MATH - Mathematics

#### MATH 103: *Preparation for Calculus*

An exploration of functions, including polynomial, rational, exponential, logarithmic, and trigonometric functions. This course is designed to prepare students for the study of calculus at Lawrence.

Units: 6

Prerequisite: Minimum score on ALEKS diagnostic exam, as set by the department.

#### STAT 107: *Principles of 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. Students with credit for MATH 140 or equivalent, or BIOL 170, or AP Statistics

**should not****take this course and should instead consider STAT 255.**Units: 6

#### 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: 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 suitable AP or IB score and minimum score on ALEKS online diagnostic exam as set by department

#### 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

#### STAT 205: *Data-Scientific Programming*

An introduction to programming with emphasis on learning from data in order to gain useful insights. Topics focus on elementary programming concepts in the R language and the necessary tools to handle, analyze and interpret data. This course will be taught in a workshop format, and students will complete regular assignments and a final project that provide hands-on programming/analysis experiences.

Units: 6

Also listed as Computer Science 205

Prerequisite: One prior course MATH, STAT, or CMSC course, or BIOL 170, or consent of instructor

#### STAT 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

Also listed as Linguistics 208, Computer Science 208

Prerequisite: CMSC 150, or CMSC 205, or CMSC 210, or consent of instructor

#### 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 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

#### STAT 255: *Statistics for Data Science*

This course introduces modern statistical techniques in the context of predictive inference and modeling. Topics will include data analysis techniques such as linear and logistic regression, ANOVA, nonparametric methods, and computational approaches such as cross-validation and bootstrapping. Statistical software will be used frequently. This class will involve regular in-class and out-of-class assignments as well as exams and quizzes.

Units: 6

Prerequisite: STAT 107, BIOL 170, or MATH 140, or instructor permission

#### MATH 340: *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

Also listed as Statistics 340

Prerequisite: MATH 200, and either MATH 220 or MATH 230

#### STAT 340: *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

Also listed as Mathematics 340

Prerequisite: MATH 200, and either MATH 220 or MATH 230

#### 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.

#### STAT 390: *Tutorial Studies in Statistics*

Advanced work in statistics 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.

#### STAT 391: *Directed Study in Statistics*

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 350

#### STAT 405: *Advanced Data Computing*

This course builds on CMSC/STAT 205, providing a deeper exploration of statistical computing in R. Topics might include efficient programming techniques, parallelization, statistical algorithms, advanced data visualization, and creation of R packages. Statistical software will be used extensively in this course. The class will be taught in an interactive lab-based format. There will be regular assignments and a project.

Units: 6

Also listed as Computer Science 405

Prerequisite: CMSC 205

#### 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 250, one CMSC (computer science) course recommended

#### 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 230 and 250, or 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/STAT 340

#### STAT 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/STAT 340

#### STAT 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/STAT 340

#### STAT 455: *Advanced Statistical Modeling*

This course expands on STAT 255, and introduces more sophisticated models, meant to capture complicated correlation structure in data. Topics might include generalized linear models, mixed-effects models, hierarchical models, spatial models, and time series. The course will involve the use of statistical software. There will be regular assignments, exams, and possibly projects.

Units: 6

Prerequisite: STAT 255 or instructor permission

#### MATH 505: *Group Theory*

The study of groups, group actions, and important examples of finite and matrix groups. Specific topics will
include group homomorphisms, quotient groups, isomorphism theorems, Lagrange's Theorem, the Orbit-
Stabilizer Theorem, and the Sylow Theorems.

Units: 6

Prerequisite: MATH 230 and MATH 250; or MATH 300 and consent of instructor

#### MATH 510: *Real Analysis*

A study of concepts in mathematical analysis, including convergence of sequences and series, continuity, differentiation, integration, and metric spaces.

Units: 6

Prerequisite: MATH 230 and MATH 250

#### 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 230 and 250, or 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 230 and MATH 250, or 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 230 and 250

#### 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 230 and 250

#### 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 in Algebra and 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).

*Topic for Fall 2021: Computational Algebraic Geometry*

This course gives an introduction to computational algebraic geometry. Often, we study a geometric space by studying the functions on that space, and in this course we see the interplay between the two. Some highlights include the Hilbert Basis Theorem, Elimination Theory, and the Nullstellensatz. We will see applications in both robotics and the invariant theory of finite groups.

Units: 6

Prerequisite: MATH 230 and MATH 250; or 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 230 and 250

#### 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 230 and 250

#### 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*

Units: 1 TO 98

Prerequisite: Counter Registration Required.

#### STAT 591: *Directed Study in Statistics*

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*

Units: 1 TO 98

Prerequisite: Counter Registration Required.

#### MATH 699: *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.