Course Information Book as of 3-4-2015

Department of Mathematics

Tufts University

Fall 2015

This booklet contains the complete schedule of courses offered by the Math Department in the Fall 2015 semester, as well as descriptions of our upper-level courses (from Math 61 [formally Math 22] and up). For descriptions of lower-level courses, see the University catalog.

If you have any questions about the courses, please feel free to contact one of the instructors.

Course renumbering Course schedule Math 61-01 Math 61-02 Math 70 Math 87 Math 135 Math 145 Math 150 Math 151 Math 161 Math 167 Math 211 Math 215 Math 226 Math 250-01 Math 250-02 Mathematics Major Concentration Checklist Applied Mathematics Major Concentration Checklist Mathematics Minor Checklist Jobs and Careers, Math Society, and SIAM Block Schedule

p. 2 p. 3 p. 5 p. 6 p. 7 p. 8 p. 9 p. 10 p. 11 p. 12 p. 13 p. 14 p. 15 p. 16 p. 17 p. 18 p. 19 p. 20 p. 21 p. 22 p. 23 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 March 4, 2015]

FALL 2015

* course coordinator

COURSE # (OLD #) COURSE TITLE BLOCK ROOM INSTRUCTOR

4 (same) Fundamentals of Mathematics C Zachary Faubion

19-01 9 Mathematics of Social Choice G+MW Linda Garant02 K+MW Linda Garant

21-01 10 Introductory Statistics C Staff02 G+MW Patricia Garmrian

30-01 5 Introduction to Calculus A Kim Ruane

32-01 11 Calculus I B Richard Weiss *02 B Hao Liang03 F Gail Kaufmann04 F TBA05 H TBA06 H TBA

34-01 12 Calculus II C Xiaozhe Hu02 D Gail Kaufmann03 F Mary Glaser *

36-01 50-01 Applied Calculus II B+TRF TBA02 E+ MWF Zachary Faubion *03 F+MWF James Adler *

39 17 Honors Calculus I-II E+ MWF Zbigniew Nitecki

42-01 13 Calculus III B TBA02 C George McNinch *03 D Kye Taylor04 E TBA05 F Mauricio Gutierrez06 H TBA

51-01 38 Differential Equations B Kye Taylor02 C Yusuf Mustopa04 H Mauricio Gutierrez *

61-01 22 Discrete Mathematics D Moon Duchin02 5M Genevieve Walsh

70-01 46 Linear Algebra C Hao LIang02 D+ Mary Glaser03 E George McNinch04 H TBA

TUFTS UNIVERSITY DEPARTMENT OF MATHEMATICS[Schedule as of March 4, 2015]

FALL 2015

* course coordinator

COURSE # (OLD #) COURSE TITLE BLOCK ROOM INSTRUCTOR

87 50 Mathematical Modeling and Computing D James Adler

135-01 (same) Real Analysis I F+TRF Todd Quinto02 G Loring Tu

145-01 (same) Abstract Algebra I D Richard Weiss02 J Kim Ruane

150 (same) Neuroscience F Christoph Börgers

151 (same) Applied Advanced Calculus H+ Christoph Börgers

161 (same) Probability E+ MWF Patricia Garmirian

167 (same) Differential Geometry G Zbigniew Nitecki

211 (same) Analysis E Moon Duchin

215 (same) Algebra I+MW Loring Tu

226 (same) Numerical Analysis F+TF Xiaozhe Hu

250-01 (same) Numerical Anaylsis II G+MW Misha Kilmer

250-02 (same) Complex Algebraic Curves D+ Yusuf Mustopa

Math 61 Discrete MathematicsCourse Information

Fall 2015

Block: D, Mon 9:30-10:20, Tue/Thu 10:30-11:20Instructor: Moon DuchinEmail: moon.duchin@tufts.eduOffice: Bromfield-Pearson 113Office hours: (Spring 2015) Thu 11-1 and by cheerful appointmentPrerequisites: Calc I, Comp 11, or instructor’s consent

Text: Rosen, Discrete Mathematics and its Applications, 7th ed. (Probably!)

Course description:This course is an introduction to discrete mathematics. In mathematics, “discrete” is

a counterpart of “continuous,” describing things that vary in jumps rather than smoothly.Calculus studies (mostly) continuous phenomena; the mathematics of the discrete largelyfits into the research area called combinatorics, where the core questions have to do withcounting or enumeration. Our topics will include sets, permutations, equivalence relations,properties of the integers, graph theory, and propositional logic.

