Course Information Book Department of Mathematics Tufts ...

Course Information Book Department of Mathematics Tufts University Fall 2013 This booklet contains the complete schedule of courses offered by the Math Department in the Fall 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 Course schedule

p. 2 p. 3

Math 63

p. 5

Math 70

p. 6

Math 72

p. 7

Math 87

p. 8

Math 135

p. 9

Math 145

p. 10

Math 150

p. 11

Math 151

p. 12

Math 161

p. 13

Math 163

p. 14

Math 211

p. 15

Math 215

p. 16

Math 217

p. 17

Math 226

p. 18

Math 250-01

p. 19

Math 250-02

p. 20

Mathematics Major Concentration Checklist

p. 21

Applied Mathematics Major Concentration Checklist

p. 22

Mathematics Minor Checklist

p. 23

Jobs and Careers, Math Society, and SIAM

p. 24

Block Schedule

p. 25

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


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


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


61

22

Linear Algebra

46

70

Number Theory

63

41

Special Topics

50

50

Linear Algebra

70

46

Abstract Linear Algebra

54

72


72

54

DEPARTMENT OF MATHEMATICS [Schedule as of April 25, 2013]

TUFTS UNIVERSITY

COURSE # 4

(OLD #)

COURSE TITLE

(same) Fundamentals of Mathematics

BLOCK D

FALL 2013

ROOM

INSTRUCTOR Zachary Faubion

19-01 02 03

9

Mathematics of Social Choice

C G+ MW H+ TR

Jonathan Ginsberg Alex Babinski Linda Garant

21-01 02

10

Introductory Statistics

E+ MW G+ MW

Alberto Lopez Martin Patricia Garmirian

30-01 02

5


D J

Gail Kaufmann TBA

32-01 02 03 04 05 06 07

11

Calculus I

B F F F F H H

Molly Hahn * TBA TBA Linda Garant Gail Kaufmann TBA Christine Offerman

34-01 02 03 04

12

Calculus II

B D F H

Mauricio Gutierrez Mauricio Gutierrez Mary Glaser * TBA

36-01 02 03

50-01

Applied Calculus II

B+ TRF E+ MWF F+ TRF

TBA Emily Stark Zachary Faubion

39-01

17


E+ MWF

Zbigniew Nitecki

42-01 02 03 04 05 06

13

Calculus III

B C D F H J

TBA Joe McGrath TBA Scott MacLachlan TBA Mathew Wolak

51-01 02 03

38

Differential Equations

C F H

George McNinch Mauricio Gutierrez* Misha Kilmer

61-01 02

22


I+ MW D+

Alberto Lopez Martin Lenore Cowen [Computer Science]

DEPARTMENT OF MATHEMATICS [Schedule as of April 25, 2013]

TUFTS UNIVERSITY

COURSE #

(OLD #)

COURSE TITLE

BLOCK

FALL 2013

ROOM

INSTRUCTOR

63-01

41

Number Theory

D

70-01 02

46

Linear Algebra

D+ TR G+ MW

72-01

54


J+ TR

TBA

87-01

50

Mathematical Modeling and Computing

D+ TR

Scott MacLachlan

Richard Weiss Mary Glaser Montserrat Teixidor

135-01 02

(same) Real Analysis I

E+ MW G

Fulton Gonzalez Loring Tu

145-01 02

(same) Abstract Algebra I

E+ MW J+ TR

George McNinch Moon Duchin

150-01

(same) Surfaces

G+ MW

Zbigniew Nitecki

151-01

(same) Applications of Advanced Calculus

H+ TR

Christoph Börgers

161-01

(same) Probability

E+ MWF

Patricia Garmirian

163-01

(same) Computational Geometry

G+ MW

Greg Aloupis [Computer Science]

211-01

(same) Analysis

D+ TR

215-01

(same) Algebra

C

217-01

(same) Geometry and Topology

226-01

(same) Numerical Analysis

J+ TR

Misha Kilmer

250-01

(same) Geometric Literacy

F+ TR

Moon Duchin

I+ MW

Loring Tu

250-02

250-03

(same)

Topics in Topology: Group Actions and Cohomology

(same) Computational and Applied Mathematics

G+ MW

TBA

Molly Hahn Richard Weiss Fulton Gonzalez

Scott MacLachlan

Math 61

Discrete Mathematics Course Information

Spring 2013

Block: I+ (Mon Wed 3:00 – 4:15) Instructor: Alberto López Mart´ın Email: [email protected] Office: Bromfield-Pearson 106 Office hours: (Spring 2013) Mondays and Wednesdays 3:00 – 4:00 p.m. Phone: (617) 627-2357 Prerequisites: Math 32, Comp 11, or consent. Required for: Computer Science majors. Recommended for Math majors. Text: Discrete Mathematics and its Applications, 7th edition, by K. Rosen (McGraw Hill) Course description: Discrete mathematics is not the name of a branch of mathematics; it is the description of several branches that all have a common feature: they deal with objects that can assume only distinct, separated values. The term “discrete mathematics” is therefore used in opposition to “continuous mathematics,” which are the branches in math dealing with objects that can vary smoothly (and which includes, for example, Calculus). While discrete objects can often be characterized by integers, continuous objects require real numbers. We will start by studying propositional logic and learning how to read, understand, and write proofs. We will then use our newly-acquired skills to study other fields that are considered parts of discrete mathematics, such as Combinatorics (the study of how discrete objects combine with one another and the probabilities of various outcomes) and Graph Theory. We will also examine sets, permutations, equivalence relations and properties of the integers. Your grade will be based on two midterm exams and a final exam, as well as homework. An old riddle. . . The Prussian city of Königsberg spanned both banks of the river Pregel, and two islands in it. Seven bridges connected both banks and both islands with each other. A popular pastime among the citizens of Königsberg was to attempt a solution to the following problem: How to walk across both banks and both islands by crossing each of the seven bridges only once? . . . and the modern use of discrete math Google Maps gives driving directions between (almost) any two points in the US quite accurately. For that, Google uses many basic algorithms from Graph Theory – for example, Dijkstra’s algorithm – to find the shortest path between two nodes in a graph. Google’s search engine ranks pages based on how important they are to the world. This is done with a complex math algorithm that uses Linear Algebra and Graph Theory!

