Mathematics and Statistics

Topics include limits and continuity. Definition, computation and applications of the derivative. An introduction to integration, including the fundamental theorem of calculus. Of the 4 credits, approximately 1 credit will be committed to a parallel track of instruction that introduces and/or reviews topics in algebra, trigonometry, exponential and log functions and graphing as they are being encountered in the calculus curriculum.

Topics include limits and continuity. Definition, computation and applications of the derivative. An introduction to integration, including the fundamental theorem of calculus.

A continuation of Mathematics 125, covering techniques for computing indefinite integrals, applications of the definite integral, infinite sequences and series, Taylor polynomials and power series.

This course introduces students to basic tools for describing and summarizing data as well as methods of statistical inference such as confidence intervals and hypothesis tests.  The randomization approach used in the course allows students to develop a deeper understanding of the fundamental idea of statistical inference. A web-based statistical applet is used throughout the course. This course does not count toward the Mathematics major or Data Science minor. Students considering these should enroll in Mathematics 247 instead.

On occasion, the mathematics and statistics department will offer courses on introductory topics in mathematics and statistics that are not generally covered in other introductory courses. Possible topics include Introduction to Number Theory, Chaos and Applied Discrete Probability. See course schedule for any current offerings.

An introduction to the approaches and tools of exploratory data analysis and visualization. Through a series of projects, we explore large data sets through methods like cleaning, filtering, sorting, boolean selections and merging. As large amounts of data typically are stored in lists, we use algorithmic thinking to transform raw data into usable form. We develop hypotheses and supporting visualizations to tell the story of the data. We learn and practice technical communication in both oral and written form. Through a series of readings and discussions, we learn best practices for the ethical use of data and how to identify problematic uses of data in society. May be elected as Computer Science 215.

This course provides a mathematical foundation for formal study of algorithms and the theory of computing. It also introduces functional programming, a powerful computing paradigm that is distinct from the imperative and object-oriented paradigms introduced in Computer Science 167. Students will practice formal reasoning over discrete structures through two parallel modes: mathematical proofs and computer programs. We will introduce sets and lists, Boolean logic, and proof techniques. We will explore recursive algorithms and data types along with mathematical and structural induction. We consider relations and functions as mathematical objects and develop idioms of higher-order programming. We consider applications useful in computer science, particularly counting sets. May be elected as Computer Science 220.

Topics include three dimensional geometry, partial derivatives, gradients, extreme value theory for functions of more than one variable, multiple integration, line integrals, and various topics in vector analysis.

This course first considers the solution set of a system of linear equations. The ideas generated from systems of equations are then generalized and studied in a more abstract setting, which considers topics such as matrices, determinants, vector spaces, inner products, linear transformations, and eigenvalues.  

This course includes first and second order linear differential equations and applications. Other topics may include systems of differential equations and series solutions of differential equations.

An introduction to statistics for students who have taken at least one course in calculus. This course focuses on introducing statistical concepts and inference through active learning assignments. Students learn about the process of statistical investigations. This includes data collection and exploration, methods of statistical inference,  and the ability to draw appropriate conclusions.  The widely-used statistical software R will be used in addition to web-based applets.

This course follows introductory statistics by investigating more complex statistical models and their application to real data. The topics may include simple linear regression, multiple regression, non-parametric methods, and logistic regression. A statistical software package will be used. Familiarity with R is expected.

An introduction to some of the concepts and methodology of advanced mathematics, including a brief introduction to number theory. Emphasis is on the notions of rigor and proof. This course is intended for students interested in majoring in mathematics and statistics; students should plan to complete it no later than the spring semester of the sophomore year.

A reading project in an area of mathematics and statistics not covered in regular courses or that is a proper subset of an existing course. The topic, selected by the student in consultation with the staff, is deemed to be introductory in nature with a level of difficulty comparable to other mathematics and statistics courses at the 200-level. May be repeated for a maximum of six credits.

This independent study in geometry will include a review of high school geometry, a few topics in advanced Euclidean geometry, a reading of Books I and II of Euclid's Elements, and an introduction to hyperbolic geometry. The grading for the course will be based on a journal (40%), a two-hour written midterm exam (30%), and a one-hour oral final exam (30%). Since the student will be working independently on the material, a disciplined work ethic is required.

Students will meet weekly to discuss problem-solving techniques. Each week a different type of problem will be discussed. Topics covered will include polynomials, combinatorics, geometry, probability, proofs involving induction, parity arguments, and divisibility arguments. The main focus of the course will be to prepare students for the William Lowell Putnam Mathematics Competition, a national examination held the first Saturday in December. Students who place in the top 500 on this exam nationwide have their names listed for consideration to mathematics graduate programs. Graded credit/no credit. May be repeated for a maximum of four credits.

Which problems can be solved computationally? Which cannot? Why? We can prove that computers can perform certain computations and not others. This course will investigate which ones, and why. Topics will include formal models of computation such as finite state automata, push-down automata, and Turing machines, as well as formal languages such as context-free grammars and regular expressions. May be elected as Computer Science 320, and must be elected as Computer Science 320 to apply toward the total credit requirement in Computer Science.   

