# Mathematics Study Abroad Course Descriptions

The courses listed below are pre-approved biological science courses offered at various Notre Dame study abroad locations.

## Perth, Australia

### MATH 24450. Multivariable Calculus and Linear Algebra

(3-0-3)
Topics in this unit include vector fields, line integrals, surface integrals, volume integrals, the theorems of Green, Stokes and Gauss, eigenvalues and eigenvectors of linear transformations, change of basis, diagonalization, and complex vector spaces. Students are able to develop geometric intuitions and the ability to articulate these intuitions within a formalism at an appropriate level; understand and appreciate the power and beauty of mathematical abstraction; communicate effectively with others; present mathematical results in a logical and coherent fashion; and undertake continuous learning, aware that an understanding of fundamentals is necessary for effective application.

### MATH 24610. Linear Algebra

(3-0-3)
This unit is required for students intending to major in Applied or Pure Mathematics or Mathematical Statistics, students in Engineering and students in some of the physical sciences. It is also suitable for other students wishing to have a strong mathematics component in their degrees. Topics include methods of proof, logic and mathematical induction; infinite sequences, bounded and monotone sequences; limits and continuity of functions; differentiability; integration; the fundamental theorem of calculus; Taylor polynomials; infinite series, absolute and conditional convergence, power series, ratio and comparison tests; vector geometry; systems of linear equations and Gaussian elimination; matrix algebra; subspaces, linear independence, bases and dimension, the rank-nullity theorem for matrices; eigenvalues and eigenvectors.

### MATH 34210. Intro to Operations Research

(3-0-3)

Taught as MATH2224 - 'Operations Research' at a host institution. This unit covers linear programs; scheduling; simplex algorithm; game theory; linear duality; network linear programs and network simplex algorithm; and nonlinear optimization.

### MATH 34650. Differential Equations

(3-0-3)
Topics include vector fields, conservative fields and potentials; line integrals, fundamental theorem and path dependence; Green's theorem, proof and applications; divergence and curl; surface integrals of scalar functions, parametric and orientable surfaces, theorems of Stokes and Gauss; Laplace transforms and solutions of ODEs; Fourier series, calculation of coefficients, convergence, Gibbs phenomenon, odd and even extensions, differentiation and integration, amplitude and phase; partial differential equations, origin of wave, diffusion and Laplace equations in physical problems, separation of variables, principle of superposition, boundary and initial conditions and use of Fourier series, differences between types of solutions. Throughout, the unit emphasis is placed on the interpretation of concepts and calculations in term of possible applications.

## Santiago, Chile

### MATH 34750. Real Analysis

(4.5-0-4.5)
Taught as MAT2515 - 'Analisis Real' at a host institution. In this course fundamental concepts of mathematical analysis are taught through the study of metric spaces and the convergence of functions.

### MATH 44844. Special Topics: Set Theory

(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.

## Hong Kong, People's Republic of China

### MATH 34540. Mathematical Statistics

(3-0-3)
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.

## Cambridge University, England

### MATH 34310. Coding Theory

(V-0-V)
Taught at a host institution. An introduction to the theory of error-correcting codes. Possible 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 44844. Special Topics: Set Theory

(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.

## London, England

### MATH 34810. Honors Algebra III

(3-0-3)
The modern axiomatic approach to mathematics is demonstrated in the study of the fundamental theory of abstract algebraic structures. Group theory, subgroups, generators, Lagrange's theorem. Normal subgroups, homomorphisms, isomorphism theorems. Ring theory, integral domains. Ideals, homomorphisms and isomorphism theorems. Polynomial rings, Euclidean algorithm, fields of fractions.

### MATH 34820. Honors Algebra IV

(3-0-3)
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. Taught at Queen Mary University in London, England

### MATH 34860. Honors Analysis II

(3-0-3)
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 44844. Special Topics: Set Theory

(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.

## Oxford University, England

### MATH 34390. Introduction to Numerical Methods

(3-0-3)
An introduction to numerical methods for solving algebraic and differential equations. Topics include numerical solution of systems of linear equations, approximating functions with polynomials and splines, solutions of nonlinear equations, numerical integration, numerical solution of ordinary differential equations and eigenvalue problems. Some computer programming is required.

### MATH 34810. Honors Algebra III

(3-0-3)
The modern axiomatic approach to mathematics is demonstrated in the study of the fundamental theory of abstract algebraic structures. Group theory, subgroups, generators, Lagrange's theorem. Normal subgroups, homomorphisms, isomorphism theorems. Ring theory, integral domains. Ideals, homomorphisms and isomorphism theorems. Polynomial rings, Euclidean algorithm, fields of fractions.

