Embed Size (px)
Citation preview
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 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
TUFTS UNIVERSITY DEPARTMENT OF MATHEMATICS
[Schedule as of November 6, 2012]
SPRING 2013
NEW COURSE # (OLD #) COURSE TITLE BLOCK ROOM INSTRUCTOR
14-01 (Old 6) Introduction to Finite Mathematics B Richard Weiss*
02 C TBA
03 F Alex Babinski
16-01 (Old 8) Symmetry D Gail Kaufmann
RA 16RA Recitation Friday 10:30-11:20 EF Jonathan Ginsberg
RB 16RB Recitation Friday 1:30-2:20 GF Jonathan Ginsberg
19-01 (Old 9) The Mathematics of Social Choice E+WF Christine Offerman
02 H+ Linda Garant
21-01 Introductory Statistics G+MW Kei Kobayashi
30-01 (Old 5) Introduction to Calculus C George McNinch*
02 B TBA
03 F Linda Garant
32-01 (Old 11) Calculus I B Marjorie Hahn
02 C Stephanie Friedhoff
03 D Kim Ruane*
04 F Gail Kaufmann
34-01 (Old 12) Calculus II B Mauricio Gutierrez
02 F Moon Duchin*
03 F Marjorie Hahn
04 F Mary Glaser
05 H Thomas Barthelme
06 H TBA
36-01 (Old 50) Applied Calculus II EM, E+WF Zachary Faubion
02 G+MW,GF TBA
42-01 (Old 13) Calculus III B TBA
02 C Zachary Faubion
03 D Fulton Gonzalez*
04 F Mauricio Gutierrez
05 H TBA
44-01 (Old 18) Honors Calculus III E+ Zbigniew Nitecki
51-01 (Old 38) Differential Equations B Jens Christensen
02 C Boris Hasselblatt*
03 C Zbigniew Nitecki
04 D Thomas Barthelme
05 H Jens Christensen
TUFTS UNIVERSITY DEPARTMENT OF MATHEMATICS
[Schedule as of November 6, 2012]
SPRING 2013
61-01 (Old 22) Discrete Mathematics K+ Alberto Lopez Martin
02 [Math 61 is the same as Comp 61] J Lenore Cowen
[Computer Science]
70-01 (Old 46) Linear Algebra D+ Mary Glaser
02 E+MW Misha Kilmer
03 G+MW 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 Mathematical Neuroscience L+ Christoph Borgers
02 Linear Algebra II I+ Misha Kilmer
03 Point-Set Topology D+ Genevieve Walsh
158-01 Complex Variables C Moon Duchin
162-01 Statistics E+MW Kei Kobayashi
163-01 Computational Geometry 10+M Diane Souvaine
[Math 163 is the same as Comp163] [Computer Science]
212-01 Topics in Analysis L+ Fulton Gonzalez
216-01 Representations of Finite Groups E+WF George McNinch
250-01 Numerical Methods for PDE'S G Scott MacLachlan
02 Differential Forms in Algebraic Topology H+ Loring Tu
03 Research Topics in Geometry J+ Genevieve Walsh
04 Seminar in Computational & Applied Mathematics TBA Scott MacLachlan
252-01 Nonlinear PDE's D+ James Adler
Math 61 Discrete MathematicsCourse Information
Spring 2013
Block: K+ (Mon Wed 4:30 – 5:15)Instructor: Alberto Lopez MartınEmail: [email protected]: Bromfield-Pearson 106Office 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; itis the description of several branches that all have a common feature: they deal with objectsthat can assume only distinct, separated values. The term “discrete mathematics” is thereforeused in opposition to “continuous mathematics,” which are the branches in math dealing withobjects that can vary smoothly (and which includes, for example, Calculus). While discreteobjects can often be characterized by integers, continuous objects require real numbers.
We will start by studying propositional logic and learning how to read, understand, andwrite proofs. We will then use our newly-acquired skills to study other fields that are consideredparts of discrete mathematics, such as Combinatorics (the study of how discrete objects combinewith one another and the probabilities of various outcomes) and Graph Theory. We will alsoexamine 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 Konigsberg spanned bothbanks of the river Pregel, and two islands in it.Seven bridges connected both banks and both is-lands with each other. A popular pastime amongthe citizens of Konigsberg was to attempt a solu-tion to the following problem: How to walk acrossboth banks and both islands by crossing each of theseven bridges only once?
. . . and the modern use of discrete math
Google Maps gives driving directions between (almost) any two points in the US quiteaccurately. 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 isdone with a complex math algorithm that uses Linear Algebra and Graph Theory!
