Courses

Course Catalog

2014-2015:

2013-2014:


Course Descriptions

Mathematics | Statistics

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 ]

MATH 113(F)The Beauty of Numbers

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 ]

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: Lauren Childs, Eyvindur Palsson

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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: Lori Pedersen, Cesar Silva

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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: Alejandro Sarria, Mihai Stoiciu

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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 ]

MATH 175Mathematical 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 ]

MATH 180The 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 ]

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, induction, infinity, graph theory, probability, 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 ]

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 ]

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 ]

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: Julie Blackwood, Leo Goldmakher

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MATH 285 TMathematics Education

Not offered this year

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. Anyone interested in this course should contact Prof Pacelli early in the fall semester if possible. [ more ]

MATH 307(S)Mathematical Modeling: Dynamics of Infectious Disease

Mathematical models are an essential tool in the study of infectious diseases and have aided in the development, implementation, and evaluation of public health interventions. This course will introduce key concepts and methods of mathematical modeling of infectious diseases and discuss the role of these models in elucidating the spread of infectious pathogens in dynamic populations. In addition to developing a basis in the theory of mathematical modeling, this course will include exploration of existing models through computer simulation. Students will learn to construct, analyze, and simulate models and interpret results within the biological context. The course will consist of a combination of lecture, discussion and computer lab sessions in R. Familiarity with computer programming is helpful but not necessary. [ more ]

Taught by: Lauren Childs

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MATH 308 TAnalysis 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 ]

MATH 309Introduction 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= -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 ]

MATH 313(F)Introduction to Number Theory

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 ]

MATH 316Protecting Information: Applications of Abstract Algebra and Quantum Physics

Not offered this year

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 ]

MATH 317Introduction to Operations Research

Not offered this year

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 ]

MATH 318 TNumerical 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 ]

MATH 319Integrative Bioinformatics, Genomics, and Proteomics Lab

Not offered this year

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 ]

MATH 321Knot Theory

Not offered this year

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 ]

MATH 323(S)Applied Topology

In topology, one studies properties of an object that are preserved under rubber-like deformations, where one is allowed to twist and pull, but one cannot tear or glue. Hence a sphere is considered the same as a cube, but distinct from the surface of a doughnut. In recent years, topology has found applications in chemistry (knotted DNA molecules), economics (stability theory), Geographic Information Systems, cosmology (the shape of the Universe), medicine (heart failure), robotics and electric circuit design, just to name some of the fields that have been impacted. In this course, we will learn the basics of topology, including point-set topology, geometric topology and algebraic topology, but all with the purpose of applying the theory to a broad array of fields. [ more ]

MATH 325Set 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 ]

MATH 326(S)Differential Geometry

Differential Geometry is the study of curvature. In turn, curvature is the heart of geometry. The goal of this course is to start the study of curvature, concentrating on the curvature of curves and of surfaces, leading the deep Gauss-Bonnet Theorem that links curvature with topology. [ more ]

MATH 327Computational 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

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MATH 331The little Questions

Not offered this year

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 ]

MATH 335Game 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 ]

MATH 341(F)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 ]

MATH 347 TOrigami

Not offered this year

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 ]

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 ]

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: Eyvindur Palsson

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MATH 354Graph 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 ]

MATH 355(F, S)Abstract Algebra

Algebra gives us tools to solve equations. The integers, the rationals, and the real numbers have special properties which make algebra 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 abstract algebraic structures such as groups, rings, and fields, as well as the concepts of factor group, quotient ring, homomorphism, isomorphism, and various types of field extensions. This course introduces students to abstract rigorous mathematics. [ more ]

MATH 357 TPhylogenetics

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 ]

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 ]

MATH 367(S)Homological Algebra

Though a relatively young subfield of mathematics, Homological Algebra has earned its place by supplying powerful tools to solve questions in the much older fields of Commutative Algebra, Algebraic Geometry and Representation Theory. This class will introduce theorems and tools of Homological Algebra, grounding its results in applications to polynomial rings and their quotients. We will focus on some early groundbreaking results and learn some of Homological Algebra's most-used constructions. Possible topics include tensor products, chain complexes, homology, Ext, Tor and Hilbert's Syzygy Theorem. [ more ]

Taught by: Haydee Lindo

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MATH 372(F)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. We will discuss these and other topics as time permits (such as the Riemann Mapping Theorem, Special Functions, and the Central Limit Theorem). [ more ]

MATH 373Investment 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 ]

MATH 374 TTopology

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 ]

MATH 378 TAlgebraic Geometry

Not offered this year

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 ]

