# Course Descriptions

**Please note:** The course descriptions displayed here are current as of Sunday, July 12, 2020, but the official Course Catalog should be used for all official planning.

#### MATH 107: *Elementary Statistics*

For students in all disciplines. Provides the background needed to evaluate statistical arguments found in newspapers, magazines, reports, and journals and the logic and techniques necessary to perform responsible elementary statistical analysis. Topics include basic data analysis, one-variable regression, experimental and sampling design, random variables, sampling distributions, and inference (confidence intervals and significance testing). This course may not be taken on a Satisfactory/Unsatisfactory basis.
Units: 6.

Prerequisite: Completion of 54 units in Lawrence courses or consent of instructor

#### MATH 120: *Applied Calculus I*

A course in the applications of mathematics to a wide variety of areas, stressing economics and the biological sciences. Topics may include recursive sequences and their equilibria, the derivative of a function, optimization, fitting abstract models to observed data. Emphasis placed on algebraic and numerical techniques and on understanding the role of mathematical thinking. Mathematics 120 and 130 do not prepare students for more advanced courses in mathematics.
Units: 6.

Prerequisite: Three years of high school mathematics;

#### MATH 130: *Applied Calculus II*

A continuation of math 120. Topics may include the indefinite and definite integral, elementary linear algebra including matrix arithmetic and solving linear equations, vectors, partial derivatives, Lagrange multipliers. Both algebraic and numerical computations.
Units: 6.

Prerequisite: MATH 120 or the equivalent

#### MATH 140: *Calculus*

Functions, limits, derivatives, the Mean Value Theorem, definition and properties of integrals, the Fundamental Theorem of Calculus, and applications to related rates, curve sketching, and optimization problems.
Units: 6.

Prerequisite: Four years of high school mathematics and minimum score on ALEKS online diagnostic exam, as set by the department.

#### MATH 155: *Multivariable Calculus*

Techniques of integration, vector algebra in the plane and space, matrix algebra, functions of several variables, partial derivatives, double and triple integration, optimization.
Units: 6.

Prerequisite: MATH 140 or equivalent, appropriate score on the departmental placement exam

#### MATH 191: *Directed Study in Mathematics*

Directed study follows a syllabus set primarily by the instructor to meet the needs or interests of an individual student or small group of students. The main goal of directed study is knowledge or skill acquisition, not research or creative work.
Units: 1 TO 98.

Prerequisite: Counter Registration Required.

#### MATH 200: *Complex Sequences and Series*

Complex numbers, sequences, convergence, series, power series, additional topics chosen from analysis, geometry, differential equations, and applied mathematics
Units: 6.

Prerequisite: MATH 155

#### MATH 208: *Machine Learning*

An overview of techniques used to discover structural patterns and make predictions using complex datasets that are prevalent in today's world. The central machine learning tasks of classification, clustering, and regression will be explored, along with methods for training models and evaluating predictions. This course will be taught in a workshop format. Assignments will involve the use of statistical software.
Units: 6.

Prerequisite: One course in mathematics or computer science, or BIOL 170, or consent of instructor

#### MATH 210: *Differential Equations with Linear Algebra*

A study of differential equations and related techniques in linear algebra. Topics include first-order equations and their applications, existence and uniqueness of solutions, second-order linear equations and their applications, series solutions, systems of first-order equations, vector spaces and dimension, linear transformations, and eigenvalues.
Units: 6.

Prerequisite: MATH 160, or MATH 150 and consent of instructor

#### MATH 217: *Applied Statistical Methods*

A second course in statistics that covers analyses needed to solve more complicated data-driven problems. Time permitting, topics include multiple regression, analysis of variance, categorical data analysis, nonparametric tests, bootstrap methods, and permutation tests. Class meetings are a mixture of lecture, discussion, and use of statistical software to investigate real data.
Units: 6.

Prerequisite: AP examination credit in statistics or MATH 107, or BIOL 170, or PSYC 170

#### MATH 220: *Applied Combinatorics*

An introduction to logic, proofs by mathematical induction, and elementary combinatorics. Additional topics include recurrence relations, generating functions, and the principle of inclusion-exclusion.
Units: 6.

Prerequisite: MATH 150

#### MATH 223: *Quantitative Decision-Making*

The students will learn how to develop formal, quantitative approaches to structuring difficult problems, particularly those problems involving probabilistic factors. We will develop and practice the steps of defining a problem, gathering data, formulating a model, performing numerical calculations, evaluating numerical information, refining the model, analyzing the model's alternatives, and communicating the results.
Units: 6.

Also listed as Economics 223

Prerequisite: Sophomore standing

#### MATH 230: *Discrete Mathematics*

An introduction to mathematical reasoning and proof in the context of discrete structures relevant to the study of computer science. Topics include induction, sets, relations and functions, graph theory, combinatorics, and probability.
Units: 6.

