Math 2000: Discrete Mathematics
Instructor: Professor Anne Shepler
Prerequisites: MATH 1710 (may be taken concurrently)
Introduction to proof writing, logic, sets, relations and functions, induction and recursion, combinatorics and counting techniques, discrete probability, graphs and trees. The course includes an exploratory project on misuses of logic in modern society and current scientific research.
Math 2000 is a new Discovery course intended for math majors entering UNT in Fall 2013 or later. Students who began at UNT prior to Fall 2013 should consult an advisor before enrolling in Math 2000.
Math 4430: Introduction to Graph Theory
Instructor: Professor Douglas Brozovic
Prerequisites: MATH 3000 OR MATH 2770 OR CSCE 2100
Graphs are mathematical objects that can be used to model a remarkably wide range of problems, both applied and theoretical. The goal of this class is to provide you with an introduction to graphs and especially to some of their applications. This will be driven by the construction and analysis of examples of graphs, as well as by linking various types of graphs and graph theoretical properties to relevant applications. You will be constructing, identifying, analyzing, and computing (no computer programming skills are necessary) examples (and properties) of graphs—and the vast majority of your grade will be based on your performance in these areas.
Some topics that I expect to discuss include (but are not limited to) Colorings of Graphs, De Bruijn sequences and associated graphs, Gray codes and associated graphs, Route problems (“the Seven Bridges of Königsberg”, efficient routing problems), trees, and optimal matching problems. I fully expect that this course will be an enjoyable introduction to a remarkable and extremely relevant topic. Computer science students who have completed CSCE2100 are welcome.
Math 5270: Mathematical Theory of Computation
Professor Su Gao
In this course we introduce the basic notion and results of Computability Theory. In the first part of the course we will cover different models of computation, the Church-Turing Thesis, recursive enumerability, the arithmetical hierarchy, definability in arithmetic, and relative computability. As an application of the basic ideas and techniques of Computability Theory, we will give an introduction of the Computational Complexity Theory. The highlight of this introduction will be to introduce the complexity classes of P and NP, and state the P vs. NP problem rigorously. The P vs. NP problem is one of seven Millennium Problems of Mathematics that carried a $1 million prize for its solution given by the Clay Institute. For this part of the course we will use the following textbook:
Herbert B. Enderton, Computability Theory: An Introduction to Recursion Theory. Academic Press, Burlington, MA, 2010. xii+192 pp. ISBN: 0-12-384958-6
As the author wrote: “The only prerequisite for reading this book is a willingness to tolerate a certain level of abstraction and rigor. The goal here is to prove theorems, not to calculate numbers or write computer programs.” Otherwise, this part is suitable for all math and computer science graduate students and advanced undergraduate students. In the second part of the course we will discuss topics of Higher Recursion Theory such as the constructive ordinals and hyperarithmetic hierarchy. These are considered the beginning of Effective Descriptive Set Theory and will be helpful for graduate students who eventually pursue research in logic at UNT. For this part the reference book is the following:
Gerald E. Sacks, Higher Recursion Theory. Perspectives in Mathematical Logic. Springer-Verlag, Berlin, 1990. xvi+344 pp. ISBN: 3-540-19305-7
This book is available electronically through the UNT library or Project Euclid.
Math 5410: Functions of a Complex Variable
Instructor: Professor William Cherry
Prerequisites: Although the course has no formal prerequisites, and in particular no previous knowledge of complex numbers or complex functions will be assumed, students should have had some prior exposure to a proof based analysis or calculus course on the level of Math 3610. In particular, students should have had some exposure to uniform convergence and Taylor's Theorem.
The course will cover the basics of functions of one complex variable. The emphasis on the course will be proofs of the foundations of complex analysis. However, there will also be some discussion of applications to fields such as electrostatics and fluid dynamics. The course is also aimed at preparing mathematics graduate students for the departmental exam in complex analysis. Topics will include: an introduction to the complex numbers, polar coordinates, stereographic projection, complex functions defined as power series, the elementary transcendental functions, fractional linear transformations, complex derivatives, the Cauchy-Riemann equations, complex integration, the Cauchy Integral Formula and Cauchy's Theorem, the Calculus of Residues, conformal mappings and their applications, and perhaps some discussion of harmonic functions. Geometric intuition is a very important part of complex function theory, and topics will be presented from the geometric viewpoint whenever possible. The section on Cauchy's Theorem will include an introduction to the topology of the plane, but no prior knowledge in topology will be assumed. For more information, see wcherry.math.unt.edu/math5410.
View the Course Descriptions from the University Catalog