Course Catalog
20242025:
Course Descriptions

MATH 102(F) TUT Foundations in Quantitative Skills
This course will strengthen a student's foundation in quantitative reasoning in preparation for the science curriculum and QFR requirements. The material will be at the college algebra/precalculus level, and covered in a tutorial format with students working in small groups with the professor. Access to this course is limited to placement by a quantitative skills counselor. [ more ]
Taught by: Julie Blackwood
Catalog detailsMATH 110 LEC Logic 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 113(S) LEC The Beauty of Numbers
This course will be an introduction to number theory and mathematical thinking and logic, with emphasis throughout on mathematics as a way of thinking and approaching the world. Have you ever wondered what keeps your credit card information safe every time 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. This course is meant to give you an appreciation for numbers and mathematics and to enhance your logical reasoning skills. Although most people will not use calculus or geometry in their jobs or everyday lives, mathematics enhances our abilities to think logically and reason effectively. This skill is useful in all aspects of life. Number theory, in particular, is a great area of mathematics that allows one to jump in right away without a lot of prerequisite knowledge. We will look at examples, look for patterns, make conjectures, and we will spend a lot of time learning how to rigorously prove those conjectures. [ more ]
Taught by: Allison Pacelli
Catalog detailsMATH 115 LEC Mathematical 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 LEC The Mathematics of Pandemics: From the Spread of Infections to CostBenefit Analyses of Responses
Last offered Fall 2020
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). Prerequisites: None for Math 119; for Math 312 linear algebra is recommended. [ more ]
MATH 120 LEC The 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 metagoals 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) LEC 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 "maxmin" 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: Lori Pedersen, Leo Goldmakher
Catalog detailsMATH 140(F, S) LEC Calculus II
Calculus answers two basic questions: how fast is something changing (the derivative) and how much is there (the integral). This course is about integration. and the miracle that unites the derivative and the integral (the Fundamental Theorem of Calculus.) Understanding calculus requires in part the understanding of methods of integration.This course will also solve equations involving derivatives ("differential equations") for population growth or pollution levels. Exponential and logarithmic functions and trigonometric and inverse functions will also 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) LEC 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: Ralph Morrison, Palak Arora
Catalog detailsMATH 151(F) LEC 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 the theorems of vector calculus. Students with the equivalent of BC 3 or higher should enroll in MATH 151, as well as students who have taken the equivalent of an integral calculus and who have already been exposed to infinite series. For further clarification as to whether MATH 150 or MATH 151 is appropriate, please consult a member of the math/stat department. [ more ]
Taught by: Susan Loepp
Catalog detailsMATH 197 IND Independent Study: Mathematics
Last offered Fall 2023
Directed 100level independent study in Mathematics. [ more ]
Taught by: Cesar Silva
Catalog detailsMATH 198 IND Independent Study: Mathematics
Last offered Spring 2024
Directed 100level independent study in Mathematics. [ more ]
Taught by: Cesar Silva
Catalog detailsMATH 200(F, S) LEC Discrete Mathematics
In contrast to calculus, which is the study of continuous processes, this course examines the structure and properties of finite sets. Topics to be covered include mathematical logic, elementary number theory, mathematical induction, set theory, functions, relations, elementary combinatorics and probability, and graphs. Emphasis will be given on the methods and styles of mathematical proofs, in order to prepare the students for more advanced math courses. [ more ]
Taught by: Palak Arora, Lori Pedersen
Catalog detailsMATH 209 LEC Differential 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) LEC 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. An optional session in Mathematica will be offered for students who are not already familiar with this computational tool. [ more ]
Taught by: Frederick Strauch
Catalog detailsMATH 250(F, S) LEC 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: Christina Athanasouli, Ivo Terek
Catalog detailsMATH 285 TUT Mathematics 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 biweekly 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 29 Math Enrichment in Public Schools
Last offered NA
We will explore math education and enrichment at all levels. Class will include a combination of lecture and discussion in a traditional classroom setting at Williams. Students will also teach lessons at the Williamstown Elementary School math club and Adventures in Learning series throughout the month of January. We will likely have one event with the Mt Greylock high school math club as well. We will present and teach various math enrichment topics including primes, Fibonacci numbers, infinity, the math of voting, fair division, logic, and more.Some proof writing experience is recommended, see instructor if you're not sure you satisfy the prerequisites. [ more ]
Taught by: TBA
Catalog detailsMATH 293 TUT Undergraduate 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 297 IND Independent Study: Mathematics
Last offered Fall 2023
Directed 200level independent study in Mathematics. [ more ]
Taught by: Cesar Silva
Catalog detailsMATH 298 IND Independent Study: Mathematics
Last offered Spring 2024
Directed 200level independent study in Mathematics. [ more ]
Taught by: Cesar Silva
Catalog detailsMATH 303 LEC Introduction to Dynamics, pAdics, and Measure
Last offered Fall 2021
At its most basic level a dynamical system consists of a set of points and a transformation or map acting on the set (i.e., sending points in the set to other points in the set). In this setting we can already ask about the existence, and prevalence, of periodic points (points that come back to themselves). One can also ask about the orbit of a point: the set of points that is obtained as one iteratively applies the transformation the point. An important dynamical notion that comes up here is that of chaos. The course will start by studying basic dynamical systems using notions from calculus. Then we will introduce the padic numbers and use them to study dynamical systems. The course will end with an exploration of the notion of measure and its connection with dynamical systems. [ more ]
MATH 306 LEC Fractals 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 LEC Computational Linear Algebra
Last offered Spring 2023
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, roundoff 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 ]
Taught by: Chad Topaz
Catalog detailsMATH 309(F, S) LEC Differential Equations
