course information book department of mathematics tufts...
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Course Information Book
Department of Mathematics
Tufts University
Fall 2014
This booklet contains the complete schedule of courses offered by the Math Department in the Fall
2014 semester, as well as descriptions of our upper-level courses (from Math 61 [formally Math 22]
and up). For descriptions of lower-level courses, see the University catalog.
If you have any questions about the courses, please feel free to contact one of the instructors.
Course renumbering
Course schedule
Math 50
Math 51-01
Math 61
Math 70
Math 87
Math 126
Math 135
Math 145
Math 150
Math 161
Math 211
Math 213
Math 215
Math 217
Math 251
Mathematics Major Concentration Checklist
Applied Mathematics Major Concentration Checklist
Mathematics Minor Checklist
Jobs and Careers, Math Society, and SIAM
Block Schedule
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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 February 20, 2014] FALL 2014
COURSE # (OLD #) COURSE TITLE BLOCK ROOM INSTRUCTOR
4 (same) Fundamentals of Mathematics G Zachary Faubion
19-01 9 Mathematics of Social Choice C TBA
02 G+TR Linda Garant
03 J+TR TBA
21-01 10 Introductory Statistics B+TR TBA
02 G+ MW Patricia Garmirian
30-01 5 Introduction to Calculus C Genevieve Walsh*
02 D Gail Kaufmann
32-01 11 Calculus I B Christoph Börgers
02 B Hao Liang
03 F Linda Garant
04 F Gail Kaufmann*
05 F Kim Ruane
06 H TBA
34-01 12 Calculus II B Marjorie Hahn*
02 C TBA
03 F Mary Glaser
04 H TBA
36-01 50-01 Applied Calculus II B+ TRF TBA
02 E+ MWF Zachary Faubion
03 F+ TRF TBA
39-01 17 Honors Calculus I-II E+ MWF Zbigniew Nitecki
42-01 13 Calculus III B TBA
02 C James Adler
03 D Fulton Gonzalez*
04 E TBA
05 F TBA
06 F TBA
07 H TBA
50-01 (same) Graphs and Surfaces E+WF Genevieve Walsh
51-01 38 Differential Equations D Christoph Börgers
02 F Misha Kilmer*
TUFTS UNIVERSITY DEPARTMENT OF MATHEMATICS
[Schedule as of February 20, 2014] FALL 2014
COURSE # (OLD #) COURSE TITLE BLOCK INSTRUCTOR
61-01 22 Discrete Mathematics D+TR Mary Glaser
02 E+WF Bruce Boghosian
03 G+MW TBA
04 J+TR TBA
70-01 46 Linear Algebra H+ Kim Ruane
02 C George McNinch
03 D Hao Liang
04 E TBA
87-01 50 Mathematical Modeling and Computing E+WF James Adler
126 (same) Numerical Analysis H+TR TBA
135-01 (same) Real Analysis I F+TR Todd Quinto
02 I Loring Tu
145 (same) Abstract Algebra I D+TR Montserrat Teixidor
150-01 (same)
Random Walks, Brownian Motion, the
Heat Equation and Beyond D+TR Marjorie Hahn
161-01 (same) Probability E+ MWF Patricia Garmirian
211 (same) Analysis G+MW Zbigniew Nitecki
213 (same) Complex Analysis J+TR Fulton Gonzalez
215 (same) Algebra E George McNinch
217 (same) Geometry and Topology K+MW Loring Tu
251 (same) Linear Partial Differential Equations C Bruce Boghosian
Math 50-01 Graphs and SurfacesCourse Information
Fall 2014
Block: E+WF, Wed Fri 10:30 - 11:45Instructor: Genevieve WalshEmail: [email protected]
Office: Robinson 156Office hours: (Spring 2014) Mon Wed 11:45 - 12:30 and 2:45-3:30Prerequisites: Geometric intuition, willingness to present mathematical ideas in class.Texts: Mostly Surfaces by R. E. Schwartz, AMS (2010).