How can we be confident that an algorithm is correct before we implement it?  How can we compare the efficiency of different algorithms? We present rigorous techniques for design and analysis of efficient algorithms. We consider problems such as sorting, searching, graph algorithms, and string processing. Students will learn design techniques such as linear programming, dynamic programming, and the greedy method, as well as asymptotic, worst-case, average-case and amortized runtime analyses. Data structures will be further developed and analyzed. We consider the limits of what can be efficiently computed. May be elected as Computer Science 327, and must be elected as Computer Science 320 to apply toward the total credit requirement in Computer Science.  

Essential for prospective high school mathematics teachers, this course includes a study of Euclidean geometry, a discussion of the flaws in Euclidean geometry as seen from the point of view of modern axiomatics, a consideration of the parallel postulate and attempts to prove it, and a discussion of the discovery of non-Euclidean geometry and its philosophical implications.

Operations research is a scientific approach to determining how best to operate a system, usually under conditions requiring the allocation of scarce resources. This course will consider deterministic models, including those in linear programming (optimization) and related subfields of operations research. May be elected as Computer Science 339.

Statistical concepts and statistical methodology useful in descriptive, experimental, and analytical study of biological and other natural phenomena.  Course covers major design structures, including blocking, nesting and repeated measures (longitudinal data), and statistical analysis associated with these structures.

A formal introduction to probability and randomness. The topics of the course include but are not limited to conditional probability, Bayes’ Theorem, random variables, the Central Limit Theorem, expectation and variance. Both discrete and continuous probability distribution functions and cumulative distribution functions are studied.

This course explores the process of machine learning through the lens of empirical modeling. We will develop the theory and algorithms that underpin the process of learning interesting things about data. Algorithms we’ll develop typically include: singular value decomposition and eigenfaces, the n-armed bandit, projections and linear regression, data clustering (k-means, Neural Gas, Kohonen’s SOM), linear neural networks, optimization algorithms, autoencoders and deep networks. The course will involve some computer programming, so previous programming experience is helpful. May be elected as Computer Science 350. Prerequisite: Mathematics 240.

Topics in elementary combinatorics, including: permutations, combinations, generating functions, the inclusion-exclusion principle, and other counting techniques; graph theory; and recurrence relations.

An introduction to mathematics commonly used in engineering and physics applications. Topics may include: vector analysis and applications; matrices, eigenvalues, and eigenfunctions; boundary value problems and spectral representations; Fourier series and Fourier integrals; solution of partial differential equations of mathematical physics.

Complex analysis is the study of functions defined on the set of complex numbers. This introductory course covers limits and continuity, analytic functions, the Cauchy-Riemann equations, Taylor and Laurent series, contour integration and integration theorems, and residue theory.

This course will cover graph theory concepts and linear programming. In the first section of the course, we will go over the basic definitions and concepts studied in graph theory, as well as optimization in graphs. In the second section, we start with an introduction to linear programming and its fundamental results. We use these results to prove well known results in graph theory and show how the two topics are connected.

A reading project in an area of mathematics and statistics not covered in regular courses or that is a proper subset of an existing course. The topic, selected by the student in consultation with the staff, is deemed to be introductory in nature with a level of difficulty comparable to other mathematics and statistics courses at the 200-level. May be repeated for a maximum of six credits.

This course studies the mathematical theory of statistics with a focus on the theory of estimation and hypothesis tests. Topics may include properties of estimators, maximum likelihood estimation, convergence in probability, the central limit theorem, order statistics, moment generating functions, and likelihood ratio tests. A statistical software package will be used.

Provides a rigorous study of the basic concepts of real analysis, with emphasis on real-valued functions defined on intervals of real numbers. Topics include sequences, continuity, differentiation, integration, and infinite series. 

The content varies from instructor to instructor with typical topics chosen from series of functions, topology of the real line, basic concepts of metric spaces, the calculus of vector-valued functions, and more advanced integration theory.

An introduction to numerical approximation of algebraic and analytic processes. Topics include numerical methods of solution of equations, systems of equations and differential equations, and error analysis of approximations. May be elected as Computer Science 467.

On occasion, the mathematics and statistics department will offer courses on advanced topics in mathematics and statistics that are not found in other course offerings. Possible topics include topology, number theory, and problem-solving. See course schedule for any current offerings.

An introduction to groups, rings and fields, including subgroups and quotient groups, homomorphisms and isomorphisms, subrings and ideals.

Topics may include fields, simple groups, Sylow theorems, Galois theory, and modules.

A reading or research project in an area of mathematics and statistics not covered in regular courses. The topic is to be selected by the student in consultation with the staff. Maximum of six credits.

Preparation of the senior project required of all graduating mathematics majors. Each student will be matched with a faculty member from the mathematics and statistics department who will help supervise the project. Course objectives include developing students’ abilities to independently read, develop, organize, and communicate mathematical ideas, both orally and in writing. A final written and oral report on the project is completed.

Preparation of an honors thesis. Required of and limited to senior honors candidates in mathematics. Students will be a part of the Mathematics 497 Senior Project class, but their work will be held to a higher standard.