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**MATH 3740 - Vector Calculus****Instructor: Huguette TranMW: 2:00 PM - 3:20 PM**

Math majors can use Math 3740 toward either the analysis or geometry breadth or depth requirement. This course is strongly recommended for students pursuing a certificate in actuarial science, and is also useful for students interested in science and engineering applications.

At the end of this course the students should be able to calculate the divergence and curl of a vector field and the gradient of a function. Students will be able to apply the change of variables theorem including polar and spherical coordinates to calculate many types of integrals. Students will also be able to calculate line integrals and they will be able to parametrize a variety of surfaces. The students will also be able to calculate the surface area of these surfaces and they will be able to apply Green's, Stokes', and Gauss' theorem to simplify these integrals. The students will also be able to calculate the Gauss and mean curvatures of a number of surfaces and be able to describe what these quantities tell us about the surface in question. If time permits we will discuss Lagrange multipliers and the implicit function theorem.

**MATH 4060 - Foundations of GeometryInstructor: John KruegerTR: 3:30-4:50Prerequisites:** MATH 3000

From previous classes students are familiar with geometry in the setting of the Cartesian plane and geometric objects are defined algebraically from equations. But for thousands of years mathematicians studied geometry without the use of numbers or algebra. This class explores Euclid's axiomatic approach to geometry, in which the subject is developed logically from a small set of axioms. Of particular importance is the parallel postulate, whose analysis led to the so-called non-euclidean geometries. This class offers students exposure to one of the most historically important and beautiful areas of mathematics.

**Math 4090: Senior Seminar** **[Cancelled because of low enrollment]****Fall 2014 seminar topic: Diophantine Approximation****Instructor: Lior Fishman****TR 11-12:20**

Satisfies UNT's Core Curriculum** Capstone **Experience Requirement**Intended Primarily for Math Majors considering continuing their educations in graduate school.****Prerequisites: **Senior standing, Math 2700, Math 3410, Math 3510, Math 3610, and one of Math 3680 or Math 4610. Note for Fall '14, the 3410, 3680, and 4610 prerequisites are not essential and students who have not had those courses may enroll by seeing Rita in GAB 443.**Textbook:** Diophantine Approximation – W. M. Schmidt (any edition will be fine).

The main objective of this course is to introduce and prepare students for math research in the 21^{st} century. Senior seminar is a new capstone option intended primarily for math majors who are considering continuing their mathematics education in graduate school. By examining a mathematical topic of contemporary interest, students will be introduced to the culture and techniques of mathematical research and will get a better sense of what is involved in a graduate mathematics education. In Fall 2014, we shall explore Diophantine approximation theory, starting with classical results and methods such as Dirichlet’s and Minkowski’s theorems, continued fractions and Farey series. We shall then read recently published research papers in the field. Students will be required to present selected topics in class, preparing them for independent study and presentations. Students interested in computer programing will be encouraged to explore some questions relating to the course’s material, and the data gathered may help us formulate and hopefully even prove some conjectures.

**Math 4100 - Fourier AnalysisInstructor: Nick AnghelTR: 12:30 PM – 1:50 PMPrerequisites**: MATH 1720, MATH 2700, and MATH 3410.

Everybody knows what a Taylor series is. By analogy, think of a Fourier series, the main object of study in this course, as a series where the trigonometric functions sine and cosine, rather than monomial powers, are used for the purpose of representing more general functions. Clearly, they are well-suited for describing oscillatory processes. Fourier invented them for solving otherwise complicated problems, involving heat conduction or string vibration.

This course is an introduction to Fourier analysis and its applications based on G. F. Folland’s book with the same name. It is intended for students in mathematics, physics and engineering at the advanced undergraduate level.

After defining rigorously the classical concept of Fourier series and describing precisely the class of functions admitting Fourier expansions, we will apply the concept to two important classes of boundary value problems in partial differential equations, the heat and wave equations.

A second part of the course will deal with function expansions associated to orthogonal sets of function. As such this course also qualifies as an introduction to Functional Analysis. The orthogonal polynomials of Hermite, Legendre, and Laguerre will be particularly scrutinized.