Math 70 Linear AlgebraCourse Information
Spring 2013
Block: E+ (MW 10:30-11:45)Instructor: Misha [email protected]
Office: Bromfield-Pearson 114Office hours: (Fall 2012)On leave - by appointment only
Phone: (617) 627-2005
Block: G+ (MW 1:30-2:45)Instructor: Alberto [email protected]
Office: Bromfield-Pearson 106Office hours: (Fall 2012)M 3:00-4:00, T 12:00-1:30,or by appointmentPhone: (617) 627-2357
Block: D+ (TR 10:30-11:45)Instructor: Mary [email protected]
Office: Bromfield-Pearson 4Office hours: (Fall 2012)T 1:00-2:00, R 2:30-4:30,or by appointmentPhone: (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 transfor-mations. In Math 70 we start by studying systems of linear equations. For example, 2x+3y = 5 is alinear equation in the unknowns x and y, whereas ex = y is nonlinear. The study of linear equationsquickly 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 helpyou solve linear equations efficiently, and more importantly, they fit together in a beautiful wholethat will give you a deeper understanding of mathematics.
Linear algebra arises everywhere in mathematics (you will use it in almost every one of our upperlevel courses) as well as in physics, chemistry, economics, biology, and a range of other fields. Evenwhen a problem involves nonlinear equations, as is often the case in applications, linear systems stillplay a central role, since the most common methods for studying nonlinear systems approximatethem by linear systems.
Among the applications of linear algebra are solutions of linear systems of equations, determiningconic 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 entersin 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) andare urged to take it as early as possible, as it is a prerequisite for most upper-level mathematicscourses. The course is also useful to majors in computer science and engineering, as well as thosein the natural and social sciences. In addition to computation and problem-solving, the course willintroduce the students to axiomatic mathematics and simple proofs.
Math 72 Abstract Linear AlgebraCourse Information
Spring 2013
Block: A (Monday, Wednesday 8:30-9:20, Thursday 9:30-10:20)Instructor: Richard WeissEmail: [email protected]: Bromfield-Pearson 116Office hours: (Spring 2013) Tuesday, Wednesday, Friday 1:00-2:00Phone: (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 lineartransformations. Linear algebra provides a fundamental link between our geometric intuitionof the world around us and the powerful tools of algebra.
The results and concepts of linear algebra are essential in virtually every branch of highermathematics, from the most pure to the most applied. In fact, one could almost define highermathematics as the study of ideas that are based on linear algebra, except that this definitionwould include most of physics as well!
This course is the honors version of Math 70. From the beginning, we will adopt a moresophisticated point of view. In particular, proofs and “theory” will play a greater role thanin Math 70.
This course is recommended for all those wanting to learn linear algebra who also enjoya little extra challenge. Math 72 of course counts as a replacement for Math 70 whereverMath 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 AlgebraCourse Information
Spring 2013
Block: J+ (Tuesday, Thursday 3:00-4:15 )Instructor: Christoph BorgersEmail: [email protected]: Bromfield-Pearson 215Office hours: (Fall 2012) Monday 1:30–3:00, Thursday 1:30–3:00, and by appointmentPhone: (617) 627-2366
Prerequisites: Math 70 or 72, and willingness and ability to do significant Matlab pro-gramming 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 problemsin this course: (1) systems of equations: Ax = b, where b is a given vector, A is a givenrectangular 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 becomputed. The importance of these problems could hardly be over-stated. Large linearsystems of equations are central parts of almost all computational problems in the physicalsciences and in engineering. Eigenvalue problems are the key to understanding the stabilityof physical systems. Even though these problems are so fundamental, their computationalsolution 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 reductionmethod 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 millionequations in equally many unknowns would be considered of moderate size. For systems ofthis size, row reduction is hopelessly inefficient, and one must invent faster methods, typicallyexploiting special properties of the systems to be solved.
General eigenvalue problems cannot be solved explicitly if the matrix A is larger than4×4. In fact, if one could solve all n×n eigenvalue problems, one could solve all polynomialequations of degree n (you will learn why in the course), but it is known that there is noexplicit formula for solving all polynomial equations of degree n > 4. One must thereforedevelop algorithms for computing approximations to the solutions of eigenvalue problems.
We will study the most fundamental algorithms for linear systems and eigenvalue prob-lems 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 TuEmail: [email protected]
Office: Bromfield-Pearson 206Office hours: (Fall 2012) Tue 12:50–1:15 (Math 135 only), Thu 2:30-3:30, Fri 1:00–2:00Phone: (617) 627-3262
Prerequisites: Math 135 or consent.
