Courses
Below are extended descriptions of certain courses to help you plan your academic pathway in mathematics.
1000
Extended Description: In this course we aim to develop an understanding for essential topics in calculus, such as limits, derivatives, and integrals. We will be following the book “Applied Calculus for the Managerial, Life, and Social Sciences” by S.T. Tan, 10th Edition to present these ideas in the context of the real-world problems they inform. This textbook is recommended but not required.
Extended Description: In MATH 1200, we mainly focused on derivatives. While we started by studying derivatives in the context of instantaneous velocity and instantaneous rates of change, we saw that derivatives had a wide variety of other applications, namely to finding extrema, curve sketching, and approximating functions. In MATH 1201, we will begin by studying how to work backwards. In particular, if we have a derivative (for example, velocity), we’ll ask how can we recover the original function (position). In doing so, we’ll introduce the concept of an integral and study what’s known as the Fundamental Theorem of Calculus. But like with derivatives, we’ll find a wide variety of other applications of integrals, including to probability, differential equations, and the study of areas and volumes.
In this course, we want to not only introduce you to the fundamental ideas and techniques of calculus, but also to develop and improve your ability to clearly write a mathematical argument, your ability to approach challenging, multi-step problems, and your ability to work with others in a team.
Extended Description: There are three introductory Calculus sequences in the Department of Mathematics: 1100, 1200–1201, and 1300–1301. The MATH 1300–1301 sequence is an intensive, mathematically rigorous introduction to calculus. We will study formal mathematical definitions, use sound logical reasoning to justify mathematical statements, solve multi-step problems, and emphasize communication and understanding of deep mathematical ideas. While we will spend some time on computations, our main goal is to increase your understanding of the mathematical thinking involved in calculus, and to help you delve deeper into the meaning of truth in mathematics. The three sequences each occupy their own place on the spectrum between theory and practice; if you are unsure which sequence is best for you, again please reach out.
2000
Extended Description: Multivariable calculus is a central topic in the curriculum that provides a foundation for more advanced mathematics and develops skills that are essential for further study in science, engineering, economics, and other disciplines. Multivariable Calculus deals with the functions of multiple variables, whereas single-variable calculus deals with the function of one variable. The differentiation and integration process are similar to the single-variable calculus. It majorly deals with three-dimensional objects or higher dimensions. The typical operations involved in the multivariable calculus are:
• Limits and Continuity
• Partial Differentiation
• Multiple Integration
In single-variable calculus, operations like differentiation and integration are made to functions of a single variable. In multivariate calculus, it is required to generalize these to multiple variables, and the domain is therefore multi-dimensional. Care is therefore required in these generalizations, because of two key differences between 1D and higher-dimensional spaces:
- There are infinite ways to approach a single point in higher dimensions, as opposed to two (from the positive and negative direction) in 1D;
- There are multiple extended objects associated with the dimension; for example, for a 1D function, it must be represented as a curve on the 2D Cartesian plane, but a function with two variables is a surface in 3D, while curves can also live in 3D space.
Topics covered include: vectors, lines, planes, curves, and surfaces in space; vector-valued functions and functions of several variables; arc length and surface area; partial derivatives and gradient vectors; double integrals and triple integrals; vector calculus, line and surface integrals, curl, divergence, fundamental theorems for double integrals, triple integrals, line integrals and surface integral (Green’s Theorem, Stokes’ Theorem, and the Divergence Theorem).
Extended Description: Textbook: “Differential Equations & Linear Algebra, 4th Edition” by Edwards, Penney, and Calvis. We will cover most sections of Chapters 1, 3, 4-7 and 10. The topics include scalar differential equations, Laplace transforms, systems of differential equations, Gauss-elimination, algebra of matrices, determinants, vector spaces, linear operators, eigenvalues and eigenvectors.
Extended Description: There are few, if any, areas of mathematics that do not use linear algebra in some capacity, and having a strong foundation in linear algebra is guaranteed to make your future mathematical life much easier.
At the most basic level, linear algebra is a tool for solving systems of linear equations. For example, in the first week of the course we will learn techniques to find all possible values of 𝑥 and 𝑦 such that both 𝑥 + 2𝑦 = 4 and 2𝑥 − 𝑦 = 5.
