# Extended Course Descriptions

On this page are links to extended descriptions, more detailed than those in the Undergraduate Catalog, of some of our upper-level courses.

### Math 3010 (formerly 200) – Intensive Problem Solving and Exposition

The main objectives of Math 3010 are:

- to teach the students how to properly write proofs in a formal, mathematical language
- to expose the students to several different areas of mathematics, sampling through problems the intrinsic beauty of this field
- to expand the experience of students with problem solving, offering problems that are more challenging than the standard exercises found in a textbook
- to review and build upon some topics covered in other math classes such as calculus, linear algebra, abstract algebra, combinatorics, number theory, geometry, real analysis (however, prior knowledge of advanced math math topics is not required)

Also, this is a very good class to take for students who want to participate in math competitions such as the Virginia Tech Regional Math Contest or the Putnam Problem Solving Competition.

### Math 3700 (formerly 215) – Discrete Mathematics

Most of the course deals with developing and applying techniques of counting. Here are some questions that can be answered by those techniques:

- The manager of a bookstore wants to arrange some books on a shelf. No two of the books look alike. There are 6 red physics books, 3 blue physics books, 5 green physics books, 4 red art books, 3 blue art books, and 2 green art books. In how many ways can these books be arranged so that all books of the same color are grouped together and within each color all books on the same subject are grouped together?
- A “bridge hand” is a set of 13 cards chosen from a standard deck. How many bridge hands include cards from exactly 2 suits?
- How many ways are there to fill a box with a dozen doughnuts chosen from five varieties with the requirement that at least one dougnut of each variety is picked?
- Given 5 pairs of gloves and 5 people, how many ways are there for each of the people to choose 2 gloves with no one getting a matching pair?
- Consider a hotel containing 8 rooms, and let r be a positive integer. Determine the number of ways in which r people can be placed into these rooms so that Room 1 contains at least 1 person, Rooms 2 and 5 each contain an odd number of people, and Room 8 contains at least 1 person.
- A Girl Scout troop with 7 members is selling boxes of cookies. Six of the girls are each given 1 box to sell. The seventh girl plans to sell cookies to 2 families, and she knows that one of those families will buy 2 boxes or none at all, while the other family will buy 5 boxes or none at all. The leader of the troop will prepare a ledger listing the first 6 girls alphabetically and the seventh girl last, and stating the number of boxes sold by each girl. If 10 boxes are sold, how many ledgers are possible? The course also includes a brief introduction to graph theory, which is the mathematical theory of networks. Some of the counting techniques developed in the earlier part of the course are applied in the graph theory.

### Math 2820 (formerly 218) – Introduction to Probability and Mathematical Statistics

Math 2820 is an introductory course in Probability and Mathematical Statistics for students who have already completed Math 2300. It assumes no prior background in either Probability or Statistics. The first half of the course deals with “classical” Probability theory and Combinatorics. The second half of the course is an introduction to Mathematical Statistics. Included is material on random variables, estimation, and hypothesis testing. Be aware that Math 2820 is not a “data” course–its purpose is to develop the mathematical foundations associated with Statistics. For anyone intending to pursue a career in science, the material in Math 2820 is essential. Understanding how to analyze and interpret data begins with understanding the mathematical and probabilistic underpinnings of Statistics.

### Math 2821 (formerly 219) – Introduction to Applied Statistics

Math 2821 is the sequel to Math 2820, but it has a much different objective. Whereas Math 2820 is a “foundations” course, Math 2821 is decidedly a “data” course that focuses on experimental design. Most of the semester is spent on statistical inference procedures of one kind or another. Included are one-sample, two-sample, and paired t tests, chi square tests, and the analysis of variance for k-sample designs, randomized block designs, and factorial designs. A fair amount of time is also devoted to various inference procedures associated with regression data. In short, Math 2821 is a course that covers basic Applied Statistics for students who have already had a semester of Mathematical Statistics (Math 2820).