Ordinary differential equations (ODEs) frequently arise as models of phenomena in the natural and social sciences. This course presents core ideas of ODEs from an applied standpoint. Topics covered early in the course may include numerical solutions, separation of variables, integrating factors, and constant coefficient linear equations. 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 information about the behavior of the ODEs without explicitly knowing the solution. [ more ]
Taught by: Julie Blackwood, Bhagya Athukorallage
Catalog detailsMATH 311 TUT Advanced topics in applied mathematics
Last offered Fall 2022
Applied mathematics is an expansive field that uses mathematical methods to explore problems that arise in biology, physics, engineering, and many other disciplines. In this course, we will explore a diversity of methods that may include stochastic processes, optimization, signal processing, and numerical analysis. We will also explore how these methods can be utilized to understand questions in other disciplines. [ more ]
Taught by: Julie Blackwood
Catalog detailsMATH 312 LEC The Mathematics of Pandemics: From the Spread of Infections to CostBenefit Analyses of Responses
Last offered Fall 2020
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). Prerequisites: None for Math 119; for Math 312 linear algebra is recommended. [ more ]
MATH 313(S) LEC 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. This course will include a significant focus on mathematical proof writing and problem solving skills. This includes writing clear and rigorous mathematical proofs, clearly explaining mathematical ideas verbally and in writing, determining how to approach certain types of problems, looking for patterns and making conjectures, and asking good questions about the implications of certain ideas and theorems. [ more ]
Taught by: Allison Pacelli
Catalog detailsMATH 314 LEC Cryptography
Last offered Fall 2023
We will discuss some classical ciphers, current assymetric cryptosystems (DES, AES, Rijndael), public key cryptosystems (RSA, DiffieHellman key exchange, ElGamal), and Error Correcting Codes. We will devote a substantial part of the semester covering the necessary mathematical background from number theory and asymptotic analysis. Time permitting, we may also discuss some special topics, such as primality testing (including the polynomialtime AKS algorithm), quantum computers, hash functions, digital signatures, zeroknowledge proofs, information theory, and elliptic curve cryptography. [ more ]
Taught by: Leo Goldmakher
Catalog detailsMATH 315 TUT Methods for Solving Diophantine Equations
Last offered Spring 2021
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 projectbased 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 316 LEC Protecting 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 317 LEC Introduction to Operations Research
Last offered Fall 2022
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 runtime 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 200level or higher CSCI, MATH or STATS course, or permission of the instructor. [ more ]
Taught by: Steven Miller
Catalog detailsMATH 318 TUT Numerical 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 SEM Integrative Bioinformatics, Genomics, and Proteomics Lab
Last offered Spring 2023
What can computational biology teach us about cancer? In this labintensive experience for the Genomics, Proteomics, and Bioinformatics program, computational analysis and wetlab investigations will inform each other, as students majoring in biology, chemistry, computer science, mathematics/statistics, and physics contribute their own expertise to explore how evergrowing gene and protein datasets can provide key insights into human disease. In this course, we will take advantage of one wellstudied system, the highly conserved Rasrelated 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 throughput 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 phosphorylationstate specific antisera to test our hypotheses. Proteomic analysis will introduce the students to de novo structural prediction and threading algorithms, as well as datamining 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 ]
Taught by: Lois Banta
Catalog detailsMATH 321 LEC Knot Theory
Last offered Spring 2024
Take a piece of string, tie a knot in it, and glue the ends together. The result is a knotted circle, known as a knot. For the last 100 years, mathematicians have studied knots, asking such questions as, "Given a nasty tangled knot, how do you tell if it can be untangled without cutting it open?" Some of the most interesting advances in knot theory have occurred in the last ten years.This course is an introduction to the theory of knots. Among other topics, we will cover methods of knot tabulation, surfaces applied to knots, polynomials associated to knots, and relationships between knot theory and chemistry and physics. In addition to learning the theory, we will look at open problems in the field. [ more ]
Taught by: Colin Adams
Catalog detailsMATH 323 LEC Applied Topology
Last offered Spring 2016
In topology, one studies properties of an object that are preserved under rubberlike 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 pointset topology, geometric topology and algebraic topology, but all with the purpose of applying the theory to a broad array of fields. [ more ]
MATH 325 LEC Set Theory
Last offered Fall 2019
Set theory is the traditional foundational language for all of mathematics. We will be discussing the ZermeloFraenkel 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 326(F) LEC 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 to the deep GaussBonnet Theorem, which links curvature with topology. [ more ]
Taught by: Ivo Terek
Catalog detailsMATH 327 LEC Computational 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 328 LEC Combinatorics
Last offered Spring 2024
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 ]
Taught by: Daniel Condon
Catalog detailsMATH 329 TUT Discrete Geometry
Last offered Spring 2023
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 higherdimensional 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 ]
Taught by: Ralph Morrison
Catalog detailsMATH 331(F) LEC The little Questions
Using math competitions such as the Putnam Exam as a springboard, in this class we follow the dictum of the Ross Program and "think deeply of simple things". The two main goals of this course are to prepare students for competitive math competitions, and to get a sense of the mathematical landscape encompassing elementary number theory, combinatorics, graph theory, and group theory (among others). While elementary frequently is not synonymous with easy, we will see many beautiful proofs and "aha" moments in the course of our investigations. Students will be encouraged to explore these topics at levels compatible with their backgrounds. [ more ]
Taught by: Steven Miller
Catalog detailsMATH 332 LEC Topics in Applied Linear Algebra
Last offered Fall 2023
This course focuses on applications of Linear Algebra. We will start with a review of the material covered in Math 250, then move on to more advanced topics and applications. We will cover Singular Value Decomposition (SVD), QR factorization, Cholesky factorization, Least Squares problems, the Taylor approximation, the Regression model, Clustering techniques, as well as Linear Dynamical Systems and some of their applications. [ more ]