MATH 382(S)Harmonic Analysis

Harmonic Analysis is a diverse field which includes Fourier Analysis, one of the major tools of modern mathematics. Applications range from mathematical topics such as partial differential equations and number theory to more applied ones such as signal processing and medical imaging. The course will begin with an introduction to the Fourier Transform and will cover a wide variety of topics including singular integral operators, maximal operators and wavelets as the semester progresses. Along the way applications from partial differential equations and ergodic theory will arise with a highlight being the almost everywhere convergence of Fourier series. [ more ]

Taught by: Eyvindur Palsson

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MATH 389Advanced Analysis

Not offered this year

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 ]

MATH 394(F)Galois Theory and Modules

Some equations -- such as x^5 - 1 = 0 -- are easy to solve. Others -- such as x^5 - x - 1 = 0 -- are very hard, if not impossible (using standard mathematical operations). Galois discovered a deep connection between field theory and group theory that led to a criterion for checking whether or not a given polynomial can be easily solved. His discovery also led to many other breakthroughs, for example proving the impossibility of squaring the circle or trisecting a typical angle using compass and straightedge. From these not-so-humble beginnings, Galois theory has become a fundamental tool in modern mathematics, from topology to number theory. In this course we will develop the theory and explore its applications to other areas of math. [ more ]

Taught by: Leo Goldmakher

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MATH 397(F)Independent Study: Mathematics

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

MATH 398(S)Independent Study: Mathematics

Directed 300-levelindependent study in Mathematics. [ more ]

MATH 401(F)Functional Analysis

Functional analysis can be viewed as linear algebra on infinite-dimensional spaces. It is a central topic in Mathematics, which brings together and extends ideas from analysis, algebra, and geometry. Functional analysis also provides the rigorous mathematical background for several areas of theoretical physics (especially quantum mechanics). We will introduce infinite-dimensional spaces (Banach and Hilbert spaces) and study their properties. These spaces are often spaces of functions (for example, the space of square-integrable functions). We will consider linear operators on Hilbert spaces and investigate their spectral properties. A special attention will be dedicated to various operators arising from mathematical physics, especially the Schrodinger operator. [ more ]

MATH 402Measure Theory and Probability

Not offered this year

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 ]

MATH 403(F)Measure and Ergodic Theory

An introduction to measure theory and ergodic theory. Measure theory is a generalization of the notion of length and area, has been used in the study of stochastic (probabilistic) systems. The course covers the construction of Lebesque and Borel measures, measurable functions, and Lebesque integration. Ergodic theory studies the probabilistic behavior of dynamical systems as they evolve through time, and is based on measure theory. The course will cover basic notions, such as ergodic transformations, 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. The Ergodic Theorem will also be covered, and will be used to illustrate notions and theorems from measure theory. [ more ]

MATH 404Ergodic 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 ]

MATH 410 T(S)Mathematical Ecology

Using mathematics to study natural phenomena has become ubiquitous over the past couple of decades. In this tutorial, we will study mathematical models comprised of both deterministic and stochastic differential equations that are developed to understand ecological dynamics and, in many cases, evaluate the dynamical consequences of policy decisions. We will learn how to understand these models through both standard analytic techniques such as stability and bifurcation analysis as well as through simulation using computer programs such as MATLAB. Possible topics include fisheries management, disease ecology, control of invasive species, and predicting critical transitions in ecological systems. [ more ]

MATH 411Commutative Algebra

Not offered this year

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 ]

MATH 416Advanced 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 ]

MATH 419(S)Algebraic Number Theory

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 ]

MATH 420 T(F)Analytic Number Theory

How many primes are smaller than x? How many divisors does an integer n have? How many different numbers appear in the N x N multiplication table? Over the course of the past 150 years, tremendous progress has been made towards resolving these and similar questions in number theory, relying on tools and methods from analysis. The goal of this tutorial is to explain and motivate the ubiquitous appearance of analysis in modern number theory--a surprising fact, given that analysis is concerned with continuous functions, while number theory is concerned with discrete objects (integers, primes, divisors, etc). Topics to be covered include: asymptotic analysis, partial and Euler-Maclaurin summation, counting divisors and Dirichlet's hyperbola method, the randomness of prime factorization and the Erdos-Kac theorem, the partition function and the saddle point method, the prime number theorem and the Riemann zeta function, primes in arithmetic progressions and Dirichlet L-functions, the Goldbach conjecture and the circle method, gaps between primes, and other topics as time and interest allow. [ more ]

Taught by: Leo Goldmakher

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MATH 425Soap 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 ]

MATH 427Tiling 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 ]

