# Descriptions of 4000 level courses

In addition to the courses on this list is a fun class that does not count towards the math major (but does count toward the required 39 upper division hours). MATH 3220, Advanced Problem Solving, is a pass/fail one-hour course which can be repeated up to three times. Various methods and strategies of problem-solving are covered. During the fall semester, students often work past Putnam exam problems in preparation for the annual competition in December. (Prerequisite or corequisite: MATH 3000 or MATH 3100 or MATH 3200 or MATH 3500(H).)

 COURSE PREREQ DESCRIPTION NEXT OFFERED MATH 4000–4010: Modern Algebra and Geometry MATH 3000, MATH 3200 A first course in modern algebra, emphasizing its interplay with number theory, geometry, and other applications. You will learn about the isometries of the plane, which primes can be written as the sum of two squares, compass and straightedge constructions (e.g., the famous fact that one cannot trisect a general angle using compass and straightedge), group actions and counting arguments, symmetry groups, elements of Galois theory (e.g., the amazing fact that there can be no formula expressing the roots of the general polynomial of degree at least five in terms of radicals), and will be introduced to projective geometry. 4000: fall, spring, summer; 4010: spring MATH 4100: Real Analysis MATH 3100, MATH 3200 The conceptual underpinnings of calculus, studied in metric spaces: limits, numerical sequences and series, continuity, differentiable functions, integrals, and different notions of convergence of sequences and series of functions. fall semester MATH 4150: Complex Variables either MATH 2270 or MATH 2500 and MATH 3100 or MATH 3510 A beautiful blend of theory and computation, this course explores the magical effect of introducing complex numbers into calculus. The central point is that when a complex function is differentiable, then it is represented by a convergent power series; this, in turn, is understood by representing its values by an integral (an appropriate "weighted average" of nearby values). Even more pleasant are the unexpected applications to "real" questions: using residues, you will find that evaluating definite integrals like and is not so difficult; applications to physics and engineering will be discussed as time allows. summer fall  spring MATH 4200: Point Set Topology either MATH 3100 and MATH 3200 or MATH 2410 A study of continuity in the setting of abstract metric and topological spaces. The notions of connectedness and compactness (which are key to the intermediate value theorem and maximum value theorem, respectively, from introductory calculus) are studied in more general settings. Can there be a space-filling curve? Can there be a function that is continuous at precisely the irrational points? the rational points? spring MATH 4220: Differential Topology either MATH 4120 or MATH 3510 and either MATH 4100 or MATH 4200 The course focuses on differentiable manifolds, a generalization of the curves and surfaces studied in multivariable calculus). Among the results proved are the Brouwer fixed point theorem (generalizing the intermediate value theorem from calculus), the fundamental theorem of algebra ("every nonconstant complex polynomial has a root"), the hairy ball theorem ("you cannot comb the hair on a hairy billiard ball") and the surprising meteorological fact that at any instant there is a pair of antipodal (opposite) points on the surface of the earth with identical barometric pressure and identical temperature. And intersection theory allows you to see immediately that a sphere and a torus are not "equivalent." Not currently scheduled. MATH 4250: Differential Geometry either MATH 2270 or MATH 2500 and MATH 3000 or MATH 3510 The notion of curvature is quite intuitive for curves (e.g., straight lines have none) and less so for surfaces (why can we use a sheet of paper to make a cylinder but not the surface of a doughnut?). You will find your multivariable calculus and linear algebra to be powerful tools in discussing such issues. Other topics of note are the Fary-Milnor Theorem (which tells us that total curvature of at least 4π is required to make a knot), the Gauss-Bonnet Theorem (which relates the total curvature of a surface to its topology), and an introduction to non-Euclidean geometry. spring MATH 4400: Number Theory MATH 4000 A careful study of the simplest of all mathematical structures, the integers. Topics include: congruences, distribution of primes, factorization of integers, "Pythagorean triples" (e.g., 3,4,5), and Fermat's Last Theorem (which states that there are no positive integral solutions of xn + yn = zn when n>2, proved in a tour de force by Andrew Wiles in 1995). In addition, various real-world applications, such as cryptology, will be discussed. spring MATH 4450: Cryptology and Computational Number Theory MATH 4000 For centuries, cryptology, the science of secret