Course description: This course is designed for early math majors who want to learnabout some interesting topics in combinatorics, geometry and topology. We will begin withgraphs, discussing and exploring Eulerian graphs, Hamiltonian graphs, the Euler character-istic, and the genus of a graph. We’ll also examine some infinite graphs and discuss theirproperties. We will then move to the definition of surfaces, the classification of surfaces, andsome of their properties. Finally, we will begin to explore the geometry of surfaces. Timeand interest permitting, we will discuss some topics in surface theory.
Your grade will be based on bi-weekly homework sets, class participation, and twoprojects/presentations. It is understood that there may be students at vastly different levelsin this class, and you will be able to choose your projects accordingly.
Math 51, Sec. 1 Differential EquationsCourse Information
Fall 2014
Please note: This page is only about Section 1, an experimental section of Math 51. It hasmore stringent prerequisites than the regular Math 51, which will be offered as Math 51,Section 2 in the fall. It also uses a different book. The two sections have somewhat differentemphasis. We offer Section 1 with the Applied Mathematics majors in mind, but all whohave fulfilled the prerequisites are welcome.
Block: D, Mon 9:30–10:20, Tue/Thu 10:30–11:20Instructor: Christoph BorgersEmail: [email protected]: Bromfield-Pearson 215Office hours: (Spring 2014) Mon 12–1:30, Wed 12–1:30Pre- and Corequisites: Math 42 or 44 (Multivariable Calculus) is a prerequisite. Math70 or 72 (Linear Algebra) is a pre- or corequisite. That is, you should have taken it, or youshould take it in parallel with this course.
Text: Differential Equations, Fourth Edition, by Paul Blanchard, Robert Devaney, andGlenn Hall. Brooks/Cole, 2011, ISBN 978-1133109037. This book is not the one used forSection 2.
Course description: Differential equations relate the rates of change of quantities totheir values. Most mathematical models in the natural sciences are differential equations.This is why we teach calculus, which underlies the study of differential equations, to so manystudents.
This course will put strong emphasis on modeling, that is, on understanding how thedifferential equations that we study arise in the natural sciences. You will learn about themost elementary ways of modeling the growth of populations of people, as well as populationsof tumor cells, and about simple models of epidemics and of populations competing forresources, resonance and the Tacoma Narrows Bridge (a suspension bridge that collapsedinto Puget Sound in 1940 — you can find video footage on youtube), and much more.
Explicit solutions of differential equations can be computed in very simple cases only, butthe methods used for doing so are useful and important. Linear algebra plays a central rolehere. There will also be strong emphasis on qualitative analysis, that is, on understandingproperties of the solutions without computing them explicitly; this, too, involves linearalgebra, in particular eigenvalues and eigenvectors, in a central way.
We will also discuss numerical methods for solving differential equations, unquestion-ably the most important approach to solving differential equations. The study of numericalmethods for differential equations is a large and sophisticated branch of contemporary math-ematics. In this course, you will learn the beginnings of that field.
Math 61 Discrete MathematicsCourse Information
Fall 2014
Block: D+TRInstructor: Mary GlaserEmail: [email protected]: Bromfield-Pearson 004
Block: E+WFInstructor: Bruce BoghosianEmail: [email protected]
Block: G+MWInstructor: TBA
Block: J+TRInstructor: TBA
Math 70 Linear AlgebraCourse Information
Fall 2014
Block: D (M 9:30-10:20 TTh 10:30-11:20)Instructor: Hao LiangEmail: [email protected]: Bromfield-Pearson 109Office hours: (Spr ’14) W 2:30–4:30pmFri 2:00 3:00pm
Block: C (TWF 9:30-10:20)Instructor: George McNinchEmail: [email protected]: Bromfield-Pearson 112Office hours: (Spr ’14) T 10:30–11:45amand W 13:30–15:00pm
Block: I+ (WM 3:00-4:20)Instructor: Yusuf Mustopa
Block: A (WM 8:30-9:20 Th 9:30 10-10:20)Instructor: Kim RuaneEmail: [email protected]: Robinson 155Office hours: (Spr ’14) M 1:30–2:30pm,W 3:30–4:30pm, Th 10:30–11:30am
Prerequisites: Math 34 or 39 or consent.
