tu
Email: loring.tu@tufts.edu
Office: Bromfield-Pearson 206
Office hours: (Fall 2012) Tue 12:50–1:15 (Math 135 only), Thu 2:30-3:30, Fri 1:00–2:00
Phone: (617) 627-3262
Prerequisites: Math 135 or consent.
Text: (Tentative) Patrick Fitzpatrick, Advanced Calculus, 2nd edition, American Mathematical Society, 2009. (ISBN-10: 0821847910)
Course description:
This course is a continuation of Math 135. In Math 135 we laid the foundation of real
analysis by studying the topology of a metric space and the concept of continuity. Math 136
applies these tools to give a rigorous treatment of derivatives and integrals of real-valued
and vector-valued functions on Rn .
The derivative of a function f : Rn → Rm is defined to be a linear transformation, representable by a matrix of partial derivatives. Using this definition, we prove rules for differentiation (including the product rule and the chain rule), the mean-value theorem, Taylor’s
theorem, and conditions for a real-valued function to have maxima and minima. All these
are results you’ve encountered in calculus courses, but now you will know when and why
they are true.
Two new results are the implicit function theorem, giving conditions under which an
equation can be solved locally, and the inverse function theorem, giving conditions under
which a function is locally invertible.
The second half of the course makes precise the notions of “area” and “volume” and
develops the theory of Riemann integrals. As before, we prove some familiar theorems such
as Fubini’s theorem and the change of variables formula.
If time permits, we will end the course with Fourier series, which has important applications to engineering and physics.
As in Math 135, apart from laying the theoretical foundation of calculus, a companion
goal of the course is to hone your ability to formulate precisely mathematical ideas and to
read and write proofs.
Math 146
Abstract Algebra II
Course Information
Spring 2013
Block: G+ MW (Monday, Wednesday 1:30-2:45)
Instructor: Montserrat Teixidor i Bigas
Email: mteixido@tufts.edu
Office: Bromfield-Pearson 115
Office hours: (Fall 2012) Tuesdays and Thursdays 1-2, Fridays 11-12 and by appointment
Phone: (617) 627-2358
Prerequisites: Math 145 or consent.
Text: John B. Fraleigh A first course in abstract algebra, 7th edition, Addison Wesley 2002
ISBN 0-201-76390-7.
Course description:
This course is a continuation of Math 145. We look at rings more in depth and move
to fields. Rings and Fields recapture in abstract properties of the sets of numbers (integers,
rational, real...) as well as polynomials. The main focus of the course is to understand how
to find the solution to polynomial equations. We will see the beautiful interplay between
the roots of polynomials and the groups of symmetries that we studied in 145. We will show
that there are counterparts to the quadratic formula for degrees 3 and 4 but that no such
formula can exist for degree 5 and higher. In the same frame of mind, we will look at why
the complex numbers are the best set of numbers one could hope for and how a straightedge
and compass can help you with your arithmetic.
As Abstract Algebra is a basic tool in most other areas of pure mathematics, particularly
Geometry, Topology and various subareas of Physics, this course will help you get ready for
advanced Math and Physics courses at the graduate level and/or a world of fun discovery
that you may want to engage on your own.
There will be two midterms exams and one final and weekly homework assignments.
Math 150-01
Mathematical Neuroscience
Course Information
Spring 2013
Block: L+ (Tuesday, Thursday 4:30–5:45)
Instructor: Christoph Börgers
Email: cborgers@tufts.edu
Office: Bromfield-Pearson 215
Office hours: (Fall 2012) Monday 1:30–3:00, Thursday 1:30–3:00, and by appointment
Phone: (617) 627-2366
Prerequisites: Math 38
Text: None. Notes will be distributed electronically.
Course description: This is a course on differential equations in neuroscience, a field
that began 60 years ago with a paper by the physiologists A. L. Hodgkin and A. F. Huxley
titled “A Quantitative Description of Membrane Current and its Application to Conduction
and Excitation in Nerve” (Journal of Physiology 1952). At the time it was well known 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 known as the Hodgkin-Huxley equations. Eleven years after their famous paper
appeared, in 1963, Hodgkin and Huxley were awarded a Nobel Prize.
