# Courses

## Course Catalog

2020-2021:

### Course Descriptions

• #### MATH 102 T(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 110Logic and Likelihood

Last offered Fall 2018

How best can we reason in the face of uncertainty? We will begin with an examination of rationality and the reasoning process including a survey of formal logic. Starting with uncertainty from a psychological and philosophical viewpoint, we will move to a careful theory of likelihood and how to reason with probabilistic models. The course will conclude with a consideration of observation and information, how to test hypotheses, and how we update our beliefs to incorporate new evidence. [ more ]

#### MATH 113The Beauty of Numbers

Last offered Spring 2020

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 115Mathematical Politics: Voting, Power, and Conflict

Last offered Spring 2011

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 119(F)The Mathematics of Pandemics: From the Spread of Infections to Cost-Benefit Analyses of Responses

The goal of the class is to help students learn to ask the right questions, and to gather and analyze the data needed to answer them, to understand the covid pandemic and the worldwide responses. Through local experts and numerous guest speakers playing key roles in these problems, we will discuss numerous aspects, from mathematical models for virus propagation to analyzing the economic, educational, social and emotional consequences of lockdowns and social distancing; from moral and legal dilemnas created by the pandemic and responses to the international political scene and relations between countries. Offered as Math 119 or Math 312 (those taking as Math 312 will have some of the readings replaced with more technical modeling papers and subsequent homework). Pre-requisites: None for Math 119; for Math 312 linear algebra is recommended. [ more ]

#### MATH 120The Art of Mathematical Thinking: An Introduction to the Beauty and Power of Mathematical Ideas

Last offered Fall 2009

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 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 ]

#### 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 ]

#### 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. Note: This course will be taught via flipped-course method, an instructional strategy reversing the traditional learning environment by delivering instructional content outside of the classroom. This includes prerecorded lectures along with questions that students must watch and answer prior to attending class. Class time include synchronous meetings clarifying concepts and working in small groups through challenging problems with the support of the professor and peers. Building positive collaborative working relationships and public speaking skills will be added benefits of this course. [ more ]

#### 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 200(F, S)Discrete Mathematics

This course will help you understand the world through a mathematical lens and will develop your powers of argumentation and critical thinking. We will explore and utilize diverse areas of discrete mathematics including logic, set theory, functions and relations, combinatorics, probability, networks, and more. We also will discuss methods and styles of mathematical proofs in order to prepare you for more advanced math courses. Finally, while mathematical knowledge is often perceived as being "pure," we will highlight some ways in which it is socially constructed and hence subject to human limitations and biases. [ more ]

#### MATH 209Differential Equations

Last offered Spring 2016

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 ]

#### MATH 285 TMathematics Education

Last offered Spring 2015

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 293 TUndergraduate Research Topics in Representation Theory

Last offered Fall 2016

Central to the study of the representation theory of Lie algebras is the computation of weight multiplicities by using Kostant's weight multiplicity formula. This formula is an alternating sum over a finite group, and involves a partition function. In this tutorial, we will address questions regarding the number of terms contributing nontrivially to the sum and develop closed formulas for the value of the partition function. Techniques used include generating functions and counting arguments, which are at the heart of combinatorics and are accessible to undergraduate students. [ more ]

#### MATH 306Fractals and Chaos

Last offered Spring 2018

Early in the course we introduce the notion of dynamical systems. Then we will develop the mathematics behind iterated function systems and study the notions of fractals and chaos. There will be a lot of computer experimentation with various programs and resources which the students are expected to use to learn and discover properties of fractals. The final topics will include dimension complex dynamics and the Mandelbrot set. [ more ]

#### MATH 307(S)Computational Linear Algebra

Linear algebra is of central importance in the quantitative sciences, including application areas such as image and signal processing, data mining, computational finance, structural biology, and much more. When the problems must be solved computationally, approximation, round-off errors, convergence, and efficiency matter, and traditional linear algebra techniques may fail to succeed. We will adopt linear algebra techniques on a large scale, implement them computationally, and apply them to core problems in scientific computing. Topics may include: systems of linear and nonlinear equations; approximation and statistical function estimation; optimization; interpolation; data scraping; singular value decomposition; and more. This course could also be considered a course in numerical analysis or computational science. [ more ]

#### MATH 309(F)Differential Equations

Ordinary differential equations (ODE) frequently arise as models of phenomena in the natural and social sciences. This course presents core ideas of ODE from an applied standpoint. Topics covered early in the course may include numerical solutions, separation of variables, integrating factors, constant coefficient linear equations, and power series solutions. Later, we will focus on nonlinear ODEs, for which it is usually impossible to find analytical solutions. Tools from dynamical systems will be introduced to allow us to obtain some information about the behavior of the ODE without explicitly knowing the solution. [ more ]

#### MATH 310(F)Mathematical Biology

This course will provide an introduction to the many ways in which mathematics can be used to understand, analyze, and predict biological dynamics. We will learn how to construct mathematical models that capture essential properties of biological processes while maintaining analytic tractability. Analytic techniques, such as stability and bifurcation analysis, will be introduced in the context of both continuous and discrete time models. Additionally, students will couple these analytic tools with numerical simulation to gain a more global picture of the biological dynamics. Possible biological applications include, but are not limited to, single and multi-species population dynamics, neural and biological oscillators, tumor cell growth, and infectious disease dynamics. [ more ]