Our fundamental tool for managing such systems will be matrices. We will begin by viewing a matrix as simply a sort of spreadsheet for juggling numbers; as we become more sophisticated, we will learn to understand systems of linear equations geometrically and view matrices as functions which manipulate space of 2, 3, or more dimensions. In this sense we will start to see linear algebra as a type of universal translator which allows us to take difficult problems in one context, rephrase them in a simpler context where they can be broken down and solved, and then morph them back to solve our original problem.
To help you navigate these new ideas and explain them to others, we will frequently practice verbalizing and writing (and rewriting) mathematical arguments. I look forward to a rewarding and enjoyable course; please let me know if you ever have any questions!
Topics include: vector and matrix operations, linear transformations, properties of vector spaces, solutions of systems of linear equations, eigenvalues, and eigenvectors.
Extended Description: Differential equations are an essential tool used by scientists and engineers to model phenomena in their fields. It is important to know how to solve these equations and interpret their solutions. As such, in this course, we will…
- Learn techniques for solving linear, separable, and exact first-order differential equations as well as higher-order linear equations with constant or variable coefficients,
- Explore applications of these equations including population models, Newtonian mechanics, mixing solutions, and mass-spring systems,
- Use techniques to visualize and/or estimate solutions to differential equations, and
- Understand the Laplace transform and power series as tools for solving differential equations.
Vector algebra and geometry; linear transformations and matrix algebra. Real and complex vector spaces, systems of linear equations, inner product spaces. Functions of several variables and vector-valued functions: limits, continuity, the derivative. Extremum and nonlinear problems, manifolds. Multiple integrals, line and surface integrals, differential forms, integration on manifolds, theorems of Green, Gauss, and Stokes. Eigenvectors and eigenvalues. Emphasis on rigorous proofs. Not open to students who have earned credit for MATH 2300, 2310, 2400, 2410, or 2600 without permission. Total credit for this course and MATH 2300 or 2410 will not exceed 5 credit hours; total credit for this course and MATH 2310 or 2400 will not exceed 4 credit hours; total credit for this course and MATH 2600 will not exceed 6 credit hours. Credit hours reduced from second course taken (or from test or transfer credit) as appropriate. Open only to first-year students with a test score of 5 on the Calculus-BC Advanced Placement examination.
Extended Description: Algebra of matrices, real and complex vector spaces, linear transformations, and systems of linear equations. Eigenvalues, eigenvectors, inner product spaces, and orthonormal bases. Designed primarily for mathematics majors.
Extended Description: Textbook: Fundamentals of Differential Equations and Boundary Value Problems, 7th edition, by Nagle, Saff, and Snider.
Course Content and Goals: We will cover most of Chapters 1, 2, 4, 5, 9, and 12. Depending on time, we may cover some additional content. By the end of the course, students should have an understanding of first-order ordinary differential equations, second-order ordinary differential equations, matrix methods for linear systems, and autonomous systems. This course will include proofs (in the textbook, presented in class, and as parts of assignments). A goal of this course is for students to continue improving their critical thinking and mathematical writing.
Extended Description: Probability and statistics are an essential tool used by scientists and engineers to model phenomena in their fields. As such, in this course, we will…
- Understand the basic principles of probability including set theory, counting, and Bayes Theorem and use them in problem-solving situations,
- Understand the definitions of discrete, continuous, joint random variables, density and distribution functions,
- Define binomial, Poisson, hypergeometric, geometric, negative binomial, normal, lognormal, exponential, uniform, and gamma random variables, know their probability density and distribution functions,
- Use the Central Limit Theorem to approximate distributions with large sample sizes,
- Estimate population parameters from data sets and compute confidence intervals for these population parameters,
- Learn the basic components of hypothesis testing and perform hypothesis tests on population means, variances, and proportions,
- Perform linear regression of two variables.
MATH 2300 (or 2501) is a prerequisite for this course.
Extended Description: 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.