Text: (Tentative) Patrick Fitzpatrick, Advanced Calculus, 2nd edition, American Mathe-matical Society, 2009. (ISBN-10: 0821847910)
Course description:
This course is a continuation of Math 135. In Math 135 we laid the foundation of realanalysis by studying the topology of a metric space and the concept of continuity. Math 136applies these tools to give a rigorous treatment of derivatives and integrals of real-valuedand vector-valued functions on R
n.The derivative of a function f : Rn
→ Rm is defined to be a linear transformation, rep-
resentable by a matrix of partial derivatives. Using this definition, we prove rules for differ-entiation (including the product rule and the chain rule), the mean-value theorem, Taylor’stheorem, and conditions for a real-valued function to have maxima and minima. All theseare results you’ve encountered in calculus courses, but now you will know when and whythey are true.
Two new results are the implicit function theorem, giving conditions under which anequation can be solved locally, and the inverse function theorem, giving conditions underwhich a function is locally invertible.
The second half of the course makes precise the notions of “area” and “volume” anddevelops the theory of Riemann integrals. As before, we prove some familiar theorems suchas Fubini’s theorem and the change of variables formula.
If time permits, we will end the course with Fourier series, which has important applica-tions to engineering and physics.
As in Math 135, apart from laying the theoretical foundation of calculus, a companiongoal of the course is to hone your ability to formulate precisely mathematical ideas and toread and write proofs.
Math 146 Abstract Algebra IICourse Information
Spring 2013
Block: G+ MW (Monday, Wednesday 1:30-2:45)Instructor: Montserrat Teixidor i BigasEmail: [email protected]: Bromfield-Pearson 115Office hours: (Fall 2012) Tuesdays and Thursdays 1-2, Fridays 11-12 and by appointmentPhone: (617) 627-2358
Prerequisites: Math 145 or consent.
Text: John B. Fraleigh A first course in abstract algebra, 7th edition, Addison Wesley 2002ISBN 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 howto find the solution to polynomial equations. We will see the beautiful interplay betweenthe roots of polynomials and the groups of symmetries that we studied in 145. We will showthat there are counterparts to the quadratic formula for degrees 3 and 4 but that no suchformula can exist for degree 5 and higher. In the same frame of mind, we will look at whythe complex numbers are the best set of numbers one could hope for and how a straightedgeand compass can help you with your arithmetic.
As Abstract Algebra is a basic tool in most other areas of pure mathematics, particularlyGeometry, Topology and various subareas of Physics, this course will help you get ready foradvanced Math and Physics courses at the graduate level and/or a world of fun discoverythat 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 NeuroscienceCourse Information
Spring 2013
Block: L+ (Tuesday, Thursday 4:30–5:45)Instructor: Christoph BorgersEmail: [email protected]: Bromfield-Pearson 215Office hours: (Fall 2012) Monday 1:30–3:00, Thursday 1:30–3:00, and by appointmentPhone: (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 fieldthat began 60 years ago with a paper by the physiologists A. L. Hodgkin and A. F. Huxleytitled “A Quantitative Description of Membrane Current and its Application to Conductionand Excitation in Nerve” (Journal of Physiology 1952). At the time it was well known thatthe interior voltage of a nerve cell can spike, and that voltage spikes are the basis of neuronalcommunication. Hodgkin and Huxley explained the mechanism underlying the spikes, andreproduced it in a system of differential equations modeling the nerve cell. These equationsare now known as the Hodgkin-Huxley equations. Eleven years after their famous paperappeared, 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 thestudy of coupled systems of Hodgkin-Huxley equations. The ultimate goal of this endeavoris 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 muchhas been understood about the transitions from rest to spiking in various different types ofneurons, the propagation of signals through neuronal networks, the frequent synchronizationof activity in large groups of neurons, the origins and functions of oscillatory activity innetworks of neurons (as observed for instance in EEG measurements), and other issuesrelated to the physics of the brain.
The two main sets of tools used for studying differential equations modeling individualneurons and neuronal networks are qualitative methods for the study of differential equations,and numerical methods. This course will focus mostly on the theoretical, qualitative studyof the differential equations, but will include some computing as well.
No background in biology or computing will be assumed.