#### MATH 312(F)The Mathematics of Pandemics: From the Spread of Infections to Cost-Benefit Analyses of Responses

The goal of the class is to help students learn to ask the right questions, and to gather and analyze the data needed to answer them, to understand the covid pandemic and the worldwide responses. Through local experts and numerous guest speakers playing key roles in these problems, we will discuss numerous aspects, from mathematical models for virus propagation to analyzing the economic, educational, social and emotional consequences of lockdowns and social distancing; from moral and legal dilemnas created by the pandemic and responses to the international political scene and relations between countries. Offered as Math 119 or Math 312 (those taking as Math 312 will have some of the readings replaced with more technical modeling papers and subsequent homework). Pre-requisites: None for Math 119; for Math 312 linear algebra is recommended. [ more ]

#### MATH 313(S)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 numbers and primes in more general settings, and consider fascinating questions that are simple to understand, but can be quite difficult to answer. This course will be held virtually using an active learning method, an instructional strategy reversing the traditional learning environment by supplying instructional content outside of class time. This will include reading the textbook and completing problem sets prior to attending class. Class time will be spent clarifying concepts and working in small groups through challenging problems with the support of the professor, teaching assistants, and your peers. Building positive collaborative working relationships and public speaking skills will be added benefits of this class. [ more ]

#### MATH 314Cryptography

Last offered Spring 2020

An introduction to the techniques and practices used to keep secrets over non-secure lines of communication, including classical cryptosystems, the data encryption standard, the RSA algorithm, discrete logarithms, hash functions, and digital signatures. In addition to the specific material, there will also be an emphasis on strengthening mathematical problem solving skills, technical reading, and mathematical communication. [ more ]

#### MATH 315 T(S)Methods for Solving Diophantine Equations

A Diophantine equation is an equation with integer (or rational) coefficients that is to be solved in integers (or rational numbers). A focus of study for hundreds of years, Diophantine analysis remains a vibrant area of research. It has yielded a multitude of beautiful results and has wide ranging applications in other areas of mathematics, in cryptography, and in the natural sciences. In this project-based tutorial, we will focus on studying and implementing various methods for solving previously unsolved infinite families of Diophantine equations. Depending on their interests, students may choose one or several methods to apply to open problems in the field. Please note that this tutorial will be held virtually. [ more ]

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

Last offered Spring 2017

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

Last offered Fall 2019

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. 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 stats component approved by the instructor. Prerequisites: Linear Algebra (MATH 250) and one other 200-level or higher CSCI, MATH or STATS course. [ more ]

#### MATH 318 TNumerical Problem Solving

Last offered Fall 2016

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 319(S)Integrative Bioinformatics, Genomics, and Proteomics Lab

What can computational biology teach us about cancer? In this lab-intensive 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, and phylogenetics 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 inflammatory and MAPK signal transduction pathways 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 may also be used to study networks of interacting proteins in colon tumor cells. [ more ]

#### MATH 321Knot Theory

Last offered Spring 2019

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 323Applied Topology

Last offered Spring 2016

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

Last offered Fall 2019

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, incompleteness theorems. At one time, these issues tore at the foundations of mathematics. They are still vital for understanding the nature of mathematical truth. [ more ]

#### MATH 326Differential Geometry

Last offered Spring 2016

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 to the deep Gauss-Bonnet Theorem, which links curvature with topology. [ more ]

#### MATH 327Computational Geometry

Last offered Spring 2013

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 ]

#### MATH 328Combinatorics

Last offered Spring 2020

Combinatorics is a branch of mathematics that focuses on enumerating, examining, and investigating the existence of discrete mathematical structures with certain properties. This course provides an introduction to the fundamental structures and techniques in combinatorics including enumerative methods, generating functions, partition theory, the principle of inclusion and exclusion, and partially ordered sets. [ more ]

#### MATH 329 TDiscrete Geometry

Last offered Spring 2018

Discrete geometry is one of the oldest and most consistently vibrant areas of mathematics, stretching from the Platonic Solids of the ancient Greeks to the modern day applications of convex optimization and linear programming. In this tutorial we will learn about polygons and their higher-dimensional cousins, polyhedra and polytopes, and the various ways to describe, compute, and classify such objects. We will learn how these objects and ideas can be applied to other areas, from computation and optimization to studying areas of math like algebraic geometry. Throughout this course we will be engaging with mathematical work and literature from as old as 500 BCE and as recent as "posted to the internet yesterday." [ more ]

#### MATH 331The little Questions

Last offered Fall 2018

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 333Investment Mathematics

Last offered Fall 2012

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 334(S)Graph Theory

A graph is a collection of vertices, joined together by edges. In this course, we will study the sorts of structures that can be encoded in graphs, along with the properties of those graphs. We'll learn about such classes of graphs as multi-partite, planar, and perfect graphs, and will see applications to such optimization problems as minimum colorings of graphs, maximum matchings in graphs, and network flows. [ more ]

