Introduction to differential and integral calculus for functions of one variable. The heart of calculus is the study of rates of change. Differential calculus concerns the process of finding the rate at which a quantity is changing (the derivative). Integral calculus reverses this process. Information is given about the derivative, and the process of integration finds the "integral," which measures accumulated change. This course aims to develop a thorough understanding of the concepts of differentiation and integration, and covers techniques and applications of differentiation and integration of algebraic, trigonometric, logarithmic, and exponential functions. MATH 115 is an introductory course designed for students who have not seen calculus before.

Units: 1

Max Enrollment: 25

Prerequisites: Fulfillment of the basic skills component of the Quantitative Reasoning requirement. Not open to students who have completed MATH 116, MATH 120, MATH 205 or the equivalent. Not open to students whose placement is MATH 205 or MATH 206.

Instructor: Staff

Distribution Requirements: MM - Mathematical Modeling and Problem Solving

Typical Periods Offered: Summer; Spring; Fall

Semesters Offered this Academic Year: Spring; Fall

The course begins with applications and techniques of integration. It probes notions of limit and convergence and adds techniques for finding limits. Half of the course covers infinite sequences and series, where the basic question is, What meaning can we attach to a sum with infinitely many terms and why might we care? The course can help students improve their ability to reason abstractly and also teaches important computational techniques. Topics include integration techniques, l'Hôpital's rule, improper integrals, geometric and other applications of integration, infinite series, power series, and Taylor series. MATH 116 is the appropriate first course for many students who have had AB calculus in high school.

Units: 1

Max Enrollment: 25

Prerequisites: MATH 115, Math 115Z, or the equivalent. Not open to students who have completed MATH 120 or MATH 205. Not open to students whose math placement is MATH 206.

Instructor: Staff

Distribution Requirements: MM - Mathematical Modeling and Problem Solving

Typical Periods Offered: Summer; Spring; Fall

Semesters Offered this Academic Year: Spring; Fall

Notes:

This course is a variant of MATH 116 for students who have a thorough knowledge of the techniques of differentiation and integration, and familiarity with inverse trigonometric functions and the logarithmic and exponential functions. It includes a rigorous and careful treatment of limits, sequences and series, Taylor's theorem, approximations and numerical methods, Riemann sums, improper integrals, l'Hôpital's rule, and applications of integration.

Units: 1

Max Enrollment: 25

Prerequisites: Open by permission of the department to students who have completed a year of high school calculus. Students who have studied Taylor series should elect MATH 205. Not open to students who have completed MATH 116, MATH 205 or the equivalent.

Instructor: Diesl

Distribution Requirements: MM - Mathematical Modeling and Problem Solving

Typical Periods Offered: Fall

Semesters Offered this Academic Year: Fall

Notes:

How can a candidate in a political race win the majority of votes yet lose the election? How can two competing candidates interpret the same statistic as being in their favor? How can the geometry of the voting district disenfranchise entire groups of voters? Can we quantify the power the President of the United States has? In this course, we will look at the mathematics behind these and related questions that arise in politics. We will study topics such as fairness, voting paradoxes, social choice, game theory, apportionment, gerrymandering, and data interpretation. The goal of the class will be to illustrate the importance of rigorous reasoning in various social and political processes while providing an introduction to some fascinating mathematics.

Units: 1

Max Enrollment: 15

Crosslisted Courses: MATH 123

Prerequisites: None.

Instructor: Ismar Volic

Distribution Requirements: MM - Mathematical Modeling and Problem Solving

Typical Periods Offered: Every other year

Semesters Offered this Academic Year: Spring

Notes:

How can a candidate in a political race win the majority of votes yet lose the election? How can two competing candidates interpret the same statistic as being in their favor? How can the geometry of the voting district disenfranchise entire groups of voters? Can we quantify the power the President of the United States has? In this course, we will look at the mathematics behind these and related questions that arise in politics. We will study topics such as fairness, voting paradoxes, social choice, game theory, apportionment, gerrymandering, and data interpretation. The goal of the class will be to illustrate the importance of rigorous reasoning in various social and political processes while providing an introduction to some fascinating mathematics.