Math 150-02 Linear Algebra IICourse Information
Spring 2013
Block: I+ (Monday, Wednesday 3:00 - 4:15)Instructor: Misha KilmerEmail: [email protected]: Bromfield-Pearson 114Office hours: (Fall 2012) On leave. By appt. onlyPhone: (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 itin 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 oflinear algebra, no first semester course (i.e., 70 or 72) can possibly scratch the surface of allthe interesting linear algebra topics.
This course is recommended for all those who enjoyed their first semester course and desireto gain breadth and depth in the subject. Topics will include, but are not limited to: sym-metric (or Hermitian) positive definite matrices and their connection to the 2nd derivativetest in third semseter calculus, quadratic forms, Jordan normal form, matrix polynomials,matrix exponentials and their connection to the solution of differential equations, Perrone-Frobenius theory and its application with respect to Markov chains, inner product spaces,orthogonal projections, the singular value decomposition, interpolation as a linear algebraproblem, and various important theorems in the field. If time allows, we will also cover somerecent results (and open problems) in multilinear algebra.
There will be three exams in this course, two during the semester and one at the end, aswell as a number of assignments heavily weighted towards proofs. Every effort will be madeto connect the topics in the course with applications in other mathematics courses or in realworld problems.
* As no one text covers, at the desired level, all the topics we will cover in this course, wewill 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 TopologyCourse Information
Spring 2013
Block: D+ (Tuesday, Thursday 10:30-11:45)Instructor: Genevieve WalshEmail: [email protected]
Office: Bromfield-Pearson 213Office hours: (Fall 2012) Monday 3:00-4:30 and Thursday 9:00-10:30Phone: (617) 627-4032
Prerequisites: Some course in the mathematics department involving proofs, willingnessto 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-settopology.
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. Wewill start by covering cardinality and the axiom of choice and general topology, includingcreating our own topologies. We will describe what a basis of a topology is, the subspacetopology, and the product topology. Then we will discuss various separation and coveringproperties, and investigate maps between topological spaces. Finally, we will discuss notionsof 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 jointlywork through proving the theorems. Everyday, students from the class will present problemsand the class will discuss them and decide if they are ready to proceed. This method ofinstruction is called the modified Moore method, and it is a very fun and challenging wayto deeply engage with the material.
There will be daily homework assignments, two in-class exams, and a final.
Math 158 Complex VariablesCourse Information
Spring 2013
Block: C (Tue Wed Fri 9:30-10:20)Instructor: Moon DuchinEmail: [email protected]: Bromfield-Pearson 113Office 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 onecomplex 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 truefor real variables. (The usefulness of complex analysis, and its harmony with other fieldsof 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 inthe 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 verysmall sample of its values. This is an example of the rigidity phenomena at the heart ofcomplex analysis.
The principal topics in our study of analytic functions include elementary functions (in-cluding Mobius transformations), contour integration, Cauchy’s theorem and integral for-mula, power series, residues, conformal mappings, and analytic continuation. We may touchon 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 anemphasis on geometric intuition, as well as many computational examples and practicalcalculus-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 vistabeyond calculus.
Math 162 StatisticsCourse Information
Spring 2013
Block: E+ (Monday, Wednesday, Friday 10:30–11:45) in BP-2Instructor: Kei KobayashiEmail: [email protected]
Office: Robinson Hall 156Office hours: (Fall 2012) Tuesday 1:30-2:30Phone: 617-627-6102
Prerequisites: Math 161 or consent.
Text: Richard J. Larsen & Morris L. Marx, An Introduction to Mathematical Statistics andIts Applications, 5th Edition, Prentice Hall (2011). ISBN-13: 978-0321693945
Course description: Suppose I claim that I make 75% of my basketball free throws. Totest my claim, you ask me to shoot 40 free throws and I make only 24 of the 40. Do youbelieve my claim?
Statistics is the science of gaining information from numerical data. Our technologicalworld generates data at an enormous rate. However, all too often the data is improperlyobtained and/or improperly assessed. Important everyday decisions for individuals, corpo-rations, societies, and governments hinge on a proper understanding and assessment of data.Every facet of industry, science, engineering, economics and business benefits from a solidknowledge of statistics.
Statistics uses the major ideas and concepts from probability. Only via the use of proba-bility can a proper assessment be made of data collected for real-world problems. Math 162provides opportunities to experience and learn the statistical thinking a functioning statis-tician must develop and use constantly. It also provides preparation for actuarial exams,graduate work in applied mathematics, and courses in physical/social sciences requiring sta-tistical methodology. Topics to be covered include 1) estimating an unknown parameterof 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) makingpredictions, 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 GroupsCourse Information
Spring 2013
Block: E+ (Wed and Fri 10:30–11:45)Instructor: George McNinchEmail: [email protected]: Bromfield-Pearson 112Office hours: (Fall 2012) T and Th 2:30–3:30, F 11:00–12:00Phone: (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 graduatealgebra initiated in Math 215.
