# Mathematics Courses

## MATH 10110. Principles of Finite Mathematics

(3-0-3)
For students in arts and letters. For first-year students who lack the necessary background for MATH 10120. (Students who take this course cannot take MATH 10120.) Topics include the fundamental principles of counting systematically, probability, statistics, linear programming, optimization problems, game theory and mathematical finance, population problems, and coding information. There is a wealth of applications of these topics to contemporary social, economic, and political issues appealing to liberal arts students. Also, these topics broaden a student’s mathematical horizon in an interesting direction not covered by calculus, which deals mostly with continuous models.

## MATH 10120. Finite Mathematics

(3-0-3)
For students in arts and letters. Topics include the fundamental principles of counting systematically, probability, statistics, linear programming, optimization problems, game theory, and mathematical finance. Other topics that may be covered include population problems, difference equations and modeling, and coding information. There is a wealth of applications of these topics to contemporary social, economic, and political issues appealing to liberal arts students. Also, these topics broaden a student's mathematical horizon in an interesting direction not covered by calculus, which deals mostly with continuous models.

## MATH 10130. Beginning Logic

(3-0-3)
For students in arts and letters. Provide the students with some formal tools for analyzing arguments. By writing proofs in a formal system, students see the importance of stating the basic premises in an argument and giving intermediate
steps that lead to the conclusion. They learn strategies for thinking up proofs. They see that proof checking is, in principle, something that a machine could do. Students learn truth tables and see an effective procedure that they could apply to any argument stated in propositional logic, to determine whether the conclusion follows logically from the premises. There is nothing like truth tables for predicate logic. Students get to experience doing what mathematicians do, trying to determine whether a particular conclusion follows from some premises by searching simultaneously for a proof or a counterexample. Writing papers gives students an opportunity to explore other topics in logic of their interest.

## MATH 10170. Mathematics in Sport

(3-0-3)
There are many applications of mathematics in sport. In this course, we explore a number of topics. We see how the theory of ranking and game theory can be applied in sport. We look at applications of basic number theory and combinatorics in tournament scheduling. We also see how elementary calculus provides insight into the biomechanics of human movement.

## MATH 10240. Principles of Calculus

(3-0-3)
For students in arts and letters. Note: Credit is not given for both this course and any other calculus course. A terminal course introducing the principles of calculus. Topics include basic properties of functions, derivatives and integrals, with interesting real-life applications throughout. This course is not intended to prepare students for more advanced work in calculus.

## MATH 10250. Elements of Calculus I

(3-0-3)
For students in arts and letters, architecture, or business. A study of basic calculus as part of a liberal education. It emphasizes conceptual learning and stresses the connections between mathematics and modern society. Topics include functions, limits, derivatives, and an introduction to integral, with interesting real-life applications throughout. Students are familiarized with the many different interpretations of the derivative as a rate of change, and the integral as a total rate of change. This enables them to learn and practice modeling in a variety of situations from economics the social and the life sciences.

## MATH 10260. Elements of Calculus II for Business

(3-0-3)
Prerequisite: MATH 10250 or MATH 10350 or MATH 10550 or MATH 10850
Credit is not given for both MATH 10280 and either of the following courses: MATH 10260 and MATH 10360. For students in business. An introduction to mathematical concepts, techniques, and ideas that are useful in understanding and solving problems that arise in economics and business. Most mathematical concepts are introduced through interesting business problems. Furthermore, by using available computer technology, real-life problems that may lead to non-trivial computations and graphics are considered. Topics include integration, differential equations, Taylor polynomial approximations, unconstrained and constrained optimization for functions of several variables, probability and statistics, with interesting real-life applications throughout.

## MATH 10270. Mathematics in Architecture: Mathematical Excursions to the World’s Great Buildings

(3-0-3)
Prerequisite: MATH 10250 or MATH 10550 or MATH 10850
This is a second mathematics course for arts and letters and architecture students. As the Roman architect Vitruvius pointed out 2000 years ago, architecture is a broad enterprise bringing together virtually all the elements of the human experience: spirituality, intelligence and creativity, economics, politics and sociology, as well as aesthetics, structural engineering, and mathematics. The agenda of this course has a focus on the last three: aesthetics, structural aspects, and related mathematics. The architecture of the world’s great historic buildings will be the environment in which the narrative of this course is developed. The aesthetic and structural properties of these structures will be described following a chronological line. Whenever the opportunity presents itself, this discussion will be informed by basic modern mathematics (such as geometry, trigonometry, and calculus). While the mathematical comments about the buildings considered are standard by today’s criteria, they would (for the most part) have been beyond the reach of the architects who built them.