This course is intended as a bridge from the more computationally oriented courses likecalculus to upper level courses in mathematics and computer science. We will spend a lotof time learning to read and write correct proofs of mathematical statements and we willpractice these skills on a variety of problems.

This course, which is co-listed as Comp 61, is required for computer science majors. Itcounts toward the math major and is highly recommended for math majors (or possiblemath majors) who want a first glimpse of higher mathematics.

Another section of this course is being taught by Genevieve Walsh in the G block.

Math 61-02 Discrete MathematicsCourse Information

Spring 2015

Block: 5 (Mondays 1:30 - 4)Instructor: Genevieve WalshEmail: genevieve.walsh@tufts.eduOffice: Robinson 156Office hours: (Spring 2015), Tuesdays 10:30 - 12, Wednesday 12:30 - 2Phone: (617) 627-4032

Prerequisites: Willingness to solve problems on your own and at the blackboard and ask ques-tions in class.

Text: B. Richmond and T. Richmond , A Discrete Transition to Advanced Mathematics, AmericanMathematical Society (2004)

Course description:Discrete math is important in computer science, combinatorics, biology, engineering, economics,

game theory, and virtually every aspect of life. It is also fun.We will start off with the foundations of logic. We will discuss what it means to prove something,

and some methods for doing so. In this class you will prove things, and decide if your classmates’proofs are valid. We will study divisibility, counting, and the Pigeonhole Principle. This willaddress the question: “What happens when there are more pigeons than there are pigeonholes?”The answer is surprisingly useful in a variety of non-pigeon situations.

We will learn about permutations and combinations and the binomial coefficients. We will visitthe vast subject of graphs and trees, and if time permits, see some of the beauty of continuedfractions.

There will be two exams and a final, weekly homework assignments, and problems to present/discussin class. Participation will be a large part of this class, and it is designed to get you to think moredeeply about the material.

Math 70 Linear AlgebraCourse Information

Fall 2015

Block: D+Instructor: Mary GlaserEmail: Mary.Glaser@tufts.eduOffice: Bromfield-Pearson 4Office hours: (Spr 2015) Tues 1:30-2:30pm,Thurs 2:30-4:30pm.

Block: CInstructor: Hao LiangEmail: Hao.Liang@tufts.eduOffice: Bromfield-Pearson 109Office hours: (Spr 2015) Mon 10:30am-12:00, Thurs 3:30-5:00pm

Block: EInstructor: George McNinchEmail: George.McNinch@tufts.eduOffice: Bromfield-Pearson 112Office hours: (Spr 2015), Mon,Tues11:00am-12:00, Fri 1:30-2:30pm

Block: HInstructor: TBA

Prerequisites: Math 42, or consent of instructor.

Text: Linear Algebra and Its Applications , 4th edition, by David Lay, Addison-Wesley (Pearson),2011.

Course description: Linear algebra begins the study of systems of linear equations in severalunknowns. The study of linear equations quickly leads to the important concepts of vector spaces,dimension, linear transformations, eigenvectors and eigenvalues. These abstract ideas enable effi-cient study of linear equations, and more importantly, they fit together in a beautiful setting thatprovides a deeper understanding of these ideas.

Linear algebra arises everywhere in mathematics – it plays an important role in almost everyupper level math course –; it is also crucial in physics, chemistry, economics, biology, and a range ofother fields. Even when a problem involves nonlinear equations, as is often the case in applications,linear systems still play a central role, since “linearizing” is a common approach to non-linearproblems.

This course introduces students to axiomatic mathematics and proofs as well as fundamentalmathematical ideas. Mathematics majors and minors are required to take linear algebra (Math 70or Math 72) and are urged to take it as early as possible, as it is a prerequisite for many upper-level mathematics courses. The course is also useful to majors in computer science, economics,engineering, and the natural and social sciences.

The course will have two midterms and a final, as well as daily homework assignments.

Math 87 Mathematical Modeling and ComputationCourse Information

Fall 2015

Block: D+TTh, Tue Thurs. 10:30 -11:45 AMInstructor: James AdlerEmail: james.adler@tufts.eduOffice: Bromfield-Pearson 209Office hours: (Spring 2015) TTh 3-4:00 & Th 10:30-11:30Prerequisites: 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...). This course willalso serve as a good starting point for those interested in participating in the MathematicalContest in Modeling each Spring.

