A World of Mathematics How long will oil last? What is the fairest voting system? How can we harvest food and other resources sustainably? To explore such real-world questions we will study a variety of mathematical ideas and methods, including modeling, logical analysis, discrete dynamical systems, and elementary statistics. This is an alternative first mathematics course for students not pursuing the calculus sequence in their first semester. The only prerequisite is an interest in exploring contemporary issues using the mathematics that lies within those issues. (Approval required; This course is not open to students who have had a prior course in calculus or statistics.) 3 hrs lect./disc.

Mathematical Problem-Solving This course is designed primarily for students concurrently enrolled in MATH 121 or MATH 122 who would benefit from structured support to reinforce their mathematical backgrounds, and these students will be given priority at registration. We will emphasize problem-solving rather than a collection of procedures, using problems selected to strengthen students’ conceptual understanding of the material and their strategic competence. In an inquiry-based setting, students will practice and improve their algebra and trigonometry skills, with an emphasis on effective exposition of mathematical arguments.(This is a half credit course.)(Approval required.) 1.5 hrs. disc.

Math and Board Games Have you ever spent minutes agonizing over which move to make in a board game? Out of all the possible options, how could you possibly determine which move is best? Was there even an objectively best decision? In this course, we will explore the mathematics and underlying gameplay structures of several modern board games. In addition to playing these games during class, we’ll use math and logic to assess and quantify the value of a range of possible in-game decisions. Using formal mathematical proofs, papers, and in-class discussions, we’ll analyze the fairness and equity of strategies across a wide variety of games. We’ll finish the course by designing our own board game based on what we’ve learned!

Mathematics for Teachers What mathematical knowledge should elementary and secondary teachers have in the 21st century? Participants in this course will strengthen and deepen their own mathematical understanding in a student-centered workshop setting. We will investigate the number system, operations, algebraic thinking, measurement, data, and functions, and consider the attributes of quantitative literacy. We will also study recent research that describes specialized mathematical content knowledge for teaching.

Making Data Visual Information can be used to inform, persuade, excite, and build community identity. Being able to move between data in a spreadsheet to a story that uses data that highlights information responsibly is a critical skill. In this course we will learn about communication standards for sharing data with experts and non-experts alike. Gaining skills in programs such as Canva, Photoshop and R, we will work to build data visualizations that are accurate, interesting, and responsibly represented. In the final project we will turn to our own community using data to tell a story about experiences at Middlebury.

Introduction to Statistical Science A practical introduction to statistical methods and the examination of data sets. Computer software will play a central role in analyzing a variety of real data sets from the natural and social sciences. Topics include descriptive statistics, elementary distributions for data, hypothesis tests, confidence intervals, correlation, regression, contingency tables, and analysis of variance. The course has no formal mathematics prerequisite, and is especially suited to students in the physical, social, environmental, and life sciences who seek an applied orientation to data analysis.

Introduction to Data Science In this course students will gain exposure to the entire data science pipeline: forming a statistical question, collecting and cleaning data sets, performing exploratory data analyses, identifying appropriate statistical techniques, and communicating the results, all the while leaning heavily on open source computational tools, in particular the R statistical software language. We will focus on analyzing real, messy, and large data sets, requiring the use of advanced data manipulation/wrangling and data visualization packages. Students will be required to bring alaptop (owned or college-loaned) to class as many lectures will involve in-class computational activities. (formerly MATH216) 3 hrs lect./disc.

Calculus I Introductory analytic geometry and calculus. Topics include limits, continuity, differential calculus of algebraic and trigonometric functions with applications to curve sketching, optimization problems and related rates, the indefinite and definite integral, area under a curve, and the fundamental theorem of calculus. Inverse functions and the logarithmic and exponential functions are also introduced along with applications to exponential growth and decay.

Calculus II A continuation of MATH 0121, may be elected by first-year students who have had an introduction to analytic geometry and calculus in secondary school. Topics include a brief review of natural logarithm and exponential functions, calculus of the elementary transcendental functions, techniques of integration, improper integrals, applications of integrals including problems of finding volumes, infinite series and Taylor's theorem, polar coordinates, ordinary differential equations.

Linear Algebra Matrices and systems of linear equations, the Euclidean space of three dimensions and other real vector spaces, independence and dimensions, scalar products and orthogonality, linear transformations and matrix representations, eigenvalues and similarity, determinants, the inverse of a matrix and Cramer's rule.

Regression Theory and Applications Regression is a popular statistical technique for making predictions and for modeling relationships between variables.  In this course we will discuss the theory and practical applications of linear, log-linear, and logistic regression models.  Topics include least squares estimation, coding for categorical predictors, analysis of variance, and model diagnostics.  We will apply these concepts to real datasets using R, a statistical programming language.  (MATH 0200; and MATH 0116 or MATH 0311) 3 hrs lect./disc.