Lastly, the course will explore the concepts of Fourier and Laplace transforms, as tools of rephrasing new and difficult problems in terms of older, better understood ones.

**MATH 5460 - Differential EquationsInstructor: J. IaiaMWF: 9:00am - 9:50am**

We will begin the course with a brief review of topics covered in an undergraduate differential equations course: separable and first order linear equations, second order linear equations, power series solutions, regular singular points, and first order linear systems. Next we will prove the standard existence theorem of ODE's: i.e. if *f*(*t, y*) and *∂f*/*∂y* are continuous in a neighborhood of (*t*_{0},* y*_{0}) then there is a unique solution of *y*’(*t*) = *f*(*t*,* y*(*t*)) with *y*(*t*_{0}) = *y*_{0} in a neighborhood of (*t*_{0},* y*_{0}). We will also prove the continuous dependence of the solution of this equation with respect to parameters. After that we will discuss linear systems *y*’(*t*) = *A*(*t*)*y*(*t*) + *g*(*t*) where *A*(*t*)and *g*(*t*) are continuous and we will also discuss the case when *A*(*t*)and *g*(*t*) are periodic. After this we will examine behavior in a neighborhood of a critical point and discuss stability for linear and almost linear systems. We will also study Lyapunov functions and their use in proving stability. In the second semester of the course we will study the heat equation, wave equation, and the Laplace equation.

**Math 6200: Topics in Ergodic Theory****Instructor: Professor Mariusz Urbanski****TR 3:30 - 4:50**

The course will cover basics of topological and measure-preserving dynamical systems and ergodic theory. The following topics will be covered.

(A) Topological dynamical systems on compact metric spaces

- (A1) Topological conjugacy and topological factors.
- (A2) Periodic points, ! and limit sets.
- (A3) Minimality, transitivity, topological mixing.
- (A4) Symbol dynamical systems and subshifts of nite type.
- (A5) Rotations on the circle, translations on tori and general compact topological groups.
- (A6) Distance expanding maps and the Gauss map.
- (A7) Expansive maps.

(B) Measure-preserving dynamical systems

- (B1) Examples of measure-preserving dynamical systems.
- (B2) Bogulubov{Krylov Theorem.
- (B3) Poincare's Return Theorem.
- (B3) The concept of ergodicity, Birkho 's Ergodic Theorem, and its numerous consequences.
- (B4) Ergodicity - examples.
- (B5) Mixing properties.
- (B6) Uniquely ergodic systems.
- (B7) First return (induced) maps; Kac Lemma.

(C) Entropy of measure-preserving dynamical systems

- (C1) Shannon entropy of a probability vector; Kchinchine's treatment of information theory
- (C2) Entropy of a partition
- (C3) Kolmogorov-Sinaj entropy of a measure-preserving dynamical system; isomorphism problem.
- (C4) Tools to calculate Kolmogorov-Sinaj entropy.
- (C5) Entropy of Examples.

(D) Entropy and topological pressure of topological dynamical systems

- (D1) The defi nition.
- (D2) Basic properties.
- (D3) Entropy and pressure of examples; relation to spectral radii of matrices.
- (D4) Variational Principle.

There will be no single textbook. I will deliver pdf files of 2-3 of my books, whose parts will be used for the course.

**Math 6310: Topics in Combinatorics****Instructor: Professor Joseph Kung****MW 12:00 - 1:20**

This will be a course on algebraic and enumerative combinatorics. The focus will be on counting and homology in partially ordered sets and lattices. We will start with basic counting methods: what to do when you meet an integer sequence, recursions, generating functions, the standard counting numbers, and using databases like OEIS [online encyclopedia of integer sequences] . More advanced topics include Moebius functions, characteristic polynomials, chain and no-broken-circuit complexes. We will also study specific examples: subspace and subgroup lattices, lattices of intersections of arrangements of hyperplanes, Bruhat orders, and permutahedrons. We will definitely not be doing CATegory theory.

We will not have an official textbook, but a useful reference is R. Graham, D. Knuth, and O.Patashnik, Concrete Mathematics, Addison-Wesley, 1989, 0-201-14236-8.

View the Course Descriptions from the University Catalog