codes, has been almost solely in th realm of government and the military. With the advent of electronic communication and commerce it has become vitally important to have secure cryptographic systems available to ordinary people. Many of the modern methods that are in use or have been suggested are based on computational problems in number theory and algebra, especially factoring large numbers and the "discrete logarithm" problem (given two elements in a group with one a power of the other, find the exponent). In this course, we take a tour of various cryptographic systems, learning their strengths and weaknesses. Along the way, we study modern algorithms for recognizing prime numbers, factoring composite numbers, and computing discrete logarithms. The mathematical structures introduced will include finite fields and elliptic curves. fall alternate years MATH 4500-4510: Numerical Analysis suggested corequisite: MATH 3100; for 4500, MATH 3000 and some experience with computer programming; for 4510, MATH 4500, MATH 2700, and MATH 2270 or MATH 2500 or MATH 3510 How does one use a computer to solve a large system of linear equations? Will the computer always produce an accurate solution? How might one use a computer to design a new alphabetic font orthe shape of a new car? This course will address such questions, along with numerical methods for solving differential equations, numerical methods for determining the eigenvalues of large matrices, interpolation and approximation of functions, and splines. Some hands-on computer programming is involved to experiment with those numerical schemes, although no specific programming language is a prerequisite. 4500: fall; 4510: spring MATH 4600: Probability Theory either MATH 3510 or MATH 2270/2500 and MATH 2260/3100 Consider such questions as: What is the likelihood that a poker hand will be a straight flush? What is the likelihood that in a room of 23 people there are two people with the same birthday? Topics in discrete probability include: conditional probability, independence, expectation value, Bernoulli trials, binomial and Poisson distributions. Then one deals with continuous random variables, the law of large numbers, the central limit theorem, and applications (e.g., to random walks, diffusion, Markov processes). This course is particularly recommended for those students intending to study statistics and actuarial science. fall MATH 4670: Combinatorics MATH 3000 and MATH 3200 A study of properties of discrete sets. Problems involve enumeration of combinatorial objects (e.g., the number of permutations with forbidden positions, the number of distinct circular necklaces made from beads with some different colors). Techniques include the pigeonhole principle and its generalizations, the inclusion/exclusion principle, generating functions and recurrence relations, and Polya's counting principle. fall MATH 4690: Graph Theory MATH 3000 and MATH 3200 A graph is just a collection of vertices with certain pairs of vertices connected by edges. Graphs can be used to solve problems like finding the shortest (or least expensive) airline route between two cities, matching job candidates to available positions, efficient routing of letter carriers, minimizing the number of computer registers needed to store all variables used simultaneously in a computer program, designing sports tournaments, or even classifying the regular polyhedra. The famous Four Color Problem ("Can you color the countries on any map using just four colors in such a way that any two contiguous countries are different colors?") is readily translated into a question about graphs, and graph theory ultimately provided a solution; sometimes the easier Five Color Theorem is proved in the course. spring MATH 4700: Qualitative Ordinary Differential Equations MATH 2700 and either MATH 3000 or MATH 3500 A more conceptual approach to the study of ODE's, both single and systems; long-term behavior, stability (which involves eigenvalues), and chaos. Applications to physics, engineering, and biology are included. fall MATH 4720: Introduction to Partial Differential Equations MATH 2700, MATH 3100 and either MATH 2270 or MATH 2500 or MATH 3510 This course focuses on the three basic partial differential equations that arise in elementary physics: Laplace's equation (which characterizes harmonic functions), the heat equation, and the wave equation. What is it in the mathematics that explains why a sudden noise is heard only once (in 3D) but a sudden burst of heat has an effect forever? spring MATH 4730: Mathematics of Climate MATH 2700 This course is a mathematical study of climate, with various models and a qualitative analysis of stability and bifurcations. fall MATH 4750:Matrix and Integral Transforms with Applications MATH 3000 and MATH 3100 Some of the most fascinating applications of mathematics come from signal processing and the CT scans prevalent in medicine. In this class we will learn the mathematical