### Math 3300 (formerly 223) – Abstract Algebra

Abstract algebraic concepts such as groups, fields, rings, have evolved from various examples. Abstraction is necessary in order to understand concrete phenomena, and vice versa. The ideas that seemed abstract yesterday, seem more concrete today as they become familiar.

The concept of a group arose from examples of symmetries. The symmetries of every object or subject of scientific research (e.g. a ball, a card, a vector space) satisfies the following axioms. (1) The composition fg of two symmetries f and g is a symmetry, and (fg)h = f(gh) for arbitrary symmetries f, g and h. (2) The identity function i defined by i(x) = x is a symmetry, and we have i f = f i = f for arbitrary symmetry f. (3) The inverse g of a symmetry f (i.e. fg = gf = i) exists, and g is itself a symmetry.

Group theory has grown up from these 3 axioms. Example of an immediate consequence : Apply axiom (1) and prove that (fg)(uv) = f((gu)v) for arbitrary symmetries f, g, u and v. A harder problem: Let a group G have exactly 3 symmetries; prove that no non-identity symmetry f of G satisfies f f = i .

To solve equation the 2x = 3 one needs rational numbers. All rational numbers form a field. (This concept is also axiomatic.) The square root of 2 is not a member of this field, but we can define a larger field F generated by the square root of 2. The field of real numbers R is larger than F, but to be able to solve all quadratic equations with real coefficients, we need an even larger field than R. It is the field of complex numbers C. Every algebraic equation generates its own field, the splitting field of the equation.

Why are the equations of degrees 2, 3 and 4 solvable in radicals, but the roots of equation x^5 = 100x + 100 are not expressible in radicals? It turns out that solvability in radicals depends on the symmetries of the equation’s splitting field. (The group of symmetries for the equation of degree 5 above, has 120 symmetries. Can you believe it?)

There are numerous applications of the modern algebra and its axiomatic method in all branches of mathematics, in physics, and in the practice of engineering. The background for Math 3300 is a standard first course in linear algebra.

### Math 3600 (formerly 229) – Advanced Engineering Mathematics

This course is comprised of two parts. Part I covers vector differential calculus and vector integral calculus. Part II covers concepts in complex analysis, including complex numbers and functions; conformal mapping; complex integration; and power series. The emphasis is on problem solving. There are three in class exams and an in class final exam.

The textbook for the course is “Advanced Engineering Mathematics,” by Erwin Kreyszig.

### Math 3120 (formerly 234) – Introduction to Partial Differential Equations

The principal objective of Math 3120 is to solve boundary value problems involving partial differential equations. A good background in calculus and differential equations is essential. The heat, wave, and potential equations are developed separately by deriving the mathematical model from physical intuition. The solution method of separation of variables receives the most attention because it is widely used in applications and because it provides a uniform method for solving the most important types of partial differential equations. Other techniques developed include D’Alembert’s solution of the wave equation, series solutions, numerical methods, and Laplace transforms.

Boundary value problems in partial differential equations arise in the context of classical mechanics in higher dimensions, quantum mechanics, the physics of elasticity, and fluid mechanics. Understanding a complex natural process comes from combining or building on simpler and more basic models. A thorough knowledge of physical models, the differential equations that describe them, and the solutions to these equations, is the first step toward describing the complicated behavior of the real world.

### Math 3200 (formerly 242) – Introduction to Topology

Overview: The three main topics covered in this undergraduate mathematics course are:

- Knot theory
- Point set topology, or abstract spaces
- Surfaces

Knot theory is about loops of string in 3-dimensional space. We’ll prove that knots exist, define some of the modern polynomial invariants that distinguish knots, and find ourselves at the frontiers of research thinking about unanswered questions.

Point set topology concerns local properties of spaces needed to discuss such fundamental notions as continuity, connectedness and compactness. This is foundational material for many branches of mathematics.

Surface theory is about spheres, tori, Moebius bands, Klein bottles, projective planes and more. We’ll prove a classification theorem and learn how to distinguish one surface from another.