Math 70

Linear Algebra Course Information

Block: D+ (TR 10:30-11:45) Instructor: Mary Glaser [email protected] Office: Bromfield-Pearson 4 Office hours: (Spring 2013) T 1:00-2:00, R 2:304:30 or by appointment Phone: (617) 627-5045

Fall 2013

Block: G+ (MW 1:30-2:45) Instructor: Montserrat Teixidor [email protected] Office: Bromfield-Pearson 115 Office hours: (Spring 2013) M, W 10:00-11:30, or by appointment Phone: (617) 627-2358

Prerequisites: Math 34 (or 39) or consent. Text: Linear Algebra and its applications, 4th edition, by David Lay. Pearson-Addison-Wesley, 2011. Course description: Linear algebra is the study of matrices, vector spaces, and linear transformations. In Math 70 we start by studying systems of linear equations. For example, 2x+3y = 5 is a linear equation in the unknowns x and y, whereas ex = y is nonlinear. The study of linear equations quickly leads to useful concepts such as vector spaces, dimension (you will learn about four-, five-, and infinite-dimensional spaces), linear transformations, and eigenvalues. These concepts will help you solve linear equations efficiently, and more importantly, they fit together in a beautiful whole that will give you a deeper understanding of mathematics. Linear algebra arises everywhere in mathematics (you will use it in almost every one of our upper level courses) as well as in physics, chemistry, economics, biology, and a range of other fields. Even when a problem involves nonlinear equations, as is often the case in applications, linear systems still play a central role, since the most common methods for studying nonlinear systems approximate them by linear systems. Among the applications of linear algebra are solutions of linear systems of equations, determining conic curves and quadric surfaces not in standard form, the second-derivative test in vector calculus, computer graphics, linear economic models, and differential equations. Linear algebra also enters in further study of calculus, where the derivative is viewed as a linear transformation. Mathematics majors and minors are required to take linear algebra (Math 70 or Math 72) and are urged to take it as early as possible, as it is a prerequisite for most upper-level mathematics courses. The course is also useful to majors in computer science and engineering, as well as those in the natural and social sciences. In addition to computation and problem-solving, the course will introduce the students to axiomatic mathematics and simple proofs.

Math 72

Abstract Linear Algebra Course Information

Fall 2013

Block: J+ (Tuesday, Thursday 3:00-4:15 ) Instructor: Christoph Börgers Email: [email protected] Office: Bromfield-Pearson 215 Office hours: (Spring 2013) Mondays 12–1:30 and 4–5:30, and by appointment Phone: (617) 627-2366 Prerequisites: Calculus II (Math 34, 36, or 39) Text: Linear Algebra, 4th ed., S. Friedberg, A. Insel and L. Spence, Prentice-Hall, 2003. Course description: You have been familiar with linear functions, described for instance by y = mx + b, for a long time. Linear algebra is the study of higher-dimensional analogs, namely linear transformations. Among all transformations, the linear ones are the easiest to understand. They play a fundamental role in every branch of mathematics, from the most pure to the most applied. Even nonlinear transformations are often studied using approximations by linear transformations. (The idea making this possible is the derivative in Calculus, and its higher-dimensional generalizations.) Thinking about linear transformations, one naturally arrives at the notion of vector space. The most familiar example is three-dimensional euclidean space, which is a threedimensional vector space. Each point in it can be characterized by a triple of real numbers. More generally, the n-tuples of real numbers form an n-dimensional real vector space. We will also often work with n-tuples of complex numbers, and they form an n-dimensional complex vector space. There are vector spaces that have infinitely many dimensions, too. An example is the set of infinite sequences of (real or complex) numbers. The main focus of the course will be on linear transformations between finite-dimensional real or complex vector spaces. You will see that any such transformation can be viewed as the multiplication of vectors (columns of numbers) by a matrix (a rectangular array of numbers). This is how matrices enter the subject. This course is the honors version of Math 70. From the beginning, we will adopt a more sophisticated point of view. Proofs and theory will play a greater role than in Math 70. This course is recommended for all those wanting to learn linear algebra who also enjoy a little extra challenge. Of course Math 72 counts as a replacement for Math 70 wherever Math 70 is required as a prerequisite.

