## Course Catalog

**2014-2015:**

**2013-2014:**

### Course Descriptions

## Mathematics Courses

#### MATH 102 (F)Foundations in Quantitative Skills

This course will strengthen a student's foundation in quantitative reasoning in preparation for the science curriculum and QFR requirements. The material will be at the college algebra / precalculus level, and covered in a tutorial format with students working in small groups with the professor. Access to this course is limited to placement by a quantitative skills counselor. [ more ]

Taught by: Stewart Johnson

Catalog details#### MATH 113 The Beauty of Numbers

Not offered this year

Have you ever wondered what keeps your credit card information safe everytime you buy something online? Number theory! Number Theory is one of the oldest branches of mathematics. In this course, we will discover the beauty and usefulness of numbers, from ancient Greece to modern cryptography. We will look for patterns, make conjectures, and learn how to prove these conjectures. Starting with nothing more than basic high school algebra, we will develop the logic and critical thinking skills required to realize and prove mathematical results. Topics to be covered include the meaning and content of proof, prime numbers, divisibility, rationality, modular arithmetic, Fermat's Last Theorem, the Golden ratio, Fibonacci numbers, coding theory, and unique factorization. [ more ]

Taught by: Allison Pacelli

Catalog details#### MATH 130 (F, S)Calculus I

Calculus permits the computation of velocities and other instantaneous rates of change by a limiting process called differentiation. The same process also solves "max-min" problems: how to maximize profit or minimize pollution. A second limiting process, called integration, permits the computation of areas and accumulations of income or medicines. The Fundamental Theorem of Calculus provides a useful and surprising link between the two processes. Subtopics include trigonometry, exponential growth, and logarithms. [ more ]

Taught by: Julie Blackwood, Eyvindur Palsson

Catalog details#### MATH 140 (F, S)Calculus II

Mastery of calculus requires understanding how integration computes areas and business profit and acquiring a stock of techniques. Further methods solve equations involving derivatives ("differential equations") for population growth or pollution levels. Exponential and logarithmic functions and trigonometric and inverse functions play an important role. This course is the right starting point for students who have seen derivatives, but not necessarily integrals, before. [ more ]

Taught by: Leo Goldmakher, Eyvindur Palsson

Catalog details#### MATH 150 (F, S)Multivariable Calculus

Applications of calculus in mathematics, science, economics, psychology, the social sciences, involve several variables. This course extends calculus to several variables: vectors, partial derivatives, multiple integrals. There is also a unit on infinite series, sometimes with applications to differential equations. [ more ]

Taught by: Satyan Devadoss, Thomas Garrity

Catalog details#### MATH 151 (F)Multivariable Calculus

Applications of calculus in mathematics, science, economics, psychology, the social sciences, involve several variables. This course extends calculus to several variables: vectors, partial derivatives and multiple integrals. The goal of the course is Stokes Theorem, a deep and profound generalization of the Fundamental Theorem of Calculus. The difference between this course and MATH 150 is that MATH 150 covers infinite series instead of Stokes Theorem. Students with the equivalent of BC 3 or higher should enroll in MATH 151, as well as students who have taken the equivalent of an integral calculus and who have already been exposed to infinite series. For further clarification as to whether MATH 150 or MATH 151 is appropriate, please consult a member of the math/stat department. [ more ]

Taught by: Colin Adams

Catalog details#### MATH 175 Mathematical Politics: Voting, Power, and Conflict

Not offered this year

Who should have won the 2000 Presidential Election? Do any two senators really have equal power in passing legislation? How can marital assets be divided fairly? While these questions are of interest to many social scientists, a mathematical perspective can offer a quantitative analysis of issues like these and more. In this course, we will discuss the advantages and disadvantages of various types of voting systems and show that, in fact, any such system is flawed. We will also examine a quantitative definition of power and the principles behind fair division. Along the way, we will enhance the critical reasoning skills necessary to tackle any type of problem mathematical or otherwise. [ more ]

Taught by: Allison Pacelli

Catalog details#### MATH 180 The Art of Mathematical Thinking: An Introduction to the Beauty and Power of Mathematical Ideas

Not offered this year

What is mathematics? How can it enrich and improve your life? What do mathematicians think about and how do they go about tackling challenging questions? Most people envision mathematicians as people who solve equations or perform arithmetic. In fact, mathematics is an artistic endeavor which requires both imagination and creativity. In this course, we will experience what this is all about by discovering various beautiful branches of mathematics while learning life lessons that will have a positive impact on our lives. There are two meta-goals for this course: (1) a better perspective into mathematics, and (2) sharper analytical reasoning to solve problems (both mathematical and nonmathematical). [ more ]

Taught by: Edward Burger

Catalog details#### MATH 200 (F, S)Discrete Mathematics

As a complement to calculus, which is the study of continuous processes, this course focuses on the discrete, including finite sets and structures, their properties and applications. Topics will include basic set theory, infinity, graph theory, logic, counting, recursion, functions, and number theory. The course serves as an introduction not only to these and other topics but also to the methods and styles of mathematical proof and problem solving. In the spring semester, students will attend class plus one conference section each week. [ more ]

Taught by: Allison Pacelli, Mihai Stoiciu

Catalog details#### MATH 209 (S)Differential Equations

Historically, much beautiful mathematics has arisen from attempts to explain physical, chemical, biological and economic processes. A few ingenious techniques solve a surprisingly large fraction of the associated ordinary and partial differential equations, and geometric methods give insight to many more. The mystical Pythagorean fascination with ratios and harmonics is vindicated and applied in Fourier series and integrals. We will explore the methods, abstract structures, and modeling applications of ordinary and partial differential equations and Fourier analysis. [ more ]