Subsequently, Math 216 will investigate the representation theory of a finite group. If Gis a group, k a field, and V a (finite dimensional) vector space over k, a representation ofG on V is a group homomorphism ρ : G → GL(V ). Here GL(V ) is the group of invertiblek-linear transformations from V to itself, and if d = dimk V , then GL(V ) ' GLd(k) may beidentified 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 wherethe field k is algebraically closed of characteristic zero – e.g. k = C, the field of complexnumbers. In this setting, the theory of representations of G on k-vector spaces – oftenreferred to as the “ordinary representation theory” of G – is rather complete. We will seethat to a representation (ρ, V ) one can associate a certain class function χV : G → k – afunction constant on the conjugacy classes of G – which “contains” much of the informationabout the representation V . As a consequence, we find for example that the number ofirreducible ordinary representations of G is the same as the number of conjugacy classes inG.
Along the way, our study will lead to investigation of finite dimensional (semi)simple k-algebras and Wedderburn’s Theorem. Most of the required material is contained in [Dummit-Foote], though the course lectures will supplement material found in the text.
Math 250-01 Numerical Methods for Partial DifferentialEquations
Course Information
Spring 2013
Block: G, MWF, 1:30-2:20 PMInstructor: Scott MacLachlanEmail: [email protected]: Bromfield-Pearson 212Office hours: (Fall 2012) TF 11:00 AM - 12:30 PMPhone: 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 themathematical models for many areas of science and engineering, analytical methods for thesolution of these equations are successful only in certain special cases. For PDE models ofheterogeneous media (for example, of flow through a porous aquifer, seismic imaging of theEarth’s structure, or the electrical activation of the heart muscle), very little can be donewith analytical techniques. Instead, we turn to computational simulation, which has becomean increasingly important tool in many fields of science and engineering, replacing expensiveor impractical experimentation in these areas. The focus of this course is on the key stepof the simulation process, where approximations to the solutions of these PDEs are foundusing computers.
The course material breaks roughly into thirds. We will consider the numerical solution ofhyperbolic equations using finite-difference discretizations. For these problems, we will lookat the questions of accuracy, consistency, and stability, and how they are related through theLax Equivalence Theorem. For elliptic equations, we will consider the finite-element methodand 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 linearequations, and the numerical techniques needed to do this efficiently. In particular, we willsee how multigrid methods naturally arise in the numerical solution of elliptic equations andhow they can be used to achieve optimally scaling simulation algorithms.
This course is intended to be approachable for students both from mathematics and fromareas of science and engineering where computer simulation is commonplace. We will bothprove 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 TuEmail: [email protected]
Office: Bromfield-Pearson 206Office hours: (Fall 2012) Tue 12:50–1:15 (Math 135 only), Thu 2:30-3:30, Fri 1:00–2:00Phone: (617) 627-3262
Prerequisites: Math 217 (Manifolds) or 218 (Algebraic Topology I) or consent. Students whohave not taken Math 217 should first speak to the instructor.
Text: Raoul Bott and Loring W. Tu, Differential Forms in Algebraic Topology, 3rd correctedprinting, Springer, New York, 1995.
Course description:
The central problem in topology is to decide if two spaces are homeomorphic or if two manifolds arediffeomorphic. Algebraic topology offers a possible solution by transforming the geometric probleminto an algebraic problem. To accomplish this, one associates to each topological space an algebraicobject such as its cohomology vector space so that homeomorphic topological spaces correspondto isomorphic vector spaces. Such an association is called a functor. The cohomology functorgreatly simplifies the original problem, for the dimension alone determines the isomorphism classof a finite-dimensional vector space. This gives a simple necessary condition for two topologicalspaces to be homeomorphic.