Text: Linear Algebra and Its Applications , 4th edition, by David Lay, Addison-Wesley(Pearson), 2011.
Course description:
Linear algebra begins the study of systems of linear equations in several unknowns. Thestudy of linear equations quickly leads to the important concepts of vector spaces, dimension,linear transformations, eigenvectors and eigenvalues. These abstract ideas enable efficientstudy of linear equations, and more importantly, they fit together in a beautiful whole thatwill give you a deeper understanding of these ideas.
Linear algebra arises everywhere in mathematics – it plays an important role in almostevery upper level math course –; it is also crucial in physics, chemistry, economics, biology,and a range of other fields. Even when a problem involves nonlinear equations, as is often thecase in applications, linear systems still play a central role, since “linearizing” is a commonapproach to non-linear problems.
This course introduces students to axiomatic mathematics and proofs as well as fun-damental mathematical ideas. Mathematics majors and minors are required to take linearalgebra (Math 70 or Math 72) and are urged to take it as early as possible, as it is a pre-requisite for many upper-level mathematics courses. The course is also useful to majors incomputer science, engineering, and the natural and social sciences.
The course will have two midterms and a final, as well as daily assignments.
Math 87 Mathematical Modeling and ComputationCourse Information
Fall 2014
Block: E+WF, Wed Fri 10:30 -11:45 AMInstructor: James AdlerEmail: [email protected]: Bromfield-Pearson 209Office hours: (Spring 2014) on leavePrerequisites: Math 34 or 39, or consent.
Text: none
Course description:This course is about using elementary mathematics and computing to solve practical prob-lems. Single-variable calculus is a prerequisite; other mathematical and computational tools,such as elementary probability, matrix algebra, elementary combinatorics, and computing inMATLAB, will be introduced as they come up.
Mathematical modeling is an important area of study, where we consider how mathe-matics can be used to model and solve problems in the real world. This class will be drivenby studying real-world questions where mathematical models can be used as part of thedecision-making process. Along the way, we’ll discover that many of these questions are bestanswered by combining mathematical intuition with some computational experiments.
Some problems that we will study in this class include:
1. The managed use of natural resources. Consider a population of fish that has a nat-ural growth rate, which is decreased by a certain amount of harvesting. How muchharvesting should be allowed each year in order to maintain a sustainable population?
2. The optimal use of labor. Suppose you run a construction company that has fixednumbers of tradespeople, such as carpenters and plumbers. How should you decidewhat to build to maximize your annual profits? What should you be willing to pay toincrease your labor force?
3. Project scheduling. Think about scheduling a complex project, consisting of a largenumber of tasks, some of which cannot be started until others have finished. Whatis the shortest total amount of time needed to finish the project? How do delays incompletion of some activities affect the total completion time?
Using basic mathematics and calculus, we will address some of these issues and others, suchas dealing with the MBTA T system and penguins (separately of course...).
Math/CS 126 Numerical AnalysisCourse Information
Fall 2014
BLOCK: H+T/Th: Tuesday, Thursday 1:30-2:45pmINSTRUCTOR: TBD (new tenure-track hire)
PREREQUISITES: For Math 126/CS 126: Math 51 (old number: 38) and experience pro-gramming in a language such as C, C++, Fortran, Matlab, etc.
TEXT: TBD
COURSE DESCRIPTION:“Numerical analysis is the study of algorithms for the problems of continuous mathe-
matics.” (L. N. Trefethen, “The definition of numerical analysis”, SIAM News, November1992.) Continuous mathematics means mathematics involving the continuum of real orcomplex numbers.