Units: 1

Max Enrollment: 15

Prerequisites: None. Open to Firstyears only.

Instructor: Ismar Volic

Distribution Requirements: MM - Mathematical Modeling and Problem Solving

Other Categories: FYS - First Year Seminar

Typical Periods Offered: Every other year

Semesters Offered this Academic Year: Not Offered

Notes:

This course is intended for students who are interested in mathematics and its applications in economics and finance. The following topics will be covered: mathematical models in economics, market equilibrium, first and second order recurrences, the cobweb model, profit maximization, derivatives in economics, elements of finance, constrained optimization, Lagrangians and the consumer, microeconomic applications, business cycles, European and American options, call and put options, Black-Scholes analysis.

Units: 1

Max Enrollment: 25

Prerequisites: MATH 116 or the equivalent.

Instructor: Bu

Distribution Requirements: MM - Mathematical Modeling and Problem Solving

Semesters Offered this Academic Year: Fall

Notes:

Most real-world systems that one may want to model, whether in the natural or in the social sciences, have many interdependent parameters. To apply calculus to these systems, we need to extend the ideas and techniques of single-variable Calculus to functions of more than one variable. Topics include vectors, matrices, determinants, polar, cylindrical, and spherical coordinates, curves, partial derivatives, gradients and directional derivatives, Lagrange multipliers, multiple integrals, vector calculus: line integrals, surface integrals, divergence, curl, Green's Theorem, Divergence Theorem, and Stokes’ Theorem.

Units: 1

Max Enrollment: 25

Prerequisites: MATH 116, MATH 120, or the equivalent. Not open to students who have completed PHYS 216.

Instructor: Staff

Distribution Requirements: MM - Mathematical Modeling and Problem Solving

Typical Periods Offered: Spring; Fall

Semesters Offered this Academic Year: Spring; Fall

Notes:

Linear algebra is one of the most beautiful subjects in the undergraduate mathematics curriculum. It is also one of the most important with many possible applications. In this course, students learn computational techniques that have widespread applications in the natural and social sciences as well as in industry, finance, and management. There is also a focus on learning how to understand and write mathematical proofs and an emphasis on improving mathematical style and sophistication. Topics include vector spaces, subspaces, linear independence, bases, dimension, inner products, linear transformations, matrix representations, range and null spaces, inverses, and eigenvalues.

Units: 1

Max Enrollment: 25

Prerequisites: MATH 205 or MATH 215; or, with permission of the instructor, MATH 116, MATH 120, or the equivalent. At most two of the three courses MATH 206, MATH 210, and MATH 215 can be counted toward the major or minor.

Instructor: Fall - Lange; Spring - H. Wang, Lauer, Schultz;

Distribution Requirements: MM - Mathematical Modeling and Problem Solving

Typical Periods Offered: Spring; Fall

Semesters Offered this Academic Year: Fall; Spring

Notes:

This course is designed to examine the degree to which a function can be determined by an algebraic relationship it has with its derivative(s) --- a so-called ordinary differential equation (ODE). For instance, can one completely catalog all functions which equal their own derivative? In service of developing techniques for solving certain classes of differential equations, some fundamental notions from linear algebra and complex numbers are presented. Differential equation topics include modeling with and solving first- and second-order ODEs, separable ODEs, and a discussion of higher order and non-linear ODEs; linear algebra topics include solving systems via elementary row operations, bases, dimension, determinants, column space, and eigenvalues/vectors.

Units: 1

Max Enrollment: 25

Prerequisites: MATH 116, MATH 120, or the equivalent. Not open to students who have completed MATH 210.