Math 135 Real Analysis ICourse Information

Fall 2015

Block: F+,F (Tue Thu 12:00–1:15p.m., Friday12:00-12:50 (see footnote * below))Instructor: Todd QuintoEmail: Todd.Quinto@tufts.eduOffice: Bromfield-Pearson 204Office hours: (Spring 2015), Tues 10:00-11:00,Thursday 1:30-2:30 Fri 12:05-12:55Phone: (617) 627-3402

Block: G (Mon Wed Fri 1:30–2:20 p.m.)Problem Session: G+ Mon 2:20–2:50 p.m.Instructor: Loring TuEmail: loring.tu@tufts.eduOffice: Bromfield-Pearson 206Office hours: (Spring 2015) Tue 2:20–2:50(136 only), Tue 4:15–5:15, Thu 4:15–5:45Phone: (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: Patrick Fitzpatrick, Advanced Calculus, 2nd edition, American Mathematical Society, 2009.(ISBN 978-0-8218-4791-6)

Course description:Real analysis is the rigorous study of real functions, their derivatives and integrals. It provides 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).

*Most weeks Prof. Quinto will have class for three 50 minute periods starting at noon on Tuesdays, Thursdays,and Fridays. However, you will need to reserve 12:00-1:15 on Tues. and Thurs. since we will use at least some of thisextra time for homework and problem/proof sessions. Also, for a couple of weeks (as announced in class) we willhave class from 12:00-1:15 TR and no class on Friday.

Math 145 Abstract AlgebraCourse Information

Fall 2015

BLOCKS: D (M 9.30-10.20, TT10:30-11:20)INSTRUCTORS: Richard WeissEMAIL: rweiss@tufts.eduOFFICE: Bromfield-Pearson 116OFFICE HOURS: TTF 1:30-2:30PHONE: 7-3802

BLOCK: J (M 4:30-5:20 and TT 3:00-3:50)INSTRUCTOR: Kim RuaneEMAIL: kim.ruane@tufts.eduOFFICE: Robinson 155OFFICE HOURS: M 5-6, Tu 2-3, F 9:30-10:30PHONE: 7-2006

PREREQUISITES: Math 70 or 72 or consent.

TEXT: Abstract Algebra, 3rd edition, by John A. Beachy and William D.Blair, WavelandPress, 2006.

COURSE DESCRIPTION: Algebra is one of the main branches of mathematics. Historically,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 to solvequadratic equations. Formulas for the roots of cubic and fourth degree polynomial equa-tions were found in Italy during the Renaissance. About the time of Beethoven, however,a young French mathematician named Evariste Galois made the dramatic discovery thatfor polynomials of degree greater than four, no such formula exists. To do this, Galoishad to introduce a completely new branch of mathematics which he called group theory.

The concept of a group is now one of the central unifying themes in mathematics.Roughly speaking, group theory is the study of symmetry. In mathematics, there aredeep connections between group theory, geometry and number theory. Group theoryalso plays a central role in the efforts of physicists to describe the basic laws of nature.

The principle (but by no means the only) goal of Math 145-146 is to describe the con-nection between group theory and the roots of polynomials. Ring theory is the naturalsetting for the study of polynomials. In Math 145, we will introduce groups and ringsand investigate their basic properties.

Math 150 Mathematical and Computational NeuroscienceCourse Information

Fall 2015

Block: F (Tu, Th, Fr 12:00-12:50)Instructor: Christoph BorgersEmail: cborgers@tufts.eduOffice: Bromfield-Pearson 215Office hours: (Spring 2015) on sabbatical leave, please e-mailPhone: (617) 627-2366

Prerequisites: Math 38, and willingness to do a considerable amount of Matlab program-ming. (Programming experience is desirable but not necessary. Almost all programmingwill involve modification of Matlab code given to you.) No biological background will beassumed.

Text: None. Notes will be distributed electronically.

Course description: This is a course on differential equations in neuroscience, a fieldthat began with a series of papers by the physiologists A. L. Hodgkin and A. F. Huxley inthe Journal of Physiology in 1952. At the time it had been well known for a long time 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 called the Hodgkin-Huxley equations. Eleven years after their groundbreaking seriesof papers appeared, Hodgkin and Huxley were awarded the Nobel Prize.