## MATH 10350. Calculus A

(3-1-4)
Corequisite: MATH 12350
This is the first course of the two-semester Calculus sequence for life science and social science majors. Calculus A emphasizes the process of problem solving and application of calculus to the natural sciences, and requires students to think deeper about the concepts covered. Students will acquire basic skills needed for a quantitative approach to scientific problems. The course introduces the mathematics needed to study change in a quantity. Topics include functions, limits, continuity, rate of change of functions, integrals, graphing and their applications.

## MATH 10360. Calculus B

(3-1-4)
Prerequisite: MATH 10350 or MATH 10550 or MATH 10850
Corequisite: MATH 12360
This is the second course of the two-semester Calculus sequence for Life and Social science majors. Calculus B emphasizes the process of problem solving and application of calculus to the natural sciences, and requires students to think deeper about the concepts covered. Students will acquire basic skills needed for quantitative approach to scientific problems. The course introduces the mathematics needed to study change in a quantity. Topics include integration techniques, application of integrals to physics, geometry and ecology, solution of differential equations and their applications, and Taylor series.

## MATH 10450. Honors Mathematics I

(4-0-4)
Corequisite: MATH 12450
This is a course that studies elementary calculus, as well as the necessary geometry, trigonometry and coordinate geometry, from within its "historical flow". The flow is historical, but the emphasis is on doing the mathematics in a pedagogically effective way. This means that the material is developed from within the relevant context: that of the contributions of Greek thinkers, Copernicus, Galileo, Kepler, Descartes, Newton and Leibniz. But, it also means that the notation is modern and that the material is selected so as to cover the basics of the subject. Calculus is developed by fusing essential insights of both Leibniz and Newton together into a complete "short calculus". This starts from scratch, is free of most theoretical "baggage", and concentrates on the intuitive grasp of the basic elements. What will have emerged is a basic mathematics course that is surrounded by the important scientific concerns of the times.

## MATH 10460. Honors Mathematics II

(4-0-4)
Prerequisite: MATH 10450
Corequisite: MATH 12460
The course starts with the central aspects of Newton’s Principia Mathematica, arguably, along with Darwin’s Origin of Species, the most influential book of science ever written. The course pursues Newton’s study of centripetal force and, following his thoughts, develops the fundamental equations of planetary motion. Calculus is present, but it is hidden within Newton’s geometric approach. A quick overview of conventional differential calculus follows next as preparation for a detailed look at polar coordinates, polar functions, and polar calculus. The course concludes with the application of polar calculus to planetary motion. This modern elementary look at Newton’s account provides the students with a paradigmatic example of an application of mathematics to the study of the physical universe.

## MATH 10550. Calculus I

(3-1-4)
Corequisite: MATH 12550
For students in science and engineering. Topics include sets, functions, limits, continuity, derivatives, integrals, and applications. Also covered are transcendental functions and their inverses, infinite sequences and series, parameterized curves in the plane, and polar coordinates.

## MATH 10560. Calculus II

(3-1-4)
Prerequisite: MATH 10550 or MATH 10850
Corequisite: MATH 12560
For students in science and engineering. Topics include sets, functions, limits, continuity, derivatives, integrals, and applications. Also covered are transcendental functions and their inverses, infinite sequences and series, parameterized curves in the plane, and polar coordinates.

## MATH 10850. Honors Calculus I

(4-0-4)
Corequisite: MATH 12850
This is not your high school Calculus course. Aimed at highly engaged math students, Honors Calculus emphasizes the 'why' of mathematics as well as the 'how'. Specifically, it begins with a thorough introduction to mathematical reasoning and proofs and then proceeds to carefully develop the central topics (limits, differentiation, integration) of one variable Calculus. Whether you have had Calculus before or not, this course will challenge you in new ways. It will strongly enhance your critical thinking skills and provide you with a much more solid grounding for any future mathematics courses. Honors calculus courses are required for the honors math major, but they have much to offer for mathematically inclined students of all majors. Math majors with AP credit for Calculus 1 and 2 may count Honors Calculus 1 and 2 in place of the upper level math courses Introduction to Mathematical Reasoning (20630) and Real Analysis (30850).