Introduction to Data Science In this course students will gain exposure to the entire data science pipeline: forming a statistical question, collecting and cleaning data sets, performing exploratory data analyses, identifying appropriate statistical techniques, and communicating the results, all the while leaning heavily on open source computational tools, in particular the R statistical software language. We will focus on analyzing real, messy, and large data sets, requiring the use of advanced data manipulation/wrangling and data visualization packages. Students will be required to bring their own laptops as many lectures will involve in-class computational activities. 3 hrs lect./disc.

Statistical Learning This course is an introduction to modern statistical, machine learning, and computational methods to analyze large and complex data sets that arise in a variety of fields, from biology to economics to astrophysics. The theoretical underpinnings of the most important modeling and predictive methods will be covered, including regression, classification, clustering, resampling, and tree-based methods. Student work will involve implementation of these concepts using open-source computational tools.

Multivariable Calculus The calculus of functions of more than one variable. Introductory vector analysis, analytic geometry of three dimensions, partial differentiation, multiple integration, line integrals, elementary vector field theory, and applications.

Topics in Linear Algebra and Differential Equations Topics may include diagonalization of matrices, quadratic forms, inner product spaces, canonical forms, the spectral theorem, positive matrices, the Cayley-Hamilton theorem, ordinary differential equations of arbitrary order, systems of first-order differential equations, power series, and eigenvalue methods of solution, applications.

MATH 0226, Differential Equations This course provides an introduction into ordinary differential equations (ODEs) with an emphasis on linear and nonlinear systems using analytical, qualitative, and numerical techniques.  Topics will include separation of variables, integrating factors, eigenvalue method, linearization, bifurcation theory, and numerous applications.  In this course, we will introduce MATLAB programming skills and develop them through the semester. (MATH 0200 or by waiver)

Introduction to Numerical Analysis We will study the development, analysis, and implementation of numerical methods for approximating solutions to mathematical problems. We will begin with applications of Taylor polynomials, computer representation of numbers, and types of errors.  Other topics will include polynomial and spline interpolation, numerical integration and differentiation, rootfinding, and numerical solutions of differential equations.  Accuracy will be quantified by the concept of numerical error.  Additionally, we will study the stability, efficiency, and implementation of algorithms.  We will utilize the software MATLAB throughout to demonstrate concepts, as well as to complete assignments and projects.

Elementary Number Theory Divisibility and prime factorization. Congruences; the theorems of Lagrange, Fermat, Wilson, and Euler; residue theory; quadratic reciprocity. Diophantine equations. Arithmetic functions and Mobius inversion. Representation as a sum of squares.

Graph Theory       A graph (or network) is a useful mathematical model when studying a set of discrete objects and the relationships among them.  We often represent an object with a vertex (node) and a relation between a pair with an edge (line).  With the graph in hand, we then ask questions, such as: Is it connected? Can one traverse each edge precisely once and return to a starting vertex? For a fixed k/, is it possible to “color” the vertices using /k colors so that no two vertices that share an edge receive the same color?  More formally, we study the following topics: trees, distance, degree sequences, matchings, connectivity, coloring, and planarity.  Proof writing is emphasized.

Abstract Algebra Groups, subgroups, Lagrange's theorem, homomorphisms, normal subgroups and quotient groups, rings and ideals, integral domains and fields, the field of quotients of a domain, the ring of polynomials over a domain, Euclidean domains, principal ideal domains, unique factorization, factorization in a polynomial ring.

Probability An introduction to the concepts of probability and their applications, covering both discrete and continuous random variables. Probability spaces, elementary combinatorial analysis, densities and distributions, conditional probabilities, independence, expectation, variance, weak law of large numbers, central limit theorem, and numerous applications.

Statistics An introduction to the mathematical methods and applications of statistical inference. Topics will include: survey sampling, parametric and nonparametric problems, estimation, efficiency and the Neyman-Pearsons lemma. Classical tests within the normal theory such as F-test, t-test, and chi-square test will also be considered. Methods of linear least squares are used for the study of analysis of variance and regression. There will be some emphasis on applications to other disciplines.

Mathematical Models in the Social and Life Sciences An introduction to the role of mathematics as a modeling tool and an examination of some mathematical models of proven usefulness in problems arising in the social and life sciences. Topics will be selected from the following: axiom systems as used in model building, optimization techniques, linear and integer programming, theory of games, systems of differential equations, computer simulation, stochastic process. Specific models in political science, ecology, sociology, anthropology, psychology, and economics will be explored.