#### MATH 335Game Theory

Last offered Spring 2014

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 337Electricity and Magnetism for Mathematicians

Last offered Fall 2017

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 338Intermediate Logic

Last offered Fall 2019

In this course, we will begin with an in-depth study of the theory of first-order logic. We will first get clear on the formal semantics of first-order logic and various ways of thinking about formal proof: natural deduction systems, semantic tableaux, axiomatic systems and sequent calculi. Our main goal will be to prove things about this logical system rather than to use this system to think about ordinary language arguments. In this way the goal of the course is significantly different from that of Logic and Language (PHIL 203). Students who have take PHIL 203 will have a good background for this class, but students who are generally comfortable with formal systems need not have taken PHIL 203. We will prove soundness and completeness, compactness, the Lowenheim-Skolem theorems, undecidability and other important results about first-order logic. As we go through these results, we will think about the philosophical implications of first-order logic. From there, we will look at extensions of and/or alternatives to first-order logic. Possible additional topics would include: modal logic, the theory of counterfactuals, alternative representations of conditionals, the use of logic in the foundations of arithmetic and Godel's Incompleteness theorems. Student interest will be taken into consideration in deciding what additional topics to cover. [ more ]

#### MATH 340Applications of Mathematics to the Real World

Last offered NA

Often for real world applications one does not need to find the optimal solution, which can be extremely difficult, but instead just find something close, or at least better than what is currently being done. We will develop material and techniques from mathematics, statistics and allied fields with an eye to applications. In addition to standard homework assignments and exams there will be a group project where students will work with a local business, write a report and present the results. Pre-requisites are multivariable calculus and linear algebra, or permission of the instructor. Knowledge of some statistics or programming is beneficial but not required. [ more ]

Taught by: TBA

Catalog details

#### MATH 341(F, S)Probability

The historical roots of probability lie in the study of games of chance. Modern probability, however, is a mathematical discipline that has wide applications in a myriad of other mathematical and physical sciences. Drawing on classical gaming examples for motivation, this course will present axiomatic and mathematical aspects of probability. Included will be discussions of random variables (both discrete and continuous), distribution and expectation, independence, laws of large numbers, and the well-known Central Limit Theorem. Many interesting and important applications will also be presented, including some from classical Poisson processes, random walks and Markov Chains. [ more ]

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

Why is the product of two negative numbers positive? Why do we depict the real numbers as a line? Why is this line continuous, and what does that actually mean? More fundamentally, what is the definition of a real number? Real analysis addresses such questions, delving into the structure of real numbers and functions on them. Along the way we'll discuss sequences and limits, series, completeness, compactness, derivatives and integrals, and metric spaces. Results covered will include the Cantor-Schroeder-Bernstein theorem, the monotone convergence theorem, the Bolzano-Weierstrass theorem, the Cauchy criterion, Dirichlet's and Riemann's rearrangement theorem, the Heine-Borel theorem, the intermediate value theorem, and many others. This course is excellent preparation for graduate studies in mathematics, statistics, and economics. [ more ]

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

Real analysis or the theory of calculus-derivatives, integrals, continuity, convergence--starts with a deeper understanding of real numbers and limits. Applications in the calculus of variations or "infinite-dimensional calculus" include geodesics, harmonic functions, minimal surfaces, Hamilton's action and Lagrange's equations, optimal economic strategies, nonEuclidean geometry, and general relativity. . [ 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 361(F, S)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 368Positive Characteristic Commutative Algebra

Last offered Spring 2018

In commutative algebra, one of the most basic invariants of a ring is its characteristic. This is the smallest multiple of 1 that equals 0. Working over a ring of characteristic zero, versus a ring of characteristic p>0, dramatically changes the proof techniques available to us. This realization has had tremendous consequences in commutative algebra. One of the most useful tools in characteristic p is the Frobenius homomorphism. In this course we will study several standard notions in commutative algebra, such as regularity of a ring, Cohen-Macaulayness, and being normal and we will see how various "splittings" of the Frobenius allow us to easily detect these properties. Many of these methods are not only applicable to commutative algebra, but also to number theory and algebraic geometry. [ more ]

Taught by: Andrew Bydlon

Catalog details

#### MATH 406 TAnalysis and Number Theory

Last offered Fall 2010

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 374(S)Topology

In Real Analysis you learned about metric spaces -- any set of objects endowed with a way of measuring distance -- and the topology of sets in such spaces (open, closed, bounded, etc). In this course we flip this on its head: we explore how to develop analysis (limits, continuity, etc) in spaces where the topology is known but the metric is not. This will lead us to a bizarre and fascinating version of geometry in which we cannot distinguish between shapes that can be continuously deformed into one another. Not only does this theory turn out to be beautiful in the abstract, it has become a vital part of data analysis and is also connected to many areas of math and physics. This course is excellent preparation for graduate programs in mathematics. [ more ]