Extended Description: Applications of the theory developed in MATH 2820. The objective of this course is to learn to apply statistical procedures practically to real-world data. This includes: basics of programming in R, preprocessing and visualizing data, estimating parameters of interest, their dependence and variability, checking model assumptions, and, most importantly, reporting your findings. Pre- or co-requisite: MATH 2810 or MATH 2820.
Extended Description: 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).
3000
Extended Description: 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.
Extended Description: 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.
Extended Description: 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.
Extended Description: 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.
Extended Description: Mathematics 3130/5130. Fourier Series, convolutions, Poisson kernels, Dirichlet kernels, pointwise and mean-square convergence of Fourier series, Fourier Integral in 1-D, Fourier inversion formula and Plancherel Theorem, Poisson Summation formula, Multidimensional Fourier integral, Radon Transform, X-Ray transform, Finite Fourier analysis, Fourier analysis on Abelian groups, Time Frequency Analysis, Gabor Transform, Multiresolution Analysis, Wavelet Transform, Sampling Theory.
Extended Description: 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
Extended Description: The foundation of differential geometry and general relativity is the concept of curvature. The course will focus on understanding this and related concepts, both geometrically and computationally. The cases of curves and surfaces in Euclidean space will be emphasized before addressing the higher dimensional framework. Interdisciplinary aspects of differential geometry will be highlighted.
Extended Description: 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.
Extended Description: Explores the distinction and connections between syntax (symbols) and semantics (meaning) by formalizing the presentation of syntax and the interpretations of that syntax in propositional, equational, and first-order deductive systems.
Extended Description: Textbook: Coding Theory and Cryptography: The Essentials (2nd edition), D. R. Hankerson et al., Marcel Dekker, 2000 (required). (Note that the other sections of Math 3320 may be using a different book.)
Course outline: We will cover Chapters 1 and 2 of the textbook, a selection of material from Chapters 3 through 6, and as much as possible of Chapters 10 through 12. Some additional material may be covered, including extra material on finite fields and material on elliptic curve cryptography. Notes will be provided for anything not covered in the textbook.
Extended Description: This course will cover real and complex vector spaces, norms, inner products and Hilbert spaces, orthonormal bases, eigenvalues and eigenvectors, various kinds of operators (unitary, normal, positive, etc.), spectra of operators and spectral theorems, and several common matrix decompositions (e.g., Jordan-Holder, polar decomposition, singular value decomposition). Brief attention will be given to some applications, including some related to data science. In addition, students will practice their mathematical writing and presentation skills.
Differences Between MATH 3330 and MATH 5330: Graduate students may enroll in MATH 5330, while undergraduate students may enroll in MATH 3330. The difference between the two courses lies in the requirements for the in-class presentation component, described below.
Extended Description: 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.
Extended Description:
Numerical analysis is a crucial tool for scientists and engineers to model and understand phenomena in their respective fields. In this course, we will:
- Explore numerical approximation and its limitations.
- Investigate floating-point arithmetic and its impact on numerical computations.
- Develop methods for solving equations:
- Find solutions for both linear and non-linear equations (e.g., bisection, Newton’s, and secant methods).
- Study interpolation and approximation:
- Explore polynomial interpolation techniques (e.g., Lagrange and Newton polynomials).
- Learn curve fitting using least squares regression.
- Solve systems of linear equations:
- Implement methods like Gaussian elimination, LU factorization, and singular value decomposition.
- Study numerical integration techniques:
- Learn and apply techniques such as the trapezoidal rule, Simpson’s rule, and Gaussian quadrature.
- Solve ordinary differential equations numerically:
- Implement methods like the Taylor series and Runge-Kutta methods.
Throughout the course, we will emphasize applications of these numerical methods in science, engineering, and economics.
Extended Description: This course is designed for students in science, mathematics, and medicine as a general introduction to the subject of mathematical models in biology and medicine. The course assumes a basic knowledge of calculus and differential equations. Mathematical tools such as basic ideas in linear regression, method of least squares, differential equations, matrix analysis, and probability theory are reviewed. General approaches to modeling and computer simulation of biological processes are provided.
Examples of models presented are:
- Population Dynamics – Models of biological evolution, populations structured by age or physiological properties, models in ecology, models of interacting species, environmentally limited population growth, epidemic populations.