### MATH 34850. Honors Analysis I

(3-0-3)
Taught at Oxford University Year Long Program

### MATH 44510. Intro to Algebraic Geometry

(3-0-3)
This was a math course taken at Oxford, England. Real algebraic curves have been studied for more than two thousand years, although it was not until the introduction of the systematic use of coordinates into geometry in the seventeenth century that they could be described in a more rigorous form. Once complex numbers were recognized as acceptable mathematical objects it quickly became clear that complex algebraic curves have at once simpler and more interesting properties than real algebraic curves. In this course, algebraic curves are studied, using ideas from algebra, from topology and from complex analysis. Students will see the concepts of projective space and curves in the projective plane. They will understand the notion of non-singularity, learn some basic intersection theory (Bezout's theorem), and see the degree-genus formula for smooth curves. They will see the concept of Riemann surface, the Riemann-Roch theorem, and a detailed study via Weierstrass functions of the equivalence between smooth cubics and genus one Riemann surfaces.

### MATH 44520. Theory of Numbers

(3-0-3)
Number theory is one of the oldest parts of mathematics. The aim of this course is to introduce students to some classical and important basic ideas of the subject including the following: The ring of integers; congruences; rings of integers modulo ; the Chinese Remainder Theorem; Wilson's Theorem; Fermat's Little Theorem for prime modulus; Euler's generalization of Fermat's Little Theorem to arbitrary modulus; primitive roots; quadratic residues modulo primes; quadratic reciprocity; factorization of large integers; basic version of the RSA encryption method.

### MATH 44740. Topology

(3-0-3)
Taught at a host institution. The goal of the course is for the student to understand and appreciate the central results of general topology and metric spaces, sufficient for the main applications in geometry, number theory, analysis and mathematical physics.

### MATH 44750. Partial Differential Equations

(3-0-3)
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 44760. Differential Geometry

(3-0-3)
This course was taught in Dublin, Ireland as part of the UCD Program and is a math course offered at at Oxford, England as well. Different ways of thinking about surfaces (also called two-dimensional manifolds) are introduced in this course: first topological surfaces and then surfaces with extra structures which allow us to make sense of differentiable functions (`smooth surfaces'), holomorphic functions (`Riemann surfaces') and the measurement of lengths and areas (`Riemannian 2-manifolds'). These geometric structures interact in a fundamental way with the topology of the surfaces. A striking example of this is given by the Euler number, which is a manifestly topological quantity, but can be related to the total curvature, which at first glance depends on the geometry of the surface. The course ends with an introduction to hyperbolic surfaces modelled on the hyperbolic plane, which gives us an example of a non-Euclidean geometry (that is, a geometry which meets all Euclid's axioms except the axioms of parallels). The candidate will be able to implement the classification of surfaces for simple constructions of topological surfaces such as planar models and connected sums; be able to relate the Euler characteristic to branching data for simple maps of Riemann surfaces; understand the definition and use of Gaussian curvature; know the geodesics and isometries of the hyperbolic plane and their use in geometrical constructions.

(2-0-2)
Consent of director of undergraduate studies in mathematics is required.

### MATH 44844. Special Topics: Set Theory

(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.

## Dublin, Ireland - Trinity College

### MATH 34530. Introduction to Probability

(3-0-3)
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 34540. Mathematical Statistics

(3-0-3)
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 34650. Differential Equations

(3-0-3)
Topics include vector fields, conservative fields and potentials; line integrals, fundamental theorem and path dependence; Green's theorem, proof and applications; divergence and curl; surface integrals of scalar functions, parametric and orientable surfaces, theorems of Stokes and Gauss; Laplace transforms and solutions of ODEs; Fourier series, calculation of coefficients, convergence, Gibbs phenomenon, odd and even extensions, differentiation and integration, amplitude and phase; partial differential equations, origin of wave, diffusion and Laplace equations in physical problems, separation of variables, principle of superposition, boundary and initial conditions and use of Fourier series, differences between types of solutions. Throughout, the unit emphasis is placed on the interpretation of concepts and calculations in term of possible applications.

### MATH 34810. Honors Algebra III

(3-0-3)
The modern axiomatic approach to mathematics is demonstrated in the study of the fundamental theory of abstract algebraic structures. Group theory, subgroups, generators, Lagrange's theorem. Normal subgroups, homomorphisms, isomorphism theorems. Ring theory, integral domains. Ideals, homomorphisms and isomorphism theorems. Polynomial rings, Euclidean algorithm, fields of fractions

### MATH 34850. Honors Analysis I

(3-0-3)
Taught at Oxford University Year Long Program

### MATH 34860. Honors Analysis II

(3-0-3)
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 44480. Complex Variables

(3-0-3)
Taught at a host institution. This is a first course on the theory of functions of one complex variable, with emphasis on Taylor and Laurent expansions, residue theory, evaluation of real integrals and the summation of series.

### MATH 44740. Topology

(3-0-3)
Taught at a host institution. The goal of the course is for the student to understand and appreciate the central results of general topology and metric spaces, sufficient for the main applications in geometry, number theory, analysis and mathematical physics.