Taught by: Palak Arora
Catalog detailsMATH 333 LEC Investment 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 BlackScholes model of the value of financial instruments. This course will study deterministic and random models, futures, options, the BlackScholes Equation, and additional topics. [ more ]
MATH 334(S) LEC 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 multipartite, 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 ]
Taught by: Ralph Morrison
Catalog detailsMATH 335 LEC Decisions, Games, and Evolutionary Dynamics
Last offered Fall 2021
Given goals, options, and uncertainty, how does one make a rational choice? What happens when we interact with others who are also choosing? How might this play out over time? We will first cover the principles of of decision theory including preference, uncertainty, utility, imperfect information, and rational choice. The majority of the course will be spent on the main topics of game theory: sequential games, bimatrix games, parlor games, Nash equilibria, bargaining, repeated games, Bayesian belief, and signaling. Applying these principles to populations that evolve over time through variation, selection, and copying, we will develop basic models of the dynamics of evolution. [ more ]
MATH 337 LEC Electricity 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 338(F) SEM Intermediate Logic
In this course, we will begin with an indepth study of the theory of firstorder logic. We will first get clear on the formal semantics of firstorder 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 LowenheimSkolem theorems, undecidability and other important results about firstorder logic. As we go through these results, we will think about the philosophical implications of firstorder logic. From there, we will look at extensions of and/or alternatives to firstorder 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 ]
Taught by: Keith McPartland
Catalog detailsMATH 340 Applications 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. Prerequisites 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 detailsMATH 341(F, S) LEC 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 wellknown Central Limit Theorem. Many interesting and important applications will also be presented, including some from classical Poisson processes, random walks and Markov Chains. [ more ]
Taught by: Mihai Stoiciu, Steven Miller
Catalog detailsMATH 342 LEC Logic
Last offered Fall 2022
This course will introduce the main ideas and basic results of mathematical logic, and explain their applications to other areas of mathematics and computer science. We will begin with a study of firstorder logic, covering structures and definability, theories, models and categoricity, as well as formal proofs. We will prove GÃ¶del's completeness and compactness theorems and the LowenheimSkolem theorems. The course will briefly dive into computability theory, enough to prove GÃ¶del's Incompleteness theorems and basic undecidability results. [ more ]
Taught by: Jenna Zomback
Catalog detailsMATH 344 LEC The Mathematics of Sports
Last offered Spring 2023
The purpose of this class is to use sports as a springboard to study applications of mathematics, especially in gathering data to build and test models and develop predictive statistics. Examples will be drawn from baseball, basketball, cross country, football, hockey, soccer, track, as well as class choices. Prerequisites are linear algebra (Math 250) and either a 200 level statistics class or a 100 level programming class, or permission of the instructor. [ more ]
Taught by: Steven Miller
Catalog detailsMATH 345(S) LEC Introduction to Numerical Analysis
Numerical analysis is the study of algorithms that use numerical approximation to solve problems which arise in scientific applications. This course provides an introduction to the theory, development, and analysis of algorithms for obtaining numerical solutions. We will also use mathematical software to facilitate numerical experimentation. Topics discussed in the course include: Error Analysis and Convergence Rates of Algorithms; Root Finding for Nonlinear Equations; Approximating Functions; Numerical Differentiation and Integration; Numerical Solutions of Ordinary Differential Equations; Iterative Methods for Solving Linear Systems. [ more ]
Taught by: Christina Athanasouli
Catalog detailsMATH 349 LEC Operations of Order
Last offered Fall 2023
One of the greatest challenges in mathematics is justifying interchanging orders of operations. Most of the time you cannot switch orders. Frequently this is obvious: the square root of a sum is typically not the sum of the square roots; however, there are many important situations where orders can be reversed. The purpose of this class is to highlight some of the difficulties and dangers in such attempts. This will be a writing intensive course, where we work on content for a book that collects counterexamples and theorems in one convenient place while also showcasing the utility of switching orders. We will discuss at great lengths how to do engaging, technical writing, keeping in mind the content and the audience. Students will receive feedback from the professor and probably other professional mathematicians and editors. [ more ]
Taught by: Steven Miller
Catalog detailsMATH 350(F, S) LEC 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 do we mean when we say that? Perhaps most fundamentally, what is a real number? Real analysis addresses such questions, delving into the structure of real numbers and functions of them. Along the way we'll discuss sequences and limits, series, completeness, compactness, derivatives and integrals, and metric spaces. Results covered will include the CantorSchroederBernstein theorem, the monotone convergence theorem, the BolzanoWeierstrass theorem, the Cauchy criterion, Dirichlet's and Riemann's rearrangement theorem, the HeineBorel theorem, the intermediate value theorem, and many others. This course is excellent preparation for graduate studies in mathematics, statistics, and economics. [ more ]
Taught by: Leo Goldmakher, Cesar Silva
Catalog detailsMATH 351(S) LEC Applied Real Analysis
This course is designed to introduce students to the underpinnings of real analysis, primarily in the context of Fourier series. By the end of the semester people will be comfortable making epsilon and delta type arguments. These types of arguments are one of the main pillars of modern mathematics. In a similar way, Fourier series and their generalizations are one of the pillars of the modern digital world. [ more ]
Taught by: Palak Arora
Catalog detailsMATH 355(F, S) LEC 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 ]
Taught by: Allison Pacelli, Mihai Stoiciu
Catalog detailsMATH 361(F) LEC 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, contextfree grammars, and Turing machines. These models provide a mathematical basis for the study of computability theorythe examination of what problems can be solved and what problems cannot be solvedand the study of complexity theorythe examination of how efficiently problems can be solved. Topics include the halting problem and the P versus NP problem. [ more ]
Taught by: Shikha Singh