#### MATH 379Asymptotic Analysis in Differential Equations

Last offered Fall 2016

Asymptotic Analysis is a fascinating subfield of differential equations in which interesting and unexpected phenomena can occur. Roughly speaking, the problem is this: Given a differential equation depending on a parameter epsilon, what happens to the solutions to the equation as we let epsilon go to 0? After an extensive survey of examples, we will cover asymptotic evaluation of integrals, such as stationary phase and Laplace's method, multiple scales, WKB approximations, averaging methods, matched asymptotic expansions, and boundary layers. If time permits, we will also discuss bifurcation theory and the Nash-Moser Inverse Function Theorem. [ more ]

#### MATH 382Harmonic Analysis

Last offered Spring 2016

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 ]

Last offered Fall 2014

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 390Undergraduate Research Topics in Algebra

Last offered NA

The well-known trace map on matrices can be generalized to a map on other algebraic objects. Undergraduates, graduates students and experts in Representation Theory, Commutative Algebra and Algebraic Geometry have been driving recent developments in the theory of trace modules and finding exciting new applications in all of these these fields. This course will serve as an introduction to mathematical research with the aim of producing original research in modern trace theory. Students in this tutorial will read and synthesize research papers, discuss the formation of research questions in pure mathematics, and engage in original mathematical research. [ more ]

Taught by: TBA

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#### MATH 391(F)Introduction to computer algebra

Students will learn new mathematics in the context of computer-based exposition, experimentation, and interaction. They will gain proficiency with Sage, GAP, Macaulay2, or Mathematica, and possibly one of the more-specialized systems SnapPea, kenzo, magma, MATLAB, Perseus, coq, etc. Individuals and teams will build interactive demonstrations of mathematical theorems, which will then be appreciated by the instructor and the rest of the class. No prior programming experience is expected. [ more ]

#### MATH 392 T(S)Undergraduate Research Topics in Graph Theory

Graph theory is a vibrant area of research with many applications to the social sciences, psychology, and economics. In this project-based tutorial, students will select among the presented topics and will develop research questions and undertake original research in the field. Student assessment is based on drafts of research project manuscript and presentations. [ more ]

#### MATH 398Independent Study: Mathematics

Last offered Spring 2020

Directed 300-levelindependent study in Mathematics. [ more ]

#### MATH 401Functional Analysis

Last offered Fall 2015

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 402(F)Measure Theory and Hilbert Spaces

How large is the unit square? One might measure the number of individual points in the square (uncountably infinite), the area of the square (1), or the dimension of the square (2). But what about for more complicated sets, e.g., the set of all rational points in the unit square? What's the area of this set? What's the dimension? In this course we'll come up with precise ways to measure size -- length, area, volume, dimension -- that apply to a broad array of sets. Along the way we'll encounter Lebesgue measure and Lebesgue integration, Hausdorff measure and fractals, space-filling curves and the Banach-Tarski paradox. We will also investigate Hilbert spaces, mathematical objects that combine the tidiness of linear algebra with the power of analysis and are fundamental to the study of differential equations, functional analysis, harmonic analysis, and ergodic theory, and also apply to fields like quantum mechanics and machine learning. This material provides good preparation for graduate studies in mathematics, statistics and economics. [ more ]

#### MATH 403Measure and Ergodic Theory

Last offered Spring 2019

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 404Random Matrix Theory

Last offered Fall 2019

Initiated by research in multivariate statistics and nuclear physics, the study of random matrices is nowadays an active and exciting area of mathematics, with numerous applications to theoretical physics, number theory, functional analysis, optimal control, and finance. Random Matrix Theory provides understanding of various properties (most notably, statistics of eigenvalues) of matrices with random coefficients. This course will provide an introduction to the basic theory of random matrices, starting with a quick review of Linear Algebra and Probability Theory. We will continue with the study of Wigner matrices and prove the celebrated Wigner's Semicircle Law, which brings together important ideas from analysis and combinatorics. After this, we will turn our attention to Gaussian ensembles and investigate the Gaussian Orthogonal Ensemble (GOE) and the Gaussian Unitary Ensemble (GUE). The last lectures of the course will be dedicated to random Schrodinger operators and their spectral properties (in particular, the phenomenon called Anderson localization). Applications of Random Matrix Theory to theoretical physics, number theory, statistics, and finance will be discussed throughout the semester. [ more ]

#### MATH 407Dance of the Primes

Last offered Fall 2018

Prime numbers are the building blocks for all numbers and hence for most of mathematics. Though there are an infinite number of them, how they are spread out among the integers is still quite a mystery. Even more mysterious and surprising is that the current tools for investigating prime numbers involve the study of infinite series. Function theory tells us about the primes. We will be studying one of the most amazing functions known: the Riemann Zeta Function. Finding where this function is equal to zero is the Riemann Hypothesis and is one of the great, if not greatest, open problems in mathematics. Somehow where these zeros occur is linked to the distribution of primes. We will be concerned with why anyone would care about this conjecture. More crassly, why should solving the Riemann Hypothesis be worth one million dollars? (Which is what you will get if you solve it, beyond the eternal fame and glory.) [ more ]