There are many striking examples illustrating the importance of the neccessity of ac-curate numerical solution methods. Here are just three of them. (1) In 1991, a Norwegianoffshore oil platform called the Sleipner A sank. The resulting economic loss was esti-mated at $700 million. The post-accident investigation traced the problem to a faulty nu-merical method for predicting shear stresses. (2) Ordinary differential equations are usedto model the spread of an infection through a population. Numerical solution techniquesare usually the only practical option for solving these problems, but without an under-lying understanding of the properties of both the model and the algorithm, numericalresults are not to be trusted. (3) CT scanners are used throughout the world for medicaldiagnostics. This technology is based on numerical algorithms for the solution of certainintegral equations. Then, of course, there are the more simple questions, “How does mycalculator compute the square root of 2??”
This course is an introduction to the field, treating linear algebra fairly lightly, insteademphasizing numerical solution of nonlinear equations and differential equations, nu-merical integration techniques, and unconstrained optimization, often in the context ofapplications. (For a thorough treatment of numerical linear algebra, take Math 128/CS128 or its 200-level branch.) Computer programming will be a substantial component ofthe homework, with Matlab as the suggested tool.
Math 135 Real Analysis I
Course Information
Fall 2014
Block: F+ (Tue Thu 12:00–1:15 p.m.)Instructor: Todd QuintoEmail: [email protected]
Office: Bromfield-Pearson 204Office hours: (Spring 2014), Tues 10:15-11:15,Wed Fri 1:30-2:30Phone: (617) 627-3402
Block: I (Mon Wed 3:00–3:50 p.m., Fri3:30–4:20 p.m.)Instructor: Loring TuEmail: [email protected]
Office: Bromfield-Pearson 206Office hours: (Spring 2014) on sabbatical,please email professor.Phone: (617) 627-3262
Prerequisites: Math 42 or 44, and 70 or 72, or consent. (In the old numbering scheme, Math 13or 18, and 46 or 54, or consent.)
Text:
• Prof. Quinto’s section: TBA (Either Fitzpatrick or Marsden and Hoffman)
• Prof. Tu’s section: Patrick Fitzpatrick, Advanced Calculus, 2nd edition, American Math-ematical Society, 2009. (ISBN-10: 0821847910)
Course description:
Real analysis is the rigorous study of real functions, their derivatives and integrals. It provides thetheoretical underpinning of calculus and lays the foundation for higher mathematics, both pureand applied. Unlike Calculus I, II, and III, where the emphasis is on intuition and computation,the emphasis in real analysis is on justification and proofs.
Is this rigor really necessary? This course will convince you that the answer is an unequivocalyes, because intuition not grounded in rigor often fails us or leads us astray. This is especiallytrue when one deals with the infinitely large or the infinitesimally small. For example, it is notintuitively obvious that, although the set of rational numbers contains the set of integers as a propersubset, there is a one-to-one correspondence between them. These two sets, in this sense, are thesame size! On the other hand, there is no such correspondence between the real numbers and therational numbers, and therefore the set of real numbers is uncountably infinite.
A metric space is a set on which there is a distance function. In this course, we will studythe topology of the real line and metric spaces, compactness, connectedness, continuous mappings,and uniform convergence. The topics constitute essentially the first five chapters of the textbook.Along the way, we will encounter theorems of calculus, such as the intermediate-value theoremand the maximum-minimum theorem, but in a more general setting that enlarges their range ofapplicability. This is one of the exciting aspects of real analysis and of upper-level math in general;by studying fundamental ideas in a general setting, we understand them more deeply and gain abroader perspective on and appreciation of the ideas. We anticipate ending with the contractionmapping principle, a fundamental theorem using which one can prove the existence and uniquenessof solutions of differential equations.
In addition to introducing a core of basic concepts in analysis, a companion goal of the courseis to hone your skills in distinguishing the true from the false and in reading and writing proofs.