This course is highly recommended if you are mathematically talented and if one of the following fits:

Warning: This is a rigorous proof-based mathematics course. You will be required to understand theorems and their proofs, and discover and write proofs of your own.

Prerequisites include a completion of our calculus sequence (preferably MATH 2300) and Linear Algebra (MATH 2600). You should know the basics of mathematical logic, sets, functions and proofs.

- you are considering graduate work in mathematics
- you are considering a research career in theoretical physics, chemistry or biology and want an introduction to some modern mathematical tools
- you like geometric thinking and want to pursue that in a mathematically rigorous way
- you are an engineering student and want to see a side of mathematics different from routine calculations

### Math 3640 (formerly 247) – Probability

This course is concerned with advanced topics of probability theory, not typically covered by introductory statistics courses, such as Math1010/1011, Math2820, and Math2821. Some of the more in-depth subjects to be dealt with are: combinatorics (the mathematics of counting), probability models (including various types of distributions for random variables), limit theorems (weak and strong versions of the law of large numbers), and stochastic processes (in particular, discrete Markov processes).

Here a few typical questions that can be resolved using the above mathematical tools:

- At a party 20 men take off their hats. The hats are then mixed up and each man randomly selects one. What is the probability that no man selects his own hat?
- A pipe-smoking mathematician carries, at all times, 2 matchboxes, 1 in his left-hand pocket and 1 in his right-hand pocket. Each time he needs a match he is equally likely to take it from either pocket. Consider the moment when the mathematician first discovers that one of his matchboxes is empty. If it is assumed that both matchboxes initially contained 30 matches, what is the probability that there are at least 10 matches in the other box?
- A person goes for a run each morning. When he leaves his house for his run he is equally likely to go out either the front or the back door; similarly, when he returns he is equally likely to go to either the front or the back door. The runner owns 5 pairs of running shoes which he takes off after the run at whichever door he happens to be at. If there are no shoes at the door from which he leaves to go running he runs barefooted. What proportion of time does he run barefooted?
- Five hundred voters are selected at random and asked who they voted for. What is the probability that the candidate whom the majority of those surveyed named won the election?

### Math 3641 (formerly 248) – Mathematical Statistics

This course is concerned with several fundamental topics in advanced statistical theory, which include distribution theory, theory of parameter estimation, hypothesis testing, statistical inference, analysis of variance, and regression. Since these theoretical tools are of interest in a host of practical applications, ranging from engineering, control theory and simulation, to actuarial sciences and econometrics, some concrete applications will also be discussed in the course. Here are some situations where the above statistical methods can be used to analyze real-world problems:

- Studying the effects of the presence of four different sugar solutions (glucose, sucrose, fructose, and a mixture of the three) on bacterial growth.
- Some defendants in criminal proceedings plead guilty and are sentenced without a trial, whereas others who plead innocent are subsequently found guilty and then are sentenced. In recent years, legal scholars have speculated as to whether sentences of those who plead guilty differ in severity from sentences for those who plead innocent and are judged guilty. Do historical data suggest that the proportion of all defendants in both groups who are sent to prison is different for the two groups?
- A statistics professor suspects that one of his graders is not properly grading the weekly quizzes taken by his students. In fact, the professor suspects that the grader assigns scores at random, unlike his other graders. Can you device a statistical test that will help in deciding (based on knowing the scores of all quizzes throughout the entire semester by all graders) whether the professor’s suspicion is justified?

### Math 3310 (formerly 250) – Introduction to Mathematical Logic

### Math 3660 (formerly 256) – Mathematical Modeling in Economics

A course in mathematical modelling emphasizing models used in economics. Demand functions and profit maximization, Nash equilibrium, analysis of mergers, auction models, valuation of options, present value of income streams and interest rate risk, as well as other topics of interest to students. Statistical models, and estimation techniques. Detailed computation of models using Mathematica.

### Math 3165 (formerly 259) – Advanced Calculus

This course reviews and builds upon some topics covered in elementary calculus. The topics covered in this course are (1) differential calculus of functions of several variables; (2) vector differential calculus; (3) integral calculus of functions of several variables; and (4) vector integral calculus. A combination of problem solving, proofs, and applications is expected of students. There are three in class exams and an in class final exam.