#### MATH 408(F)L-Functions and Sphere Packing

Optimal packing problems arise in many important problems, and have been a source of excellent mathematics for centuries. The Kepler Problem (what is the most efficient way to pack balls in three-space) is a good example. The original formulation has been used in such diverse areas as stacking cannonballs on battleships to grocers preparing fruit displays, and its generalizations allow the creation of powerful error detection and correction codes. While the solution of the Kepler Problem is now known, the higher dimensional version is very much open. There has been remarkable progress in the last few years, with number theory playing a key role in these results. We will develop sufficient background material to understand many of these problems and the current state of the field. Pre-requisites are real analysis. [ more ]

#### MATH 410 TMathematical Ecology

Last offered Spring 2016

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

Last offered Spring 2019

Commutative Algebra is an essential area of mathematics that provides indispensable tools to many areas, including Number Theory and Algebraic Geometry. This course will introduce you to the fundamental concepts for the study of commutative rings, with a special focus on the notion of "prime ideals," and how they generalize the well-known notion of primality in the set of integers. Possible topics include Noetherian rings, primary decomposition, localizations and quotients, height, dimension, basic module theory, and the Krull Altitude Theorem. [ more ]

Taught by: Andrew Bydlon

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#### MATH 416Advanced Applied Linear Algebra

Last offered Fall 2012

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 419Algebraic Number Theory

Last offered Spring 2020

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 Lamé'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(S)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? Precise formulas for these quantities probably don't exist, but over the past 150 years tremendous progress has been made towards understanding these and similar questions using 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 will include some subset of the following: 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, and sieve methods and gaps between primes. [ more ]

#### MATH 421Quandles, Knots and Virtual Knots

Last offered Spring 2018

A quandle is an algebraic object that, like a group, has a "multiplication" of pairs of elements that satisfies certain axioms. But the quandle axioms are very different from the group axioms, and quandles turn out to be incredibly useful when considering the mathematical theory of knots. In this course, we will learn about this relatively new area of research (1982) and learn some knot theory and see how quandles apply to both classical knot theory and the relatively new area of virtual knot theory (1999). [ more ]

#### MATH 422Algebraic Topology

Last offered Fall 2019

Is a sphere really different from a torus? Can a sphere be continuously deformed to a point? Algebraic Topology concerns itself with the classification and study of topological spaces via algebraic methods. The key question is this: How do we really know when two spaces are different and in what senses can we claim they are the same? Our answer will use several algebraic tools such as groups and their normal subgroups. In this course we will develop several notions of "equality" starting with the existence of homeomorphisms between spaces. We will then explore several weakenings of this notion, such as homotopy equivalence, having isomorphic homology or fundamental groups, and having homeomorphic universal covers. [ more ]

Taught by: TBA

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#### MATH 424Geometry, Surfaces and Billiards

Last offered Fall 2016

Mathematical billiards is the study of a ball bouncing around in a table--a rectangle in the popular pub game, but any shape of table for us, including triangles and ellipses. The geometry of billiards is elegant, and is related to surfaces, fractals, and even continued fractions. We will study many types of billiards and surfaces, and take time to explore some beautiful examples and ideas. [ more ]

#### MATH 426 TDifferential Topology

Last offered Fall 2019

Differential topology marries the rubber-like deformations of topology with the computational exactness of calculus. This sub eld of mathematics asks and answers questions like "Can you take an integral on the surface of doughnut?" and includes far-reaching applications in relativity and robotics. This tutorial will provide an elementary and intuitive introduction to differential topology. We will begin with the definition of a manifold and end with a generalized understanding of Stokes Theorem. [ more ]

#### MATH 427Tiling Theory

Last offered Spring 2020

Since humans first used stones and bricks to tile the floors of their domiciles, 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 428Catching Robbers and Spreading Information

Last offered Spring 2020

Cops and robbers is a widely studied game played on graphs that has connections to searching algorithms on networks. The cop number of a graph is the smallest number of cops needed to guarantee that the cops can catch a robber in the graph. Similar combinatorial games such as "zero forcing" can be used to model the spread of information. The idea of "throttling" is to spread the information (or catch the robber) as efficiently as possible. This course will survey some of the main results about cops and robbers and the cop number. We will also explore recent research on throttling for cops and robbers, zero forcing, and other variants. [ more ]

#### MATH 431Nonlinear Waves, Solitons

Last offered Fall 2016

Waves arise in scientific and engineering disciplines such as acoustics, optics, fluid/solid mechanics, electromagnetism and quantum mechanics. Although linear waves are well understood, the study of nonlinear wave phenomena remains an active field of research and a source of inspiration and challenge for several areas of mathematics. We discuss traveling waves, shallow water models, wave steepening, solitons and blowup. Additional topics may include shocks, weak solutions and conservation laws. [ more ]

#### MATH 433(S)Mathematical Modeling

Mathematical modeling means (1) translating a real-life problem into a mathematical object, (2) studying that object using mathematical techniques, and (3) interpreting the results in order to learn something about the real-life problem. Mathematical modeling is used in biology, economics, chemistry, geology, sociology, political science, art, and countless other fields. This is an advanced, seminar-style, course appropriate for students who have strong enthusiasm for applied mathematics, data science, and collaborative teamwork. [ more ]