Math 87

Mathematical Modeling and Computation Course Information

Fall 2013

Block: D+ (Tuesday, Thursday 10:30-11:45) Instructor: Scott MacLachlan Email: [email protected] Office: Bromfield-Pearson 212 Office hours: (Spring 2013) Monday, Wednesday 10:30-noon Phone: (617) 627-2356 Prerequisites: Math 12 (new number: 34) or 17 (new number: 39) or consent. Text: none Course description: This course is about using elementary mathematics and computing to solve practical problems. Single-variable calculus is a prerequisite; other mathematical and computational tools, such as elementary probability, matrix algebra, elementary combinatorics, and computing in Matlab, will be introduced as they come up. Mathematical modeling is an important area of study, where we consider how mathematics can be used to model and solve problems in the real world. This class will be driven by studying real-world questions where mathematical models can be used as part of the decision-making process. Along the way, we’ll discover that many of these questions are best answered by combining mathematical intuition with some computational experiments. Some of the problems that we will study in this class include: 1. The managed use of natural resources. Consider a population of fish that has a natural growth rate, which is decreased by a certain amount of harvesting. How much harvesting should be allowed each year in order to maintain a sustainable population? 2. The optimal use of labor. Suppose you run a construction company that has fixed numbers of tradespeople, such as carpenters and plumbers. How should you decide what to build to maximize your annual profits? What should you be willing to pay to increase your labor force? 3. Project scheduling. Think about scheduling a complex project, consisting of a large number of tasks, some of which cannot be started until others have finished. What is the shortest total amount of time needed to finish the project? How do delays in completion of some activities affect the total completion time?

Math 135

Real Analysis I Course Information

Block: F+ (Mon Wed 12:00–1:15 p.m.) Instructor: Fulton Gonzalez Email: [email protected] Office: Bromfield-Pearson 203 Office hours: (Spring 2013) Mon 10:30–12 p.m., Thu 11:30 a.m.–1 p.m. Phone: (617) 627-2368

Fall 2013

Block: G (Mon Wed Fri 1:30–2:20 p.m.) Instructor: Loring Tu Email: [email protected] Office: Bromfield-Pearson 206 Office hours: (Spring 2013) Tue 12:50– 1:15 p.m. (Math 136 only), Thu 3–4 p.m., or by appointment Phone: (617) 627-3262

Prerequisites: Math 42 or 44, and 70 or 72, or consent. (In the old numbering scheme, Math 13 or 18, and 46 or 54, or consent.) Text: Patrick Fitzpatrick, Advanced Calculus, 2nd edition, American Mathematical Society, 2009. (ISBN-10: 0821847910) Course description: Real analysis is the rigorous study of real functions, their derivatives and integrals. It provides the theoretical underpinning of calculus and lays the foundation for higher mathematics, both pure and applied. Unlike Calculus I, II, and III, where the emphasis is on intuition and computation, the emphasis in real analysis is on justification and proofs. Is this rigor really necessary? This course will convince you that the answer is an unequivocal yes, because intuition not grounded in rigor often fails us or leads us astray. This is especially true when one deals with the infinitely large or the infinitesimally small. For example, it is not intuitively obvious that, although the set of rational numbers contains the set of integers as a proper subset, there is a one-to-one correspondence between them. These two sets, in this sense, are the same size! On the other hand, there is no such correspondence between the real numbers and the rational numbers, and therefore the set of real numbers is uncountably infinite. A metric space is a set on which there is a distance function. In this course, we will study the topology of the real line and metric spaces, compactness, connectedness, continuous mappings, and uniform convergence. The topics constitute essentially the first five chapters of the textbook. Along the way, we will encounter theorems of calculus, such as the intermediate-value theorem and the maximum-minimum theorem, but in a more general setting that enlarges their range of applicability. If time permits, the course will end with the contraction mapping principle, a fundamental theorem using which one can prove the existence and uniqueness of solutions of differential equations. In addition to introducing a core of basic concepts in analysis, a companion goal of the course is to hone your skills in distinguishing the true from the false and in reading and writing proofs. Math 135 is required of all pure and applied math majors. A math minor must take Math 135 or 145 (or both).

Math 145

Block: J+, Tue/Thu 3-4:15 Instructor: Moon Duchin Email: [email protected] Office: Bromfield-Pearson 113 Office hours (Spring 2013): Weds 10:30-11:30, Fri 1-2

Abstract Algebra I Course Information

Fall 2013

Block: E+, Mon/Wed 10:30-11:45 Instructor: George McNinch Email: [email protected] Office: Bromfield-Pearson 112 Office hours (Spring 2013): Tues 11-12, Thurs 1-2, Fri 1-2