Math 135 is required of all pure and applied math majors. A math minor must take Math 135or 145 (or both).
Math 145 Abstract AlgebraCourse Information
Fall 2014
Block: D+ (Tuesday, Thursday 10:30-11:45)Instructor: montserat teixidorEmail: [email protected]: Bromfield-Pearson 115Office hours: (Spring 2014) T/Th 12-1.30, Fr. 8.30-9.20 and by appointmentPhone: (617) 627-2358
Prerequisites: Linear Algebra (Math 70 or 72).
Text: A first course in Abstract Algebra seventh edition by John B. Fraleigh , AddisonWesley 2003.
Course description: Algebra, along with Analysis and Geometry is one of the mainpillars of mathematics. It has ancient roots, especially in Europe, India and China. Histori-cally, algebra was concerned with the manipulation of equations and, in particular, with theproblem of finding the roots of polynomials. This is the algebra that you know from highschool. There are clay tablets from 1700B.C. that show that the Babylonians knew how tosolve quadratic equations. The solutions to cubic and fourth degree polynomial equationswere solved in Italy during the Renaissance. About the time of Beethoven, a young Frenchmathematician Evariste Galois made the dramatic discovery that for polynomials of degreegreater than four, no similar solution exists. To do this, Galois introduced the branch ofmathematics known as group theory. This was not, of course, the end of the story. Algebrahas continued its development to the present day most notably with the classification offinite simple groups and with Andrew Wiles proof of Fermat’s Last Theorem.
The concept of a group is now one of the most important in mathematics. Roughlyspeaking, group theory is the study of symmetry. There are deep connections betweengroup theory, geometry and number theory. Groups pop up in every area of mathematics:symmetry groups can be used to find solutions to differential equations, associating groups(and rings) to topological spaces allows to distinguish among them. Groups also appear inthe attempts of physicists to describe the basic laws of nature.
In Math 145, we introduce the concepts of group and ring. Their properties mimic thearithmetic properties of numbers and polynomials. In Math 146, we describe the connectionGalois discovered between group theory and the roots of polynomials.
Math 150 Random Walks, Brownian Motion, the Heat Equationand Beyond
Course Information
Fall 2014
Block: D+TR, Tue Thurs 10:30-11:45 PMInstructor: Marjorie HahnEmail: [email protected]
Office: Bromfield-Pearson 202Office hours: T, Th, F 9:30-10Prerequisites: Math 161 preferred or another calculus based probability course plus Math42 or consent of instructor.
Text: Most likely a small inexpensive book by Greg Lawler plus handouts.
Course description:Stochastic processes are random processes that evolve in discrete or continuous time and
provide useful models for many phenomena. This is an interactive course focused on twoof the most important kinds of stochastic processes–random walks and Brownian motion.The course is meant to be accessible and of interest to both undergraduate and graduatestudents who have had both multivariate calculus and a suitable calculus based probabilitycourse. No previous exposure to stochastic processes is needed. The intent is to developtheses processes and use them to explore current areas of active research.
This course is based on materials prepared and used by Greg Lawler for several activesummer student research experiences in probability. These materials lend themselves to aseries of accessible interconnected lectures and investigations that give rise to many ideasin probability and analysis via random walks on the line, in the plane, on groups and ongraphs as well as their simulations and generalizations to Brownian motion, Markov chains,and beyond. In the process students will not only get a taste of modern day probability andcurrent areas of research, but see the development of beautiful interconnections with topicsthey have already learned, as well as many new thought provoking ideas and directions. Forinstance, are there curves that are nowhere differentiable? If so, do they arise naturally? Arethey important and abundant? Are there important uncountable subsets of the line withzero length? If so, where do they arise and how are their relative sizes compared?