It is only a slight simplification to say that the field of mathematical neuroscience is the
study of coupled systems of Hodgkin-Huxley equations. The ultimate goal of this endeavor
is to understand what is unquestionably the most astonishing form of matter found on earth
— the human brain. Not surprisingly, the field is very far from reaching this goal, but much
has been understood about the transitions from rest to spiking in various different types of
neurons, the propagation of signals through neuronal networks, the frequent synchronization
of activity in large groups of neurons, the origins and functions of oscillatory activity in
networks of neurons (as observed for instance in EEG measurements), and other issues
related to the physics of the brain.
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 numerical methods. This course will focus mostly on the theoretical, qualitative study
of the differential equations, but will include some computing as well.
No background in biology or computing will be assumed.
Math 150-02
Linear Algebra II
Course Information
Spring 2013
Block: I+ (Monday, Wednesday 3:00 - 4:15)
Instructor: Misha Kilmer
Email: misha.kilmer@tufts.edu
Office: Bromfield-Pearson 114
Office hours: (Fall 2012) On leave. By appt. only
Phone: (617) 627-2005
Prerequisites: Math 70 (old 46) or 72 (old 54) or consent.
Text: (Recommended∗ )Linear Algebra and its applications, 4th edition, by David Lay.
Pearson-Addison-Wesley, 2011.
Course description: Linear algebra arises everywhere in mathematics (you will use it
in almost every one of our upper level courses) as well as in physics, chemistry, economics,
biology, and a range of other fields. Given the clear level of the importance and breadth of
linear algebra, no first semester course (i.e., 70 or 72) can possibly scratch the surface of all
the interesting linear algebra topics.
This course is recommended for all those who enjoyed their first semester course and desire
to gain breadth and depth in the subject. Topics will include, but are not limited to: symmetric (or Hermitian) positive definite matrices and their connection to the 2nd derivative
test in third semseter calculus, quadratic forms, Jordan normal form, matrix polynomials,
matrix exponentials and their connection to the solution of differential equations, PerroneFrobenius theory and its application with respect to Markov chains, inner product spaces,
orthogonal projections, the singular value decomposition, interpolation as a linear algebra
problem, and various important theorems in the field. If time allows, we will also cover some
recent results (and open problems) in multilinear algebra.
There will be three exams in this course, two during the semester and one at the end, as
well as a number of assignments heavily weighted towards proofs. Every effort will be made
to connect the topics in the course with applications in other mathematics courses or in real
world problems.
* As no one text covers, at the desired level, all the topics we will cover in this course, we
will be relying heavily on lecture notes and materials made available on the web. Therefore,
good attendance, though not required, is highly recommended.
Math 150-03
Point-Set Topology
Course Information
Spring 2013
Block: D+ (Tuesday, Thursday 10:30-11:45)
Instructor: Genevieve Walsh
Email: genevieve.walsh@tufts.edu
Office: Bromfield-Pearson 213
Office hours: (Fall 2012) Monday 3:00-4:30 and Thursday 9:00-10:30
Phone: (617) 627-4032
Prerequisites: Some course in the mathematics department involving proofs, willingness
to present material in class.
Text: There is no text for this class. Copies of notes will be provided by the instructor,
and will also be on Trunk. Students are requested not to look at other sources on point-set
topology.
Course description: The subject of point-set topology is extremely useful in many areas,
particularly in analysis and topology. This class is a great way to prepare for math 135. We
will start by covering cardinality and the axiom of choice and general topology, including
creating our own topologies. We will describe what a basis of a topology is, the subspace
topology, and the product topology. Then we will discuss various separation and covering
properties, and investigate maps between topological spaces. Finally, we will discuss notions
of connectivity of a topological space. Throughout, specific examples will be emphasized.
This course is designed to get you to think mathematically for yourself. To that end,
there will be a running list of definitions and theorems and the goal of the class is to jointly
work through proving the theorems. Everyday, students from the class will present problems
and the class will discuss them and decide if they are ready to proceed. This method of
instruction is called the modified Moore method, and it is a very fun and challenging way
to deeply engage with the material.
There will be daily homework assignments, two in-class exams, and a final.