Prerequisites: Math 70 or 72 (Linear Algebra or Abstract Linear Algebra) Text: Abstract Algebra third edition by John A. Beachy and William D. Blair, Waveland Press 2006. Course description: Algebra is a subject that began in the study of equations—especially polynomial equations—and their roots. As modern algebra developed, it began to focus more and more on structure. For instance, the ordinary operations of addition and multiplication obey some basic rules or axioms which earn the set of integers the name of a ring. The collections of all symmetries of a pentagon, or all motions of a Rubik’s cube, form finite sets with an algebraic operation and are known as groups. A key theme in modern algebra is the roughly 200 year old insight known as Galois theory: to study the roots of a polynomial, one should study the group of symmetries of those roots that preserve all algebraic relations among them. By development of a systematic theory of algebraic structures and an emphasis on detection of essential equivalence (i.e. the question of when two algebraic structures are isomorphic, or have the “same shape”), algebra has matured into a very active field of research. The last half-century has seen some big breakthroughs in algebra and its applications, such as the solution of Fermat’s Last Theorem and the classification of finite simple groups (the building blocks of all finite groups). Algebraic topics like representation theory, algebraic geometry, and geometric group theory are still cutting edge research areas, including here at Tufts. Galois theory – the study of the algebraic structure of roots of polynomials – is the subject of Math 146. It provides tools that can already be used to derive dramatic results: for instance, polynomials of degree 2, 3, and 4 can be solved by a quadratic, cubic, and quartic formula, but there is no similar formula for polynomials of degree 5 and more! Or for another example, there is no way to “double the cube” (using a straightedge and compass to produce a cube of volume 2). Math 145 sets the stage for this investigation by introducing the concept of groups, rings, and fields, which are general names for algebraic structures that abstract the essential properties of familiar objects like numbers, groups of motions and other symmetries, and matrices.

Math 150

Surfaces and Low-Dimensional Geometry Course Information

Fall 2008

Block: G+(1:30-2:45) , MW Instructor: Zbigniew Nitecki Email: [email protected] Office: Bromfield-Pearson 214 Office hours: (Spring 2013): M 9:30-10:20, T 10:30-11:30, or by appointment Prerequisites: Math 135 or 145, or consent. Text: Low-Dimensional Geometry: from Euclidean Surfaces to Hyperbolic Knots by Francis Bonahon, American Mathematical Society, 2009, Providence, Rhode Island. Course description: One of the big developments in geometry and topology in the late 20th and early 21st centuries has been the reinvigoration of the study of three-dimensional spaces, prompted in large part by the work of the late Bill Thurston (1946-2012).1 Thurston brought geometry back into topology, formulating the Geometrization Conjecture—that every threedimensional space is made up of pieces each of which has one of eight specific geometric structures—and proving it for a large class of such spaces. The full proof of this fact was completed by Gregory Perelman in 2003; a corollary is the Poincaré Conjecture, a statement that many famous mathematicians have unsuccessfully tried to prove since its statement shortly before Poincaré’s death in 1912. This course is intended to provide at an undergraduate level the mathematical background in hyperbolic geometry, basic topology of surfaces, tesellations and group actions needed to understand the Geometrization Conjecture and some of its ramifications. Tentatively, this will be the first of a two-semester sequence (the second would be taught in the Spring semester, 2014 by Genevieve Walsh, who is an expert three-dimensional topology) that would go through the complete contents of Bohanon’s book, which is the text for the course. This (first) semester, the topics will mostly deal with surfaces. This is recent and exciting mathematics, fundamentally geometric, but bringing to bear analytic and algebraic as well as geometric points of view. I intend to combine my own lectures with student presentations in a kind of undergraduate seminar course.

1

see http://blogs.scientificamerican.com/observations/2012/08/23/the-mathematical-legacy-of-williamthurston-1946-2012/

Math 151

Applications of Advanced Calculus (Linear Partial Differential Equations) Course Information

Fall 2013

Block: H+ (Tuesday, Thursday 1:30–2:45) Instructor: Christoph Börgers Email: [email protected] Office: Bromfield-Pearson 215 Office hours: (Spring 2013) Mondays 12–1:30 and 4–5:30, and by appointment Phone: (617) 627-2366 Prerequisites: Calculus III (Math 42 or 18), Ordinary Differential Equations (Math 51) Text: J. David Logan, Applied Partial Differential Equations, 2nd edition, Springer. Course description: Most deterministic models in the sciences are partial differential equations. The weather, the global climate, air flow around airplanes, blood flow in the heart, the propagation of electro-magnetic waves, the forces in bridges and buildings, the growth of tumors, the propagation of signals in nerve cells in the brain, the spread of epidemics, and countless other phenomena are governed by partial differential equations. Quite arguably there is no branch of mathematics with greater impact on the world. The emphasis in Math 151 is on linear partial differential equations. Three fundamental, prototypical examples are (1) the diffusion equation (describing the diffusion of a pollutant in still air, for instance), (2) the Poisson equation (the fundamental equation of electrostatics), and (3) the wave equation (describing the propagation of sound for instance). You will learn about mathematical properties of the solutions of these and similar equations. In very special cases, it is possible and useful to write down explicit, analytic expressions for the solutions, and you will learn about examples of that. Fourier analysis is a central tool in this subject, and therefore an introduction to Fourier analysis is a part of the course. By far the most important solution techniques for partial differential equations are numerical methods. The study of such methods is a large and rapidly growing branch of mathematics. Although this is not a course on numerical analysis, we will study the simplest, most fundamental numerical methods when it is conceptually clarifying to do so. Math 151 is to be followed by Math 152 in the spring of 2014. Math 152 will be an introduction to nonlinear partial differential equations.