Taught by: Julie Blackwood

Catalog details#### MATH 210 (S)Mathematical Methods for Scientists

This course covers a variety of mathematical methods used in the sciences, focusing particularly on the solution of ordinary and partial differential equations. In addition to calling attention to certain special equations that arise frequently in the study of waves and diffusion, we develop general techniques such as looking for series solutions and, in the case of nonlinear equations, using phase portraits and linearizing around fixed points. We study some simple numerical techniques for solving differential equations. A series of optional sessions in Mathematica will be offered for students who are not already familiar with this computational tool. [ more ]

Taught by: Frederick Strauch

Catalog details#### MATH 250 (F, S)Linear Algebra

Many social, political, economic, biological, and physical phenomena can be described, at least approximately, by linear relations. In the study of systems of linear equations one may ask: When does a solution exist? When is it unique? How does one find it? How can one interpret it geometrically? This course develops the theoretical structure underlying answers to these and other questions and includes the study of matrices, vector spaces, linear independence and bases, linear transformations, determinants and inner products. Course work is balanced between theoretical and computational, with attention to improving mathematical style and sophistication. [ more ]

Taught by: Leo Goldmakher, Mihai Stoiciu

Catalog details#### MATH 285 T (S)Mathematics Education

This course will be a study of mathematics education, from the practical aspects of teaching to numerous ideas in current research. This is an exciting time in mathematics education. The new common core state standards have sparked a level of interest and debate not often seen in the field. In this course, we will look at a wide range of issues in math education, from content knowledge to the role of creativity in a math class to philosophies of teaching. In addition to weekly tutorial meetings that focus on some of the key questions in math education, we will also meet weekly as a group to discuss the mechanics of teaching. Each student will also be responsible for teaching bi-weekly extra sessions for MATH 200 at which they will make presentations, field questions, and offer guidance on homework questions. Students will also attend the MATH 200 lecture, and do some grading for the course. [ more ]

Taught by: Allison Pacelli

Catalog details#### MATH 308 T Analysis and Number Theory

Not offered this year

Gauss said "Mathematics is the queen of the sciences and number theory the queen of mathematics"; in this class we shall meet some of her subjects. We will discuss many of the most important questions in analytic and additive number theory, with an emphasis on techniques and open problems. Topics will range from Goldbach's Problem and the Circle Method to the Riemann Zeta Function and Random Matrix Theory. Other topics will be chosen by student interest, coming from sum and difference sets, Poissonian behavior, Benford's law, the dynamics of the 3x+1 map as well as suggestions from the class. We will occasionally assume some advanced results for our investigations, though we will always try to supply heuristics and motivate the material. No number theory background is assumed, and we will discuss whatever material we need from probability, statistics or Fourier analysis. For more information, see http://www.math.brown. edu/~sjmiller/williams/406. [ more ]

Taught by: Steven Miller

Catalog details#### MATH 309 Introduction to Complex Analysis

Not offered this year

The complex numbers are amazingly useful in mathematics, physics, engineering, and elsewhere. We'll learn the meaning of complex multiplication and exponentiation, as in Euler's famous e^{iπ }= -1. We'll study complex functions and their power series, learn how to integrate in the complex plane, including residue calculus, and how to map one domain to another (conformal mapping). We'll see the easiest proof of the Fundamental Theorem of Algebra, which says that every algebraic equation has a solution as long as you allow complex numbers. [ more ]

Taught by: Frank Morgan

Catalog details#### MATH 310 Mathematical Modeling of Ecological Systems

Not offered this year

Mathematical models are extensively used to understand biological phenomena. In this course we will study how differential and difference equations can be used to model various ecological systems ranging from predator-prey interactions to infectious disease dynamics. We will explore how to formulate these models, and methods for analyzing these systems including local and global stability analysis will be introduced. [ more ]

Taught by: Julie Blackwood

Catalog details#### MATH 313 Introduction to Number Theory

Not offered this year

The study of numbers dates back thousands of years, and is fundamental in mathematics. In this course, we will investigate both classical and modern questions about numbers. In particular, we will explore the integers, and examine issues involving primes, divisibility, and congruences. We will also look at the ideas of number and prime in more general settings, and consider fascinating questions that are simple to understand, but can be quite difficult to answer. [ more ]

Taught by: Allison Pacelli

Catalog details#### MATH 316 (S)Protecting Information: Applications of Abstract Algebra and Quantum Physics

Living in the information age, we find ourselves depending more and more on codes that protect messages against either noise or eavesdropping. This course examines some of the most important codes currently being used to protect information, including linear codes, which in addition to being mathematically elegant are the most practical codes for error correction, and the RSA public key cryptographic scheme, popular nowadays for internet applications. We also study the standard AES system as well as an increasingly popular cryptographic strategy based on elliptic curves. Looking ahead by a decade or more, we show how a quantum computer could crack the RSA scheme in short order, and how quantum cryptographic devices will achieve security through the inherent unpredictability of quantum events. [ more ]

Taught by: Susan Loepp, William Wootters

Catalog details#### MATH 317 (F)Introduction to Operations Research

In the first N math classes of your career, you can be misled as to what the world is truly like. How? You're given exact problems and told to find exact solutions. The real world is sadly far more complicated. Frequently we cannot exactly solve problems; moreover, the problems we try to solve are themselves merely approximations to the world! We are forced to develop techniques to approximate not just solutions, but even the statement of the problem. Additionally, we often need the solutions quickly. Operations Research, which was born as a discipline during the tumultuous events of World War II, deals with efficiently finding optimal solutions. In this course we build analytic and programming techniques to efficiently tackle many problems. We will review many algorithms from earlier in your mathematical or CS career, with special attention now given to analyzing their run-time and seeing how they can be improved; students will be implementing many of these algorithms on computer systems of their choice. The culmination of the course is a development of linear programming and an exploration of what it can do and what are its limitations. For those wishing to take this as a Stats course, the final project must have a substantial implementation computation (respectively, statistics) component approved by the instructor. [ more ]