Math 158
Complex Variables
Course Information
Spring 2013
Block: C (Tue Wed Fri 9:30-10:20)
Instructor: Moon Duchin
Email: moon.duchin@tufts.edu
Office: Bromfield-Pearson 113
Office Hours: TBD
Prerequisites: Math 42 or 44 or consent of the instructor.
Text: Basic Complex Analysis, 3rd Edition, by Marsden and Hoffman. WH Freeman, 1999.
Course description: This course is an introduction to the theory of functions of one
complex variable, a subject that has many important applications in physics, engineering,
and elsewhere, and which is also fundamentally geometric in a way that is not always true
for real variables. (The usefulness of complex analysis, and its harmony with other fields
of
√ mathematics, is ironic since it is founded on the “imaginary” or “impossible” number
−1...)
In this course we will primarily explore the notion of an analytic function on a region in
the complex plane. Quite simply, an analytic function is a complex-differentiable function,
and yet this alone is enough to ensure that the function is completely determined by a very
small sample of its values. This is an example of the rigidity phenomena at the heart of
complex analysis.
The principal topics in our study of analytic functions include elementary functions (including Möbius transformations), contour integration, Cauchy’s theorem and integral formula, power series, residues, conformal mappings, and analytic continuation. We may touch
on some of the many applications of complex analysis inside and outside of mathematics,
such as two-dimensional potential theory, Fourier series, and hyperbolic geometry.
The course will be taught rigorously, with plenty of theorem-proving, but also with an
emphasis on geometric intuition, as well as many computational examples and practical
calculus-style problems. It is appropriate for mathematics, science, and engineering majors,
and also as a view of more abstract mathematics for anyone hoping to glimpse the vista
beyond calculus.
Math 162
Statistics
Course Information
Spring 2013
Block: E+ (Monday, Wednesday, Friday 10:30–11:45) in BP-2
Instructor: Kei Kobayashi
Email: kei.kobayashi@tufts.edu
Office: Robinson Hall 156
Office hours: (Fall 2012) Tuesday 1:30-2:30
Phone: 617-627-6102
Prerequisites: Math 161 or consent.
Text: Richard J. Larsen & Morris L. Marx, An Introduction to Mathematical Statistics and
Its Applications, 5th Edition, Prentice Hall (2011). ISBN-13: 978-0321693945
Course description: Suppose I claim that I make 75% of my basketball free throws. To
test my claim, you ask me to shoot 40 free throws and I make only 24 of the 40. Do you
believe my claim?
Statistics is the science of gaining information from numerical data. Our technological
world generates data at an enormous rate. However, all too often the data is improperly
obtained and/or improperly assessed. Important everyday decisions for individuals, corporations, societies, and governments hinge on a proper understanding and assessment of data.
Every facet of industry, science, engineering, economics and business benefits from a solid
knowledge of statistics.
Statistics uses the major ideas and concepts from probability. Only via the use of probability can a proper assessment be made of data collected for real-world problems. Math 162
provides opportunities to experience and learn the statistical thinking a functioning statistician must develop and use constantly. It also provides preparation for actuarial exams,
graduate work in applied mathematics, and courses in physical/social sciences requiring statistical methodology. Topics to be covered include 1) estimating an unknown parameter
of the underlying population, 2) turning data into evidence, as in testing a hypothesis, 3)
determining existence and strength of a correlation between several variables, 4) making
predictions, 5) testing models for goodness-of-fit, etc.
Assessment will be based on weekly problem sets, midterms, and final exam or project
(to be determined).
Math 216-01
Representations of Finite Groups
Course Information
Spring 2013
Block: E+ (Wed and Fri 10:30–11:45)
Instructor: George McNinch
Email: george.mcninch@tufts.edu
Office: Bromfield-Pearson 112
Office hours: (Fall 2012) T and Th 2:30–3:30, F 11:00–12:00
Phone: (617) 627-6210
Prerequisites: Math 215 or consent.
Text: Abstract Algebra, 3rd Edition by Dummit and Foote (John Wiley and Sons inc.)
Course description: This course will finish the standard first-year course in graduate
algebra initiated in Math 215.
Subsequently, Math 216 will investigate the representation theory of a finite group. If G
is a group, k a field, and V a (finite dimensional) vector space over k, a representation of
G on V is a group homomorphism ρ : G → GL(V ). Here GL(V ) is the group of invertible
k-linear transformations from V to itself, and if d = dimk V , then GL(V ) ' GLd (k) may be
identified with the group of invertible d × d matrices having coefficients in k.