Instructor: Fall - Fernandez, H. Wang; Spring - Tannenhauser

Distribution Requirements: MM - Mathematical Modeling and Problem Solving

Typical Periods Offered: Spring; Fall

Semesters Offered this Academic Year: Spring; Fall

Notes:

Probability is the mathematics of uncertainty. We begin by developing the basic tools of probability theory, including counting techniques, conditional probability, and Bayes's Theorem. We then survey several of the most common discrete and continuous probability distributions (binomial, Poisson, uniform, normal, and exponential, among others) and discuss mathematical modeling using these distributions. Often we cannot calculate probabilities exactly, and we need to approximate them. A powerful tool here is the Central Limit Theorem, which provides the link between probability and statistics. Another strategy when exact results are unavailable is simulation. We examine Markov chain Monte Carlo methods, which offer a means of simulating from complicated distributions.

Units: 1

Max Enrollment: 25

Crosslisted Courses: STAT 220

Prerequisites: MATH 205

Instructor: Tannenhauser

Distribution Requirements: MM - Mathematical Modeling and Problem Solving

Semesters Offered this Academic Year: Fall

Notes:

Number theory is the study of the most basic mathematical objects: the natural numbers (1, 2, 3, etc.). It begins by investigating simple patterns: for instance, which numbers can be written as sums of two squares? Do the primes go on forever? How can we be sure? The patterns and structures that emerge from studying the properties of numbers are so elegant, complex, and important that number theory has been called "the Queen of Mathematics." Once studied only for its intrinsic beauty, number theory has practical applications in cryptography and computer science. Topics include the Euclidean algorithm, modular arithmetic, Fermat's and Euler's Theorems, public-key cryptography, quadratic reciprocity. MATH 223 has a focus on learning to understand and write mathematical proofs; it can serve as valuable preparation for MATH 305.

Units: 1

Max Enrollment: 25

Prerequisites: MATH 116, MATH 120, or the equivalent; or CS 230 together with permission of the instructor.

Instructor: Phillips

Distribution Requirements: MM - Mathematical Modeling and Problem Solving

Typical Periods Offered: Spring

Semesters Offered this Academic Year: Spring

Notes:

Number theory is the study of the most basic mathematical objects: the natural numbers (1, 2, 3, etc.). It begins by investigating simple patterns: for instance, which numbers can be written as sums of two squares? Do the primes go on forever? How can we be sure? The patterns and structures that emerge from studying the properties of numbers are so elegant, complex, and important that number theory has been called "the Queen of Mathematics." Once studied only for its intrinsic beauty, number theory has practical applications in cryptography and computer science. Topics include the Euclidean algorithm, modular arithmetic, Fermat's and Euler's Theorems, public-key cryptography, quadratic reciprocity. MATH 223 has a focus on learning to understand and write mathematical proofs; it can serve as valuable preparation for MATH 305.

Units: 1

Max Enrollment: 15

Prerequisites: Open to first-year students only.

Instructor: Lange

Distribution Requirements: MM - Mathematical Modeling and Problem Solving

Other Categories: FYS - First Year Seminar

Typical Periods Offered: Fall

Semesters Offered this Academic Year: Not Offered

Notes:

Combinatorics is the art of counting possibilities: for instance, how many different ways are there to distribute 20 apples to 10 kids? Graph theory is the study of connected networks of objects. Both have important applications to many areas of mathematics and computer science. The course will be taught emphasizing creative problem-solving as well as methods of proof, such as proof by contradiction and induction. Topics include: selections and arrangements, generating functions, recurrence relations, graph coloring, Hamiltonian and Eulerian circuits, and trees.

Units: 1

Max Enrollment: 25

Prerequisites: MATH 116, MATH 120, or the equivalent; or CS 230 together with permission of the instructor.