Taught by: Steven Miller

Catalog details#### MATH 318 T Numerical Problem Solving

Not offered this year

In the last twenty years computers have profoundly changed the work in numerical mathematics (in areas from linear algebra and calculus to differential equations and probability). The main goal of this tutorial is to learn how to use computers to do quantitative science. We will explore concepts and ideas in mathematics and science using numerical methods and computer programming. We will use specialized software, including Mathematica and Matlab. [ more ]

Taught by: Mihai Stoiciu

Catalog details#### MATH 319 (F)Integrative Bioinformatics, Genomics, and Proteomics Lab

What can computational biology teach us about cancer? In this capstone experience for the Genomics, Proteomics, and Bioinformatics program, computational analysis and wet-lab investigations will inform each other, as students majoring in biology, chemistry, computer science, mathematics/statistics, and physics contribute their own expertise to explore how ever-growing gene and protein data-sets can provide key insights into human disease. In this course, we will take advantage of one well-studied system, the highly conserved Ras-related family of proteins, which play a central role in numerous fundamental processes within the cell. The course will integrate bioinformatics and molecular biology, using database searching, alignments and pattern matching, phylogenetics, and recombinant DNA techniques to reconstruct the evolution of gene families by focusing on the gene duplication events and gene rearrangements that have occurred over the course of eukaryotic speciation. By utilizing high through-put approaches to investigate genes involved in the MAPK signal transduction pathway in human colon cancer cell lines, students will uncover regulatory mechanisms that are aberrantly altered by siRNA knockdown of putative regulatory components. This functional genomic strategy will be coupled with independent projects using phosphorylation-state specific antisera to test our hypotheses. Proteomic analysis will introduce the students to de novo structural prediction and threading algorithms, as well as data-mining approaches and Bayesian modeling of protein network dynamics in single cells. Flow cytometry and mass spectrometry will be used to study networks of interacting proteins in colon tumor cells. [ more ]

Taught by: Lois Banta

Catalog details#### MATH 321 (S)Knot Theory

Take a piece of string, tie a knot in it, and glue the ends together. The result is a knotted circle, known as a knot. For the last 100 years, mathematicians have studied knots, asking such questions as, "Given a nasty tangled knot, how do you tell if it can be untangled without cutting it open?" Some of the most interesting advances in knot theory have occurred in the last ten years.This course is an introduction to the theory of knots. Among other topics, we will cover methods of knot tabulation, surfaces applied to knots, polynomials associated to knots, and relationships between knot theory and chemistry and physics. In addition to learning the theory, we will look at open problems in the field. [ more ]

Taught by: Colin Adams

Catalog details#### MATH 322 Differential Geometry

Not offered this year

It is easy to convince oneself that the shortest distance from equatorial Africa to equatorial South America is along the equator. This illustrates the fact that "straight lines" on a sphere are described by so-called great circles. It is somewhat more difficult to describe the shortest path between two points on the surface of, for example, a doughnut, reflecting the fact that a doughnut curves in space in a more complicated way than the sphere. Differential geometry is the mathematical language describing these curvature properties. In this course we will learn this language and use it to answer many interesting questions. We will also develop the tools needed to begin the more advanced study of "Riemannian" geometry, which describes (among other things) Einstein's Relativity Theory. Topics: Curves in space, the Frenet-Serret Theorem, the first and second fundamental forms, geodesics, principal/Gaussian/mean/normal curvatures, the Theorema Egregium, the Gauss-Bonnet formula and Theorem, introduction to n-dimensional Riemannian manifolds/metrics/curvature. [ more ]

Taught by: Mihai Stoiciu

Catalog details#### MATH 325 Set Theory

Not offered this year

Set theory is the traditional foundational language for all of mathematics. We will be discussing the Zermelo-Fraenkel axioms, including the Axiom of Choice and the Continuum Hypothesis, basic independence results and, if time permits, Goedel's Incompleteness Theorem. At one time, these issues tore at the foundations of mathematics. They are still vital for understanding the nature of mathematical truth. [ more ]

Taught by: Thomas Garrity

Catalog details#### MATH 327 Computational Geometry

Not offered this year

The subject of computational geometry started just 25 years ago, and this course is designed to introduce its fundamental ideas. Our goal is to explore "visualization" and "shape" in real world problems. We focus on both theoretic ideas (such as visibility, polyhedra, Voronoi diagrams, triangulations, motion) as well as applications (such as cartography, origami, robotics, surface meshing, rigidity). This is a beautiful subject with a tremendous amount of active research and numerous unsolved problems, relating powerful ideas from mathematics and computer science. [ more ]

Taught by: TBA

Catalog details#### MATH 329 Geometry By Its History

Not offered this year

The thorough study of Euclidean geometry has been a cornerstone of a complete education for thousands of years. In this course, we trace the origins of modern geometry by studying its classical roots, including ancient Greek geometry, conic sections, triangle centers, circle theorems, trigonometry, and analytic geometry. Other topics include the impossibility of doubling the cube or trisecting an angle, non-constructable polygons, non-Euclidean geometry, and geometry in higher dimensions. [ more ]