## MATH 10860. Honors Calculus II

(4-0-4)
Prerequisite: MATH 10850
Corequisite: MATH 12860
Required of honors mathematics majors. A rigorous course in differential and integral calculus of one variable. Topics include an axiomatic formulation of the real numbers, mathematical induction, infima and suprema, functions, continuity, derivatives, integrals, infinite sequences and series, transcendental functions and their inverses, and applications. The course stresses careful mathematical definitions and emphasizes the proofs of the standard theorems of the subject.

## MATH 12350. Calculus A Tutorial

(0-1-0)
Corequisite: MATH 10350 Section: 01 CRN: 12112
Tutorial for the first course of the two-semester Calculus sequence for life science and social science majors. Calculus A emphasizes the process of problem solving and application of calculus to the natural sciences, and requires students to think deeper about the concepts covered. Students will acquire basic skills needed for a quantitative approach to scientific problems. The course introduces the mathematics needed to study change in a quantity. Topics include functions, limits, continuity, rate of change of functions, integrals, graphing and their applications.

## MATH 12360. Calculus B Tutorial

(0-1-0)
Corequisite: ATH 10360 Section: 01 CRN: 13450
Perfecting problem-solving skills in smaller group settings. Tutorial for the second course of the two-semester Calculus sequence for Life and Social science majors. Calculus B emphasizes the process of problem solving and application of calculus to the natural sciences, and requires students to think deeper about the concepts covered. Students will acquire basic skills needed for quantitative approach to scientific problems. The course introduces the mathematics needed to study change in a quantity. Topics include integration techniques, application of integrals to physics, geometry and ecology, solution of differential equations and their applications, and Taylor series. MATH 12450. Honors Mathematics Tutorial

## MATH 12450. Honors Mathematics Tutorial

(0-1-0)
Corequisite: MATH 10450 Section: 01 CRN: 14506
Perfecting problem-solving skills in smaller group settings.

## MATH 12460. Honors Mathematics II Tutorial

(0-1-0)
Corequisite: MATH 10460
Perfecting problem-solving skills in smaller group settings.

## MATH 12550. Calculus I Tutorial

(0-1-0)
Corequisite: MATH 10550 Section: 01 CRN: 12118
Perfecting problem-solving skills in smaller group settings.

## MATH 12560. Calculus II Tutorial

(0-1-0)
Corequisite: MATH 10560 Section: 01 CRN: 10657
Perfecting problem-solving skills in smaller group settings.

## MATH 12850. Honors Calculus I Tutorial

(0-1-0)
Corequisite: MATH 10850 Section: 01 CRN: 11321
Perfecting problem-solving skills in smaller group settings.

## MATH 12860. Honors Calculus II Tutorial

(0-1-0)
Corequisite: MATH 10860
Perfecting problem-solving skills in smaller group settings.

## MATH 13150. First Year Math Seminar

(3-0-3)
The goal of this new course is to give students a panoramic view of mathematics by considering a variety of topics displaying its enormous power and beauty. It aspires to present the first year students with an opportunity to participate in the excitement of discovering ideas of their own by practicing the mathematical way of thinking. This topical course will be rich, in content and context. It will stress the connections between mathematics and modern society by considering a wide variety of problems ranging from environmental and economic issues to social and political situations that can be modeled and solved by mathematical means. Also, by giving appropriate assignments and projects, it will allow students to make contributions in areas of their interest and expertise. “The Magic of Numbers” is the first theme of this seminar course.

## MATH 14360. Calculus B for Life and Social Sciences

(4-0-4)

This is a second-semester calculus course designed for biology and social science majors. It is required for all premedical students. Mathematics plays a prominent role in the understanding of complex systems in modern biology and social science. This course aims to develop basic mathematical literacy in students for this modern era. Students will acquire skills needed for a quantitative approach to scientific problems and the mathematics needed to study change in a quantity. Topics include integration techniques, solution of differential equations, geometric series, Taylor series and their applications to physics, geometry and ecology. Pre-requisites: First semester calculus or freshmen calculus. Note: this course is delivered fully online. The course design combines required live weekly meetings online with self-scheduled lectures, problems, assignments, and interactive learning materials. To participate, students will need to have a computer with webcam, reliable internet connection, and a quiet place to participate in live sessions. Students who will be on the Main campus or residing in the Michiana region may only enroll with permission from the DUS of the Department of Mathematics.