Instructor: Trenk (Fall & Spring), Chan (Spring), Grodzicki (Fall)

Distribution Requirements: MM - Mathematical Modeling and Problem Solving

Typical Periods Offered: Spring; Fall

Semesters Offered this Academic Year: Fall; Spring

Notes:

Units: 1

Max Enrollment: 25

Prerequisites:

Instructor:

Semesters Offered this Academic Year: Fall; Spring

Notes:

Real analysis is the study of the rigorous theory of the real numbers, Euclidean space, and calculus. The goal is to thoroughly understand the familiar concepts of continuity, limits, and sequences. Topics include compactness, completeness, and connectedness; continuous functions; differentiation and integration; limits and sequences; and interchange of limit operations as time permits.

Units: 1

Max Enrollment: 25

Prerequisites: MATH 205 and MATH 206.

Instructor: Volic (Fall), H. Wang (Spring)

Distribution Requirements: MM - Mathematical Modeling and Problem Solving

Typical Periods Offered: Spring; Fall

Semesters Offered this Academic Year: Fall; Spring

Notes:

**Topic for 2020-2021: Lebesgue Theory, Functional Analysis, and Differential Forms**

**Topic for 2020-2021: Lebesgue Theory, Functional Analysis, and Differential Forms**

This course is a continuation of MATH 302, focusing on further exploration of integration and related ideas. One part of the course will be devoted to Lebesgue theory where integration of more general functions over more general domains than those encountered in MATH 302 is considered. Connections between analysis and linear algebra will be explored during the functional analysis part of the class. This will include a study of normed spaces, such as function and Hilbert spaces, and linear maps between them. Applications to the Fourier series will also be discussed. Time permitting, a generalization of Stokes' Theorem, leading to an exciting interplay between integration and algebra, will also be covered.

Units: 1

Max Enrollment: 15

Prerequisites: MATH 302

Instructor: Diesl

Distribution Requirements: MM - Mathematical Modeling and Problem Solving

Typical Periods Offered: Spring

Semesters Offered this Academic Year: Spring

Notes: This is a topics course and can be taken more than once for credit as long as the topic is different each time.

In this course, students examine the structural similarities between familiar mathematical objects such as number systems, matrix sets, function spaces, general vector spaces, and mod n arithmetic. Topics include groups, rings, fields, homomorphisms, normal subgroups, quotient spaces, isomorphism theorems, divisibility, and factorization. Many concepts generalize number theoretic notions such as Fermat's little theorem and the Euclidean algorithm. Optional subjects include group actions and applications to combinatorics.

Units: 1

Max Enrollment: 25

Prerequisites: MATH 206

Instructor: Lauer (Fall), Schultz (Spring)

Distribution Requirements: MM - Mathematical Modeling and Problem Solving

Typical Periods Offered: Spring; Fall

Semesters Offered this Academic Year: Fall; Spring

Notes:

**Topic for 2020-21: **

**Topic for 2020-21: ****Algebraic Geometry**

Units: 1

Max Enrollment: 15

Prerequisites: MATH 305

Instructor: Diesl

Distribution Requirements: MM - Mathematical Modeling and Problem Solving

Semesters Offered this Academic Year: Not Offered

Notes: This is a topics course and can be taken more than once for credit as long as the topic is different each time.

This course covers some basic notions of point-set topology, such as topological spaces, metric spaces, connectedness and compactness, Heine-Borel Theorem, quotient spaces, topological groups, groups acting on spaces, homotopy equivalences, separation axioms, Euler characteristic, and classification of surfaces. Additional topics include the study of the fundamental group (time permitting).

Units: 1

Max Enrollment: 15

Prerequisites: MATH 302. Corequisite - MATH 305.

Instructor: Schultz

Distribution Requirements: MM - Mathematical Modeling and Problem Solving

Typical Periods Offered: Fall

Semesters Offered this Academic Year: Fall

Notes:

This course will introduce students to aspects of set theory and formal logic. The notion of set is one of the fundamental notions of modern mathematics. In fact, other mathematical notions, such as function, relation, number, etc., can be represented in terms of purely set theoretical notions, and their basic properties can be proved using purely set theoretic axioms. The course will include the Zermelo-Fraenkel axioms for set theory, the Axiom of Choice, transfinite arithmetic, ordinal numbers, and cardinal numbers.