Catalog detailsMATH 368 LEC Positive 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 p0, 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, CohenMacaulayness, 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 ]
MATH 406 TUT Analysis 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(F) LEC Topology
In Real Analysis you learned about metric spacesany set of objects endowed with a way of measuring distanceand 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 plays an important role in math, physics, and data analysis. This course is excellent preparation for graduate programs in mathematics. [ more ]
Taught by: Leo Goldmakher
Catalog detailsMATH 379 LEC Asymptotic 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 NashMoser Inverse Function Theorem. [ more ]
MATH 382(S) TUT Fourier Analysis
Fourier analysis is the study of waves and frequencies. More precisely, the goal of Fourier analysis is to decompose a complicated function into a simple combination of pure waves, thereby gleaning insight into the behavior of the function itself. It's difficult to overstate the impact of this branch of mathematics; it is foundational throughout theoretical mathematics (e.g., to study the distribution of prime numbers), applied mathematics (e.g., to solve differential equations), physics (e.g., to study properties of light and sound), computer science (e.g., to compute with large integers and matrices), audio engineering (e.g., to pitchcorrecting algorithms), medical science (e.g., throughout radiology), etc. The goal of this course is to cover the basic theory (fourier series, the fourier transform, the fast fourier transform) and explore a number of applications, including Dirichlet's theorem on primes in arithmetic progressions, the isoperimetric inequality, the heat equation, and Heisenberg's uncertainty principle. [ more ]
Taught by: Leo Goldmakher
Catalog detailsMATH 383 LEC Complex Analysis
Last offered Fall 2023
The calculus of complexvalued functions turns out to have unexpected simplicity and power. As an example of simplicity, every complexdifferentiable function is automatically infinitely differentiable. As examples of power, the socalled "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: Steven Miller
Catalog detailsMATH 389(S) LEC Advanced Analysis
This course further develops and explores topics and concepts from real analysis, with special emphasis on introducing students to subject matter and techniques that are useful for graduate study in mathematics or an allied field. Material will be drawn, based on student interest, from many areas, including analytic number theory, Fourier series and harmonic analysis, generating functions, differential equations and special functions, integral operators, equidistribution theory and probability, random matrix theory and probabilistic methods. This will be an intense, fast paced class which will give a flavor for graduate school. In addition to standard homework problems, students will also write reviews for MathSciNet, referee papers for journals, write programs in SAGE or Mathematica to investigate and conjecture, and read classic and current research papers. [ more ]
Taught by: Steven Miller
Catalog detailsMATH 390 Undergraduate Research Topics in Algebra
Last offered NA
The wellknown 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
Catalog detailsMATH 391 LEC Introduction to computer algebra
Last offered Fall 2020
Students will learn new mathematics in the context of computerbased exposition, experimentation, and interaction. They will gain proficiency with Sage, GAP, Macaulay2, or Mathematica, and possibly one of the morespecialized 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 TUT Undergraduate Research Topics in Graph Theory
Last offered Spring 2021
Graph theory is a vibrant area of research with many applications to the social sciences, psychology, and economics. In this projectbased 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 393 SEM Research Topics in Combinatorics
Last offered Spring 2023
Combinatorics provides techniques and tools to enumerate, examine, and investigate the existence of discrete mathematical structures with certain properties. There are numerous areas of applications including algebra, discrete geometry, and number theory. In this projectbased research course students will work in small groups to learn combinatorial techniques and tools in order to develop research questions and begin tackling unsolved problems in combinatorics. [ more ]
Taught by: TBA
Catalog detailsMATH 394 LEC Galois Theory
Last offered Spring 2024
Some equationssuch as x^51=0are easy to solve. Otherssuch as x^5x1=0are very hard, if not impossible (using finite combinations of 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 notsohumble 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: Leo Goldmakher
Catalog detailsMATH 397 IND Independent Study: Mathematics
Last offered Fall 2023
Directed 300level independent study in Mathematics. [ more ]
Taught by: Cesar Silva
Catalog detailsMATH 398 IND Independent Study: Mathematics
Last offered Spring 2024
Directed 300levelindependent study in Mathematics. [ more ]
Taught by: Cesar Silva
Catalog detailsMATH 401 LEC Functional Analysis
Last offered Spring 2022
Functional analysis can be viewed as linear algebra on infinitedimensional 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 infinitedimensional spaces (Banach and Hilbert spaces) and study their properties. These spaces are often spaces of functions (for example, the space of squareintegrable 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 LEC Measure Theory and Hilbert Spaces
Last offered Fall 2020
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, spacefilling curves and the BanachTarski 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 403(F) LEC Measure and Ergodic Theory
An introduction to measure theory and ergodic theory. Measure theory is a generalization of the notion of length and area, and 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, 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 rankone constructions. The Ergodic Theorem will also be covered, and will be used to illustrate notions and theorems from measure theory. [ more ]
Taught by: Cesar Silva
Catalog detailsMATH 404 LEC Random 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 405 LEC Representation Theory and Special Functions
Last offered Spring 2022
Representation theory is at the heart of much of modern mathematics. It provides a link between ideas of symmetries, groups and matrices. It has applications from number theory to Fourier Analysis to elementary particle theory. In part, representation theory is a method for producing interesting functions. While not having a single definition, special functions are "functions that have names.'' Over the last few hundred years, scientists have needed to define and develop certain families of functions, in order to describe different physical phenomena. These families started to be named, and include Bessel functions, Hermite functions, Laguerre functions and more generally hypergeometric functions. In recent years it has been seen that these different types of functions are best understood through the lens of symmetry and in particular via representation theory. This course will be an introduction to representation theory, starting with finite groups, while at the same time being an introduction to special functions. Thus the course will be a mix of abstract algebra, matrices, calculus and analysis. [ more ]