## Dublin, Ireland - UCD

### MATH 34710. Algebra

(3-0-3)
MATH 30100 Field Theory at UCD; This course gives an introduction to field theory, with an emphasis on the study of finite fields, which underlies many modern applications. Axioms, properties and constructions of arbitrary fields will be described. Fields will be viewed both as vector spaces and as quotients of polynomial rings. While the study of fundamental infinite fields will be given a rigorous treatment, students will also study properties of finite fields in a concrete way. Standard techniques used in the manipulation of the algebra of finite fields will be introduced. Applications of finite fields arise naturally in digital technology, such as coding theory, cryptography or sequences. Some of these applications will be described here. During the course of this module, students will: 1. become familiar with the properties of a field; 2. have a knowledge of constructions and properties of important classes of fields; 3. be furnished with an understanding of field theory so that further studies in algebra reliant on field theory can be undertaken, such as Galois theory, number theory and commutative algebra; 4. have a concrete understanding of finite fields, including manipulation of additive and multiplicative representations of elements; 5. be familiar with the Frobenius automorphism and trace map and their use as tools in the study of finite fields; 6. be familiar with important applications of finite fields related to the mathematics of communications theory. Whan taught in Perth: The theme of this unit is the theory of groups, and their use in measuring symmetry with special emphasis on geometric examples. Apart from the mathematical interest of groups, they are of great and increasing importance in chemistry, geology and physics, and those aspects that have significance in such disciplines are strongly emphasized. The fundamental notion is that of a group action. Group actions are used to elucidate the structure of a group, culminating in the Sylow theorems, which connect finite groups and number theory. The applications of group actions extend to symmetries of the plane, three-space and higher dimensional spaces, rotation and spin, crystallographic groups and symmetries of regular figures in two and three dimensions. Other algebraic structures such as rings and fields may also be explored.

### MATH 34714. Algebraic Structures

(3-0-3)
This course is intended as a first introduction to abstract algebra and its basic structures. The general objects of study are groups, rings and fields, each defined by its own set of axioms. To make these objects meaningful, we will study the best known examples, such as the field of complex numbers, the ring of quaternions, the ring of integers modulo n, the polynomial rings, permutation groups and symmetry groups.

### MATH 44480. Complex Variables

(3-0-3)
Taught at a host institution. This is a first course on the theory of functions of one complex variable, with emphasis on Taylor and Laurent expansions, residue theory, evaluation of real integrals and the summation of series.

### MATH 44570. Financial Mathematics

(3-0-3)
MST 30030 Financial Mathematics at UCD; The aim of "Financial Mathematics" is to introduce to the student to the Black-Scholes model for pricing options. The module opens by looking at various types of options and discussing their properties. The technique of constructing binomial trees to price options (based on the Cox, Ross and Rubenstein paper of 1979) is then discussed in detail. We then study the model of stock price behavior introduced by Black, Scholes and Merton in 1973, and derive the Black-Scholes model for valuing European call and put options on a non-dividend-paying stock. A brief introduction to probability theory is included in the course.

### MATH 44720. Topics in Algebra

(3-0-3)
Taught at a host institution. MATH 20260 The Mathematics of Google at UCD; The Google search engine has made accessing information easy and its speed and efficacy amazes people. This module explains how it works. An essential component is the ability to rank information according to its importance, and in Google, this relies on simple algebraic principles. Some companies hire SEOs (search engine optimizers) to try to gain competitive advantage by raising Google's estimate of their importance and the strategies used to achieve, and also to prevent, this. The mathematical areas involved are linear algebra and elementary probability theory. The presentation will in part be based on the book by Amy Langville and Carl Meyer "Google's PageRank and Beyond: The Science of Search Engine Rankings" (Princeton University Press 2006)

### MATH 44760. Differential Geometry

(V-0-V)
This course was taught in Dublin, Ireland as part of the UCD Program and is a math course offered at at Oxford, England as well. Different ways of thinking about surfaces (also called two-dimensional manifolds) are introduced in this course: first topological surfaces and then surfaces with extra structures which allow us to make sense of differentiable functions (`smooth surfaces'), holomorphic functions (`Riemann surfaces') and the measurement of lengths and areas (`Riemannian 2-manifolds'). These geometric structures interact in a fundamental way with the topology of the surfaces. A striking example of this is given by the Euler number, which is a manifestly topological quantity, but can be related to the total curvature, which at first glance depends on the geometry of the surface. The course ends with an introduction to hyperbolic surfaces modelled on the hyperbolic plane, which gives us an example of a non-Euclidean geometry (that is, a geometry which meets all Euclid's axioms except the axioms of parallels). The candidate will be able to implement the classification of surfaces for simple constructions of topological surfaces such as planar models and connected sums; be able to relate the Euler characteristic to branching data for simple maps of Riemann surfaces; understand the definition and use of Gaussian curvature; know the geodesics and isometries of the hyperbolic plane and their use in geometrical constructions.

### MATH 44844. Special Topics: Set Theory

(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.