## MATH 20210. Computer Programming and Problem Solving

(3-0-3)
An introduction to solving mathematical problems using computer programming in high-level languages such as C.

## MATH 20480. Intro to Dyn Sys for Scientist

(3 -0- 3)
Prerequisite: MATH 10260 or MATH 10360 or MATH 10560
This is a one-semester course introducing students to linear algebra and ordinary differential equations by way of their scientific usage. The course serves as a gateway to more advance mathematical methods that are commonly used in contemporary scientific studies and their literature. Students will learn how to take a mathematical approach to various scientific problems, solve the resulting equations, and interpret the mathematical solution in the original context. There will be course projects and some usage of computing software. Topics include matrix algebra, eigenvalues and eigenvectors, vector-valued functions, linear and non-linear systems of differential equations, and phase portraits. The scientific topics include age-structured population growth, the Richardson’s theory of war, and infectious disease modeling.

## MATH 20550. Calculus III

(3-1-3.5)
Prerequisite: MATH 10560 or MATH 10860
Corequisite: MATH 22550
A comprehensive treatment of differential and integral calculus of several variables. Topics include space curves, surfaces, functions of several variables, partial derivatives, multiple integrals, line integrals, surface integrals, Stokes theorem, and applications.

## MATH 20570. Mathematical Methods in Physics I

(3-0-3.5)
Prerequisite: MATH 10560 or MATH 10860
Corequisite: MATH 22570
A study of methods of mathematical physics. Topics include matrices, linear algebra (including matrices and determinants), vector and tensor analysis, vector calculus, curvilinear coordinates, series, ordinary differential equations, partial differential equations, orthogonal functions and vector spaces, special functions (including Bessel, Legendre, and Hermite), calculus of variations, Fourier series, and group theory. Weekly tutorial sessions. Cross-listed with PHYS 20451 (271).

## MATH 20580. Introduction to Linear Algebra and Differential Equations

(3-1-3.5)
Prerequisite: MATH 20550
Corequisite: MATH 22580
An introduction to linear algebra and to first- and second-order differential equations. Topics include elementary matrices, LU factorization, QR factorization, the matrix of a linear transformation, change of basis, eigenvalues and eigenvectors, solving first-order differential equations and second-order linear differential equations, and initial value problems. This course is part of a two-course sequence that continues with Math 30650 (325). Credit is not given for both Math 20580 (228) and Math 20610 (221).

## MATH 20610. Linear Algebra

(3-0-3)
Open to all students. An introduction to vector spaces, matrices, linear transformations, inner products, determinants and eigenvalues. Emphasis is given to careful mathematical definitions and understanding the basic theorems of the subject. Credit is not given for both MATH 20610 (221) and MATH 20580 (228).

## MATH 20630. Introduction to Mathematical Reasoning

(3-0-3)
This course serves as a transition to upper-level math courses. The general subject is numbers of all sorts—integers, rationals, real numbers, etc. The main point will be to treat everything the way a mathematician would. That is, we will give precise definitions of the objects we consider and careful statements of the assertions we make about them. And, most importantly, we will justify our assertions by giving mathematical proofs. Topics covered include basic language of sets, common methods of proof, integers, factorization, modular arithmetic, rational numbers, completeness, real numbers, cardinality, limits, and continuity.

## MATH 20670. Mathematical Methods in Physics II

(3-0-3.5)
Corequisite: MATH 22670
A study of methods of mathematical physics. Topics include linear algebra (including matrices and determinants), vector and tensor analysis, vector calculus, curvilinear coordinates, series, ordinary differential equations, partial differential equations, orthogonal functions and vector spaces, special functions (including Bessel, Legendre, and Hermite), calculus of variations, Fourier series, and group theory. Weekly tutorial sessions.

## MATH 20750. Ordinary Differential Equations

(3-1-3.5)
Prerequisite: MATH 20610
Corequisite: MATH 22750
An introduction to differential equations. Topics include first-order equations, n-th order linear equations, power series methods, systems of first order linear equations, non-linear systems and stability. Credit is not given for both MATH
20750 (230) and MATH 30650 (325).

## MATH 20810. Honors Algebra I

(3-0-3)
A comprehensive treatment of vector spaces, linear transformations, inner products, determinants, eigenvalues, tensor and exterior algebras, spectral decompositions of finite-dimensional symmetric operators, and canonical forms of matrices. The course stresses careful mathematical definitions and emphasizes the proofs of the standard theorems of the subject.