Real Analysis An axiomatic treatment of the topology of the real line, real analysis, and calculus. Topics include neighborhoods, compactness, limits, continuity, differentiation, Riemann integration, and uniform convergence.

Complex Analysis An introduction to functions of a complex variable. Mappings of the complex plane, analytic functions, Cauchy Integral Theorem and related topics.

Partial Differential Equations An introduction to partial differential equations (PDEs) with an emphasis on first and second-order  linear equations. Using analytical, qualitative, and numerical techniques, we will study the Laplace,  heat, and wave equations, as well as their applications. MATLAB will be used where applicable.  (MATH  0223 and either of MATH 0225 or MATH 0226) 3 hr lect.

Numerical Linear Algebra Numerical linear algebra is the study of algorithms for solving problems such as finding solutions of linear systems and eigenvalues of matrices.   Many real-life applications simplify to these scenarios and often involve millions of variables.  We will analyze shortcomings of direct methods such as Gaussian Elimination, which theoretically produces the true solution but fails in practical applications.  In contrast, iterative methods are often more practical and precise, and continually evolve with changing technology and our understanding of mathematics.  Our study will include the First Order Richardson, Steepest Descent, and Conjugate Gradient algorithms for linear systems, and the power method for eigenvalue problems.

Elementary Topology An introduction to the concepts of topology. Theory of sets, general topological spaces, topology of the real line, continuous functions and homomorphisms, compactness, connectedness, metric spaces, selected topics from the topology of Euclidean spaces including the Jordan curve theorem. (MATH 0122 or MATH 0200 or by waiver)

Combinatorics       Combinatorics is the “art of counting.”  Given a finite set of objects and a set of rules placed upon these objects, we will ask two questions.  Does there exist an arrangement of the objects satisfying the rules?  If so, how many are there?  These are the questions of existence and enumeration.  As such, we will study the following combinatorial objects and counting techniques: permutations, combinations, the generalized pigeonhole principle, binomial coefficients, the principle of inclusion-exclusion, recurrence relations, and some basic combinatorial designs.

Stochastic Processes Stochastic processes are mathematical models for random phenomena evolving in time or space. This course will introduce important examples of such models, including random walk, branching processes, the Poisson process and Brownian motion. The theory of Markov chains in discrete and continuous time will be developed as a unifying theme. Depending on time available and interests of the class, applications will be selected from the following areas: queuing systems, mathematical finance (Black-Scholes options pricing), probabilistic algorithms, and Monte Carlo simulation.

Finite Fields Seminar This course is a tutorial in the theory and applications of finite fields, which lie in the intersection of algebra and number theory.  Working in small groups, students will study the fundamental structure and properties of finite fields (also known as Galois fields).  They will then work independently, exploring applications in cryptography, coding theory, or other areas. Students will gain experience reading advanced sources and communicating their insights in expository writing and oral presentations. This course fulfills the capstone senior work requirement for the mathematics major. (MATH 0241 or MATH 0302; Approval required)  3 hrs. Sem

Advanced Probability Seminar This course is a tutorial in Probability Theory for students who have completed work in Probability and Real Analysis.  Starting from elementary results about random walks, we will explore the fundamental mathematical ideas underlying measure theoretic probability, martingales, the Weiner process, and the Itô stochastic calculus.  Working independently and in small groups, students will gain experience reading advanced sources and communicating their insights in expository writing and oral presentations.  This course fulfills the capstone senior work requirement for the mathematics major. (MATH 0310, MATH 0323, and by approval).

Statistics Capstone Seminar In this course we will work with community partners to solve real-world problems using modern statistical and data science techniques. Students will work in small groups to translate research questions into actionable analysis and visualizations. Students will select a project of interest from a subset of community partners, maintain contact and collaboration with the community partner, and present their findings in a final symposium.

Advanced Mathematical Modeling Seminar A tutorial on advanced mathematical model building and analysis for students who have completed work in Differential Equations and Probability. We will study deterministic and stochastic models of interacting populations with a focus on mathematical ecology and epidemiology. Working independently and in small groups, students will gain experience reading advanced sources and communicating their insights in expository writing and oral presentations. Fulfills the capstone senior work requirement for the mathematics major. (Approval Only) 3 hrs. Sem.

Mathematical Methods in Fluid Dynamics This course is an introduction to the mathematical models and methods used in modern fluid dynamics.  Students will derive and analyze fundamental equations of fluid flow, explore their applications, as well as examine theoretical and practical solution techniques.  Equations of study will include the Poisson, diffusion, and Navier-Stokes equations.  We will also introduce basic methods of computational fluid dynamics.  Working independently and in small groups, students will gain experience reading advanced sources and communicating their insights in expository writing and oral presentations. Fulfills the capstone senior work requirement for the mathematics major. 3 hrs. Lect./Lab

Advanced Number Theory A senior tutorial on some topics in advanced elementary number theory and an introduction to analytic number theory.  In this course we will review key areas of elementary number theory and abstract algebra followed by the study of integer partitions, continued fractions, rational approximations of irrationals, primes and primality testing, the average order of magnitude of several number theoretic functions, the Basel problem, Bernoulli numbers, and the Riemann zeta function.