#### MATH 434(F)Applied Dynamics and Optimal Control

We seek to understand how dynamical systems evolve, how that evolution depends on the various parameters of the system, and how we might manipulate those parameters to optimize an outcome. We will explore the language of dynamics by deepening our understanding of differential and difference equations, study parameter dependence and bifurcations, and explore optimal control through Pontryagin's maximum principle and Hamilton-Jacobi-Bellman equations. These tools have broad application in ecology, economics, finance, and engineering, and we will draw on basic models from these fields to motivate our study. [ more ]

#### MATH 453Partial Differential Equations

Last offered Fall 2017

Partial differential equations (PDE) arise as mathematical models of phenomena in chemistry, ecology, economics, electromagnetics, fluid dynamics, neuroscience, thermodynamics, and more. We introduce PDE models and develop techniques for studying them. Topics include: derivation, classification, and physical interpretation of canonical PDE; solution techniques, including separation of variables, series solutions, integral transforms, and characteristics; and application to problems in the natural and social sciences. [ more ]

#### MATH 456(F)Representation Theory

Representation theory has applications in fields such as physics (via models for elementary particles), engineering (considering symmetries of structures), and even in voting theory (voting for committees in agreeable societies). This course will introduce the concepts and techniques of the representation theory of finite groups, and will focus on the representation theory of the symmetric group. We will undertake this study through a variety of perspectives, including general representation theory, combinatorial algorithms, and symmetric functions. [ more ]

#### MATH 458Algebraic Combinatorics

Last offered Spring 2019

Algebraic combinatorics is a branch of mathematics at the intersection of combinatorics and algebra. On the one hand, we study combinatorial structures using algebraic techniques, while on the other we use combinatorial arguments and methods to solve problems in algebra. This course will focus on the study of symmetric functions, young tableaux, matroids, graph theory, and other related topics. [ more ]

#### MATH 459 TApplied Partial Differential Equations

Last offered Spring 2019

Partial differential equations (PDE) arise as mathematical models of phenomena in chemistry, ecology, economics, electromagnetics, epidemiology, fluid dynamics, neuroscience, and much more. Furthermore, the study of partial differential equations connects with diverse branches of mathematics including analysis, geometry, algebra, and computation. Adopting an applied viewpoint, we develop techniques for studying PDE. We draw from a body of knowledge spanning classic work from the time of Isaac Newton right up to today's cutting edge applied mathematics research. This tutorial is appropriate as a second course in differential equations. In this tutorial, students will: build and utilize PDE-based models; determine the most appropriate tools to apply to a PDE; apply the aforementioned tools; be comfortable with open-ended scientific work; read applied mathematical literature; communicate applied mathematics clearly, precisely, and appropriately; collaborate effectively. [ more ]

Last offered Fall 2017

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, as well as applications in industry. Topics include Benford's law of digit bias, random matrix theory, and Fourier analysis, and as time permits additional areas based on student interest from analytic number theory, generating functions 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 assist in writing both reviews for MathSciNet and referee reports for papers for journals, write programs to investigate and conjecture, and read classic and current research papers, and possibly apply these and related methods to real world problems. [ more ]

#### MATH 474(S)Tropical Geometry

This course offers an introduction to tropical geometry, a young subject that has already established deep connections between itself and pure and applied mathematics. We will study a rich variety of objects arising from polynomials over the min-plus semiring, where addition is defined as taking a minimum, and multiplication is defined as usual addition. We will learn how these polyhedral objects connect to other areas of mathematics like algebraic geometry, and how they can be applied to solve problems in scheduling theory, phylogenetics, and other diverse fields. [ more ]

#### MATH 475Methods in Mathematical Fluid Dynamics

Last offered Spring 2016

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 ]

#### MATH 478On Expressing Numbers

Last offered Spring 2016

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 every 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 ]

Last offered Spring 2015

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 ]

#### MATH 482Homological Algebra

Last offered Fall 2019

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 ]

#### MATH 484Galois Theory

Last offered Spring 2020

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 concept 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: Andrew Bydlon

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#### MATH 485Complex Analysis

Last offered Fall 2019

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: Andrew Bydlon

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#### MATH 487Computational Algebraic Geometry

Last offered Spring 2019

Algebraic geometry is the study of shapes described by polynomial equations. It has been a major part of mathematics for at least the past two hundred years, and has influenced a tremendous amount of modern mathematics, ranging from number theory to robotics. In this course, we will develop the Ideal-Variety Correspondence that ties geometric shapes to abstract algebra, and will use computational tools to explore this theory in a very explicit way. [ more ]

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

Mathematics senior honors thesis; this is part of a full-year thesis (493-494). 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; this is part of a full-year thesis (493-494). 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 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 ]

• #### 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 161(F, S)Introductory Statistics for Social Science

This course will cover the basics of modern statistical analysis with a view toward applications in the social sciences and sciences. Topics include exploratory data analysis, elements of probability theory, basic statistical inference, and introduction to statistical modeling. The course focuses on the application of statistics tools to solve problems, to make decisions, and the use of statistical thinking to understand the world. [ 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 202(F, S)Introduction to Statistical Modeling