Theoretical neuroscience can be viewed as the study of coupled systems of Hodgkin-Huxley equations. The goal is to understand the human brain. Not surprisingly, the fieldis very far from reaching this goal, but a lot has been understood about the transitionsfrom rest to spiking in various different types of neurons, the propagation of signals throughneuronal networks, the frequently observed synchronization of activity in large groups ofneurons, the origins and functions of oscillatory activity in neuronal networks (as observedfor instance in the EEG), and other issues related to the physics of the brain. We will studysome of the known mathematical results, and think about what they might suggest aboutbrain function and disease.

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 computational simulation. You will see examples of both in this course.

Math 151/ME 150 Applications of Advanced Calculus(Linear Partial Differential Equations)

Course Information

Fall 2015

Block: H+ (Tu, Th 1:30–2:45)Instructor: Christoph BorgersEmail: christoph.borgers@tufts.eduOffice: Bromfield-Pearson 215Office hours: (Spring 2015) on sabbatical leave, please e-mailPhone: (617) 627-2366

Prerequisites: Calculus III (Math 42 or 18), Ordinary Differential Equations (Math 51)

Text: None. Notes will be distributed electronically.

Course description: Most deterministic models in the sciences are partial differentialequations. The weather, the global climate, air flow around airplanes, blood flow in the heart,the propagation of electro-magnetic waves, the forces in bridges and buildings, the growth oftumors, the propagation of signals in nerve cells in the brain, the spread of epidemics, andcountless other phenomena are governed by partial differential equations. Quite arguablythere is no branch of mathematics with greater practical impact on the world.

The emphasis in Math 151 is on linear partial differential equations. Three fundamental,prototypical examples are (1) the diffusion equation (describing the diffusion of a pollutant instill air, for instance), (2) the Poisson equation (the fundamental equation of electrostatics),and (3) the wave equation (describing the propagation of sound for instance). You will learnabout mathematical properties of the solutions of these and similar equations. In very specialcases, it is possible and useful to write down explicit, analytic expressions for the solutions,and you will learn examples of that. Fourier analysis is a central tool in this subject, andtherefore an introduction to Fourier analysis is a part of the course.

By far the most important solution techniques for partial differential equations are nu-merical methods. The study of such methods is a large and rapidly growing branch of math-ematics. Although this is not a course on numerical analysis, we will study the simplest,most fundamental numerical methods whenever it is conceptually clarifying to do so.

Math 151 is to be followed by Math 152 in the spring of 2016. Math 152 will be anintroduction to nonlinear partial differential equations.

Math 161 ProbabilityCourse Information

Fall 2015

Block: E+ MWF 10:30-11:45 AMInstructor: Patricia GarmirianEmail: patricia.garmirian@tufts.eduOffice: 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 167 Differential GeometryCourse Information

Fall 2015

Block: G+MW, Mon, Wed 1:30-2:45PMInstructor: Zbigniew NiteckiEmail: znitecki@tufts.eduOffice: Bromfield-Pearson 214Office hours: (Spring 2015) M 9:30-10:20, WF 12:30-1:30Phone: 7-3843

Prerequisites: Math 42 or 44, and 70 or 72, or consent.

Text: Differential Geometry of Curves and Surfaces by Thomas Banchoff and StephenLovett, AK Peters, Ltd, 2010, Natick, MA.

Course description:In this course you will discover how to use the material you have learned in calculus andlinear algebra to study beautiful and important properties of curves and surfaces in three-dimensional Euclidean space. These ideas generalize to the higher-dimension analogues ofsurfaces, called manifolds. Differential geometry, formulated in terms of manifolds, is a largeand important branch of mathematics, interacting with and contributing to many fields inscience and technology, including engineering, economics, and certainly much of physics.Indeed, the language of modern theoretical physics is essentially differential-geometric.

We will start by studying the local and global properties of curves in space, and then moveon to the richer and more complex arena of surfaces. We will see how to properly define suchconcepts as curvature, parallelism, geodesic, vector field, and surface area intrinsically. Wewill examine important topological concepts such as orientability and Euler characteristic.Finally, we will unite the geometry and the topology of a compact surface by means of theall-important Gauss-Bonnet theorem.