The following should give a sense of tentative topics: simple random walk and Stirling’sformula, Simple random walk in many dimensions, Self-avoiding walk, Brownian motion,Shuffling and random permutations, Seven shuffles are enough (sort of), Markov chains onfinite sets, Markov chain Monte Carlo, Random walks and electrical networks, Uniform span-ning trees, Random walk and other simulations including applications to finance, Randomwalk and the discrete heat equation, Brownian motion and the heat equation, Self-similarityand fractals; Random fractals.
The course is meant to be interactive. Working individually and in groups, students willhave opportunities to experience a sense of the discovery that propels research.
Math 161 ProbabilityCourse Information
Fall 2014
Block: E+ MWF 10:30-11:45 AMInstructor: Patricia GarmirianEmail: [email protected]: Bromfield-Pearson 201Office hours: MW 3:00-4:30 PMPhone: (617) 627-2682
Prerequisites: Math 42 or consent.
Text: Probability and Stochastic Processes by Frederick Solomon, Prentice-Hall, Inc., 1987,Upper Saddle River, NJ.
Course description:Many things we experience in the real world are unpredictable. Consider flipping a coin. Canwe predict if it will land heads or tails? If the exact position of your finger, the compositionof the table, and the air currents were known, then we could predict the outcome of thecoin flip. However, due to the lack of information, we cannot predict the result. This lackof information is explained as being due to “randomness.” In this course, we will study theprobability distributions of outcomes of random experiments.
In this course, we will cover sample spaces associated with a random experiment, the ax-ioms of probability, combinatorics, conditional probability, independence of events, discreteand continuous random variables, joint probability distributions, the central limit theorem,and the law of large numbers. The probability distributions that we will cover includegeometric, binomial, uniform, exponential, poisson, gamma, and the normal distribution.Knowledge of single and multivariable calculus is required for this course. There will beweekly problem sets, two midterms, and a final exam.
Math 211 AnalysisCourse Information
Fall 2014
Block: G+ MW: Mon, Wed 1:30-2:45Instructor: Zbigniew NiteckiEmail: [email protected]: Bromfield-Pearson 214Office hours: (Spring 2014) MWF 1:30-2:30Prerequisites: Math 135, or consent.
Course description:The focus of this course is graduate level real analysis, which is a beautiful, coherent subject.It is comprised of three main topics that provide essential foundations for all other areas ofanalysis, as well as for geometry, topology, and applied mathematics.1 The main topicsare measure theory (general measures, Lebesgue integration, convergence theorems, productmeasures), point set topology (topologies, metrics, compactness, completeness), and Banachspaces (norms, Lp spaces, Hilbert spaces, orthonormal sets and bases, linear forms, duality,Riesz representation theorems, signed measures). The course provides excellent preparationfor the Ph.D. oral examinations in analysis.
Lectures will be blackboard style, providing motivation for the theory, emphasizing basicideas of the proofs, and providing examples. Grades will be based on weekly homework aswell as exams.
Strong undergraduates who have done well in Math 135 and/or Math 136 are encouragedto consider taking the course, especially if they intend to apply to graduate school in pureor applied mathematics.
1As Dennis Sullivan once famously remarked, “The subject is topology, the object is analysis,” referring tothe fact that much of Poincare’s work inventing algebraic topology was motivated by the study of differentialequations.
Math 213 Complex AnalysisCourse Information
Fall 2014
Block: F+TR, Tue Thu 12:00-1:15 PMInstructor: Fulton GonzalezEmail: [email protected]: Bromfield-Pearson 203Office hours: (Spring 2014) Mondays 10:30 a.m. – 12:00 p.m.,Tuesdays 3:30 – 5:00 p.m.Prerequisites: Math 211, or consent.
Possible Texts: Real and Complex Analysis, Third Edition by Walter Rudin, McGraw-Hill, 1987.Functions of a Complex Variable I, Second Edition, by John Conway, Springer, New York,1978.A Guide to Complex Variables, by Steven Krantz, Mathematical Association of America,2008. (Available online for free.)