Units: 1

Max Enrollment: 15

Prerequisites: MATH 302 or MATH 305.

Instructor: Lange

Distribution Requirements: MM - Mathematical Modeling and Problem Solving

Typical Periods Offered: Every other year

Semesters Offered this Academic Year: Not Offered

Notes:

Complex analysis is the study of the differential and integral calculus of functions of a complex variable. Complex functions have a rich and tightly constrained structure: for example, in contrast with real functions, a complex function that has one derivative has derivatives of all orders and even a convergent power series. This course develops the theory of complex functions, leading up to Cauchy's theorem and its consequences, including the theory of residues. While the primary viewpoint is calculus, many of the essential insights come from geometry and topology, and can be used to prove results such as the Fundamental Theorem of Algebra.

Units: 1

Max Enrollment: 15

Prerequisites: MATH 302

Instructor: Chang

Distribution Requirements: MM - Mathematical Modeling and Problem Solving

Semesters Offered this Academic Year: Fall

Notes:

Differential geometry has two aspects. Classical differential geometry, which shares origins with the beginnings of calculus, is the study of local properties of curves and surfaces. Local properties are those properties which depend only on the behavior of the curve or the surface in a neighborhood of point. The other aspect is global differential geometry: here we see how these local properties influence the behavior of the entire curve or surface. The main idea is that of curvature. What is curvature? It can be intrinsic or extrinsic. What's the difference? What does it mean to have greater or smaller (or positive or negative) curvature? We will answer these questions for surfaces in three-space, as well as for abstract manifolds. Topics include curvature of curves and surfaces, first and second fundamental forms, equations of Gauss and Codazzi, the fundamental theorem of surfaces, geodesics, and surfaces of constant curvature.

Units: 1

Max Enrollment: 15

Prerequisites: MATH 302.

Instructor: Kerr

Distribution Requirements: MM - Mathematical Modeling and Problem Solving

Typical Periods Offered: Every other year

Semesters Offered this Academic Year: Not Offered

Notes: Ann E. Maurer '51 Speaking Intensive Course.

Geometry is concerned with the properties of rigid shapes (either planar, or in higher dimensions). When confronted with a geometric question (e.g. computing a tangent line to a curve), one often thinks first of using the tools of calculus. However, this is not the only option. Many of the most common geometric objects (e.g. lines, circles, etc.) can be defined by polynomial equations, and can therefore be studied using algebra. In this course, we will explore this connection. Specifically, we will see how one can associate a certain ring to such a geometric object, and how questions about geometry can then be translated into questions about the algebra of this ring. This course will expand upon the material learned in Math 305, with a view toward such connections to geometry. Both computational and theoretical topics will be addressed.

Students can NOT satisfy the presentation requirement in this course.

Units: 1

Max Enrollment: 25

Prerequisites: MATH 305

Instructor: Alexander Diesl

Distribution Requirements: MM - Mathematical Modeling and Problem Solving

Typical Periods Offered: Spring

Semesters Offered this Academic Year: Spring

Notes:

Linear algebra at this more advanced level is a basic tool in many areas of mathematics and other fields. The course begins by revisiting some linear algebra concepts from MATH 206 in a more sophisticated way, making use of the mathematical maturity picked up in MATH 305. Such topics include vector spaces, linear independence, bases, and dimensions, linear transformations, and inner product spaces. Then we will turn to new notions, including dual spaces, reflexivity, annihilators, direct sums and quotients, tensor products, multilinear forms, and modules. One of the main goals of the course is the derivation of canonical forms, including triangular form and Jordan canonical forms. These are methods of analyzing matrices that are more general and powerful than diagonalization (studied in MATH 206). We will also discuss the spectral theorem, the best example of successful diagonalization, and its applications.

Units: 1

Max Enrollment: 15

Prerequisites: MATH 305.