## MATH 20820. Honors Algebra II

(3-0-3)
Prerequisite: MATH 20810
A comprehensive treatment of vector spaces, linear transformations, inner products, determinants, eigenvalues, tensor and exterior algebras, spectral decompositions of finite-dimensional symmetric operators, and canonical forms of matrices. The course stresses careful mathematical definitions and emphasizes the proofs of the standard theorems of the subject.

## MATH 20850. Honors Calculus III

(4-0-4)
Prerequisite: MATH 10860
Corequisite: MATH 22850
This is a two semester sequence integrating linear algebra and multivariable calculus, intended for students with a strong interest in and aptitude for math. Topics covered include algebra and geometry of vectors and linear transformations; continuity, differentiation and integration for functions of several variables; and various applications of all these things. Material will be covered in substantially greater depth than in similar courses (e.g. Calculus 3), with strong emphasis on both computation and careful mathematical argument (i.e. proofs).Honors Calculus 1 and 2 (Math 10850/60 is the normal prerequisite for Honors Calculus 3 and 4, but highly motivated students with exceptionally strong mathematical backgrounds may request permission (from the director of honors math) to skip the earlier sequence and enroll directly in Honors Calculus 3.

## MATH 20860. Honors Calculus IV

(4-0-4) Connolly
Prerequisite: MATH 20850
Corequisite: MATH 22860
Required of honors mathematics majors. A rigorous course in differential and integral calculus of several variables. Topics include functions of several variables, the inverse function theorem, partial derivatives, multiple integrals, line integrals, surface integrals, Stokes’ theorem, an introduction to ordinary differential equations and applications. The course stresses careful mathematical definitions and emphasizes the proofs of the standard theorems of the subject.

## MATH 22550. Calculus III Tutorial

(0-3-0)
Corequisite: MATH 20550 Section: 01 CRN: 10659
Perfecting problem-solving skills in smaller group settings.

## MATH 22570. Mathematical Methods in Physics I Tutorial

(0-1-0)
Corequisite: MATH 20570
Perfecting problem-solving skills in smaller group settings.

## MATH 22580. Linear Algebra and Differential Equations Tutorial

(0-1-0)
Corequisite: MATH 20580 Section: 01 CRN: 10662
Perfecting problem-solving skills in smaller group settings.

## MATH 22670. Mathematical Methods in Physics II Tutorial

(0-0-0)
Corequisite: MATH 20670
Required tutorial for MATH 20670.

## MATH 22750. Ordinary Differential Equations Tutorial

(0-1-0)
Corequisite: MATH 20750 Section: 01 CRN: 13452
Perfecting problem-solving skills in smaller group settings.

## MATH 22850. Honor Calculus III Tutorial

(0-1-0)
Corequisite: MATH 20850 Section: 01 CRN: 11325
Perfecting problem-solving skills in smaller group settings.

## MATH 22860. Honors Calculus IV Tutorial

(0-1-0)
Corequisite: MATH 20860
Perfecting problem-solving skills in smaller group settings.

## MATH 30210. Introduction to Operations Research

(3-0-3)
Prerequisites: MATH 20580 or MATH 20610 or MATH 20480 or MATH 20670 or PHYS 20452 or ACMS 20620 or MATH 20620 or MATH 20860
Operations research is the science (and art) of decision making. The success of a decision is frequently quantified through an achievement of certain goals (objectives) under restrictions imposed on various resources. Thus, mathematical models of operations research are frequently described as optimization problems, i.e. problems of minimization (maximization) of an objective function subject to various constraints. In this course we will consider optimization problems most frequently arising in various applications. Namely, linear programming problems, i.e. problems of minimization of a linear function subject to linear constraints. We will discuss in detail the simplex algorithm for solving linear programming problems and duality theory (optimality conditions).The Transportation Problem is a particular case of a linear programming which has various applications and specific algorithms. The Assignment Problem is an important example of so-called discrete optimization problems which can be solved using ideas of linear programming. We discuss both problems in detail.Game Theory is typically used in decision making with conflicting interests. We will discuss two person zero-sum games and show how the simplex algorithm can be used to solve them. If time permits, we will consider some applications of optimization theory to financial mathematics.