Course description: This is a course on complex function theory which will assume abit of graduate real analysis.
Complex function theory is a beautiful subject with many surprising and deep results, andwith connections to many other areas of mathematics, including number theory, representa-tion theory, and harmonic analysis.
We’ll start at the beginning: Cauchy’s theorem and integral formula, the residue theorem,and the classification of singularities. Then we’ll look at various properties of harmonicfunctions and the Poisson kernel. We’ll examine consequences of the maximum modulusprinciple, including Phragmen-Lindelof methods and interpolation theorems.
We will then explore two sister theorems: the Mittag-Leffler theorem, which posits theexistence of meromorphic function with preassigned poles, and the Weierstrass interpolationtheorem (along with infinite products), which posits the existence of holomorphic functionswith preassigned zeros.
Further topics will include normal families, conformality and the Riemann mapping theo-rem, subharmonicity and Hardy spaces, analytic continuation and the monodromy theorem,Riemann surfaces and modular functions, value distribution theory and Picard’s little andbig theorems.
Math 215 Graduate Algebra ICourse Information
Fall 2014
Block: E block MWF 10:30–11:20Instructor: George McNinchEmail: [email protected]: Bromfield-Pearson 112Office hours: (Spring 2014) Tues 10:30–11:45 and Wed 13:30–15:00Prerequisites: Math 145 or consent.
Text: D.Dummit, S. Foote, Abstract Algebra, Prentice Hall.
Course description:
Algebra is an important part of modern mathematics, both as a subject in itself, and as aubiquitous tool. Historically, algebra was concerned with the manipulation of equations and,in particular, with the problem of finding the roots of polynomials; this is the algebra youknow from high school. Modern algebra systematically studies groups, rings, fields andmodules. Roughly speaking, group theory is the study of symmetry; of course, symmetriesare important in mathematics but also in physics, chemistry and other physical sciences.The study of rings and fields is part of arithmetic – i.e. of number theory – and ofgeometry – since the geometry of a space may be investigated by study of (suitable) ringsof functions on the space. The study of modules over rings is in some sense a generalizationof linear algebra.
The course is suitable for students who have successfully taken at least one semester ofundergraduate abstract algebra and wish to acquire a solid foundation in algebra.
There will be weekly problem sets, a midterm, and a final exam.
Math 217 Geometry and TopologyCourse Information
Fall 2014
Block: K+, Mon Wed 4:30–5:45 p.m.Instructor: Loring TuEmail: [email protected]
Office: Bromfield-Pearson 206Office hours: (Spring 2014) on sabbatical, please email professor.Phone: (617) 627-3262
Prerequisites: Undergraduate real analysis (Math 135, 136) and abstract algebra (Math145), some point-set topology.
Text: Loring Tu, An Introduction to Manifolds, 2nd edition, Springer, 2011.
Course description:
Undergraduate calculus progresses from differentiation and integration of functions onthe real line to functions on the plane and in 3-space. Then one encounters vector-valuedfunctions and learns about integrals on curves and surfaces. Real analysis extends differentialand integral calculus from R
3 to Rn. This course is about the extension of calculus from
curves and surfaces to higher dimensions.The higher-dimensional analogues of smooth curves and surfaces are called manifolds.
The constructions and theorems of vector calculus become simpler in the more general settingof manifolds; gradient, curl, and divergence are all special cases of the exterior derivative,and the fundamental theorem for line integrals, Green’s theorem, Stokes’ theorem, andthe divergence theorem are different manifestations of a single general Stokes’ theorem formanifolds.
Higher-dimensional manifolds arise even if one is interested only in the three-dimensionalspace which we inhabit. For example, if we call a rotation followed by a translation an affinemotion, then the set of all affine motions in R
3 is a six-dimensional manifold. As anotherexample, the zero set of a system of equations is often, though not always, a manifold.We will study conditions under which a topological space becomes a manifold. Combiningaspects of algebra, topology, and analysis, the theory of manifolds has found applications tomany areas of mathematics and even classical mechanics, general relativity, and quantumfield theory.