Over the past one hundred years our forefathers have discovered many useful functors, amongwhich are the fundamental group, higher homotopy groups, homology groups, and cohomologyrings. There are various versions of cohomology—singular, de Rham, and Cech, among others.This leads naturally to two new problems: (1) How does one compute these functors? (2) Whatare the relations that exist among the different functors? Algebraic topology, as it is usuallypracticed, concerns itself with these questions in the continuous category, with topological spacesand continuous maps. Its great generality makes it applicable in a variety of settings, but alsorenders 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 ofthe less digestible aspects of algebraic topology. We will work primarily but not exclusively withsmooth manifolds. This has the advantage of simplifying the theory as well as the constructions; forexample, all three versions of cohomology coincide on smooth manifolds. It allows us to study topicsthat are normally relegated to a second course in algebraic topology, such as Poincare duality, theThom isomorphism, and spectral sequences, without assuming a knowledge of homology. Thesetopics 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, cohomologywith compact support, Mayer–Vietoris sequence, Poincare lemma on the cohomology of Rn, degreeof a proper map, Brouwer fixed-point theorem, hairy-ball theorem, Poincare duality for a manifold,Kunneth formula for the cohomology of a product, Thom isomorphism for the cohomology ofa vector bundle, Euler class, the nonorientable case, Cech cohomology, isomorphism between deRham and Cech cohomology, presheaves, Hopf index theorem on the zeros of a vector field, spectralsequences, cohomology with integer coefficients, and de Rham’s theorem.
Math 252 Nonlinear Partial Differential EquationsCourse Information
Spring 2012
BLOCK: D+: Tuesday, Thursday 10:30-11:45 a.m.INSTRUCTOR: James AdlerEMAIL: [email protected]: Bromfield-Pearson 209OFFICE HOURS: Fall 2012 Tues. 1:30-3:00pm and Wed. 3:00am-4:30pmPHONE: 7-2354
PREREQUISITES: Math 251, or permission of the instructor.
TEXT: TBD
COURSE 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, theirsolution can be difficult and problem-dependent. In this course we’ll learn the tools to askquestions like: what is causing the turbulence on my flight?; why do the bubbles in a Guinnessgo up not down?; and, of course, why on earth is traffic backed up so badly on I-93!!??. Inaddition, all students are given the chance at one million dollars!!∗
This course will cover topics such as the method of characteristics for quasilinear equa-tions, which is later generalized to other kinds of nonlinearity, similarity solutions, andperturbation methods. A variety of examples will be used to study various phenomenathat arise in nonlinear PDEs, including shocks, jump conditions, etc. We will considervarious 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, investigatingquestions 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 Prob-lems, http://www.claymath.org/millennium/Navier-Stokes_Equations, solving this problemwill 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. Math 42: Calculus III or 4. Math 145: Abstract Algebra I
Math 44: Honors Calculus 5. Math 136: Real Analysis II or
2. Math 70: Linear Algebra or Math 146: Abstract Algebra II
Math 72: Abstract Linear Algebra
3. Math 135: Real Analysis I
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. Math 42: Calculus III or 4. Math 87: Mathematical Modeling
Math 44: Honors Calculus III 5. Math 158: Complex Variables
2. Math 70: Linear Algebra or 6. Math 135: Real Analysis I
Math 72: Abstract Linear Algebra 7. Math 136: Real Analysis II
3. Math 51: Differential Equations
II. One of the following:
1. ______________
Math 145: Abstract Algebra I Math 61/Comp 22: Discrete Mathematics
Comp 15: Data Structures 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. Math 42 (old: 13): Calculus III or Math 44 (old: 18): Honors Calculus
2. Math 70 (old: 46): Linear Algebra or 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 job-
hunting 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 Mon Mon Tue Tue Wed Wed Thu Thu Fri Fri 150/180 Minute ClassesClasses and Seminars
8:05-9:20 (A+,B+) A+ B+ A+ B+ B+0+ 1+ 2+ 3+ 4+
8:30-9:20 (A,B) A B A 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) D 0 C 1 C 2 A 3 C 4
10:30-11:20 (D,E) E D E D E10:30-11:45 (D+,E+) E+ D+ E+ D+ E+
12:00-12:50 (F) Open F Open F F12:00-1:15 (F+) F+ F+ F+
1:30-2:20 (G,H) G 5 H 6 G 7 H 8 G 9 1:30-4:00 (5,6,7,8,9)
1:30-2:45 (G+,H+) G+ H+ G+ H+ 1:20-4:20 (5+,6+,7+,8+,9+)2:30-3:20 (H on Fri) H (2:30-3:20)
3:00-3:50 (I,J) I J I J3:00-4:15 (J+,I+) I+ J+ I+ J+ I (3:30-4:20)3:30-4:20 (I on Fri) 5+ 6+ 7+ 8+ 9+
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+
10+ 11+ 12+ 13+6:00-6:50 ,(M N) N/M N M N 6:00-7:15 (M+,N+) M+ N+ M+ N+
10 11 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.