Data come from a variety of sources: sometimes from planned experiments or designed surveys, sometimes by 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 to answer questions about the world -- and about its limitations. We'll emphasize applications over theory and analyze real data sets throughout the course. [ more ]

#### STAT 231 TStatistical Design of Experiments

Last offered Fall 2012

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 302(F, S)Applied Statistical Modeling

Data may come from various sources and studies with different purpose of analysis. Statistical modeling provides a unified framework to embrace different data types, and focuses on the goals of understanding relationships, assessing differences and making predictions. We will explore different types of statistical models (linear regression, ANOVA, logistic regression etc), and focus on their conditions, the interactive modeling process, as well as the statistical inference tools for drawing conclusions from them. Throughout the course, real datasets will be modeled for interesting questions about the world, and the limitations will be addressed as well. [ more ]

#### STAT 310(F)Data Visualization

This course is about preparing, visualizing, reporting and presenting different types of data. We will start with creating common plots (e.g., barcharts, histograms, density plots, boxplots, time series and lattice plots), but also discuss visualizing results of statistical models, such as linear or logistic regression models. We will use the ggplot library in R but then switch to the plotly library for interactive graphs with mouse-over and click events. Using R's shiny and DT libraries, we will learn how to create and publish web-apps and dashboards that explore datasets and support online filtering. We will end the class with creating web apps that contain multiple graphs or maps which react to user inputs (such as selecting which variables to plot) or provide real time monitoring of streaming data. Throughout, we will use version control software (Github) to organize and keep track of our code. This course will be taught in a semi-flipped style. While the instructor will introduce certain topics, students will often be responsible for reading material ahead of time and then work individually or in pairs to reproduce material or implement it on their own data. [ more ]

#### STAT 315(S)Applied Machine Learning

How does Netflix recommend films based on your viewing history? How does Facebook group its users and send out targeted ads? How did Google select from thousands of search terms to predict flu? Machine learning (ML) is a rapidly growing field that is concerned with algorithms and models to find patterns in data and solve these practical problems at the intersection between statistics, data science and computer science. This course provides a broad introduction to ideas and methods in machine learning, with emphasis on statistical intuitions and practical data analysis. Topics including regularized regression, SVM, supervised/unsupervised learning, text analysis, neural networks will be covered. Students will use R extensively throughout the course while getting introduced to some ML tools in Python. [ more ]

#### STAT 335(S)Biostatistics and Epidemiology

Epidemiology, a public health discipline, is the study of disease and disability in human populations. Biostatistics focuses on the development and application of statistical methods to address questions that arise in medicine, public health, or biology. This course will begin with epidemiological study designs and core concepts in epidemiology, followed by key statistical methods in public health research. Topics will include multiple regression, analysis of categorical data (two sample methods, sets of 2x2 tables, RxC tables, and logistic regression), survival analysis (Cox proportional hazards model), and a brief introduction to regression with correlated data. [ more ]

#### STAT 341(F, S)Probability

The historical roots of probability lie in the study of games of chance. Modern probability, however, is a mathematical discipline that has wide applications in a myriad of other mathematical and physical sciences. Drawing on classical gaming examples for motivation, this course will present axiomatic and mathematical aspects of probability. Included will be discussions of random variables (both discrete and continuous), distribution and expectation, independence, laws of large numbers, and the well-known Central Limit Theorem. Many interesting and important applications will also be presented, including some from classical Poisson processes, random walks and Markov Chains. [ more ]

#### STAT 342Introduction to Stochastic Processes

Last offered Fall 2019

Stochastic processes are mathematical models for random phenomena evolving in time or space. Examples include the number of people in a queue at time t or the accumulated claims paid by an insurance company in an interval of time t. This course introduces the basic concepts and techniques of stochastic processes used to construct models for a variety of problems of practical interest. The theory of Markov chains will guide our discussion as we cover topics such as martingales, random walks, Poisson process, birth and death processes, and Brownian motion. [ more ]

#### STAT 344(F)Statistical Design of Experiments

How do you get informative research results? By doing the right experiment in the first place. We'll explore the techniques used to plan experiments that are both efficient and statistically sound, the analysis of the resulting data, and the conclusions we can draw from that analysis. We'll look at classical tools like one- and two-way ANOVA and fractional factorial designs, but we'll also look at optimal design, and see how these two frameworks differ in their philosophy and in what they can do. Throughout the course, we'll make extensive use of R to work with real-world data. [ more ]

#### STAT 346(F, S)Regression Theory and Applications

This course focuses on the building of empirical models through data in order to predict, explain, and interpret scientific phenomena. Regression modeling is the most widely used method for analyzing and predicting a response data and for understand the relationship with explanatory variables. This course provides both theoretical and practical training in statistical modeling with particular emphasis on simple linear, logistic 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 356(S)Time Series Analysis

Time series--data collected over time--crop up in applications from economics to engineering to transit. But because the observations are generally not independent, we need special methods to investigate them. This course will include exploratory methods and modeling for time series, including smoothing, ARIMA and state space models, and a foray into the frequency domain. We will emphasize applications to a variety of real data. [ more ]