Taught by: Michael Biro

Catalog details#### MATH 331 (F)The little Questions

Using math competitions such as the Putnam Exam as a springboard, in this class we follow the dictum of the Ross Program and "think deeply of simple things". The two main goals of this course are to prepare students for competitive math competitions, and to get a sense of the mathematical landscape encompassing elementary number theory, combinatorics, graph theory, and group theory (among others). While elementary frequently is not synonymous with easy, we will see many beautiful proofs and "a-ha" moments in the course of our investigations. Students will be encouraged to explore these topics at levels compatible with their backgrounds. [ more ]

Taught by: Steven Miller

Catalog details#### MATH 335 Game Theory

Not offered this year

Game theory is the study of interacting decision makers involved in a conflict of interest. We investigate outcomes, dynamics, and strategies as players rationally pursue objective goals and interact according to specific rules. Game theory has been used to illuminate political, ethical, economical, social, psychological, and evolutionary phenomenon. We will examine concepts of equilibrium, stable strategies, imperfect information, repetition, cooperation, utility, and decision. [ more ]

Taught by: Stewart Johnson

Catalog details#### MATH 341 (S)Probability

While probability began with a study of games, it has grown to become a discipline with numerous applications throughout mathematics and the sciences. Drawing on gaming examples for motivation, this course will present axiomatic and mathematical aspects of probability. Included will be discussions of random variables, expectation, independence, laws of large numbers, and the Central Limit Theorem. Many interesting and important applications will also be presented, potentially including some from coding theory, number theory and nuclear physics. [ more ]

Taught by: Steven Miller

Catalog details#### MATH 347 T (S)Origami

Origami is the art and study of folding and unfolding. Although ancient in origin, there has been a tremendous resurgence of interest recently, resulting in stunning sculptures and marvelously intricate pop-up books. The applications of origami have grown as well, from NASA's James Webb space telescope to cutting-edge protein folding models. This is a beautiful subject with a tremendous amount of active research, relating powerful ideas from studio art, computer science, and mathematics. This tutorial is designed to introduce the foundations of origami design from a mathematical viewpoint: 1D linkages, 2D crease patterns and cut-theorems, 3D unfolding polyhedra. No experience in paper folding is necessary. [ more ]

Taught by: Satyan Devadoss

Catalog details#### MATH 350 (F, S)Real Analysis

Real analysis is the theory behind calculus. It is based on a precise understanding of the real numbers, elementary topology, and limits. Topologically, nice sets are either closed (contain their limit points) or open (complement closed). You also need limits to define continuity, derivatives, integrals, and to understand sequences of functions. [ more ]

Taught by: Cesar Silva

Catalog details#### MATH 351 (S)Applied Real Analysis

Real Analysis is the underlying theory of calculus and is hence the study of limits. This course will study limits in the context of Fourier Analysis, which is one of the major tools of modern mathematics. (Applications of Fourier Analysis range from the abstract study of the distribution of prime numbers to the quite practical construction of mpeg files in an iPod.) By the end of the semester we will be using the theoretical machinery of limits as applied to Fourier series to understand some of the basic differential equations of science, such as the wave equation and, possibly, the heat equation. [ more ]

Taught by: Thomas Garrity

Catalog details#### MATH 354 Graph Theory

Not offered this year

Investigation of the structure and properties of graphs with emphasis both on certain classes of graphs such as multi-partite, planar, and perfect graphs and on application to various optimization problems such as minimum colorings of graphs, maximum matchings in graphs, network flows, etc. [ more ]

Taught by: Mark Mixer

Catalog details#### MATH 355 (F, S)Abstract Algebra

Algebra gives us the tools to solve equations. Sets such as the integers or real numbers have special properties which make algebra work or not work according to the circumstances. In this course, we generalize algebraic processes and the sets upon which they operate in order to better understand, theoretically, when equations can and cannot be solved. We define and study the abstract algebraic structures called groups, rings and fields, as well as the concepts of factor group, quotient ring, homomorphism, isomorphism, and various types of field extensions. [ more ]

Taught by: Susan Loepp, Mihai Stoiciu

Catalog details#### MATH 357 T Phylogenetics

Not offered this year

Phylogenetics is the analysis and construction of information trees based on shared characteristics. The foundational problem asks, given some data from objects, how can a tree be constructed which shows the proper relationships between the objects? This is a beautiful subject with a tremendous amount of cutting-edge research, relating powerful ideas from statistics, computer science, biology, and mathematics, having a wide range of applications, from literature, to linguistics, to visual graphics. This course is designed to introduce fundamental ideas of this subject from a mathematical viewpoint, touching and expanding upon the interests of the enrolled students. [ more ]

Taught by: Satyan Devadoss

Catalog details#### MATH 361 (F)Theory of Computation

This course introduces a formal framework for investigating both the computability and complexity of problems. We study several models of computation including finite automata, regular languages, context-free grammars, and Turing machines. These models provide a mathematical basis for the study of computability theory--the examination of what problems can be solved and what problems cannot be solved--and the study of complexity theory--the examination of how efficiently problems can be solved. Topics include the halting problem and the P versus NP problem. [ more ]

Taught by: Brent Heeringa

Catalog details#### MATH 372 (S)Complex Analysis

The calculus of complex-valued functions turns out to have unexpected simplicity and power. As an example of simplicity, every complex-differentiable function is automatically infinitely differentiable. As examples of power, the so-called "residue calculus" permits the computation of "impossible" integrals, and "conformal mapping" reduces physical problems on very general domains to problems on the round disc. The easiest proof of the Fundamental Theorem of Algebra, not to mention the first proof of the Prime Number Theorem, used complex analysis. [ more ]