The textbook for the course is “Advanced Calculus,” by Wilfred Kaplan.

### Math 3100 (formerly 260) – Introduction to Analysis

Introduction to Analysis is the study of the foundations of Calculus and is designed to bridge the gap between the intuitive calculus normally offered at the undergraduate level and the sophisticated analysis courses the student encounters at the senior or first-year-graduate level. This course is an introduction to the theory of the real number system, sequences and limits, continuity of function, derivatives, and the Riemann integral. Much of the course material will be familiar from calculus, but the focus here is on understanding and writing mathematical proofs.

This course is good preparation for graduate study in mathematics, and is a good course for secondary mathematics teachers and actuaries who wish to have a solid understanding of calculus.

### Math 3110 (formerly 261) – Complex Variables

In this course we study some of the amazing and beautiful properties of the complex number system and analytic functions of a complex variable. For example, we find that the trigonometric functions, the hyperbolic functions and the exponential function can be expressed in terms of one another when considered as functions of a complex variable. Complex numbers and analytic functions are widely used in engineering and science. We will cover several applications of the complex function theory including the Fourier transform and boundary value problems from electrostatics and steady-state heat flow in the plane. The central tools in our study will be Cauchy’s integral theorem and the theory of residues.

This course is suitable for upper level undergraduate students and graduate students in mathematics, engineering, and science.

### Math 4200/4201 (formerly 272) – Topology

Topology is a graduate level course that attracts some advanced undergraduates. The majority of students in this course will be graduate students in mathematics. It would be a good idea to have had either 3300, 3200 or 3165 before taking on this course. The goal of the course is to prepare one for graduate study in mathematics. Solid foundations for further study in Analysis, Geometry, Geometric Topology and Algebraic Topology are established in this course. If you plan to go to graduate school in mathematics or physics, this course would give you a fantastic start. Beyond the core material (see the undergraduate catalog) the course is designed to develop a level of mathematical maturity consistent with that of beginning graduate students in very good graduate programs. The ability to prove more and more sophisticated problems is cultured. A yardstick for measuring the difficulty of a problem is the number of distinct ideas necessary to solve the problem. As the course progresses, students are asked to solve and present problems that require increasingly more ideas.

### Math 4700 (formerly 274) – Combinatorics

Math 4700 is an introduction to combinatorics, the art of counting. Besides being an interesting area of pure mathematics, combinatorics also has applications to the analysis of algorithms in computer science, and to basic probability theory. The tools we use, particularly generating functions, occur in other parts of mathematics such as statistics. There are surprising connections to mathematical analysis (formal power series) and to algebra (group actions when counting objects with symmetries), although no particular background in these areas will be assumed.

In this course we cover five fundamental counting techniques. First, we start with the basic theory of permutations and combinations, subject to various restrictions. Second, we examine how power series can be used to represent counting information in the form of generating functions, and how manipulation of these generating functions can solve counting problems. Third, we look at recurrence relations, where the number of objects of a given size can be expressed in terms of the numbers of objects of smaller sizes. Fourth, we see how the theory of inclusion-exclusion can be used to count objects with combinations of properties. Finally, we apply group theory to count objects with symmetry in Polya’s Theorem.

The course is suitable for graduate students and upper-level undergraduates.

### Math 4710 (formerly 275) – Graph Theory

A graph is a set of points (called vertices) with some connections (called edges) between them. They are particularly helpful in understanding the behavior of networks but they can arise in many different contexts, both theoretical and applied. The theorems and techniques developed in the course are useful in electrical engineering, operations research, and computer science among other areas. They are also incredibly interesting in and of themselves.