Math 161

Probability Course Information

Fall 2013

Block: E+ (Monday, Wednesday, Friday 10:30-11:45) Instructor: TBA Email: Office: TBA Office hours: Phone: TBA Prerequisites: Math 42 or consent. Text: To be determined. Course description: While it has been motivated by simple applications such as coin flipping and weather prediction, probability theory is a well-established branch of mathematics which analyzes random events observed in a wide variety of fields, including natural science, social science, computer science, medicine, engineering, and finance. Math 161 is an introductory calculus-based probability course. Students taking the course should have a working knowledge of single variable calculus plus multiple integrals and partial derivatives. Math 162, which builds on Math 161, is a course on statistics. The emphasis in this course is on both concepts and applications. The goal is to learn the basic ideas of probability theory in an intuitive, yet mathematical way, and get a feeling for some of the many ways in which the ideas are used. The concepts to be studied are essential for further study of stochastic processes or statistics. Topics to be covered include sample spaces and events, conditioning and independence, discrete and continuous random variables and their distributions, expectations and variances, jointly distributed random variables, conditional probability and conditional expectations, Chebyshev’s inequality, the weak and strong laws of large numbers, moment generating functions, and the central limit theorem. A highlight of the course is the central limit theorem. It is among the most famous and important theorems in all of mathematics. It explains why the distribution of so many random quantities in nature can be described using 2 the function (2π)−1/2 e−x /2 , whose graph is the famous “bell curve.” Assessment will be based on exams and weekly problem sets.

Math 163

Computational Geometry Course Information

Fall 2013

Block G+: Mondays and Wednesdays, 1:30-2:45 p.m. Instructor: Greg Aloupis Office hours: TBA Office: Halligan Hall, Rm. 107A Email: [email protected] Prerequisite: Comp 160 or consent Recommended Texts: Mark de Berg, Marc van Kreveld, Mark Overmars, Otfried Schwarzkopf. Computational Geometry: Algorithms and Applications. Springer-Verlag. Third Edition. 2010. ISBN: 3642096816 Joseph O’Rourke. Computational Geometry in C. Cambridge University Press. 2nd edition. 2001. ISBN: 0521649765. Course Description: Computational geometry is concerned with the design and analysis of algorithms for solving geometric problems and the analysis of the underlying complexity of the problems themselves. Applications can be found in such fields as VLSI design, computer graphics, robotics, computer-aided design, pattern recognition, and statistics. This course will study key problems in computational geometry and the design and analysis of algorithms for their solution. Topics include convex hulls, polygonal triangulation, art gallery theorems, range searching, point location, plane sweep, duality, arrangements of lines, Voronoi diagrams and Delaunay triangulations, intersection problems, decomposition and partitioning, proximity graphs, data depth, and combinatorial geometry. The ultimate aim will be to identify general paradigms and data structures of particular importance to solving computational geometry problems, and thereby provide the participants with a solid foundation in the field. Course work includes written homework assignments, two exams, and a project consisting of creating a web page demonstrating an approved topic related to geometry. Implementations are not strictly required; projects on theory are accepted.

Math 211

Analysis Course Information

Fall 2013

B LOCK : D+TR, 10:30-11:45 I NSTRUCTOR : Marjorie Hahn E MAIL : [email protected] O FFICE : Bromfield-Pearson 202 O FFICE HOURS : (Spring 2013) T, Th 9:30-10:30 or by appointment P HONE : (617) 627-2363 (send an e-mail rather than leaving a voice message) P REREQUISITES : Math 135 or consent. T EXT: (Probably) Real Analysis, Modern Techniques and Their Applications, 2nd Edition by Gerald B. Folland, John Wiley & Sons, 1999. C OURSE 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 of analysis, including functional analysis, measure-theoretic probability theory, ergodic theory, dynamical systems, differential equations, harmonic analysis, complex analysis, as well as for geometry, topology, and applied mathematics. The main topics are measure theory (general measures, Lebesgue integration, convergence theorems, product measures), point set topology (topologies, metrics, compactness, completeness), and Banach spaces (norms, Lp spaces, Hilbert spaces, orthonormal sets and bases, linear forms, duality, Riesz representation theorems, signed measures). The course provides excellent preparation for the Ph.D. oral examinations in analysis. Most students feel well-prepared to take the exams in January, less than a month after completing the course. Lectures will be blackboard style, providing motivation for the theory, emphasizing basic ideas of the proofs, and providing examples. Grades will be based on weekly homework as well as exams. Strong undergraduates who have done well in Math 135 and/or Math 136 are encouraged to consider taking the course, especially if they intend to apply to graduate school in pure or applied mathematics.

Math 215

Algebra Course Information

Fall 2013

B LOCK C: TWF 9:30-10:20 I NSTRUCTOR : Richard Weiss E MAIL : [email protected] O FFICE : Bromfield-Pearson 116 O FFICE HOURS : TWF 1:30-1:30 P HONE : 7-3802 P REREQUISITES : Math 146 or the equivalent. T EXT: Abstract Algebra, 3rd ed. by David Dummit and Richard Foote C OURSE DESCRIPTION : We will study the basic properties of groups and commutative rings (the first nine chapters of our textbook) with an emphasis on the notion of a group acting on a set. We will then go on to field theory, as time permits. These are topics covered in a typical undergraduate course on abstract algebra, but we will look at them much more thoroughly and from a more sophisticated point of view.