Taught by: Cesar Silva

Catalog details#### MATH 373 Investment Mathematics

Not offered this year

Over the years financial instruments have grown from stocks and bonds to numerous derivatives, such as options to buy and sell at future dates under certain conditions. The 1997 Nobel Prize in Economics was awarded to Robert Merton and Myron Schloles for their Black-Scholes model of the value of financial instruments. This course will study deterministic and random models, futures, options, the Black-Scholes Equation, and additional topics. [ more ]

Taught by: Frank Morgan

Catalog details#### MATH 374 T Topology

Not offered this year

Topology is the study of when one geometric object can be continuously deformed and twisted into another object. Determining when two objects are topologically the same is incredibly difficult and is still the subject of a tremendous amount of research, including recent work on the Poincare Conjecture, one of the million-dollar millennium-prize problems. The first part of the course on point-set topology establishes a framework based on "open sets" for studying continuity and compactness in very general spaces. The second part on homotopy theory develops refined methods for determining when objects are the same. We will prove for example that you cannot twist a basketball into a doughnut. [ more ]

Taught by: Frank Morgan

Catalog details#### MATH 378 T (F)Algebraic Geometry

Algebraic Geometry has been at the heart of mathematics for at least two hundred years. While starting with a humble study of circles, it has influenced a tremendous amount of modern mathematics, ranging from number theory to robotics. Algebraic Geometry uses tools from almost all areas of mathematics; key for this course will be abstract algebra and multivariable calculus. We will study conics, cubics (books are written about the geometry of cubics; the depth of ideas involved with these curves is amazing) and higher degree curves. In particular, we will study Bezout's Theorem and Riemann-Roch for curves. Simultaneously with learning about curves, we will also cover the more abstract ideas behind affine and projective varieties. Emphasis will be placed on both "big picture" concepts and the underlying technical details. [ more ]

Taught by: Thomas Garrity

Catalog details#### MATH 389 (F)Advanced Analysis

This course further develops and explores topics and concepts from real analysis, with special emphasis on introducing students to subject matter and techniques that are useful for graduate study in mathematics or an allied field. Material will be drawn, based on student interest, from many areas, including analytic number theory, Fourier series and harmonic analysis, generating functions, differential equations and special functions, integral operators, equidistribution theory and probability, random matrix theory and probabilistic methods. This will be an intense, fast paced class which will give a flavor for graduate school. In addition to standard homework problems, students will also write reviews for MathSciNet, referee papers for journals, write programs in SAGE or Mathematica to investigate and conjecture, and read classic and current research papers. [ more ]

Taught by: Steven Miller

Catalog details#### MATH 394 T Galois Theory and Modules

Not offered this year

In the 1830's Evariste Galois developed a beautiful theory relating the structure of field extensions to the structure of a group. By understanding this relationship, one can often translate a problem about field extensions to a question about groups that is easier to answer. In this course, we will study Galois Theory and modules. A module is a generalization of vector spaces; in particular, a module can be thought of as a vector space with the weaker condition that the set of scalars are elements of a ring instead of a field. Possible topics covered will include field theory, galois theory, quotient modules, direct sums, free modules, and exact sequences. [ more ]

Taught by: Susan Loepp

Catalog details#### MATH 397 (F)Independent Study: Mathematics

Directed 300-level independent study in Mathematics. [ more ]

Taught by: Stewart Johnson

Catalog details#### MATH 398 (S)Independent Study: Mathematics

Directed 300-levelindependent study in Mathematics. [ more ]

Taught by: Stewart Johnson

Catalog details#### MATH 402 (F)Measure Theory and Probability

The study of measure theory arose from the study of stochastic (probabilistic) systems. Applications of measure theory lie in biology, chemistry, physics as well as in economics. In this course, we develop the abstract concepts of measure theory and ground them in probability spaces. Included will be Lebesgue and Borel measures, measurable functions (random variables). Lebesgue integration, distributions, independence, convergence and limit theorems. This material provides good preparation for graduate studies in mathematics, statistics and economics. [ more ]

Taught by: Cesar Silva

Catalog details#### MATH 404 Ergodic Theory

Not offered this year

Ergodic theory studies the probabilistic behavior of dynamical systems as they evolve through time. This course will be an introduction to the basic notions in ergodic theory. The course starts with an introduction to measure theory: (sigma-algebras, measurable sets and measurable transformations and Lebesgue integration). Then we will cover ergodic, weak mixing, mixing, and Bernoulli transformations, and transformations admitting and not admitting an invariant measure. There will be an emphasis on specific examples such as group rotations, the binary odometer transformations, and rank-one constructions. We will aslo cover some notions from topological dynamics. For the textbook: http://www.ams.org/bookstore-getitem/item=STML-42 [ more ]

Taught by: Cesar Silva

Catalog details#### MATH 411 (F)Commutative Algebra

Commutative algebra has applications ranging from algebraic geometry to coding theory. For example, one can use commutative algebra to create error correcting codes. It is perhaps most often used, however, to study curves and surfaces in different spaces. To understand these structures, one must study polynomial rings over fields. This course will be an introduction to commutative algebra. Possible topics include polynomial rings, localizations, primary decomposition, completions, and modules. [ more ]

Taught by: Susan Loepp

Catalog details#### MATH 416 Advanced Applied Linear Algebra

Not offered this year

In the first N math classes of your career, it's possible to get an incomplete picture as to what the real world is truly like. How? You're often given exact problems and told to find exact solutions. The real world is sadly far more complicated. Frequently we cannot exactly solve problems; moreover, the problems we try to solve are themselves merely approximations to the world. We're forced to develop techniques to approximate not just solutions, but even the statement of the problem. In this course we discuss some powerful methods from advanced linear algebra and their applications to the real world, specifically linear programming (and, if time permits, random matrix theory). Linear programming is used to attack a variety of problems, from applied ones such as the traveling salesman problem, determining schedules for major league sports (or a movie theater, or an airline) to designing efficient diets to feed the world, to pure ones such as Hales' proof of the Kepler conjecture. [ more ]