This course is intended especially, but not exclusively, for students in mathematics,physics, economics, and engineering, who have an interest in higher-level geometry.

Math 211 AnalysisCourse Information

Fall 2015

Block: E+MW Mon/Wed 10:30-11:45Instructor: Moon DuchinEmail: moon.duchin@tufts.eduOffice: Bromfield-Pearson 113Office hours: (Spring 2015) Thu 11-1 and by cheerful appointmentPrerequisites: graduate student standing or instructor’s consent.

Course description:This is an introductory graduate-level course covering central topics in real and functionalanalysis. Topics will include measure theory (general measures, Lebesgue integration, con-vergence theorems, product measures), metrics and topology, and Banach spaces (norms, Lp

spaces, Hilbert spaces, orthonormal sets and bases, linear forms, duality, Riesz representationtheorems, signed measures).

The course provides preparation for the Ph.D. oral examinations in analysis. Grades willbe based on weekly homework as well as exams.

Undergraduates who have done well in Math 135-136 are strongly encouraged to considertaking the course, especially if they intend to apply to graduate school in pure or appliedmathematics.

Math 215 Algebra

Course Information

Fall 2015

Block: I+MW, Mon Wed 3:00–4:15 PMInstructor: Loring TuEmail: loring.tu@tufts.edu

Office: Bromfield-Pearson 206Office hours: (Spring 2015) Tue 2:20–2:50 (136 only), Tue 4:15–5:15, Thu 4:15–5:45Phone: (617) 627-3262Prerequisites: Math 146 or the equivalent.

Text: Abstract Algebra, 3rd ed., by David S. Dummit and Richard M. Foote, John Wiley & Sons,2004, New York. (ISBN 0-471-43334-9)

Course description:

Algebra is the study of patterns and structures. For this reason it is the underpinning of manyareas of mathematics. This course concentrates on four basic algebraic structures: groups, rings,modules, and fields. Since one of the purposes of the course is to prepare the students for thealgebra core exam, we will follow closely its syllabus, reproduced below.

1. Generalities:

Quotients and isomorphism theorems for groups, rings, and modules.

2. Groups:

The action of a group on a set; applications to conjugacy classes and the class equation.

The Sylow theorems; simple groups.

Simplicity of the alternating group for n ≥ 5.

3. Rings and Modules:

Polynomial rings, Euclidean domains, principal ideal domains.

Unique factorization; the Gauss lemma and Eisenstein’s criteria for irreducibility.

Free modules; the tensor product.

Structure of finitely generated modules over a PID; applications (finitely generatedabelian groups, canonical forms of linear transformations).

4. Fields:

Algebraic, transcendental, separable, and Galois extensions, splitting fields.

Finite fields, algebraic closures.

The fundamental theorem of Galois theory for a finite extension of a field of arbitrarycharacteristic.

Many of these topics are covered in Math 145 and 146, but we will study them in greater depth.It is unlikely that we can cover all of these topics in the first semester. Students wishing to takethe algebra core exam in January will need to do some additional study on their own.

This course is suitable for undergraduates who have done very well in Math 145 and/or 146,especially those wishing to attend graduate school.

Math 226 Numerical AnalysisCourse Information

Fall 2015

Block: F+ (Tuesday, Friday 12:00 pm -1:15 pm)Instructor: Xiaozhe HuEmail: Xiaozhe.Hu@tufts.eduOffice: Bromfield-Pearson 212Office hours: (Spring 2015), Tuesday, Thursday 9:30 am- 11:30 am, or by appointment

Prerequisites: Math 126 and some programming ability in a language such as C, C++,Fortran, Matlab, etc.

Text: TBD

Course description:“Numerical analysis is the study of algorithms that uses numerical approximations for

the problems of mathematical analysis” – WikipediaNumerical analysis creates, analyzes, and implements algorithms for numerical solving

real-world problems, such as fluid dynamics, image processing, social sciences, petroleumengineering, and many other areas. The overall goal is the design and analysis of numericaltools to give efficient and reliable approximations to hard problems. For example, computingthe trajectory of a rocket, predicting the weather, and deciding the price of airplane tickets.