Topics to be covered included manifolds and submanifolds, smooth maps, tangent spaces,vector bundles, vector fields, Lie groups and their Lie algebras, differential forms, exteriordifferentiation, orientations, and integration. If time permits, there may be a short intro-duction to fundamental groups and covering spaces.
There will be weekly problem sets, a midterm, and a final. Undergraduates who havedone well in the prerequisite courses should find this course within their competence.
Math 251 Linear Partial Differential EquationsCourse Information
Fall 2014
Block: CInstructor: Bruce BoghosianEmail: [email protected]
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 50 (in the new numbering scheme) before taking these upper level
courses.
II. Two additional 100-level math courses.
III. Three additional mathematics courses numbered 50 or higher (in the new numbering
scheme); up to two of these courses may be replaced by courses in related fields
including:
Chemistry 133, 134; Computer Science 15, 126, 160, 170; Economics 107, 108, 154, 201,
202; Electrical Engineering 18, 107, 108, 125; Engineering Science 151, 152; Mechanical
Engineering 137, 138, 150, 165, 166; Philosophy 33, 103, 114, 170; Physics 12, 13 any
course numbered above 30; Psychology 107, 108, 140.
1. ______________ 2. ______________
3. ______________
Advisor’s signature: Date:
Chair’s signature: Date:
Note: It is the student’s responsibility to return completed, signed degree sheets
to the Office of Student Services, Dowling Hall.
(form revised September 23, 2013)
APPLIED MATHEMATICS MAJOR CONCENTRATION CHECKLIST
(To Be Submitted with University Degree Sheet)
Name:
I.D.#:
E-Mail Address:
College and Class:
Other Major(s):
(Note: Submit a signed checklist with your degree sheet for each major.)
Please list courses by number. For transfer courses, list by title, and add “T”. Indicate
which courses are incomplete, in progress, or to be taken.
Note: If substitutions are made for courses listed as “to be taken”, it is the student’s responsibility
to make sure the substitutions are acceptable.
Thirteen courses beyond Calculus II. These courses must include:
I. Seven courses required of all majors. (Check appropriate boxes.)
1. 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 61: 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:
Chair’s signature: Date:
Note: It is the student’s responsibility to return completed, signed degree sheets
to the Office of Student Services, Dowling Hall.
(form revised December 23, 2013)
DECLARATION OF MATHEMATICS
MINOR Name:
I.D.#:
E-mail address: College and Class:
Major(s)
Faculty Advisor for Minor (please print)
MATHEMATICS MINOR CHECKLIST Please list courses by number. For transfer courses, list by title and add “T”. Indicate which
courses are incomplete, in progress, or to be taken. Courses numbered under 100 will be renumbered starting in the Fall 2012 semester. Courses are
listed here by their new number, with the old number in parentheses. Note: If substitutions are made for courses listed as “to be taken”, it is the student’s
responsibility to make sure that the substitutions are acceptable.
Six courses distributed as follows:
I. Two courses required of all minors. (Check appropriate boxes.)
1. 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 (in the new
numbering scheme).
These four courses must include Math 135: Real Analysis I or 145: Abstract Algebra
(or both).
Note that Math 135 and 145 are only offered in the fall.
1. ______________ 2. ______________
3. ______________ 4. ______________
Advisor’s signature Date
Note: Please return this form to the Mathematics Department Office
(Form Revised September 23, 2013)
Jobs and Careers
The Math Department encourages you go to discuss your career plans with your professors. All of
us would be happy to try and answer any questions you might have. Professor Quinto has built
up a collection of information on careers, summer opportunities, internships, and graduate
schools and his web site (http://equinto.math.tufts.edu) is a good source.
Career Services in Dowling Hall has information about writing cover letters, resumes and 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.