We begin by investigating the representations of a finite group G in the case where
the field k is algebraically closed of characteristic zero – e.g. k = C, the field of complex
numbers. In this setting, the theory of representations of G on k-vector spaces – often
referred to as the “ordinary representation theory” of G – is rather complete. We will see
that to a representation (ρ, V ) one can associate a certain class function χV : G → k – a
function constant on the conjugacy classes of G – which “contains” much of the information
about the representation V . As a consequence, we find for example that the number of
irreducible ordinary representations of G is the same as the number of conjugacy classes in
G.
Along the way, our study will lead to investigation of finite dimensional (semi)simple kalgebras and Wedderburn’s Theorem. Most of the required material is contained in [DummitFoote], though the course lectures will supplement material found in the text.
Math 250-01
Numerical Methods for Partial Differential
Equations
Course Information
Spring 2013
Block: G, MWF, 1:30-2:20 PM
Instructor: Scott MacLachlan
Email: scott.maclachlan@tufts.edu
Office: Bromfield-Pearson 212
Office hours: (Fall 2012) TF 11:00 AM - 12:30 PM
Phone: 617-627-2356
Prerequisites: Math 135 and 151/251 or consent.
Text: No required text. Course notes will be given as handouts.
Course description: While partial differential equations (PDEs) naturally arise in the
mathematical models for many areas of science and engineering, analytical methods for the
solution of these equations are successful only in certain special cases. For PDE models of
heterogeneous media (for example, of flow through a porous aquifer, seismic imaging of the
Earth’s structure, or the electrical activation of the heart muscle), very little can be done
with analytical techniques. Instead, we turn to computational simulation, which has become
an increasingly important tool in many fields of science and engineering, replacing expensive
or impractical experimentation in these areas. The focus of this course is on the key step
of the simulation process, where approximations to the solutions of these PDEs are found
using computers.
The course material breaks roughly into thirds. We will consider the numerical solution of
hyperbolic equations using finite-difference discretizations. For these problems, we will look
at the questions of accuracy, consistency, and stability, and how they are related through the
Lax Equivalence Theorem. For elliptic equations, we will consider the finite-element method
and its analysis using variational formulations and approximation theory in Sobolev spaces.
Finally, we will see the connection between PDEs and the solution of large systems of linear
equations, and the numerical techniques needed to do this efficiently. In particular, we will
see how multigrid methods naturally arise in the numerical solution of elliptic equations and
how they can be used to achieve optimally scaling simulation algorithms.
This course is intended to be approachable for students both from mathematics and from
areas of science and engineering where computer simulation is commonplace. We will both
prove mathematical theorems and write simulation codes.
Math 250-02
Differential Forms in Algebraic Topology
Course Information
Spring 2013
Block: H+ (Tuesday, Thursday, 1:30–2:45 p.m.)
Instructor: Loring Tu
Email: loring.tu@tufts.edu
Office: Bromfield-Pearson 206
Office hours: (Fall 2012) Tue 12:50–1:15 (Math 135 only), Thu 2:30-3:30, Fri 1:00–2:00
Phone: (617) 627-3262
Prerequisites: Math 217 (Manifolds) or 218 (Algebraic Topology I) or consent. Students who
have not taken Math 217 should first speak to the instructor.
Text: Raoul Bott and Loring W. Tu, Differential Forms in Algebraic Topology, 3rd corrected
printing, Springer, New York, 1995.
Course description:
The central problem in topology is to decide if two spaces are homeomorphic or if two manifolds are
diffeomorphic. Algebraic topology offers a possible solution by transforming the geometric problem
into an algebraic problem. To accomplish this, one associates to each topological space an algebraic
object such as its cohomology vector space so that homeomorphic topological spaces correspond
to isomorphic vector spaces. Such an association is called a functor. The cohomology functor
greatly simplifies the original problem, for the dimension alone determines the isomorphism class
of a finite-dimensional vector space. This gives a simple necessary condition for two topological
spaces to be homeomorphic.