Math 217

Geometry and Topology Course Information

Fall 2013

Block: G+, Mon Wed 1:30–2:45 p.m. Instructor: Fulton Gonzalez Email: [email protected] Office: Bromfield-Pearson 206 Office hours: (Spring 2013) Mon 10:30 a.m. – 12:00 p.m., Thu 11:30 a.m. – 1:00 p.m. Phone: (617) 627-2368 Prerequisites: Undergraduate real analysis (Math 135, 136) and abstract algebra (Math 145), 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 on the real line to functions on the plane and in 3-space. Then one encounters vector-valued functions and learns about integrals on curves and surfaces. Real analysis extends differential and integral calculus from R3 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 setting of 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, and the divergence theorem are different manifestations of a single general Stokes’ theorem for manifolds. Higher-dimensional manifolds arise even if one is interested only in the three-dimensional space which we inhabit. For example, if we call a rotation followed by a translation an affine motion, then the set of all affine motions in R3 is a six-dimensional manifold. As another example, 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. Combining aspects of algebra, topology, and analysis, the theory of manifolds has found applications to many areas of mathematics and even classical mechanics, general relativity, and quantum field 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, exterior differentiation, orientations, and integration. If time permits, we may also cover integral manifolds and the Frobenius theorem. There will be ten problem sets, a midterm, and a final. Undergraduates who have done well in the prerequisite courses should find this course within their competence.

Math 226

Numerical Analysis Course Information

Fall 2011

B LOCK : J+TR: 3:00-4:15, Tues. and Thurs. I NSTRUCTOR : Misha Kilmer E MAIL : [email protected] O FFICE : Bromfield-Pearson 114 O FFICE HOURS : Spring 2013 M, W, 1-2; Th 1:30-2:30 (some exceptions) P HONE : 7-2500 P REREQUISITES : Permission of Instructor T EXT: TBD C OURSE DESCRIPTION : “Numerical analysis is the study of algorithms for the problems of continuous mathematics.” (L. N. Trefethen, “The definition of numerical analysis”, SIAM News, November 1992.) Continuous mathematics means mathematics involving the continuum of real or complex numbers. There are many striking examples illustrating for accurate numerical solution methods. (1) On Feb. 25, 1991, an American Patriot Missile failed to intercept an Iraqi scud missile, resulting in the deaths of 28 soldiers. Ultimately, the problem with the tracking and interception of the scud was traced to issues related to computation in finite precision arithmetic. (2) Ordinary differential equations are used to model the spread of an infection through a population. Numerical solution techniques are usually the only practical option for solving these problems, but without an underlying understanding of the properties of both the model and the algorithm, numerical results are not to be trusted. (3) CT scanners are used throughout the world for medical diagnostics. This technology is based on numerical algorithms for the solution of certain integral equations. This course is intended as a rigorous first graduate-level course in Numerical Analysis. Students are assumed to have background in ordinary differential equations and analysis at the undergraduate level, as well as previous programming experience. We will cover a large portion of the Numerical Analysis qualifying exam core topics, which include numerical integration and orthogonal polynomials, interpolation, numerical solution of differential equations, unconstrained optimization, and solution of nonlinear equations. Some programming will be required.

Math 250

Geometric Literacy Course Information

Fall 2013

Block: TBD Instructor: Moon Duchin Email: [email protected] Office: Bromfield-Pearson 113 Office Hours (Spring 2013): Wed 10:30-11:30, Fri 1-2 Prerequisites: Graduate standing or consent of the instructor. Text: none; individual modules may have notes or chapters associated to them. Course description: This course is modeled on the very successful long-running course of the same name at the University of Chicago, subtitled “What every good geometer should know”; however, many of the topics covered here should be useful to mathematicians in neighboring areas as well, from algebraic geometry to applied math. The topics are broken into modules, interspersing more elementary background with topics close to current research, and should be understandable to second-year graduate students. The modular structure is designed to make the topics self-contained, and should allow students to follow more advanced modules before formally learning all the prerequisites. Possible modules for Fall 2013 • Haar measure (2 weeks) • Lattices in Lie groups (4 weeks) • The 84(g − 1) theorem on symmetries of surfaces (2 weeks) • Bass-Serre theory: groups acting on trees (3 weeks) • Description of the eight 3D geometries; overview of Geometrization Thm (3 weeks) • Billiards and flat surfaces (5 weeks) • Mostow rigidity (4 weeks) • Introduction to Teichm¨ uller theory (5 weeks) • Nilpotent geometry: the Heisenberg group (4 weeks)