MATH 407 LEC Dance of the Primes
Last offered Fall 2023
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 ]
Taught by: Thomas Garrity
Catalog detailsMATH 408(F) LEC LFunctions 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 threespace) 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. Prerequisites are real analysis. [ more ]
Taught by: TBA
Catalog detailsMATH 409 LEC The Little Questions
Last offered Fall 2022
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 `aha' moments in the course of our investigations. Students will be encouraged to explore these topics at levels compatible with their backgrounds. [ more ]
Taught by: Steven Miller
Catalog detailsMATH 410 TUT Mathematical 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 411(S) LEC Commutative Algebra
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 wellknown notion of primality in the set of integers. Commutative algebra has applications ranging from algebraic geometry to coding theory. For example, one can use commutative algebra to create error correcting codes. It is perhaps most often used, however, to study curves and surfaces in different spaces. To understand these structures, one must study polynomial rings over fields. This course will be an introduction to commutative algebra. Possible topics include polynomial rings, localizations, primary decomposition, completions, and modules. [ more ]
Taught by: Susan Loepp
Catalog detailsMATH 412(S) LEC 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 may include, but are not limited to, single and multispecies population dynamics, neural and biological oscillators, tumor cell growth, and infectious disease dynamics. [ more ]
Taught by: Julie Blackwood
Catalog detailsMATH 413 LEC Computational Algebraic Geometry
Last offered Spring 2023
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 IdealVariety Correspondence that ties geometric shapes to abstract algebra, and will use computational tools to explore this theory in a very explicit way. [ more ]
Taught by: Ralph Morrison
Catalog detailsMATH 415 LEC Advanced Matrix Analysis
Last offered Fall 2023
This course will start with a review of various attributes of matrices (determinants, rank, etc), as well as eigenvalues, eigenvectors, and their properties. Then we will move on to study special matrices and their decompositions, along with similarities, and Jordan canonical forms. In the third segment, we will define norms on vectors and matrices and study their analytic properties. Finally, we will discuss another important class of matrices  positive definite and semidefinite matrices. If time permits, we will also cover positive and negative matrices and their properties. [ more ]
Taught by: Palak Arora
Catalog detailsMATH 416 LEC Advanced 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 419 LEC Algebraic Number Theory
Last offered Fall 2023
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 LamÃ©'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 numbertheoretic notions in more abstract algebraic settings, and we will see a beautiful interplay between groups, rings, and fields. [ more ]
Taught by: Allison Pacelli
Catalog detailsMATH 420 TUT Analytic Number Theory
Last offered Spring 2021
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 theorya 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 EulerMaclaurin summation, counting divisors and Dirichlet's hyperbola method, the randomness of prime factorization and the ErdosKac theorem, the partition function and the saddle point method, the prime number theorem and the Riemann zeta function, primes in arithmetic progressions and Dirichlet Lfunctions, the Goldbach conjecture and the circle method, and sieve methods and gaps between primes. [ more ]
MATH 421 LEC Quandles, 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 422 LEC Algebraic 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 ]
MATH 424 LEC Geometry, Surfaces and Billiards
Last offered Fall 2016
Mathematical billiards is the study of a ball bouncing around in a tablea 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(F) LEC Differential Topology
Differential topology marries the rubberlike 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 farreaching 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 ]
Taught by: Ivo Terek
Catalog detailsMATH 427 LEC Tiling Theory
Last offered Spring 2023
Since people 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, types of tilings, random tilings, the classification of tilings and aperiodic tilings. We will also look at tilings of the sphere, tilings of the hyperbolic plane, and tilings in in higher dimensions, including "knotted tilings". [ more ]
Taught by: Colin Adams
Catalog detailsMATH 428 LEC Catching 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 431 LEC Nonlinear 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 SEM Mathematical Modeling
Last offered Spring 2021
Mathematical modeling means (1) translating a reallife problem into a mathematical object, (2) studying that object using mathematical techniques, and (3) interpreting the results in order to learn something about the reallife problem. Mathematical modeling is used in biology, economics, chemistry, geology, sociology, political science, art, and countless other fields. This is an advanced, seminarstyle, course appropriate for students who have strong enthusiasm for applied mathematics, data science, and collaborative teamwork. [ more ]
MATH 434 LEC Applied Dynamics and Optimal Control
Last offered Spring 2024
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 overall outcome. The primary focus of this course will be optimal control using Pontryagin's maximum principle and HamiltonJacobiBellman 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. The course will begin with a solid review of modeling with dynamical systems, and deepening our understanding of differential and difference equations, parameter dependence, and bifurcations. [ more ]
Taught by: Stewart Johnson
Catalog detailsMATH 435 SEM Chipfiring Games on Graphs
Last offered Fall 2021
Starting with a graph (a collection of nodes connected by edges), place an integer number of poker chips on each vertex. Move these chips around according to "chipfiring moves", where a vertex donates a chip along each edge. These simple and intuitive games quickly lead to challenging mathematics with applications ranging from dynamical systems to algebraic geometry. In this course we'll build up a mathematical framework for studying chipfiring games, drawing on linear algebra and group theory. We'll discover algorithms for winning these games, and study their complexity; and we'll prove graphtheoretic versions of famous results like the RiemannRoch theorem. A key component of this course will be research projects that draw on open questions about chipfiring. [ more ]
MATH 441 LEC Information Theory and Applications
Last offered Fall 2021