Over the past one hundred years our forefathers have discovered many useful functors, among
which are the fundamental group, higher homotopy groups, homology groups, and cohomology
rings. There are various versions of cohomology—singular, de Rham, and Čech, among others.
This leads naturally to two new problems: (1) How does one compute these functors? (2) What
are the relations that exist among the different functors? Algebraic topology, as it is usually
practiced, concerns itself with these questions in the continuous category, with topological spaces
and continuous maps. Its great generality makes it applicable in a variety of settings, but also
renders many of its constructions difficult and inaccessible to the novice.
The guiding principle in this course is to use differential forms as an aid in exploring some of
the less digestible aspects of algebraic topology. We will work primarily but not exclusively with
smooth manifolds. This has the advantage of simplifying the theory as well as the constructions; for
example, all three versions of cohomology coincide on smooth manifolds. It allows us to study topics
that are normally relegated to a second course in algebraic topology, such as Poincaré duality, the
Thom isomorphism, and spectral sequences, without assuming a knowledge of homology. These
topics will respond to the two problems mentioned above: (1) techniques of computation, (2)
theorems relating the various functors.
More specifically, the topics to be covered are the following: de Rham cohomology, cohomology
with compact support, Mayer–Vietoris sequence, Poincaré lemma on the cohomology of Rn , degree
of a proper map, Brouwer fixed-point theorem, hairy-ball theorem, Poincaré duality for a manifold,
Künneth formula for the cohomology of a product, Thom isomorphism for the cohomology of
a vector bundle, Euler class, the nonorientable case, Čech cohomology, isomorphism between de
Rham and Čech cohomology, presheaves, Hopf index theorem on the zeros of a vector field, spectral
sequences, cohomology with integer coefficients, and de Rham’s theorem.
Math 252
Nonlinear Partial Differential Equations
Course Information
Spring 2012
B LOCK : D+: Tuesday, Thursday 10:30-11:45 a.m.
I NSTRUCTOR : James Adler
E MAIL : james.adler@tufts.edu
O FFICE : Bromfield-Pearson 209
O FFICE HOURS : Fall 2012 Tues. 1:30-3:00pm and Wed. 3:00am-4:30pm
P HONE : 7-2354
P REREQUISITES : Math 251, or permission of the instructor.
T EXT: TBD
C OURSE DESCRIPTION :
The course is on the analysis and solution of nonlinear partial differential equations
(PDEs). Nonlinear PDES are used to describe many physical phenomena, however, their
solution can be difficult and problem-dependent. In this course we’ll learn the tools to ask
questions like: what is causing the turbulence on my flight?; why do the bubbles in a Guinness
go up not down?; and, of course, why on earth is traffic backed up so badly on I-93!!??. In
addition, all students are given the chance at one million dollars!!∗
This course will cover topics such as the method of characteristics for quasilinear equations, which is later generalized to other kinds of nonlinearity, similarity solutions, and
perturbation methods. A variety of examples will be used to study various phenomena
that arise in nonlinear PDEs, including shocks, jump conditions, etc. We will consider
various equations that describe different physics, including reaction-diffusion equations
(Fisher’s equation), convection-diffusion equations (Burger’s equation), kdV equations,
hyperbolic equations (gas dynamics), and other hydrodynamic equations (Navier-Stokes,
phase dynamics, etc.) if time permits.
As this is a graduate course, we will study these topics at a deeper level, investigating
questions of existence and uniqueness, linear stability theory, and energy minimization.
∗
The Navier-Stokes equations are a nonlinear system of equations that describe fluid flow. As of yet,
there is no proof for the existence and uniqueness of these equations. Listed as one of the Millenium Problems, http://www.claymath.org/millennium/Navier-Stokes_Equations, solving this problem
will earn you one million dollars and lots of fame...
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 51 before taking these upper level courses.
II. Two additional 100-level math courses.
III. Three additional mathematics courses numbered 51 or higher; 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:
Note: It is the student’s responsibility to return completed, signed
degree sheets to the Office of Student Services, Dowling Hall.
(Form Revised September 8, 2012)
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 22: 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:
Note: It is the student’s responsibility to return completed, signed degree
sheets to the Office of Student Services, Dowling Hall.
(Form Revised February 8, 2012)
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.
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 May 7th, 2012)
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.