Math 250-02

Group Actions and Cohomology Course Information

Fall 2013

Block: I+ (Mon Wed 3:00–4:15 p.m.) Instructor: Loring Tu Email: [email protected] Office: Bromfield-Pearson 206 Office hours: (Spring 2013) Tue 12:50–1:15 p.m. (Math 136 only), Thu 3–4 p.m., or by appointment Phone: (617) 627-3262 Prerequisites: Math 217 or consent. Some knowledge of homology or cohomology is desirable. Text: Raoul Bott and Loring Tu, Elements of Equivariant Cohomology, notes available from the instructor. Course description: A group action on a space is an expression of the symmetries of the space. A topological space with an action of a topological group G is called a G-space. In this course we are interested in the algebraic topology of G-spaces in much the same way that the cohomology functor encodes the algebraic topology of topological spaces. Surprisingly, the topology of a G-space is intimately related to another fundamental notion of twentieth-century topology—that of a fiber bundle. A fiber bundle is a space that is locally a product with another space called its fiber. In 1977, the Nobel-prize winning physicist C. N. Yang wrote, “Gauge fields are deeply related to some profoundly beautiful ideas of contemporary mathematics, ideas that are the driving forces of part of the mathematics of the last forty years, ... , the theory of fiber bundles.” Examples of fiber bundles abound in all areas of geometry and topology: covering spaces, tangent bundle, normal bundle, complex vector bundles, circle bundles, sphere bundles, frame bundles, projective bundles. All the fiber bundles arise from one particular type called principal bundles. Principal bundles are fiber bundles with a group G as fiber and moreover, G acts freely on each fiber by multiplication. The first part of the course will be on principal bundles and fiber bundles. We will study their classification by a universal bundle and a classifying space. In order to have a good theory, we need to restrict our spaces to those that are constructed by attaching cells; these are called CW complexes. We will briefly discuss higher homotopy groups of CW complexes. The second part of the course will be on the cohomology of G-spaces. Here the best candidate is the equivariant cohomology of Armand Borel and Henri Cartan. Just as differential forms can be used to compute the singular cohomology of a manifold, so one can use equivariant differential forms to compute the equivariant cohomology of a G-manifold. This is based on the work of André Weil and Henri Cartan. Finally, we prove the Atiyah–Bott–Berline–Vergne equivariant localization theorem. Using this, an integral on an manifold with a group action can often be converted to a finite sum over the fixed points of the action. This theorem is a far-reaching extension of the Hopf index theorem that computes the Euler characteristic in terms of the zeros of a vector field, and provides a powerful tool for computing integrals on a manifold.

MATHEMATICS MAJOR CONCENTRATION CHECKLIST For students matriculating Fall 2012 and after (and optionally for others) (To be submitted with University Degree Sheet) Name:

I.D.#:

E-Mail Address:

College and Class:

Other Major(s): (Note: Submit a signed checklist with your degree sheet for each major.) Please list courses by number. For transfer courses, list by title, and add “T”. Indicate which courses are incomplete, in progress, or to be taken. Note: If substitutions are made, it is the student’s responsibility to make sure the substitutions are acceptable to the Mathematics Department. Ten courses distributed as follows: I. Five courses required of all majors. (Check appropriate boxes.) 1. 2. 3.

Math 42: Calculus III or 4. Math 44: Honors Calculus 5. Math 70: Linear Algebra or Math 72: Abstract Linear Algebra Math 135: Real Analysis I

Math 145: Abstract Algebra I Math 136: Real Analysis II or Math 146: Abstract Algebra II

We encourage all students to take Math 70 or 72 before their junior year. To prepare for the proofs required in Math 135 and 145, we recommend that students who take Math 70 instead of 72 also take another course above 51 before taking these upper level courses. II. Two additional 100-level math courses. III. Three additional mathematics courses numbered 51 or higher; up to two of these courses may be replaced by courses in related fields including: Chemistry 133, 134; Computer Science 15, 126, 160, 170; Economics 107, 108, 154, 201, 202; Electrical Engineering 18, 107, 108, 125; Engineering Science 151, 152; Mechanical Engineering 137, 138, 150, 165, 166; Philosophy 33, 103, 114, 170; Physics 12, 13 any course numbered above 30; Psychology 107, 108, 140. 1. ______________ 2. ______________ 3. ______________ Advisor’s signature:

Date:

Note: It is the student’s responsibility to return completed, signed degree sheets to the Office of Student Services, Dowling Hall. (Form Revised September 8, 2012)

APPLIED MATHEMATICS MAJOR CONCENTRATION CHECKLIST (To Be Submitted with University Degree Sheet) Name:

I.D.#:

E-Mail Address:

College and Class:

Other Major(s): (Note: Submit a signed checklist with your degree sheet for each major.) Please list courses by number. For transfer courses, list by title, and add “T”. Indicate which courses are incomplete, in progress, or to be taken. Note: If substitutions are made for courses listed as “to be taken”, it is the student’s responsibility to make sure the substitutions are acceptable. Thirteen courses beyond Calculus II. These courses must include: I. Seven courses required of all majors. (Check appropriate boxes.) 1. 2. 3.

Math 42: Calculus III or 4. Math 44: Honors Calculus III 5. Math 70: Linear Algebra or 6. Math 72: Abstract Linear Algebra 7. Math 51: Differential Equations

II. One of the following: 1. ______________ Math 145: Abstract Algebra I Comp 15: Data Structures

Math 87: Mathematical Modeling Math 158: Complex Variables Math 135: Real Analysis I Math 136: Real Analysis II

Math 61/Comp 22: Discrete Mathematics Math/Comp 163: Computational Geometry

III. One of the following three sequences: 1. ______________ Math 126/128: Numerical Analysis/Numerical Algebra Math 151/152: Applications of Advanced Calculus/Nonlinear Partial Differential Equations Math 161/162: Probability/Statistics IV. An additional course from the list below but not one of the courses chosen in section III: 1. ______________ Math 126 Math 128 Math 151 Math 152 Math 161 Math 162 V. Two electives (math courses numbered 61 or above are acceptable electives. With the approval of the Mathematics Department, students may also choose as electives courses with strong mathematical content that are not listed as Math courses.) 1. ______________ 2. _____________ Advisor’s signature:

Date:

Note: It is the student’s responsibility to return completed, signed degree sheets to the Office of Student Services, Dowling Hall. (Form Revised February 8, 2012)

DECLARATION OF MATHEMATICS MINOR Name:

I.D.#:

E-mail address:

College and Class:

Major(s) Faculty Advisor for Minor (please print)

MATHEMATICS MINOR CHECKLIST Please list courses by number. For transfer courses, list by title and add “T”. Indicate which courses are incomplete, in progress, or to be taken. Courses numbered under 100 will be renumbered starting in the Fall 2012 semester. Courses are listed here by their new number, with the old number in parentheses.