## MATH 30310. Coding Theory

(3-0-3)
This course provides an elementary treatment of the theory of error-correcting codes. Topics include an introduction to finite fields and vectors over finite fields, linear codes, encoding and decoding with a linear code, Hamming codes, perfect codes, codes based on Latin squares, cyclic codes, MDS codes, weight enumerators.

## MATH 30530. Introduction to Probability

(3-0-3)
Prerequisite: MATH 20850 or MATH 20550
An introduction to the theory of probability, with applications to the physical sciences and engineering. Topics include discrete and continuous random variables, conditional probability and independent events, generating functions, special discrete and continuous random variables, laws of large numbers and the central limit theorem. The course emphasizes computations with the standard distributions of probability theory and classical applications of them.

## MATH 30540. Mathematical Statistics

(3-0-3)
Prerequisite: MATH 30540 Mathematical Statistics
An introduction to mathematical statistics. Topics include distributions involved in random sampling, estimators and their properties, confidence intervals, hypothesis testing including the goodness-of-fit test and contingency tables, the general linear model and analysis of variance.

## MATH 30610. Introduction to Financial Mathematics

(3 -0- 3)
Prerequisite: MATH 30530 and (MATH 20610 or MATH 20550)
The course serves as a preparation for first actuarial exam in financial mathematics, known as Exam FM or Exam 2. The first part of the course deals with pricing of fixed income securities, such as bonds and annuities. The second part of the course can serve as an introduction to derivative securities such as options and futures. Although the amount of material for both parts is almost the same, exam FM devotes usually about 2/3 of its questions to Part 1. Therefore, about 2/3 of the course is devoted to Part 1. Topics covered: interest rates, annuities, loans and bonds, forwards, options, hedging, and swaps

## MATH 30650. Differential Equations

(3-0-3)
Prerequisite: MATH 20580
A second course in differential equations. Topics include higher order linear equations, numerical methods, Laplace transforms, linear systems, non-linear systems and stability, and an introduction to partial differential equations and Fourier series.

## MATH 30705. Algebraic Structures

(3-0-3)
An introduction to groups, rings and fields, homomorphisms, ideals, polynomial rings and extensions fields Emphasis is given to careful mathematical definitions and understanding the basic theorems of the subject.

## MATH 30710. Algebra

(3-0-3)
Prerequisite: MATH 20630 or MATH 20610
An introduction to groups, rings and fields. Topics include permutations, divisibility, modular arithmetic, cryptography, cyclic and dihedral groups, Lagrange's theorem, homomorphisms, ideals, integral and Euclidean domains, extension fields.

## MATH 30750. Real Analysis

(3-0-3)
Prerequisite: MATH 20630
A rigorous treatment of differential and integral calculus. Topics include a review of sequences and continuity, differentiability, Taylor’s theorem, integration, the fundamental theorem of Calculus, point-wise and uniform convergence, and power series. Additional topics are likely and will depend on the instructor. Emphasis throughout will be on careful mathematical definitions and thorough understanding of basic results.

## MATH 30810. Honors Algebra III

(3-0-3)
Prerequisite: MATH 20820
A comprehensive treatment of groups, polynomials, rings, homomorphisms, isomorphism theorems, field theory, and Galois theory. The course stresses careful mathematical definitions and emphasizes the proofs of the standard theorems of the subject.

## MATH 30820. Honors Algebra IV

(3-0-3)
Prerequisite: MATH 30810
Required of honors mathematics majors. A comprehensive treatment of groups, polynomials, rings, homomorphisms, isomorphism theorems, field theory, and Galois theory. The course stresses careful mathematical definitions and emphasizes the proofs of the standard theorems of the subject.

## MATH 30850. Honors Analysis I

(3-0-3)
Prerequisite: MATH 20860
Required of honors mathematics majors. An advanced course in mathematical analysis in one and several variables. Topics include an axiomatic formulation of the real and complex number systems, compactness, connectedness, metric spaces, limits, continuity, infinite sequences and series, differentiation, the Riemann-Stieltjes integral, the Stone-Weierstrass theorem, the implicit function theorem, differential forms, partitions of unity, simplexes and chains, and Stokes’ theorem.