What is information? And how do we communicate information effectively? This course will introduce students to the fundamental ideas of Information Theory including entropy, communication channels, mutual information, and Kolmogorov complexity. These ideas have surprising connections to a fields as diverse as physics (statistical mechanics, thermodynamics), mathematics (ergodic theory and number theory), statistics and machine learning (Fisher information, Occam's razor), and electrical engineering (communication theory). [ more ]
MATH 442 LEC Introduction to Descriptive Set Theory
Last offered Fall 2022
Descriptive set theory (DST) combines techniques from analysis, topology, set theory, combinatorics, and other areas of mathematics to study definable (typically Borel) subsets of Polish spaces. The first part of this course will cover the topics necessary to understand the main objects of study in DST: we will develop comfort with pointset topology (enough to juggle with Polish spaces and Borel sets), and set theory (just wellorderings and cardinality). The second part of the course will feature selected topics in descriptive set theory: for example, trees, the perfect set property, Baire category, and infinite games. [ more ]
Taught by: Jenna Zomback
Catalog detailsMATH 443 Introduction to Optimal Transport Theory
Last offered NA
This course will introduce you to the fascinating world of transportation optimization, a field that has important applications in many areas of science and engineering, such as economics, image processing, and machine learning. We will start by exploring the discrete Optimal Transport (OT) problem, which involves finding the most efficient way to transport a set of objects from one location to another. While the discrete OT problem can be formulated as a linear programming problem, finding an optimal solution to this problem can be computationally expensive, especially for largescale problems. To overcome this computational challenge, a popular approach is to use entropy regularization. We will also investigate the entropy regularized OT problem, which provides us with an approximation of optimal transport, with lower computational complexity and easy implementation. In the second half of the course, we will delve into the continuous case, which allows us to consider transport between infinitely many locations. We will study the famous MongeKantorovich problem, which involves finding the optimal transportation plan that minimizes the total cost of moving a given amount of mass from one location to another, subject to various constraints. Throughout the course, we will use a combination of theoretical and practical approaches to understand and apply the concepts we cover. By the end of the course, you will have a strong foundation in OT theory, which will prepare you for further studies in this exciting and rapidly evolving field. Recommended Textbooks / Articles: Topics in Optimal Transportation  CÃ©dric Villani Optimal Transport for Applied Mathematicians  Filippo Santambrogio Computational Optimal Transport  Gabriel PeyrÃ©, Marco Cuturi (https://arxiv.org/abs/1803.00567) [ more ]
Taught by: TBA
Catalog detailsMATH 445 LEC Topics in Numerical Analysis
Last offered Spring 2024
Numerical analysis is a field of mathematics that focuses on developing algorithms and computational methods to solve problems that cannot be solved exactly. In this senior seminar course on numerical analysis we will cover advanced topics such as numerical solutions of Partial Differential Equations, Random Numbers and Monte Carlo simulation, Fast Fourier Transform and signal processing, as well as applications or the Singular Value Decomposition for matrices. The course will start with a review of basic concepts from calculus, linear algebra, and differential equations. Students who have taken Introduction to Numerical Analysis (Math 345) are welcome to take this course. [ more ]
Taught by: Bhagya Athukorallage
Catalog detailsMATH 453(S) LEC Partial Differential Equations
In this course, we further explore the world of differential equations. Mainly, we cover topics in partial differential equations. Partial Differential Equations (PDEs) are fundamental to the modeling of many natural phenomena, arising in many fields, including fluid mechanics, heat and mass transfer, electromagnetic theory, finance, elasticity, and more. The goals of this course are to discuss the following topics: classification of PDEs in terms of order, linearity and homogeneity; physical interpretation of canonical PDEs; solution techniques, including separation of variables, series solutions, integral transforms, and the method of characteristics. [ more ]
Taught by: Bhagya Athukorallage
Catalog detailsMATH 456 LEC Representation Theory
Last offered Fall 2020
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 457 LEC Partition Theory
Last offered Spring 2024
The partitions of a positive integer are the different ways of writing it as a sum of positive integers. For example, 5 has seven partitions, three of which are 5=1+1+1+1+1, 5=2+3, and 5=5. (Can you find the rest?) Partition theory is a rich area of combinatorics with applications to algebra and mathematical physics. In this class we will focus on enumerative and bijective methods to answer questions such as: How can we calculate the number of partitions of a number efficiently? Why is the number of partitions of N into strictly odd numbers always the same as the number of its partitions into distinct numbers? Why does a 2dimensional partition look like a stack of cubes, and what does that have to do with tilings? [ more ]
Taught by: Daniel Condon
Catalog detailsMATH 458 SEM Algebraic Combinatorics
Last offered Spring 2022
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. In this collaborative projectbased course, students will select among the presented topics, develop research questions, and undertake original research in the field. Student assessment is based building positive and supportive collaborative working relationships with their peers, drafts of research project manuscript, and oral presentations. [ more ]
MATH 459 TUT Applied 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 PDEbased models; determine the most appropriate tools to apply to a PDE; apply the aforementioned tools; be comfortable with openended scientific work; read applied mathematical literature; communicate applied mathematics clearly, precisely, and appropriately; collaborate effectively. [ more ]
MATH 466 LEC Advanced Applied Analysis
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 LEC Tropical Geometry
Last offered Spring 2021
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 minplus 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 478 LEC On 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 ]
MATH 479 LEC Additive Combinatorics
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 sumproduct theorems, the structure of sets of small doubling (e.g. the FreimanRuzsa 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 higherorder Fourier analysis, the polynomial method, and the ergodic approach to Szemeredi's theorem. [ more ]
MATH 481 LEC Measure theory and Hilbert spaces
Last offered Spring 2023
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 sizelength, area, volume, dimension, etc.that apply to a broad array of sets. Along the way we'll encounter Lebesgue measure and Lebesgue integration, Hausdorff measure and fractals, spacefilling curves and the BanachTarski 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 excellent preparation for graduate studies in mathematics, statistics and economics. [ more ]