Note: If substitutions are made for courses listed as “to be taken”, it is the student’s responsibility to make sure that the substitutions are acceptable. Six courses distributed as follows: I. Two courses required of all minors. (Check appropriate boxes.) 1. 2.

Math 42 (old: 13): Calculus III or Math 70 (old: 46): Linear Algebra or

Math 44 (old: 18): Honors Calculus Math 72 (old: 54): Abstract Linear Algebra

II. Four additional math courses with course numbers Math 50 or higher. These four courses must include Math 135: Real Analysis I or 145: Abstract Algebra (or both). Note that Math 135 and 145 are only offered in the fall. 1. ______________

2. ______________

3. ______________

4. ______________

Advisor’s signature

Date

Note: Please return this form to the Mathematics Department Office (Form Revised May 7th, 2012)

Jobs and Careers The Math Department encourages you go to discuss your career plans with your professors. All of us would be happy to try and answer any questions you might have. Professor Quinto has built up a collection of information on careers, summer opportunities, internships, and graduate schools and his web site (http://equinto.math.tufts.edu) is a good source. Career Services in Dowling Hall has information about writing cover letters, resumes and jobhunting in general. They also organize on-campus interviews and networking sessions with alumni. There are job fairs from time to time at various locations. Each January, for example, there is a fair organized by the Actuarial Society of Greater New York. On occasion, the Math Department organizes career talks, usually by recent Tufts graduates. In the past we had talks on the careers in insurance, teaching, and accounting. Please let us know if you have any suggestions.

The Math Society The Math Society is a student run organization that involves mathematics beyond the classroom. The club seeks to present mathematics in a new and interesting light through discussions, presentations, and videos. The club is a resource for forming study groups and looking into career options. You do not need to be a math major to join! See any of us about the details. Check out http://ase.tufts.edu/mathclub for more information. The SIAM Student Chapter Students in the Society for Industrial and Applied Mathematics (SIAM) student chapter organize talks on applied mathematics by students, faculty and researchers in industry. It is a great way to talk with other interested students about the range of applied math that’s going on at Tufts. You do not need to be a math major to be involved, and undergraduates and graduate students from a range of fields are members. Check out http://neumann.math.tufts.edu/~siam for more information.

BLOCK SCHEDULE 50 and 75 Minute Classes

Mon

8:05-9:20 (A+,B+)

A+

Mon

Tue B+

0+ 8:30-9:20 (A,B)

Tue

A

Wed

Wed

A+

Thu

Fri

B+

1+ B

Thu

2+ A

Fri

150/180 Minute Classes and Seminars

B+ 3+

4+

B

B

8:30-11:30 (0+,1+,2+,3+,4+) 9-11:30 (0,1,2,3,4)

9:30-10:20 (A,C,D)

10:30-11:20 (D,E) 10:30-11:45 (D+,E+)

D

0

C

1

C

2

A

3

E

D

E

D

E

E+

D+

E+

D+

E+

12:00-12:50 (F)

Open

F

12:00-1:15 (F+)

Open

F+

1:30-2:20 (G,H)

G

1:30-2:45 (G+,H+)

G+

5

H

6

H+

G

7

G+

F

F

F+

F+

H

8

G

4

9

H+

1:30-4:00 (5,6,7,8,9) 1:20-4:20 (5+,6+,7+,8+,9+)

2:30-3:20 (H on Fri) 3:00-3:50 (I,J) 3:00-4:15 (J+,I+) 3:30-4:20 (I on Fri)

C

H (2:30-3:20)

I I+

J J+ 5+

I I+ 6+

J J+ 7+

4:30-5:20 ( K,L)

J/K

L

K

L

4:30-5:20 (J ( on Mon) 4:30-5:45 (K+,L+i)

K+

L+

K+

L+

6:00-6:50 (M,N)

N/M

6:00-7:15 (M+,N+)

M+

10+

11+ N N+

10

12+ M

9+

13+ N

M+ 11

I (3:30-4:20)

8+

N+ 12

13

7:30-8:15 (P,Q)

Q/P

Q

P

Q

6:00-9:00 (10+,11+,12+,13+)

7:30-8:45 (P+,Q+)

P+

Q+

P+

Q+

6:30-9:00 (10,11,12,13)

Notes * A plain letter (such as B) indicates a 50 minute meeting time. * A letter augmented with a + (such as B+) indicates a 75 minute meeting time. * A number (such as 2) indicates a 150 minute class or seminar. A number with a + (such as 2+) indicates a 180 minute meeting time. * Lab schedules for dedicated laboratories are determined by department/program. * Monday from 12:00-1:20 is departmental meetings/exam block. * Wednesday from 12:00-1:20 is the AS&E-wide meeting time. * If all days in a block are to be used, no designation is used. Otherwise, days of the week (MTWRF) are designated (for example, E+MW). * Roughly 55% of all courses may be offered in the shaded area. * Labs taught in seminar block 5+-9+ may run to 4:30. Students taking these courses are advised to avoid courses offered in the K or L block.