- Random Movement in Space and Time – Models of biological membranes, diffusion processes, transfer of oxygen across the placenta, transport phenomena.
- Disposition of Drugs and Inorganic Toxins – Models of lead contamination, early embryogenesis, gas exchange, the digestive system, the circulatory system, bone formation, kidney function, pharmacokinetics.
- Neurophysiology – Models of synapses and inter-neuronal connections, conduction of action potentials, the Fitzhugh-Nagumo equations.
- Biochemistry of Cells – Models of atomic bonding in biochemistry, biopolymers, molecular information transfer, enzymes and their functions.
- Disease Pathogenesis – Models of the immune system, viral infections, bacterial infections, in-host HIV infection, tumor growth.
- The COVID-19 Pandemic – Models of asymptomatic and symptomatic transmission, models of vaccination programs.
Extended Description: This course is concerned with advanced topics of probability theory, not typically covered by introductory statistics courses, such as MATH 1010/1011, MATH 2820, and MATH 2821. 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 people take off their hats. The hats are then mixed up and each man randomly selects one. What is the probability that no person selects their 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 she leaves his house for her run she is equally likely to go out either the front or the back door; similarly, when she returns she is equally likely to go to either the front or the back door. The runner owns 5 pairs of running shoes which she takes off after the run at whichever door she happens to be at. If there are no shoes at the door from which she leaves to go running he runs barefooted. What proportion of time does she 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?
Extended Description: 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 devise 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?
Extended Description: Applications of actuarial (probabilistic) models to insurance. Covers a subset of the topics covered by the SOA ALTAM exam.
Extended Description: 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.
Extended Description: Texts: There is no official textbook for the course, but lectures will be influenced by the following texts, which are freely available on the authors’ websites:
• The Elements of Statistical Learning Theory (by Hastie, Tibshirani, Friedman)
• Pattern Recognition and Machine Learning (by Bishop)
Prerequisites: Linear Algebra and Probability/Statistics. Graduate Workload Differential: MATH 3670 is open to undergraduate enrollment only; graduate students may register for MATH 5670. Students registered for 5670 will, in addition to the normal course requirements for 3670, be required to read and present to the instructor a suitable research paper; this presentation will count as part of the homework grade.
Extended Description:
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.
Extended Description: This course is an introduction to number theory, where we will rely solely on the fundamental properties of the integers. Number theory concerns questions, many old or even unsolved, about basic properties of prime numbers and integers. Complex and intricate phenomena will become apparent soon, such as the Fundamental Theorem of Arithmetic, Fermat’s Little Theorem, multiplicative functions, quadratic reciprocity, Hensel’s lemma, Pythagorean triples, and sums of squares. Number theory is one of the oldest subjects of mathematics, and still a very active area of research. Two of the famous seven Millenium problems deal with (more advanced) number theoretic questions, and both of them are still unsolved. Important applications of Number Theory include cryptography or web security, and we will lay the foundations to approach these areas.
Extended Description: The course deals with a powerful and relatively new class of approximating functions called Spline Functions, which have many applications in mathematics, science, and engineering. These include data fitting and approximation, numerical quadrature, curve and surface design and representation, and the numerical solution of both ODE’s and PDE’s, including very recent developments in IGA. The purpose of this seminar is to investigate the best available algorithms for solving problems numerically based on splines. There will be no written homework problems (and no exams), and students will not be doing proofs. However, they will be asked to carry out a number of programming exercises. It should be of interest to anyone interested in applications of mathematics and should be particularly valuable to students in all branches of science and engineering. Familiarity with linear algebra, differential equations, and MATLAB is expected.
4000
Extended Description: 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.
Extended Description: The course is a continuation of MATH 4300/6300. We will pick up where that course ended by reviewing covering basic field theory and continue with Galois theory. The next major topic after this will be module theory. Time permitting, after this we will cover additional topics of interest, such as some basic algebraic number theory. In particular, we will roughly follow the outline of Chapters 10-14 of Dummit and Foote for the first two parts of the course for the remainder of the core topics of the course.
Extended Description: 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.
Extended Description: 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.
Extended Description: 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.
Extended Description: 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.
Extended Description: 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.