## MATH 30860. Honors Analysis II

(3-0-3)
Prerequisite: MATH 30850
Required of honors mathematics majors. An advanced course in mathematical analysis in one and several variables. Topics include an axiomatic formulation of the real and complex number systems, compactness, connectedness, metric spaces, limits, continuity, infinite sequences and series, differentiation, the Riemann-Stieltjes integral, the Stone-Weierstrass theorem, the implicit function theorem, differential forms, partitions of unity, simplexes and chains, and Stokes’ theorem.

## MATH 40210. Basic Combinatorics

(3-0-3)
Combinatorics is the study of objects that are fundamentally discrete (made up of distinct and separated parts) as opposed to continuous. For example, instead of studying differential equations to see how a system evolves as time passes continuously, we study recurrence relations to see how some quantity changes as time increments unit by unit? in place of the functions on the real line studied in analysis, combinatorics looks at functions on a finite set. Combinatorics has been increasing in importance in recent decades, in part because of the advent of digital computers (which operate and store data discretely), and in part because of the recent ubiquity of large discrete networks (social, biological, ecological, ...). Typical objects studied in combinatorics include graphs (networks consisting of nodes, some pairs of which are joined), permutations (arrangements of distinct objects in various different orders), and finite sets and their subsets. There are many aspects of combinatorics, such as enumerative, structural (e.g., when is it possible to travel around a network, visiting each edge once and only once?), extremal (e.g., if you know which pairs of people in a class don't like each other, what's the smallest number of groups the class can be broken up into, with no two enemies together in a group?), and algorithmic (e.g., if you know the cost of connecting each possible pair of a set of towns, how can you find the most economical network that fully connects up the towns?). In this course, we will explore each of these aspects of combinatorics, and maybe some more as time permits.

## MATH 40220. Introduction to Algebraic Combinatorics

(3-0-3)
Algebraic Combinatorics is a subject involving the use of algebraic techniques and intuition to inform us about a variety of combinatorial structures. However, it is important to note that the information flows both ways---sometimes the combinatorial structures give us surprising insights into the underlying algebraic structures. Possible topics include, amongst others, using monomial ideals to study the invariants of simplicial complexes and graphs, enumeration via linear algebra, partially ordered sets and lattices, and counting the faces of polyhedra and polytopes.

## MATH 40390. Numerical Analysis

(3-0-3)
Prerequisite: (MATH 20750 or MATH 20860 or MATH 30650) and (MATH 20610 or MATH 20810) and (ACMS 20210 or CSE 20232 or CSE 20211)
An introduction to the numerical solution of ordinary and partial differential equations. Topics include the finite difference method, projection methods, cubic splines, interpolation, numerical integration methods, analysis of numerical errors, numerical linear algebra and eigenvalue problems, and continuation methods.

## MATH 40480. Complex Variables

(3-0-3)
Prerequisite: MATH 20550 or MATH 20850
An introduction to the theory of functions of one complex variable. Topics include analytic functions, Cauchy integral theorems, power series, Laurent series, poles and residues, applications of conformal mapping, and Schwarz-Christoffel transformations.

## MATH 40510. Intro to Algebraic Geometry

(3-0-3)
Prerequisite: MATH 30710 or MATH 30810
Algebraic geometry is the study of systems of polynomial equations and their vanishing loci. It has important components that lie in the realm of geometry, of algebra and of computation (among others) and countless applications. This course tries to give a flavor of these different aspects of the field and how they fit together. Indeed, much of the fascination of this subject comes from the myriad ways in which arguments squarely in one realm give surprising consequences that fall squarely in a different realm.

## MATH 40520. Theory of Numbers

(3-0-3)
Prerequisite: MATH 30705 or MATH 20820 or MATH 30710
An introduction to elementary number theory. Topics include the Euclidean algorithm, congruencies, primitive roots and indices, quadratic residues, quadratic reciprocity, distribution of primes, and Waring’s problem.

## MATH 40570. Mathematical Methods in Financial Economics

(3-0-3)
Prerequisite: MATH 30530 and (MATH 20750 or MATH 30650) and (MATH 30750 or MATH 30850) or FIN 30600 or FIN 70670
An introduction to financial economic problems using mathematical methods, including the portfolio decision of an investor and the determination of the equilibrium price of stocks in both discrete and continuous time, will be discussed.
The pricing of derivative securities in continuous time including various stock and interest rate options will also be included. Projects reflecting students’ interests and background are an integral part of this course.

