
Mathematics 301, Numerical Analysis
This lab is an introduction to mathematical computation. It will start with tutorials on the two software packages that we will be using: Mathematica and its new open source alternative, Sage. The course will then discuss algorithms for finding the zeros of nonlinear functions, solving linear systems quickly, and approximating eigenvalues. The bulk of the course will be devoted to curve fitting by means of polynomial interpolation, splines, bezier curves, and least squares. Corequisites: Mathematics 242 and any Computer Science course or basic programming experience.

Mathematics 303, Computational Geometry
This class will cover a variety of topics from computational geometry. Focus will be given to computational complexity of the algorithms presented as well as appropriate data structures. Topics may include Voronoi Diagrams, convex hull calculations, line segment intersections and more. Prerequisites: Mathematics 212, Mathematics 242 and some programming knowledge.

Mathematics 312, Advanced Calculus
This is a proofsbased introduction to the theory of numbers and covers the fundamentals of quadratic number fields. Topics include factorization, class group, unit group, Diophantine approximation, zeta functions, and applications to cryptography. Prerequisite: Mathematics 212 or permission of the instructor.

Mathematics 314, Modeling Realizable Phenomena
In nearly every aspect of human knowledgeconceptual, qualitative, statistical, analytic, numericalmodeling plays a prominent role. Modeling strategies vary widely depending on the phenomenon being studied, the nature and type of the salient state variables, the problem's dimensions, and additional simplifying constraints. In this course, we will learn some modeling approaches and styles, considering both situations when the governing equations are unknown, and when the governing equations are known but are imperfect. Topics may include statistical modeling, timeseries, spatial modeling and kriging, modeling in terms of special complete sets, auto and crosscorrelation functions, Markov chains, differential equations, data assimilation and combining data with models. Prerequisite: Mathematics 212 and Mathematics 242.

Mathematics 316, Combinatorics
Combinatorial mathematics is the study of how to combine objects into finite arrangements. Topics covered in this course are chosen from enumeration and generating functions, graph theory, matching and optimization theory, combinatorial designs, ordered sets, and coding theory. Prerequisite: Mathematics 261 or permission of the instructor.

Mathematics 317, Graph Theory
Graph theory is a branch of mathematics that has applications in areas ranging from operations research to biology. This course is a survey of the theory and applications of graphs. Topics are chosen from among connectivity, trees, Hamiltonian and Eulerian paths and cycles; isomorphism and reconstructability; planarity, coloring, colorcritical graphs, and the fourcolor theorem; intersection graphs and vertex and edge domination; matchings and network flows; matroids and their relationship with optimization; and random graphs. Several applications of graph theory are discussed in depth. Prerequisite: Mathematics 261 or permission of the instructor.

Mathematics 318, Number Theory
This is a proofsbased introduction to the theory of numbers and covers the fundamentals of quadratic number fields. Topics include factorization, class group, unit group, Diophantine approximation, zeta functions, and applications to cryptography. Prerequisite: Mathematics 261.

Mathematics 321, Partial Differential Equations
This course is an introduction to the theory of partial differential equations. The primary focus is the derivation and solutions of the main examples in the subject rather than on the existence and uniqueness theorems and higher analysis. Topics include hyperbolic and elliptic equations in several variables, Dirichlet problems, the Fourier and Laplace transform, Green's functions. Prerequisite: Mathematics 211.

Mathematics 322, Operations Research
Operations research is a scientific approach to decision making that seeks to determine how best to design and operate a system, usually under conditions requiring the allocation of scarce resources. This course introduces students to the branch of operations research known as deterministic optimization, which tackles problems such as how to schedule classes with a limited number of classrooms on campus, determine a diet that is both rich in nutrients and low in calories, and create an investment portfolio that meets your investment needs. Techniques covered include linear programming, network flows, integer/combinatorial optimization, and nonlinear programming. Emphasis is on the importance of problem formulation as well as how to apply algorithms to realworld problems. Prerequisites: working knowledge of multivariable calculus and basic linear algebra.

Mathematics 323, Dynamical Systems
An introduction to the theory of discrete dynamical systems. Topics to be covered include iterated functions, bifurcations, chaos, fractals and fractal dimension, complex functions, Julia sets, and the Mandelbrot set. We will make extensive use of computers to model the behavior of dynamical systems. Prerequisite: Mathematics 213, or Mathematics 212 and Mathematics 242.

Mathematics 328, Probability
A calculusbased introduction to probability with an emphasis on computation and applications. Topics include continuous and discrete random variables, combinatorial methods, conditional probability, joint distributions, expectation, variance, covariance, laws of large numbers, and the Central Limit Theorem. Students will gain practical experience using mathematical software to run probability simulations. Prerequisite: Mathematics 212 or Mathematics 241, or permission of the instructor.

Mathematics 333, Abstract Algebra II
This course is a continuation of Mathematics 332. After a review of group theory, we will cover selected topics from ring, field and module theory. The second half of the course will include applications to Representation Theory and/or Galois Theory, as time permits. Prerequisite: Mathematics 332.

Mathematics 335, Advanced Linear Algebra
This course covers the advanced theory of abstract vector spaces over arbitrary fields. It will start with a discussion of dual spaces, direct sums, quotients, tensor products, spaces of homomorphisms and endomorphisms, inner product spaces, and quadratic forms. It will then move on to multilinear algebra, discussing symmetric and exterior powers, before turning to the Jordan canonical form and related topics. Other more advanced topics may include Hilbert spaces, modules, algebras, and matrix Lie groups. Prerequisite: Mathematics 342. Corequisite: Mathematics 332.