#### STAT 359Statistical Computing

Last offered Spring 2018

This course introduces a variety of computational and data-centric topics of applied statistics, which are broadly useful for acquiring, manipulating, visualizing, and analyzing data. We begin with the R language, which will be used extensively throughout the course. Then we'll introduce a variety of other useful tools, including the UNIX environment, scripting analyses using bash, databases and the SQL language, alternative data formats, techniques for visualizing high-dimensional data, and text manipulation using regular expressions. We'll also cover some modern statistical techniques along the way, which are made possible thanks to advances in computational power. This course is strongly computer oriented, and assignments will be project-based. [ 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 introductory statistics 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 362Design of Experiments

Last offered Fall 2017

How do you get informative research results? By doing the right experiment in the first place. We'll look at the techniques used to plan experiments that are both efficient and statistically sound, the analysis of the resulting data, and the conclusions we can draw from that analysis. Using a framework of optimal design, we'll examine the theory both of classical designs and of alternatives when those designs aren't appropriate. On the applied side, we'll make extensive use of R to work with real-world data. [ more ]

#### STAT 365Bayesian Statistics

Last offered Fall 2018

The Bayesian approach to statistical inference represents a reversal of traditional (or frequentist) inference, in which data are viewed as being fixed and model parameters as unknown quantities. Interest and application of Bayesian methods have exploded in recent decades, being facilitated by recent advances in computational power. We begin with an introduction to Bayes' Theorem, the theoretical underpinning of Bayesian statistics which dates back to the 1700's, and the concepts of prior and posterior distributions, conjugacy, and closed-form Bayesian inference. Building on this, we introduce modern computational approaches to Bayesian inference, including Markov chain Monte Carlo (MCMC), Metropolis-Hastings sampling, and the theory underlying these simple and powerful methods. Students will become comfortable with modern software tools for MCMC using a variety of applied hierarchical modeling examples, and will use R for all statistical computing. [ more ]

#### STAT 368Modern Nonparametric Statistics

Last offered Spring 2020

Many statistical procedures and tools are based on a set of assumptions, such as normality or other parametric models. But, what if some or all of these assumptions are not valid and the adopted models are miss-specified? This question leads to an active and fascinating field in modern statistics called nonparametric statistics, where few assumptions are made on data's distribution or the model structure to ensure great model flexibility and robustness. In this course, we start with a brief overview of classic rank-based tests (Wilcoxon, K-S test), and focus primarily on modern nonparametric inferential techniques, such as nonparametric density estimation, nonparametric regression, selection of smoothing parameter (cross-validation), bootstrap, randomization-based inference, clustering, and nonparametric Bayes. Throughout the semester we will examine these new methodologies and apply them on simulated and real datasets using R. [ more ]

#### STAT 372(F)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 ]

#### STAT 410Statistical Genetics

Last offered Fall 2019

Genetic studies explore patterns of genetic variation in populations and the effect of genes on diseases or traits. This course provides an introduction to statistical and computational methods for genetic studies. Topics will include Mendelian traits (such as single nucleotide polymorphisms), genome-wide association studies, pathway-based analysis, and methods for population genetics. Students will be introduced to some of the major computational tools for genetic analysis, including PLINK and R/Bioconductor. The necessary background in genetics and biology will be provided alongside the statistical and computational methods. [ more ]

#### STAT 440Categorical Data Analysis

Last offered Fall 2017

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 442(S)Statistical Learning 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 458Spatio-Temporal Data

Last offered Fall 2018

Everything happens somewhere and sometime. But the study of data collected over multiple times and locations requires special methods, due to the dependence structure that relates different observations. In this course, we'll look at exploring, analyzing, and modeling this kind of information--introducing standard methods for purely time-series and purely spatial data, and moving on to methods that incorporate space and time together. Topics will include autocovariance structures, empirical orthogonal functions, and an introduction to Bayesian hierarchical modeling. We'll use R to apply these techniques to real-world datasets. [ more ]

#### STAT 462Modern Nonparametric Statistics

Last offered Spring 2015

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 465(F)Bayesian Statistics

Interest and application of Bayesian methods have exploded in recent decades, being facilitated by recent advances in computational power. Indeed, the Bayesian approach is now recognized across scientific disciples as a modern and powerful tool. We begin with an introduction to Bayes' Theorem, the theoretical underpinning of Bayesian statistics which dates back to the 1700's, and the concepts of prior and posterior distributions, conjugacy, and closed-form Bayesian inference. Building on this, we introduce modern computational approaches to performing Bayesian inference, including Markov chain Monte Carlo (MCMC), Metropolis-Hastings sampling, and the theory underlying these simple and powerful methods, before moving on to multivariate sampling methods and methodology. Students will become comfortable with modern software tools for MCMC using a variety of applied hierarchical modeling examples, and will use R for all statistical computing. The course will culminate in an independent Bayesian research project. [ 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 Statistics. [ 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 Statistics. [ more ]

#### STAT 499(F, S)Statistics Colloquium

Statistics senior colloquium. Meets every week for an hour both fall and spring. Senior statistics majors must participate. This colloquium is in addition to the regular four semester-courses taken by all students. [ more ]