Taught by: TBA
Catalog detailsMATH 482 LEC Homological 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 mostused constructions. Possible topics include tensor products, chain complexes, homology, Ext, Tor and Hilbert's Syzygy Theorem. [ more ]
MATH 487 LEC Computational 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 IdealVariety 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, S) HON Senior Honors Thesis: Mathematics
Mathematics senior honors thesis; this is part of a fullyear thesis (493494). Each student carries out an individual research project under the direction of a faculty member that culminates in a thesis. See description under The Degree with Honors in Mathematics. [ more ]
Taught by: Cesar Silva
Catalog detailsMATH 494 HON Senior Honors Thesis: Mathematics
Last offered Spring 2024
Mathematics senior honors thesis; this is part of a fullyear thesis (493494). Each student carries out an individual research project under the direction of a faculty member that culminates in a thesis. See description under The Degree with Honors in Mathematics. [ more ]
Taught by: Cesar Silva
Catalog detailsMATH 497(F, S) IND Independent Study: Mathematics
Directed 400level independent study in Mathematics. [ more ]
Taught by: Cesar Silva
Catalog detailsMATH 498 IND Independent Study: Mathematics
Last offered Spring 2024
Directed 400level independent study in Mathematics. [ more ]
Taught by: Cesar Silva
Catalog detailsMATH 499(F, S) LEC 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 semestercourses taken by all students. [ more ]
Taught by: Julie Blackwood
Catalog details 
STAT 101(F, S) LEC Elementary Statistics and Data Analysis
It is impossible to be an informed citizen in today's world without an understanding of data. Whether it is opinion polls, unemployment rates, salary disparities, the efficacy of vaccines, etc., we need to be able to interpret and gain information from statistics. This course will introduce the common methods used to analyze and present data with an emphasis on interpretation and informed decision making. [ more ]
Taught by: Mihai Stoiciu, Anna Plantinga
Catalog detailsSTAT 161(F, S) LEC Introductory Statistics for Social Science
This course will cover the basics of modern statistical analysis with a view toward applications in the social sciences. Topics include exploratory data analysis, linear regression, basic statistical inference, and elements of probability theory. The course focuses on the application of statistical tools to solve problems, to make decisions, and the use of statistical thinking to understand the world. [ more ]
Taught by: Duncan Clark
Catalog detailsSTAT 197 IND Independent Study: Statistics
Last offered Fall 2023
Directed 100level independent study in Statistics. [ more ]
Taught by: Bernhard Klingenberg
Catalog detailsSTAT 198 IND Independent Study: Statistics
Last offered Spring 2024
Directed 100level independent study in Statistics. [ more ]
Taught by: Richard De Veaux
Catalog detailsSTAT 201(F, S) LEC Statistics and Data Analysis
Statistics can be viewed as the art and science of turning data into information. Real world decisionmaking, 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 ]
Taught by: Elizabeth Upton
Catalog detailsSTAT 202(F, S) LEC 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 ]
Taught by: Anna Neufeld
Catalog detailsSTAT 297 IND Independent Study: Statistics
Last offered Fall 2023
Directed 200level independent study in Statistics. [ more ]
Taught by: Bernhard Klingenberg
Catalog detailsSTAT 298 IND Independent Study: Statistics
Last offered Spring 2024
Directed 200level independent study in Statistics. [ more ]
Taught by: Richard De Veaux
Catalog detailsSTAT 302 LEC Applied Statistical Modeling
Last offered Spring 2021
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 LEC Data Visualization
Last offered Fall 2020
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 mouseover and click events. Using R's shiny and DT libraries, we will learn how to create and publish webapps 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 semiflipped 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 LEC Applied Machine Learning
Last offered Spring 2021
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 319 LEC Statistical Computing
Last offered Spring 2022
This course introduces a variety of computational and datacentric 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 highdimensional 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 projectbased. [ more ]
STAT 335(S) LEC Introduction to Biostatistics and Epidemiology
Epidemiology is the study of disease and disability in human populations, while 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 if time permits, a brief introduction to regression with correlated data. [ more ]
Taught by: Anna Plantinga
Catalog detailsSTAT 341(F, S) LEC 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 wellknown Central Limit Theorem. Many interesting and important applications will also be presented, including some from classical Poisson processes, random walks and Markov Chains. [ more ]
Taught by: Mihai Stoiciu, Steven Miller
Catalog detailsSTAT 342(S) LEC Introduction to Stochastic Processes
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 ]
Taught by: Elizabeth Upton
Catalog detailsSTAT 344(S) LEC Statistical Design of Experiments
When you hear the word experiment you might be picturing white lab coats and pipettes, but businesses, especially ecommerce, are constantly experimenting as well. How do you get the most out of both scientific and business investigations? 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. We'll learn how to analyze the data that come from these experiments and the conclusions we can draw from that analysis. We'll look at both classical tools like fractional factorial designs as well as 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 both R and JMP software to work with realworld data. [ more ]
Taught by: Richard De Veaux
Catalog detailsSTAT 346(F, S) LEC 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 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 ]
Taught by: Xizhen Cai
Catalog detailsSTAT 355(F) LEC Multivariate Statistical Analysis
To better understand complex processes, we study how variables are related to one another and how they work in combination. In addition, we want to make inferences about more than one variable at a 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 datasets with more than multiple variables. Topics covered will include data visualization techniques for data sets with more variables, clustering algorithms, parametric and nonparametric techniques to estimate joint distributions, techniques for combining variables, performing dimension reduction, and making inferences. [ more ]
Taught by: Xizhen Cai
Catalog detailsSTAT 356 LEC Time Series Analysis
Last offered Fall 2022
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 descriptive methods and checking for significance, and a foray into the frequency domain. We will emphasize applications to a variety of real data, explored using R. [ more ]
Taught by: TBA
Catalog detailsSTAT 358 LEC Introduction to Categorical Data Analysis