Mathematics 340, Coding Theory
The digital transmission of information is considered to be extremely reliable, and yet it suffers from the same sorts of interference, corruption, and data loss that plague analog transmission. The reliability of digital transmission comes from the use of sophisticated techniques that encode the digital data so that errors can be easily detected and corrected. This theory of error correcting codes, while having broad applications ranging from CDs to the internet to high definition television, requires some surprisingly beautiful mathematics. We will interpret strings of bits as vectors in an abstract vector space, which allows us to manipulate binary data using linear algebra over finite fields. This class will introduce students to the basics of error correcting codes, as well as touching on the mathematics of data compression and encryption. If time permits, we will also discuss quantum error correction. Although this course will mention encryption, the emphasis will NOT be on cryptography. This course will not involve any programming. Prerequisites: Mathematics 242 and either Mathematics 261 or CMSC 242 (Discrete Mathematics).

Mathematics 351, Point Set Topology
An introduction to point set topology. Topics include topological spaces, metric spaces, compactness, connectedness, continuity, homomorphisms, separation criteria, and, possibly, the fundamental group. Prerequisite: Mathematics 261 or permission of the instructor.

Mathematics 352, Differential Geometry
This course explores the mathematics of curved spaces, particularly curved surfaces embedded in threedimensional Euclidean space. Originally developed to study the surface of the earth, differential geometry is an active area of research, and it is fundamental to physics, particularly general relativity. The basic issue is to determine whether a given space is indeed curved, and if so, to quantitatively measure its curvature using multivariable calculus. This course also introduces geodesics, curves of minimal length. The course culminates with the GaussBonnet theorem, giving a link between the geometry and topology of surfaces. Prerequisite: Mathematics 212 and Mathematics 261, or permission of the instructor.

Mathematics 362, Complex Analysis
This course will cover the basic theory of functions of one complex variable. Topics will include the geometry of complex numbers, holomorphic and harmonic functions, Cauchy's theorem and its consequences, Taylor and Laurent series, singularities, residues, elliptic functions and/or other topics as time permits. Prerequisite: Mathematics 212, Mathematics 261, and one prior 300level mathematics course is recommended, or permission of the instructor.

Mathematics 384, Computational Algebraic Geometry
This course is an introduction to computational algebraic geometry and commutative algebra. We will explore the idea of solving systems of polynomial equations by viewing the solutions to these systems as both algebraic and geometric objects. We will also see how these objects can be manipulated using the Groebner basis algorithm. This course will include a mixture of theory and computation as well as connections to other areas of mathematics and to computer science. Prerequisite: Mathematics 332.

Mathematics 405, Mathematical Logic
An introduction to mathematical logic. Topics include firstorder logic, completeness and compactness theorems, model theory, nonstandard analysis, decidability and undecidability, incompleteness, and Turing machines. Prerequisite: Mathematics 332 or permission of the instructor.

Mathematics 417, Algebraic Number Theory
In this course we will study algebraic number fields (finite extensions of the rational numbers) from an algebraic and an analytic viewpoint, motivated by the special cases of quadratic and cyclotomic fields. The goal of the course is to develop an understanding of the deep connections between algebra, analysis, and arithmetic. Topics will include: rings of integers, factorization, ideal class group, unit group, zeta and Lfunctions, Dirichlet's theorem. Prerequisite: Mathematics 332.

Mathematics 432, Advanced Topics in Abstract Algebra
This course continues the study of abstract algebra begun in Mathematics 332. Topics are chosen by the instructor, and may include some additional group theory, Galois theory, modules, group representations, and commutative algebra. Prerequisite: Mathematics 332 or permission of the instructor.

Mathematics 436, Representation Theory
This course will cover the basic theory of representations of finite groups in characteristic zero: Schur's Lemma, Mashcke's Theorem and complete reducibility, character tables and orthogonality, direct sums and tensor products. The major goals of the course will be to understand the representation theory of the symmetric group and the general linear group over a finite field. If time permits, the theory of Brauer characters and modular representations will be introduced. Corequisites: Mathematics 242 and Mathematics 332.

Mathematics 453, Modern Geometry
Geometry is an ancient subject, but it has received a modern makeover in the past two centuries, where the type of geometry now studied is broader than just Euclidean geometry, and where the approach is now analytic rather than axiomatic. In this course we will look at Euclidean, nonEuclidean (hyperbolic and elliptic) and projective geometries, making use of tools from linear algebra and abstract algebra. Prerequisites: Mathematics 242 and Mathematics 332 (which can be taken simultaneously with this course), or permission of instructor.

Mathematics 454, Knot Theory
Knot theory is an active branch of contemporary mathematics that, similarly to number theory, involves many problems that are easy to state but difficult to solve; unlike number theory, knot theory involves a lot of visual reasoning. Knot theory is a branch of topology, but it also has applications to aspects of biology, chemistry and physics (though for lack of time these applications will not be discussed in class). This course is an introduction to the theory of knots and links. Topics include methods of knot tabulation, knot diagrams, Reidemeister moves, invariants of knots, knots and surfaces, and knot polynomials. Prerequisites: Mathematics 351 or Mathematics 361, or permission of instructor.

Mathematics 461, Real Analysis II
This course continues the study of real analysis begun in Math 361. Topics include functions of several variables, metric spaces, Lebesgue measure and integration, and, time permitting, additional topics such as the Inverse and Implicit Function Theorems, differential forms and Stokes’ Theorem. Prerequisites: Mathematics 361.