The Polynomial Method A tutorial in the Polynomial Method for students who have completed work in Abstract Algebra and at least one of Combinatorics, Graph Theory, and Number Theory.  We will study Noga Alon’s Combinatorial Nullstellensatz and related theorems, along with their applications to combinatorics, graph theory, number theory, and incidence geometry.  Working independently and in small groups, students will gain experience reading advanced sources and communicating their insights in expository writing and oral presentations.  Fulfills the capstone senior work requirement for the mathematics major.  (Approval required; MATH 0302 and one of the following: MATH 0241, MATH 0247, or MATH 0345).

Introduction to Networks In this course we will explore the ubiquity of networks and the beautiful mathematics that helps us understand them. Together we will cover the basics of graph theory, introduce real world social, informational, and biological networks, explore how information (or a virus) can diffuse or cascade through a network, and learn about popular social and graph phenomena like the six degrees of separation and the friendship paradox. We will utilize jupyter notebooks and python libraries to build a toolset for studying networks and you will have the opportunity to analyze an empirical network using the ideas and tools you develop over the course of this class. No previous coding or mathematical experience is necessary: we will cover all concepts together.

Data Science Across Disciplines In this course, we will gain exposure to the entire data science pipeline—obtaining and cleaning large and messy data sets, exploring these data and creating engaging visualizations, and communicating insights from the data in a meaningful manner. During morning sessions, we will learn the tools and techniques required to explore new and exciting data sets. During afternoon sessions, students will work in small groups with one of several faculty members on domain-specific research projects in Sociology, Neuroscience, Animation, Art History, or Environmental Science. This course will utilize the R programming language. No prior experience with R is necessary. ENVS: Students will engage in research within environmental health science—the study of reciprocal relationships between human health and the environment. High-quality data and the skills to make sense of these data are key to studying environmental health across diverse spatial scales, from individual cells through human populations. In this course, we will explore common types of data and analytical tools used to answer environmental health questions and inform policy. FMMC: Students will explore how to make a series of consequential decisions about how to present data and how to make it clear, impactful, emotional or compelling. In this hands-on course we will use a wide range of new and old art making materials to craft artistic visual representations of data that educate, entertain, and persuade an audience with the fundamentals of data science as our starting point. NSCI/MATH: Students will use the tools of data science to explore quantitative approaches to understanding and visualizing neural data. The types of neural data that we will study consists of electrical activity (voltage and/or spike trains) measured from individual neurons and can be used to understand how neurons respond to and process different stimuli (e.g., visual or auditory cues). Specifically, we will use this neural data from several regions of the brain to make predictions about neuron connectivity and information flow within and across brain regions. SOCI: Students will use the tools of data science to examine how experiences in college are associated with social and economic mobility after college.  Participants will combine sources of "big data" with survey research to produce visualizations and exploratory analyses that consider the importance of higher education for shaping life chances.  HARC: Students will use the tools of data science to create interactive visualizations of the Dutch textile trade in the early eighteenth century. These visualizations will enable users to make connections between global trade patterns and representations of textiles in paintings, prints, and drawings.

Alex Lyford

Becky Tang

Charlotte Morris-Wright

Daniel Velleman

David Dorman

Emily Malcolm-White

Emily Proctor

Francis Swenton

Gyula Zsombok

Ivy Zhang

Jeanne Albert

Jennifer Crodelle

John Schmitt

Joseph Crodelle

Kara Karpman

Katherine O'Brien

Michaela Kubacki

Michael Olinick

Peter Schumer

Priscilla Bremser

Stephen Abbott

William Peterson

Emily Proctor is the best! If you're going to take linear, take it with her. She was consistently accommodating to my various needs throughout the semester and made the subject enjoyable and fun. There is homework due every class period, but it is not difficult. 

This was Becky Tang's first semester teaching, as she has just graduated from grad school. I thought the pace of the course was way too fast, with little to no introduction to the material at the beginning of the semester. Labs assigned on Friday's and due on Sunday at midnight were the worst part of this course. 

I felt like this course was not taught in the proper way. We were just given Theorems to read and were expected to understand them and prove them. There was no intuition behind this teaching style. The exams were unnecessarily hard with tough proofs that were harshly graded(almost no partial credit is ever given). Overall the experience was not great and the class felt like is was much harder than it was supposed to.