MATH 433Mathematical Modeling

Not offered this year

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 ]

MATH 436 TChaos 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 ]

MATH 437Electricity 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 ]

MATH 453Partial Differential Equations

Not offered this year

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 475(S)Methods in Mathematical Fluid Dynamics

The mathematical study of fluids is an exciting field with applications in areas such as engineering, physics and biology. The applied nature of the subject has led to important developments in aerodynamics and hydrodynamics. From ocean currents and exploding supernovae to weather prediction and even traffic flow, several partial differential equations (pde) have been proposed as models to study fluid phenomena. This course is designed to both, introduce students to some of the techniques used in mathematical fluid dynamics and lay down a foundation for future research in this and other related areas. Briefly, we start with the method of characteristics, a useful tool in the study of pde. Symmetry and geometrical arguments, special solutions, energy methods, particle trajectories, and techniques from ordinary differential equations (ode) are also discussed. A special focus will be on models from hydrodynamics. These include the KdV and the Camasss Holm equations (and generalizations thereof), and the Euler equations of ideal fluids. Mainly, we will be concerned with models whose solutions depend on time and one spatial variable, although depending on student interest and time, we may also investigate higher-dimensional models. [ more ]

Taught by: Alejandro Sarria

Catalog details

MATH 478(S)On Expressing Numbers

The real numbers are overall mysterious. Attempts even to describe different real numbers can quickly lead to deep, open questions in mathematics. For example, writing numbers via their decimal expansions leads to the result that a number is rational precisely when the decimal expansion is eventually periodic. There is an entirely different method for describing real numbers: continued fractions, which go back thousands of years. Here very real number can be captured by a sequence of integers (just like for the decimal expansion) but now eventually periodicity corresponds to the number being a square root. The mathematics of continued fractions, and especially their higher dimensional generalizations, lead to a great deal of mathematics. We will be using tools from linear algebra, functional analysis, dynamical systems, ergodic theory and algebraic number theory to explore the best way to express a real number. [ more ]

MATH 479Additive Combinatorics

Not offered this year

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 ]

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 ]

MATH 497(F)Independent Study: Mathematics

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

MATH 498(S)Independent Study: Mathematics

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

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 ]


Statistics Courses

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

It is impossible to be an informed citizen in the world today without an understanding of data and information. Whether opinion polls, unemployment rates, salary differences between men and women, the efficacy of vaccines or consumer webdata, we need to be able to separate the signal from the noise. We will learn the statistical methods used to analyze and interpret data from a wide variety of sources. The goal of the course is to help reach conclusions and make informed decisions based on data. [ more ]

STAT 201(F, S)Statistics and Data Analysis

Statistics can be viewed as the art and 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. This is an accelerated introductory statistics course that involves computational programming and incorporates modern statistical techniques. [ more ]

STAT 202Introduction to Statistical Modeling

Not offered this year

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 ]

STAT 231 TStatistical 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 ]

STAT 317Introduction to Operations Research

Not offered this year

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 ]

STAT 340(S)Categorical Data Analysis

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 ]

STAT 341Bayesian 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 ]

STAT 341(F)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 ]

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. Regression modeling is the standard method for analyzing continuous response data and their relationship with explanatory variables. This course provides both theoretical and practical training in statistical modeling with particular emphasis on simple linear and multiple regression, using R to develop and diagnose models. The course covers the theory of multiple regression and diagnostics from a linear algebra perspective with emphasis on the practical application of the methods to real data sets. The data sets will be taken from a wide variety of disciplines. [ more ]

STAT 355(F)Multivariate Statistical Analysis

To better understand complex processes, we study how variables are related to one another, and how they work in combination. Therefore, we want to make inferences about more than one variable at time? 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. Topics covered will include data visualization techniques for high dimensional data sets, parametric and non-parametric techniques to estimate joint distributions, techniques for combining variables, as well as classification and clustering algorithms. [ more ]

STAT 360(S)Statistical Inference

How do we estimate unknown parameters and express the uncertainty we have in our estimate? Is there an estimator that works best? Many topics from Stat 201 such as random variables, the central limit theorem, point and interval estimation and hypotheses testing will be revisited and put on a more rigorous mathematical footing. The focus is on maximum likelihood estimators and their properties. Bayesian and computer intensive resampling techniques (e.g., the bootstrap) will also be considered. [ more ]

STAT 372Longitudinal Data Analysis: Modeling Change over Time

Not offered this year

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 ]

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 ]

STAT 462Modern Nonparametric Statistics

Not offered this year

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 ]

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 ]

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 ]