Taught by: Steven Miller

Catalog details#### MATH 419 Algebraic Number Theory

Not offered this year

We all know that integers can be factored into prime numbers and that this factorization is essentially unique. In more general settings, it often still makes sense to factor numbers into "primes," but the factorization is not necessarily unique! This surprising fact was the downfall of Lame's attempted proof of Fermat's Last Theorem in 1847. Although a valid proof was not discovered until over 150 years later, this error gave rise to a new branch of mathematics: algebraic number theory. In this course, we will study factorization and other number-theoretic notions in more abstract algebraic settings, and we will see a beautiful interplay between groups, rings, and fields. [ more ]

Taught by: Allison Pacelli

Catalog details#### MATH 425 Soap Bubbles and Geometric Measure Theory

Not offered this year

A single round soap bubble is the least-area way to enclose a given volume of air, as ws proved in 1884 by Schwarz. A double soap bubble is the least-area way to enclose and separate two given volumes of air, as was proved in 2000 as the culmination of a decade of work by many, including Williams faculty and students. Because it is hard to control ahead of time the complicated ways ("singularities") in which pieces of soap film theoretically might come together, the study of such physical problems had to wait for the development of a more general and inclusive kind of geometry, now known as Geometric Measure Theory. (These same tools can be applied to all kinds of singularities from fractures in materials to black holes in the universe. [ more ]

Taught by: Frank Morgan

Catalog details#### MATH 427 Tiling Theory

Not offered this year

Since humankind first utilized stones and bricks to tile the floors of their abodes, tiling has been an area of interest. Practitioners include artists, engineers, designers, architects, crystallographers, scientists and mathematicians. This course will be an investigation into the mathematical theory of tiling. The course will focus on tilings of the plane, including topics such as the symmetry groups of tilings, the topology of tilings, the ergodic theory of tilings, the classification of tilings and the aperiodic Penrose tilings. We will also look at tilings in higher dimensions, including "knotted tilings". [ more ]

Taught by: Colin Adams

Catalog details#### MATH 432 Lie Algebras

Not offered this year

A Lie algebra is a vector space endowed with a multiplication operation known as a bracket. They have applications to a wide variety of mathematical fields such as geometry, representation theory, combinatorics, and mathematical physics. This course will cover the basic theory of Lie algebras, including solvable and nilpotent Lie algebras, Cartan subalgebras, the Killing form, root systems, the Weyl group, Dynkin diagrams, and Cartan matrices. Special attention will be paid to examples that highlight the importance of Lie algebras in modern mathematics. [ more ]

Taught by: Edward Hanson

Catalog details#### MATH 433 (S)Mathematical Modeling

Mathematical modeling is concerned with translating a natural phenomenon into a mathematical form. In this abstract form the underlying principles of the phenomenon can be carefully examined and real-world behavior can be interpreted in terms of mathematical shapes. The models we investigate include feedback phenomena, phase locked oscillators, multiple population dynamics, reaction-diffusion equations, shock waves, and the spread of pollution, forest fires, and diseases. We will employ tools from the fields of differential equations and dynamical systems. The course is intended for students in the mathematical, physical, and chemical sciences, as well as for students who are seriously interested in the mathematical aspects of physiology, economics, geology, biology, and environmental studies. [ more ]

Taught by: Julie Blackwood

Catalog details#### MATH 436 T Chaos and Fractals

Not offered this year

This course is an introduction to chaotic dynamical systems. The topics will include bifurcations, the quadratic family, symbolic dynamics, chaos, dynamics of linear systems, and some complex dynamics. [ more ]

Taught by: Cesar Silva

Catalog details#### MATH 437 Electricity and Magnetism for Mathematicians

Not offered this year

Maxwell's equations are four simple formulas, linking electricity and magnetism, that are among the most profound equations ever discovered. These equations led to the prediction of radio waves, to the realization that a description of light is also contained in these equations and to the discovery of the special theory of relativity. In fact, almost all current descriptions of the fundamental laws of the universe are deep generalizations of Maxwell's equations. Perhaps even more surprising is that these equations and their generalizations have led to some of the most important mathematical discoveries (where there is no obvious physics) of the last 25 years. For example, much of the math world was shocked at how these physics generalizations became one of the main tools in geometry from the 1980s until today. It seems that the mathematics behind Maxwell is endless. This will be an introduction to Maxwell's equations, from the perspective of a mathematician. [ more ]

Taught by: Thomas Garrity

Catalog details#### MATH 453 (S)Partial Differential Equations

Partial differential equations are often used to model the most basic natural phenomena. Examples include the flow of liquids, the spread of heat and the radiation of electromagnetic waves. These type of equations have lead to advances such as the prediction of radio waves, the discovery of the special theory of relativity and are essential to the theory of quantum mechanics. In this course we will introduce the theory of partial differential equations. A special focus will be on three classical equations: the wave equation, the Laplace equation and the heat equation. Classical techniques and theorems will be covered such as the Method of Characteristics, the Cauchy-Kovalevski Theorem and Fourier Transform techniques. [ more ]