Prerequisite: MATH 155

#### MATH 240: *Probability*

An introduction to probability and its applications. Topics will include combinatorial and axiomatic probability, conditional probability and Bayes' Theorem, random variables, expectation and variance, discrete and continuous probability distributions, joint and conditional distributions, and limit laws.
Units: 6.

Prerequisite: MATH 160, or MATH 150 and consent of instructor

#### MATH 250: *Linear Algebra*

The study of vector spaces, linear transformations, matrices, and applications. Topics include linear independence, dimension, rank-nullity, change of basis, eigenvectors and eigenvalues, determinants, and inner products.
Units: 6.

Prerequisite: MATH 200

#### MATH 300: *Foundations of Algebra*

An introduction to the rigorous study of mathematics. Topics include elementary theory of sets and mappings, number theory, equivalence relations, finite groups, homomorphisms, quotient groups, and rings.
Units: 6.

Prerequisite: MATH 210, 220, or 240

#### MATH 310: *Foundations of Analysis*

A study of the concepts that underlie mathematical analysis: the completeness of the real numbers, convergence, continuity, derivatives, integrals, infinite series, and, if time permits, an introduction to metric spaces or Fourier series.
Units: 6.

Prerequisite: MATH 300

#### MATH 350: *Ordinary Differential Equations*

A study of ordinary differential equations and applications. Topics include standard techniques for special types of equations, linear and non-linear systems, existence and uniqueness, and qualitative behavior.
Units: 6.

Prerequisite: MATH 250

#### MATH 390: *Tutorial Studies in Mathematics*

Advanced work in mathematics on topics not covered in regular offerings.
Units: 1 TO 98.

Prerequisite: Counter Registration Required.

#### MATH 391: *Directed Study in Mathematics*

Directed study follows a syllabus set primarily by the instructor to meet the needs or interests of an individual student or small group of students. The main goal of directed study is knowledge or skill acquisition, not research or creative work.
Units: 1 TO 98.

Prerequisite: Counter Registration Required.

#### MATH 395: *Internship In Mathematics*

The academic component of the internship includes readings related to the substance of the internship, discussions with the faculty supervisor, and a written report appropriate to the discipline. Course grades are based on this academic work.
Units: 1 TO 98.

#### MATH 399: *Independent Study in Mathematics*

Guided independent study of an advanced topic in undergraduate mathematics or supervised work on an undergraduate research project, generally culminating in a final presentation and/or paper.
Units: 1 TO 98.

Prerequisite: Counter Registration Required.

#### MATH 400: *Partial Differential Equations*

A survey of techniques used in modeling physical systems, with particular emphasis on partial differential equations and methods used to attack problems that do not have clean or simple solutions. Topics include techniques for solving partial differential equations exactly, the Fourier transform, perturbation theory, variational methods, Monte Carlo techniques, and finite difference schemes.
Units: 6.

Prerequisite: MATH 300 or consent of instructor

#### MATH 410: *Linear Algebra*

A study of vector spaces, linear transformations, and their representations. The focus will be on algebraic and coordinate-free methods, and topics will include dimension, dual spaces, determinants, canonical forms, inner product spaces, and the spectral theorem.
Units: 6.

Prerequisite: MATH 300

#### MATH 420: *Numerical Analysis*

Computer approximated (numerical) solutions to a variety of problems with an emphasis on error analysis. Interpolation, evaluation of polynomials and series, solution of linear and non-linear equations, eigenvectors, quadrature (integration), and differential equations.
Units: 6.

Prerequisite: MATH 300 and CMSC 210 or CMSC 150

#### MATH 430: *Statistical Modeling*

An exploration of methods to select, fit, evaluate and compare statistical models, while also providing an introduction to statistical inference. Lectures will develop the necessary theory for regression models while maintaining the focus on applications. Students will complete regular assignments as well as a midterm and final exam.
Units: 6.

Prerequisite: MATH 240

#### MATH 435: *Optimization*

The study of local and global maximums and minimums of function, given various sorts of constraints. Linear problems and the simplex algorithm, general non-linear problems and the Kuhn-Tucker conditions, convex problems. Perturbation of problem parameters and duality. Applications to a wide variety of fields, including economics, game theory, and operations research.
Units: 6.

Prerequisite: MATH 310

#### MATH 440: *Probability Theory*

The mathematics of chance: probability, discrete and continuous random variables and their distributions, moments, jointly distributed random variables, conditional distributions, the Central Limit Theorem, and weak and strong convergence.
Units: 6.