In this course we will study various properties of graphs. We will analyze how easy they are to disconnect. We will consider when they have closed paths that go through all of their vertices. These are called Hamiltonian cycles and are closely related to the Traveling Salesman Problem. We will also consider how many different colors are needed to color the vertices so that no two vertices of the same color have an edge between them. This problem is the general case of the Four Color Theorem, a problem first discussed in the mid-19th century but now relevant to the assigning of cell phone frequencies.

This course is suitable for upper-level undergraduates as well as graduate students in mathematics and engineering.

### Math 4310 (formerly 280) – Set Theory

### Math 4600 (formerly 286) – Numerical Analysis

Partial differential equations are the key mathematical tool for describing many physical processes involving functions of several variables. Most such real-world problems cannot be solved with the help of classical PDE methods. We need algorithms that can be run on a computer. This topic is so important that, despite 50 years of research, the search for even faster and more accurate methods continues.

The purpose of this course is to acquaint the student with some of the best available numerical methods for solving BVP’s. Both finite difference and variational methods are treated.

The course can be considered as a natural sequel to our introductory course in numerical mathematics, MATH 3620. The main prerequisite is Math 3620 or some other basic numerical analysis course, but some familiarity with PDE’s would be useful (although classical methods for exact solution of BVP’s will not play much of a role other than to serve to create test examples). Students should have good programming skills in order to write programs testing various methods. The course is open to both advanced undergraduates and graduate students.

The course is a must for students who want to solve real-world problems in any area of science or engineering, but the methods treated here also find applications in a variety of other fields, ranging from Business to Medicine.

### Math 4630 (formerly 287) – Nonlinear Optimization

Math 4630 is an introduction to the theory and practice of nonlinear optimization. Optimization occurs whenever you wish to find the best way to do something, and all areas of science and engineering, and even many areas in the humanities, make use of it. Nonlinear optimization deals with the most general continuous optimization problems, which occur very frequently, for example, in engineering applications.

The course begins with an introduction to how real world problems can be modelled as mathematical optimization problems. We discuss the basic theory of unconstrained optimization, including the important idea of convexity. We look at methods for 1-dimensional unconstrained optimization, such as Newton’s method, the bisection method, the Golden Section method, and interpolation methods. We then look at methods for n-dimensional unconstrained optimization, including methods that use second derivatives (Newton’s), first derivatives (steepest descent, quasi-Newton, conjugate gradient) and no derivatives. Approaches such as line search and trust regions are discussed. We investigate the basic theory of constrained optimization, particularly the Karush-Kuhn-Tucker conditions. Then we examine methods for constrained optimization, including Sequential Quadratic Programming and barrier/penalty function methods.

Note that you do NOT need to have taken Math 4620, Linear Optimization, in order to take Math 4630. Prerequisites are multivariable calculus, linear algebra, and a computer programming course.

The course is suitable for graduate students and upper-level undergraduates.

### Math 4620 (formerly 288) – Linear Optimization

Math 4620 is an introduction to the theory and practice of linear optimization. Optimization occurs whenever you wish to find the best way to do something, and all areas of science and engineering, and even many areas in the humanities, make use of it. Linear optimization is a special well-solved case of the general optimization problem, and has applications to many disciplines.

The first half of the course concentrates on the basic theory of linear programming, developed in conjunction with a discussion of the classic algorithm for solving linear programs, known as the Simplex Method. We look at linear program models for real problems, feasibility and boundedness, the Simplex Method (both the basic method and the two-phase method), duality, complementary slackness, the Dual Simplex Method, and sensitivity analysis.

The second half of the course investigates several further aspects of linear programming. First, we look at newer methods for linear programming, the Ellipsoid Method and Interior Point Methods. Then we will look at applications of linear programming to solving problems of combinatorial optimization, such as network problems like shortest path or maximum flow. In particular we cover as many as we can of integer programming and total unimodularity, cutting planes, branch-and-bound, and the primal-dual algorithm.

Note that you do NOT need to have taken Math 4630, Nonlinear Optimization, in order to take Math 4620. Prerequisites are linear algebra, and a computer programming course.

The course is suitable for graduate students and upper-level undergraduates.

*Updated GS, March 2011*

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