Taught by: Eyvindur Palsson

Catalog details#### MATH 479 (S)Additive Combinatorics

Lying at the interface of combinatorics, ergodic theory, harmonic analysis, number theory, and probability, Additive Combinatorics is an exciting field which has experienced tremendous growth in recent years. Very roughly, it is an attempt to classify subsets of a given field which are almost a subspace. We will discuss a variety of topics, including sum-product theorems, the structure of sets of small doubling (e.g. the Freiman-Ruzsa theorem), long arithmetic progressions (e.g. Roth's theorem), structured subsets of sumsets, and applications to computer science (e.g. to pseudorandomess). Depending on time and interest, we may also discuss higher-order Fourier analysis, the polynomial method, and the ergodic approach to Szemeredi's theorem. [ more ]

Taught by: Leo Goldmakher

Catalog details#### MATH 493 (F)Senior Honors Thesis: Mathematics

Mathematics senior honors thesis. Each student carries out an individual research project under the direction of a faculty member that culminates in a thesis. See description under *The Degree with Honors in Mathematics*. [ more ]

Taught by: Stewart Johnson

Catalog details#### MATH 494 (S)Senior Honors Thesis: Mathematics

Mathematics senior honors thesis. Each student carries out an individual research project under the direction of a faculty member that culminates in a thesis. See description under *The Degree with Honors in Mathematics*. [ more ]

Taught by: Stewart Johnson

Catalog details#### MATH 497 (F)Independent Study: Mathematics

Directed 400-level independent study in Mathematics. [ more ]

Taught by: Stewart Johnson

Catalog details#### MATH 498 (S)Independent Study: Mathematics

Directed 400-level independent study in Mathematics. [ more ]

Taught by: Stewart Johnson

Catalog details#### MATH 499 (F, S)Senior Colloquium

Mathematics senior colloquium. Meets every week for two hours both fall and spring. Senior majors must participate at least one hour a week. This colloquium is in addition to the regular four semester-courses taken by all students. [ more ]

Taught by: Stewart Johnson

Catalog details## Statistics Courses

#### STAT 101 (F, S)Elementary Statistics and Data Analysis

It is nearly impossible to live in the world today without being inundated with data. Even the most popular newspapers feature statistics to catch the eye of the passerby, and sports broadcasters overwhelm the listener with arcane statistics. How do we learn to recognize dishonest or even unintentionally distorted representations of quantitative information? How are we to reconcile two medical studies with seemingly contradictory conclusions? How many observations do we need in order to make a decision? It is the purpose of this course to develop an appreciation for and an understanding of the interpretation of data. We will become familiar with the standard tools of statistical inference including the t-test, the analysis of variance, and regression, as well as exploratory data techniques. Applications will come from the real world that we all live in. [ more ]

Taught by: Brianna Heggeseth, Stewart Johnson

Catalog details#### STAT 201 (F, S)Statistics and Data Analysis

Statistics can be viewed as the art (science?) of turning data into information. Real world decision-making, whether in business or science is often based on data and the perceived information it contains. Sherlock Holmes, when prematurely asked the merits of a case by Dr. Watson, snapped back, "Data, data, data! I can't make bricks without clay." In this course, we will study the basic methods by which statisticians attempt to extract information from data. These will include many of the standard tools of statistical inference such as hypothesis testing, confidence intervals, and linear regression as well as exploratory and graphical data analysis techniques. [ more ]

Taught by: Richard De Veaux, Wendy Wang

Catalog details#### STAT 202 (S)Introduction to Statistical Modeling

Data come from a variety of sources sometimes from planned experiments or designed surveys, but also arise by much less organized means. In this course we'll explore the kinds of models and predictions that we can make from both kinds of data as well as design aspects of collecting data. We'll focus on model building, especially multiple regression, and talk about its potential as well as its limits to answer questions about the world. We\'ll emphasize applications over theory and analyze real data sets throughout the course. [ more ]

Taught by: Brianna Heggeseth

Catalog details#### STAT 231 T Statistical Design of Experiments

Not offered this year

What does statistics have to do with designing and carrying out experiments? The answer is, surprisingly perhaps, a great deal. In this course, we will study how to design an experiment with the fewest number of observations possible to achieve a certain power. We will also learn how to analyze and present the resulting data and draw conclusions. After reviewing basic statistical theory and two sample comparisons, we cover one and two-way ANOVA and (fractional) factorial designs extensively. The culmination of the course will be a project where each student designs, carries out, analyzes, and presents an experiment of interest to him or her. Throughout the course, we will use the free statistical software program R to carry out the statistical analysis. [ more ]

Taught by: Andrey Glubokov

Catalog details#### STAT 317 (F)Introduction to Operations Research

In the first N math classes of your career, you can be misled as to what the world is truly like. How? You're given exact problems and told to find exact solutions. The real world is sadly far more complicated. Frequently we cannot exactly solve problems; moreover, the problems we try to solve are themselves merely approximations to the world! We are forced to develop techniques to approximate not just solutions, but even the statement of the problem. Additionally, we often need the solutions quickly. Operations Research, which was born as a discipline during the tumultuous events of World War II, deals with efficiently finding optimal solutions. In this course we build analytic and programming techniques to efficiently tackle many problems. We will review many algorithms from earlier in your mathematical or CS career, with special attention now given to analyzing their run-time and seeing how they can be improved; students will be implementing many of these algorithms on computer systems of their choice. The culmination of the course is a development of linear programming and an exploration of what it can do and what are its limitations. For those wishing to take this as a Stats course, the final project must have a substantial implementation computation (respectively, statistics) component approved by the instructor. [ more ]

Taught by: Steven Miller

Catalog details#### STAT 341 (S)Probability

While probability began with a study of games, it has grown to become a discipline with numerous applications throughout mathematics and the sciences. Drawing on gaming examples for motivation, this course will present axiomatic and mathematical aspects of probability. Included will be discussions of random variables, expectation, independence, laws of large numbers, and the Central Limit Theorem. Many interesting and important applications will also be presented, potentially including some from coding theory, number theory and nuclear physics. [ more ]