Instructor: Lange

Distribution Requirements: MM - Mathematical Modeling and Problem Solving

Typical Periods Offered: Every other year

Semesters Offered this Academic Year: Spring

Notes:

Graph Theory has origins both in recreational mathematics problems (i.e., puzzles and games) and as a tool to solve practical problems in many areas of society. Topics covered will include trees and distance, connectivity and paths, network flow, graph coloring, directed graphs, and tournaments. In addition, students will gain a sense of what it means to do research in graph theory.

Units: 1

Max Enrollment: 15

Prerequisites: MATH 225 and either MATH 305 or MATH 302. Students lacking one of these prerequisites may enroll with permission of the instructor.

Instructor: Trenk

Distribution Requirements: MM - Mathematical Modeling and Problem Solving

Typical Periods Offered: Every other year

Semesters Offered this Academic Year: Fall

Notes: Majors can fulfill the major presentation requirement in this course in 2020-21.

This course covers questions of enumerations, existence, and construction in combinatorics, building on the fundamental ideas introduced in MATH 225. Topics include: famous number families, combinatorial and bijective proofs, counting under equivalence, combinatorics on graphs, combinatorial designs, error-correcting codes, and partially ordered sets.

Units: 1

Max Enrollment: 15

Prerequisites: MATH 225. Corequisite - MATH 305.

Instructor: Trenk

Distribution Requirements: MM - Mathematical Modeling and Problem Solving

Typical Periods Offered: Every other year

Semesters Offered this Academic Year: Not Offered

Notes:

In this course, students will leverage their prior mathematical knowledge to communicate complex mathematical ideas to audiences ranging from the general public to other mathematicians. Each week, students will research a new topic and produce a piece of writing explaining this topic in a specific context. Assignments may include research abstracts, book reviews, interviews with mathematicians, newspaper articles, and technical documentation. Class time will be devoted to discussing the mathematical content behind each assignment as well as workshopping students' writing. This course will give students the opportunity to ground (and expand on) the mathematics they have learned and make connections across the discipline. Moreover, this course's unique format will help students develop their research and independent learning skills.

Units: 1

Max Enrollment: 12

Prerequisites: Math 302 and Math 305, or permission of the instructor.

Instructor: Lange

Distribution Requirements: MM - Mathematical Modeling and Problem Solving

Other Categories: CSPW - Calderwood Seminar in Public Writing

Typical Periods Offered: Spring

Semesters Offered this Academic Year: Spring

Notes:

**Topic for 2020-21: **

**Topic for 2020-21: ****Mathematical Methods for the Physical Sciences**** (Fall)**

Units: 1

Max Enrollment: 15

Prerequisites: MATH 302 or MATH 305.

Instructor: Chang (Fall);

Distribution Requirements: MM - Mathematical Modeling and Problem Solving

Typical Periods Offered: Fall

Semesters Offered this Academic Year: Fall

Notes: Majors can fulfill the major presentation requirement in this course in 2020-21**. **This is a topics course and can be taken more than once for credit as long as the topic is different each time.

Units: 1

Max Enrollment: 25

Prerequisites: Permission of the instructor. Open to juniors and seniors.

Typical Periods Offered: Spring; Fall

Semesters Offered this Academic Year: Fall; Spring

Units: 1

Max Enrollment: 25

Prerequisites: Permission of the department.

Typical Periods Offered: Spring; Fall

Semesters Offered this Academic Year: Fall; Spring

Notes: Students enroll in Senior Thesis Research (360) in the first semester and carry out independent work under the supervision of a faculty member. If sufficient progress is made, students may continue with Senior Thesis (370) in the second semester.

Units: 1

Max Enrollment: 25

Prerequisites: MATH 360 and permission of the department.

Typical Periods Offered: Spring; Fall

Semesters Offered this Academic Year: Fall; Spring

Notes: Students enroll in Senior Thesis Research (360) in the first semester and carry out independent work under the supervision of a faculty member. If sufficient progress is made, students may continue with Senior Thesis (370) in the second semester.