## MATH 40720. Topics in Algebra

(3-0-3)
Prerequisites: MATH 30710 or MATH 30810
This will be an introduction to Lie algebras and representation theory. Lie algebras are some of the most important objects in mathematics and its applications to physics, and also applications to many other areas. We will start with studying finite dimensional Lie algebras as represented by matrices, but will hopefully find ourselves by the end of the semester also exploring a very active area of current research where we will look at infinite dimensional Lie algebras and how they arise in physical models such as string theory, which stills holds many mathematical mysteries. We will likely use the text "Introduction to Lie Algebras" by Erdman and Wildon, as well as other resources. The only prerequisite will be some familiarity with linear algebra.

## MATH 40730. Mathematical and Computational Modeling in Biology and Physics

(3-0-3)
Prerequisite: MATH 20750 or MATH 30650 or ACMS 20750
Introductory course on applied mathematics and computational modeling with emphasis on modeling of biological problems in terms of differential equations and stochastic dynamical systems. Students will be working in groups on several projects and will present them in class in the end of the course.

## MATH 40740. Topology

(3-0-3)
Prerequisite: MATH 20630 or MATH 10850
Topology is the study of when one geometric object can be continuously deformed and twisted into another object. The course starts with point-set topology, a framework based on "open sets" for studying continuity and compactness in very general spaces. These general notions are then applied to study actual geometric objects: surfaces (classify all possible surfaces with no boundary) and knots (which knots can you untangle?). Other possible topics might include fundamental group (can you shrink any loop in a space to a point) and homology (study of higher dimensional "loops").

## MATH 40750. Partial Differential Equations

(3-0-3)
Prerequisite: MATH 20750 or MATH 20820 or MATH 30650
An introduction to partial differential equations. Topics include Fourier series, solutions of boundary value problems for the heat equation, wave equation and Laplace’s equation, Fourier transforms, and applications to solving heat, wave and Laplace’s equations in unbounded domains.

## MATH 40760. Differential Geometry

(3-0-3)
Prerequisite: MATH 20750 or MATH 20860 or MATH 30650
An introduction to differential geometry. Topics include analysis of curves and surfaces in space, the first and second fundamental forms of surfaces, torsion, curvature and the Gauss-Bonnet theorem.

## MATH 40910. Topics in Mathematical Logic

(3-0-3)
Prerequisite: MATH 30710 or MATH 30810
An introduction to mathematical logic and Gödel’s Incompleteness Theorem. The course will include topics from model theory, computability, and set theory, as time permits.

## MATH 40960. Topics in Geometry or Topology

(3-0-3)
Prerequisite: MATH 20630 or MATH 10860
This course covers topics in geometry and topology.

## MATH 43900. Problem Solving in Mathematics

(0-0-1)
The main goal of this course is to develop problem-solving strategies in mathematics.

(V-0-V)
Consent of director of undergraduate studies in mathematics is required.

(V-0-V)
This course offers students the opportunity to study and do research on a topic of their interest with faculty members of Mathematics department. It is a variable credit hour course, with a maximum of 4 credits per semester, arranged individually for each student. This is a repeatable for credit course.

## MATH 48844. Special Topics

(V-0-V)
This topical course is intended for students attending international study programs. It is a variable credit hour course, with a maximum of 4 credits per semester, arranged individually for each student. It is a repeatable for credit course.

## MATH 48900. Thesis

(V-0-V)
Seniors in the mathematics program have the option of writing a senior thesis on a more advanced subject than is provided in the normal undergraduate courses. A program of readings on the topic must be begun with a faculty advisor by the spring semester of the junior year.

## MATH 50570. Mathematical Methods in Financial Economics

(3-0-3)
Prerequisite: (MATH 30530 and (MATH 20750 or MATH 30650) and (MATH 30750 or MATH 30850)) or FIN 30600 or FIN 70670
An introduction to financial economic problems using mathematical methods, including the portfolio decision of an investor and the determination of the equilibrium price of stocks in both discrete and continuous time, will be discussed. The pricing of derivative securities in continuous time including various stock and interest rate options will also be included. Projects reflecting students’ interests and background are an integral part of this course.

## MATH 50780. Summer Honors Topics—Seminar for Undergraduate Mathematical Research

(3-0-3)
This course permits students to pursue a special topic in advanced mathematics. It is offered as a part of SUMR (the Seminar for Undergraduate Mathematical Research). The consent of the Director of Undergraduate Studies in Mathematics is required.