Taught by: Steven Miller

Catalog details#### STAT 341 Bayesian Statistics

Not offered this year

The probability of an event can be defined in two ways: (1) the long-run frequency of the event, or (2) the belief that the event will occur. Classical statistical inference is built on the first definition given above, while Bayesian statistical inference is built on the second. This course will introduce the student to methods in Bayesian statistics. Topics covered include: prior distributions, posterior distributions, conjugacy, and Bayesian inference in single-parameter, multi-parameter, and hierarchical models. The computational issues associated with each of these topics will also be discussed. [ more ]

Taught by: Wendy Wang

Catalog details#### STAT 346 (F)Regression and Forecasting

This course focuses on the building of empirical models through data in order to predict, explain, and interpret scientific phenomena. The main focus will be on multiple regression as a technique for doing this. We will study both the mathematics of regression analysis and its applications, including a discussion of the limits to such analyses. The applications will range from a broad range of disciplines, such as predicting the waiting time between eruptions of the Old Faithful geyser, forecasting housing prices or modeling the probability of failure of a scientific experiment. [ more ]

Taught by: Wendy Wang

Catalog details#### STAT 355 Multivariate Statistical Analysis

Not offered this year

In elementary statistics courses, one typically studies how to analyze data and make inferences when only one population variable is of interest. But what if one wanted to make inferences about more than one variable in the population? In such cases, elementary statistical methods might not apply. In this course, we study the tools and the intuition that are necessary to analyze and describe such data sets. Specific topics covered will include the multivariate normal distribution, multivariate analysis of variance, principal component analysis, factor analysis, canonical correlation, and clustering. [ more ]

Taught by: Carsten Botts

Catalog details#### STAT 360 (F)Statistical Inference

This course will introduce students to advanced mathematical concepts and techniques for a deeper understanding of statistical inference. Many topics from STAT 201 such as random variables, the central limit theorem or how to test and estimate unknown parameters will be revisited and put on a more rigorous footing. In addition, emphasis will be placed on simulation and resampling (e.g., permutation and bootstrap) approaches to statistical inference and implemented with the statistical software R. [ more ]

Taught by: Richard De Veaux

Catalog details#### STAT 372 (S)Longitudinal Data Analysis: Modeling Change over Time

This course explores modern statistical methods for drawing scientific inferences from longitudinal data, i.e., data collected repeatedly on experimental units over time. The independence assumption made for most classical statistical methods does not hold with this data structure because we have multiple measurements on each individual. Topics will include linear and generalized linear models for correlated data, including marginal and random effect models, as well as computational issues and methods for fitting these models. We will consider many applications in the social and biological sciences. [ more ]

Taught by: Brianna Heggeseth

Catalog details#### STAT 397 (F)Independent Study: Statistics

Directed independent study in Statistics. [ more ]

Taught by: Stewart Johnson

Catalog details#### STAT 398 (S)Independent Study: Statistics

Directed independent study in Statistics. [ more ]

Taught by: Stewart Johnson

Catalog details#### STAT 440 Categorical Data Analysis

Not offered this year

This course focuses on methods for analyzing categorical response data. In contrast to continuous data, categorical data consist of observations classified into two or more categories. Traditional tools of statistical data analysis are not designed to handle such data and pose inappropriate assumptions. We will develop methods specifically designed to address the discrete nature of the observations and consider many applications in the social and biological sciences as well as in medicine, engineering and economics. All methods can be viewed as extensions of traditional regression models and ANOVA. [ more ]

Taught by: Bernhard Klingenberg

Catalog details#### STAT 442 (S)Computational Statistics and Data Mining

In both science and industry today, the ability to collect and store data can outpace our ability to analyze it. Traditional techniques in statistics are often unable to cope with the size and complexity of today's data bases and data warehouses. New methodologies in Statistics have recently been developed, designed to address these inadequacies, emphasizing visualization, exploration and empirical model building at the expense of traditional hypothesis testing. In this course we will examine these new techniques and apply them to a variety of real data sets. [ more ]

Taught by: Richard De Veaux

Catalog details#### STAT 462 (S)Modern Nonparametric Statistics

Many statistical procedures and tools are based on a set of assumptions, such as normality. But, what if some or all of these assumptions are not valid? This question leads to the consideration of distribution-free analysis, an active and fascinating field in modern statistics called nonparametric statistics. In this course we aim to make inference for population characteristics while making as few assumptions as possible. Besides the classical rank or randomization-based tests, this course especially focuses on various modern nonparametric inferential techniques, such as nonparametric density estimation, nonparametric regression, selection of smoothing parameter (cross validation and unbiased risk estimation), bootstrap and jackknife, and Minimax theory. Throughout the semester we will examine these new methodologies and apply them on simulated and real data sets using R. [ more ]

Taught by: Wendy Wang

Catalog details#### STAT 493 (F)Senior Thesis: Statistics

Each student carries out an individual research project under the direction of a faculty member that culminates in a thesis. See description under *The Degree with Honors in Mathematics.* [ more ]

Taught by: Stewart Johnson

Catalog details#### STAT 494 (S)Senior Thesis: Statistics

Each student carries out an individual research project under the direction of a faculty member that culminates in a thesis. See description under *The Degree with Honors in Mathematics.* [ more ]

Taught by: Stewart Johnson

Catalog details#### STAT 497 (F)Independent Study: Statistics

Directed independent study in Statistics. [ more ]

Taught by: Stewart Johnson

Catalog details#### STAT 498 (S)Independent Study: Statistics

Directed independent study in Statistics. [ more ]

Taught by: Stewart Johnson

Catalog details