Prerequisite: MATH 310

#### MATH 445: *Mathematical Statistics*

Development of the mathematical theory of statistics and its application to the real world. The course will focus on the principles of estimation and testing from both the frequentist and Bayesian perspectives. Resampling methods (permutation tests and bootstrap intervals) will also be explored.
Units: 6.

Prerequisite: MATH 240

#### MATH 450: *Bayesian Statistics*

A study of the Bayesian statistical philosophy, contrasting it with the traditional frequentist approach taught in other statistics courses. Topics include Bayes' Theorem, prior and posterior probability distributions, hierarchical models, and Markov Chain Monte Carlo methods. The course will involve a mixture of lecture, discussion, and use of statistical software. Requirements include exams, a project, and assignments involving the use of statistical software.
Units: 6.

Prerequisite: MATH 240

#### MATH 525: *Graph Theory*

A survey of graph theory that balances the abstract theory of graphs with a wide variety of algorithms and applications to “real world” problems. Topics include trees, Euler tours and Hamilton cycles, matchings, colorings, directed graphs, and networks.
Units: 6.

Prerequisite: MATH 300

#### MATH 530: *Topics in Geometry*

The axiomatic development of euclidean and non-euclidean geometry, including the historical and philosophical issues raised by the “non-euclidean revolution.” Additional topics, such as projective or differential geometry and convexity, may be included.
Units: 6.

Prerequisite: MATH 300

#### MATH 535: *Complex Analysis*

An introduction to functions of a complex variable, the Cauchy-Riemann equations, conformal mappings, Cauchy’s theorem, Cauchy’s integral formula, Taylor and Laurent series, and a sampling, as time and interest permit, of the corollaries to Cauchy’s theorem.
Units: 6.

Prerequisite: MATH 310

#### MATH 545: *Rings and Fields*

Modern algebra with topics selected from group theory, ring theory, field theory, classical geometric construction problems, and Galois theory. Emphasis on the use of mathematical abstraction to illuminate underlying relationships and structure.
Units: 6.

Prerequisite: MATH 300

#### MATH 550: *Topics in Analysis*

Selected topics in analysis covering a wide variety of spaces and leading to applications of classical importance. In recent years, topics have included fixed point theory, inverse and implicit function theorems, abstract theory of differential equations, Lebesgue measure and integration, Fourier series and transforms.
Units: 6.

Prerequisite: MATH 310

#### MATH 555: *Topics Algebra & Combinatorics*

A study of interconnections between abstract algebra (especially finite group theory) and combinatorics (especially graph theory). Topics will include classical results (such as the matrix-tree theorem), as well as recent subjects and advances (such as the abelian sandpile model and the Riemann-Roch theorem for graphs).
Units: 6.

Prerequisite: MATH 300

#### MATH 560: *Topology*

A study of metric and topological spaces, including continuity, compactness, connectedness, product and quotient spaces. Additional topics may include Zorn’s Lemma, separation properties, surfaces, the fundamental group, and fixed point theorems.
Units: 6.

Prerequisite: MATH 310

#### MATH 565: *Number Theory*

A study of the integers, including unique factorization, congruences, and quadratic reciprocity. Other topics may include finite fields, higher reciprocity laws, and algebraic number theory.
Units: 6.

Prerequisite: MATH 300

#### MATH 590: *Tutorial Studies in Mathematics*

Advanced work in mathematics on topics not covered in regular offerings.
Units: 1 TO 98.

Prerequisite: Counter Registration Required.

#### MATH 591: *Directed Study in Mathematics*

Directed study follows a syllabus set primarily by the instructor to meet the needs or interests of an individual student or small group of students. The main goal of directed study is knowledge or skill acquisition, not research or creative work.
Units: 1 TO 98.

Prerequisite: Counter Registration Required.

#### MATH 599: *Independent Study in Mathematics*

Guided independent study of an advanced topic in undergraduate mathematics or supervised work on an undergraduate research project, generally culminating in a final presentation and/or paper.
Units: 1 TO 98.

Prerequisite: Counter Registration Required.

#### MATH 600: *History of Mathematics*

A study of the history of mathematics from the ancient Greeks through the present, emphasizing the role of mathematics in scientific advances, the work of great mathematicians, and the modern branching of the subject into a multitude of specialties.
Units: 6.

Prerequisite: MATH 310

#### MATH 690: *Tutorial Studies in Mathematics*

Advanced work in mathematics on topics not covered in regular offerings.
Units: 1 TO 98.

Prerequisite: Counter Registration Required.

#### MATH 691: *Directed Study in Mathematics*

Directed study follows a syllabus set primarily by the instructor to meet the needs or interests of an individual student or small group of students. The main goal of directed study is knowledge or skill acquisition, not research or creative work.
Units: 1 TO 98.

Prerequisite: Counter Registration Required.