The course will focus on the development of numerical algorithms and the primary ob-jective of the course is to understand the construction of numerical algorithms and, moreimportantly, the applicability and limits of their usages. We will discuss the mathematicalfoundations of numerical algorithms and also pay attention to their computational complex-ity, computer storage, and finite precision arithmetic. The emphasis will be the accuracy,numerical stability, efficiency, robustness, and scalability. The topics include: approximationtheory, numerical integration, solving linear and nonlinear equations, numerical optimiza-tion, and numerical solution to differential equations. In-class examples, homework, andprojects will feature some of the practical applications of these algorithms to solve real-world problems. Homework problems will consist of written problems as well as computerprogramming assignments in Matlab (or your favorite programming language).

This course is intended as a rigorous first graduate level course in Numerical Analysis.We will cover a large portion of the Numerical Analysis qualifying exam core topics, whichinclude numerical integration and orthogonal polynomials, interpolation, numerical solutionof differential equations, unconstrained optimization, and solution of nonlinear equations.

Math 250-NAII Numerical Analysis IICourse Information

Fall 2015

Block: G+MW, 1:30-2:45Instructor: Misha KilmerEmail: misha.kilmer@tufts.eduOffice: Bromfield-Pearson 103Office hours: (Spring 2015) By AppointmentPrerequisites: 226 and 228, or permission

Text: TBD

Course description:In this course, we will cover, in depth, a selection of important topics from Numerical Analysisand Numerical Linear Algebra. It is assumed that students have already had our 226/228sequence (or equivalent). Homework will consist of proofs as well as coding assignments, sothe ability and willingness to program are musts. It is likely there will be multiple sources,and not a single textbook for the course, given the variety of topics.

An list of likely topics includes: Matrix polynomials and invariant subspaces; eigenvalueand singular value perturbation bounds; perturbation of invariant subspaces. Detailed inves-tigation of short-term recurrence Krylov methods for symmetric and unsymmetric systems,and the connection to the topics previously listed as well as preconditioning. Quasi-Newtonmethods. Trust-region methods for non-linear least squares problems. Designing time-stepping methods for realistic problems. Introduction to Lattice-Boltzmann method. Timepermitting, intro to parallel computing.

MATH 250 (COMPLEX ALGEBRAIC CURVES) FALL 2015

Instructor: Yusuf Mustopa

E-mail: Yusuf.Mustopa@tufts.edu

Office Hours: Bromfield-Pearson 106, TBD

Course Meeting Time: Block D+ (Tuesday and Thursday, 10:30AM-11:45AM).

Prerequisites: A course in complex variable theory (e.g. Math 158), or instructor’s consent.

Text: Miranda’s book Algebraic Curves and Riemann Surfaces (Graduate Studies in Mathematics vol. 5,AMS) will be our primary reference.

Description: The theory of complex algebraic curves—which originated with the study of elliptic integrals—ranks among the greatest successes of late-19th/early-20th-century mathematics. It has substantial connec-tions with areas ranging from number theory to partial differential equations, and is a pillar of modernalgebraic geometry as well as a subject of research interest in its own right.

This course introduces complex algebraic curves as compact oriented surfaces with complex structure,and provides a tour of their geometry whose viewpoint shifts from complex-analytic to algebro-geometricas the semester progresses. The following is a list of topics we plan to cover; more advanced topics will beincluded if time and interest allow.

• Complex structures on 2-dimensional manifolds• Examples of compact Riemann surfaces• Holomorphic and meromorphic functions• Group actions and monodromy• Elementary complex projective geometry• Integration and the residue theorem• Divisors and maps to projective space• Riemann-Roch Theorem• The canonical embedding• Castelnuovo’s bound for curves in projective 3-space• Inflectionary behavior and Weierstrass points• Abel’s Theorem

MATHEMATICS MAJOR CONCENTRATION CHECKLIST

For students matriculating Fall 2012 and after (and optionally for others)

(To be submitted with University Degree Sheet)

Name:

I.D.#:

E-Mail Address:

College and Class:

Other Major(s):

(Note: Submit a signed checklist with your degree sheet for each major.)

Please list courses by number. For transfer courses, list by title, and add “T”. Indicate

which courses are incomplete, in progress, or to be taken.

Note: If substitutions are made, it is the student’s responsibility to make sure the substitutions are

acceptable to the Mathematics Department.

Ten courses distributed as follows:

I. Five courses required of all majors. (Check appropriate boxes.)

1. 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 https://sites.google.com/site/tuftssiam/ 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.