Last offered Spring 2022
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 (such as linear regression) are not designed to handle such data and pose inappropriate assumptions. We will develop methods specifically designed for modeling categorical data, with applications in the social and biological sciences as well as in medical research, engineering and economics. This course has two parts. The first part will discuss statistical inference for parameters of categorical distributions (Bernoulli, Binomial, Multinomial, Poisson) and for measures of association arising in contingency tables (difference and ratio of proportions and odds ratios). Inferential methods covered include Wald, score and likelihood ratio tests and confidence intervals, as well as the bootstrap. The longer second part will focus on statistical modeling of categorical response data via generalized linear models, with a heavy focus on logistic regression models with both quantitative and categorical predictors and their interactions. Model fitting and inference will be based on maximum likelihood and carried out via R. [ more ]
STAT 360(F) LEC 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 ]
Taught by: Richard De Veaux
Catalog detailsSTAT 365 LEC Bayesian Statistics
Last offered Spring 2024
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 closedform Bayesian inference. Building on this, we introduce modern computational approaches to Bayesian inference, including Markov chain Monte Carlo (MCMC), MetropolisHastings 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 ]
Taught by: TBA
Catalog detailsSTAT 368 LEC Modern 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 missspecified? 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 rankbased tests (Wilcoxon, KS test), and focus primarily on modern nonparametric inferential techniques, such as nonparametric density estimation, nonparametric regression, selection of smoothing parameter (crossvalidation), bootstrap, randomizationbased 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 LEC Longitudinal Data Analysis
Last offered Spring 2023
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. As time permits, we will also investigate joint modeling of longitudinal and timetoevent data. We will consider many applications in the social and biological sciences. [ more ]
Taught by: Anna Plantinga
Catalog detailsSTAT 397 IND Independent Study: Statistics
Last offered Fall 2023
Directed independent study in Statistics. [ more ]
Taught by: Bernhard Klingenberg
Catalog detailsSTAT 398 IND Independent Study: Statistics
Last offered Spring 2024
Directed independent study in Statistics. [ more ]
Taught by: Richard De Veaux
Catalog detailsSTAT 410 LEC Statistical 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), genomewide association studies, pathwaybased 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 440 LEC Categorical Data Analysis
Last offered Spring 2024
This course focuses on methods for analyzing categorical response data. Traditional tools of statistical data analysis for continuous response data 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. The first part of the course will discuss statistical inference for parameters of categorical distributions and arising in contingency tables. The longer second part will focus on statistical modeling via generalized linear models for binary, multinomial, ordinal and count response variables, using maximum likelihood. [ more ]
Taught by: TBA
Catalog detailsSTAT 441 LEC Information Theory and Applications
Last offered Fall 2021
What is information? And how do we communicate information effectively? This course will introduce students to the fundamental ideas of Information Theory including entropy, communication channels, mutual information, and Kolmogorov complexity. These ideas have surprising connections to a fields as diverse as physics (statistical mechanics, thermodynamics), mathematics (ergodic theory and number theory), statistics and machine learning (Fisher information, Occam's razor), and electrical engineering (communication theory). [ more ]
STAT 442(S) LEC 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 ]
Taught by: Anna Neufeld
Catalog detailsSTAT 458(F) LEC Generalized Linear Models Theory and Applications
This course will explore generalized linear models (GLMs)the extension of linear models, discussed in Stat346, to response variables that have specific nonnormal distributions, such as counts and proportions. We will consider the general structure and theory of GLMs and see their use in a range of applications. As time permits, we will also examine extensions of these models for clustered data such as mixed effects models and generalized estimating equations. [ more ]
Taught by: Elizabeth Upton
Catalog detailsSTAT 462 LEC Modern 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 distributionfree 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 randomizationbased 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 LEC Bayesian Statistics
Last offered Spring 2024
Prior knowledge being constantly updated by empirical observations  the essence of Bayesian thinking provides a natural, intuitive, and more importantly, mathematically sounded, probabilistically principled way to characterize the process of learning. With some of its key ideas formulated based on Bayes' Theorem dating back to 18th century, Bayesian inference is one of oldest schools of statistics (more than a century earlier than the Frequentist!). Yet it was not until the recent developments in sampling algorithms and computational powers that Bayesian inference gained its revival. Bayesian, and Bayesianbased methods, with their flexibilities in modeling (generative) process of data, interpretability with posterior probability statements, and coherent principles to incorporate empirical evidence a priori, have played key roles in modern data analysis, especially for those "big data" with enhanced complexity and connectivity. This course is designed to provide students a comprehensive understanding to what is Bayesian and the how's and why's. Students will be introduced to classic Bayesian models, basic computational algorithms/methods for Bayesian inference, as well as their applications in various fields, and comparisons with classic Frequentist methods. As Bayesian inference finds its roots and merits particularly in application, this course puts great emphasis on enhancing students' skills in statistical computation (mostly with R) and data analysis. [ more ]
Taught by: Shaoyang Ning
Catalog detailsSTAT 493(F, S) HON 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 ]
Taught by: Richard De Veaux
Catalog detailsSTAT 494 HON Senior Thesis: Statistics
Last offered Spring 2024
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 ]
Taught by: Cesar Silva
Catalog detailsSTAT 497(F, S) IND Independent Study: Statistics
Directed independent study in Statistics. [ more ]
Taught by: Richard De Veaux
Catalog detailsSTAT 498 IND Independent Study: Statistics
Last offered Spring 2024
Directed independent study in Statistics. [ more ]
Taught by: Richard De Veaux
Catalog detailsSTAT 499(F, S) SEM 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 semestercourses taken by all students. [ more ]
Taught by: Richard De Veaux
Catalog details