background, including eigenvector decompositions (building on the spectral theorem), the discrete and continuous Fourier transforms, singular value decomposition (SVD) of matrices, and the Radon transform (with applications to tomography). spring MATH 4780: Mathematical Biology MATH 2270 or MATH 2500 and MATH 4700 This class will cover various discrete, continuous, and probabilistic models for biological systems, reaction kinetics, epidemics, and tumors. It should be of particular interest to students considering further work in biology or medical school. spring MATH 4790:Mathematics of Option Pricing either MATH 2270 or MATH 2500 and MATH 3000 or MATH 3510 Consider a financial portfolio that contains some options (European, American, or Exotic). This course will explain the process of computing derivative prices in terms of underlying equity prices, emphasizing the mathematics of investments that replicate equities under arbitrage-free trading, and eventually leading to the Nobel-prize winning Black-Scholes formula. Spreadsheets are used to compute option trees; the necessary probability theory and linear algebra concepts will be developed as needed. spring MATH 4950: Research in Mathematics Students involved in VIGRE undergraduate research may earn from 1 to 3 credit hours. (May be taken for at most 6 credit hours.) fall, spring, summer COURSES OFFERED ON A LESS REGULAR BASIS MATH 4050: Advanced Linear Algebra MATH 4000 This course expands on the culminating topic of MATH 3000: diagonalizability and the spectral theorem. Linear transformations and matrices in the complex setting are emphasized: hermitian and unitary matrices provide the appropriate setting of the spectral theorem. And, in general, when diagonalizability fails, Jordan and rational canonical forms are a powerful tool, in both theoretical and applied mathematics. TBA MATH 4110: The Lebesgue Integral and Applications MATH 4100 or MATH 4200 It was discovered toward the end of the nineteenth century that the Riemann integral we all learn in calculus is woefully ill-equipped to deal with problems that arise in real-life applications. For example, the limit of a uniformly bounded sequence of integrable functions needn't even be integrable. The Lebesgue integral is defined in such a way as to have all the "nice" properties, and we see in this course that it arises naturally in dealing with questions from Fourier analysis and probability. TBA MATH 4120: Multivariable Analysis MATH 3510, MATH 4100, or MATH 4200 A rigorous treatment of multivariable differential and integral calculus, including the inverse and implicit function theorems, an introduction to differentiable manifolds, differential forms, and the n-dimensional Stokes' Theorem, the ultimate version of the Fundamental Theorem of Calculus. TBA MATH 4130: Introduction to Functional Analysis MATH 4100 and MATH 3000 or 3500 Introduction to linear algebra and analysis in the infinite-dimensional setting. Topics include complete metric spaces, Banach spaces, Hilbert spaces, and applications to physics and other disciplines. TBA MATH 4300: Introduction to Algebraic Curves MATH 4000 By studying the ring of polynomials in several variables, one sees a rich interplay between algebra and geometry. This course gives a deeper treatment of the algebraic curves, from both a local and a global approach. What does "rationality" of conics have to do with parametrizing Pythagorean triples; why is the cubic curve y2=x3+ax+b not "rational" and why does this lead to "elliptic integrals"? TBA MATH 4630: Mathematical Analysis of Computer Algorithms MATH 3000 and MATH 3200 Computing takes time. Some problems take a very long time, others can be done quickly. Some problems seem to take a long time, and then someone discovers a faster way to accomplish the same task. The study of the amount of computational effort required to perform certain kinds of tasks is the study of complexity, a relatively new branch of mathematics. This course is involved with answering, via complexity, the question: "how quickly can one compute the answer to a question?" TBA MATH 4760:Mathematics and Music MATH 2260 and MATH 2270 or MATH 2500 or MATH 2700, and an acquaintance with musical notation Music is filled with mathematics! This course begins with elementary aspects of the theory of sound, Fourier series, the wave equation, and vibrations of strings and drums (Bessel functions), then explores the continued fractions underlying different scales and temperaments, and culminates in a discussion of synthesis techniques. Not currently scheduled. MATH 4850: History of Mathematics senior standing in mathematics A study of (periods of) the history of mathematics by means of mathematical problems. Either a term paper or project will be required. spring MATH 4900: Topics in Mathematics Covering different material not included in the usual curriculum. (May be taken at most twice for credit towards the major.) spring (Game Theory)