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**MATH 4060 – Foundations of Geometry****Instructor: William Cherry****TR 3:30 - 4:50 PM****Prerequisites:** MATH 3000; Prior or concurrent enrollment in either MATH 3510 or MATH 3610 is strongly recommended.**Texts: **Robin Hartshorne, *Geometry: Euclid and Beyond,* Springer Verlag, 2005; ISBN 9780387986500. *Euclid's Elements,* The Thomas L. Heath translation edited by Dana Densmore, Green Lion Press, 2002. ISBN 9781888009194**Course Website:** wcherry.math.unt.edu/math4060

Early interest in geometry was almost certainly motivated by a desire to improve building techniques and to be able to create interesting shapes for temples, altars, toys, and machines. Scholars in ancient Greece "abstracted" the study of shapes into an idealized form. The most widely read textbook in history is a geometry text by Euclid known as the *Elements.* Thousands of years later, much of what we learn in high school geometry is based on the ideas in Euclid's text. Euclid's goal was to create a logical foundation for idealized geometry. He wanted to provide proofs for geometric facts based on as few axioms as possible. The course will begin with a careful study of parts of Euclid and an exploration of what he was trying to do and why. The course will then move on to the contributions of 19th and 20th century scholars who built on Euclid's work. The second half of the course will look at alternative geometries where the so-called parallel postulate need not hold.

The course will emphasize the role of proof in geometry, its historical development, and the philosophical implications that proof had on the development of scientific thought. Students will begin by reading Euclid and continually develop their own proof writing skills and geometric intuition.

**MATH 4090 - Senior SeminarInstructor: Lior FishmanTR 11:00 AM - 12:20 PM**

The main objective of this course is to introduce and prepare students for math research in the 21

**MATH 4430 - Introduction to Graph TheoryInstructor: Elizabeth DrellichTR 12:30 PM - 1:50 PM**

When you send an email, how many computers must it go through before it reaches your friend's inbox?

Why does the mail truck drive that particular route around the neighborhood?

How many telephone poles can get knocked down without your house losing electricity?

And of course, how many people are you away from Kevin Bacon?

All of these questions are Graph Theory questions. In Graph Theory a graph is a collection of vertices or nodes connected by edges. These vertices might represent computers, houses, telephone poles, people, or any number other real world objects. The goal of the class is to survey the introductory concepts in the eld, including, but not limited to, distance in a graph (the email question), Euler paths (the mail truck), vertex connectivity (the telephone pole question) as well as many classic problems like the four color theorem and the bridges of Konigsberg:

Can Snowball cross all seven bridges exaclty once and end where she started?

**MATH 5210 - Numerical Analysis - Scientific Computation****Instructor: Jianguo (Jay) Liu****TR 3:30 PM - 4:50 PM****Prerequisites:** Advanced linear algebra and multivariable calculus. Computer programming experience will be helpful.

This course (one of the two courses) will prepare you for the Applied Mathematics Qualifying Exam. It will cover the fundamentals of matrix computation and optimization including the following topics:

- Error analysis and sensitivity analysis
- Design of computer programs
- Matrix decompositions and iterative methods for linear systems
- Optimality conditions for optimization
- Data fitting
- Monte Carlo simulation
- A few case studies will be presented, including image deblurring and data classification.

**MATH 6150 - Functional AnalysisInstructor: Bunyamin SariTR 11:00 AM - 12:20 PM**

This course is an introduction to Functional Analysis and to common functional analytic techniques in Analysis. Since the basic principles of functional analysis (Hahn-Banach, Open mappings, etc.) are covered in Real Analysis series, we will only give a short review of those. Then we will cover the basics of

- Weak and weak* topologies, and duality
- Bases and basic sequences
- Structure of Hilbert and the classical Banach spaces $C(K)$
*,*$c_0$, $\ell_p$, $1\le p\le\infty$. - The algebra of bounded linear operators

The material is likely to be somewhat adjusted depending on participants' background and interests.

No textbook is required. Royden's book that is used in Real Analysis is still useful because it covers some of the material. Some recommended texts (there are plenty other, however)

- B. Bollobas, Linear Analysis
- N. Carothers, A short course on Banach space theory
- Fabian et al, Banach space theory
- Conway, A course in Functional Analysis

Grading will be based on attendance and a class presentation. Each participant selects a project and writes a short but concise exposition (less than 5 pages or so) and present in class. In most cases, a project consists of presenting a single theorem, or explaining a widely used notion. I will meet with you in the second week to help you select a project that is likely be helpful to your own thesis research. I will provide references and clear instructions on what to include in the project.

**MATH 6610 – Topology and GeometryInstructor: Mariusz UrbanskiTR 3:30 PM - 4:50 PM**

The course will cover basic properties of (finite) topological dimension in the class of separable metrizable spaces. These will include three definitions/characterizations of topological dimension: small inductive one, large inductive, and covering dimension. Also, a characterization by partitions will be given. Furthermore: zero-dimensional spaces, the countable sum theorem, Cartesian product theorem, universal space, compactication, and embedding theorem will be proved. It will be shown that the dimension of R

Also selected topics from the theory of transfinite topological dimension will be covered. In particular small and large, now differing, transfinite dimensions will be treated. Countable-dimensional and strongly countable-dimensional spaces will be introduced and studied. Basic theorems will be proved. The class of Smirnov spaces will be defined and its dimension-wise properties will be dealt with.

If time permits, Hausdor dimension will be introduced and its relations to topological dimension will be emphasized. If still some time is left (rather unlikely) transfinite Hausdor dimension will be dealt with.

The textbook is: "Theory of Dimensions; Finite and Infinite" by R. Engelking. It can be purchesed directly from the publisher Heldermann Verlag at http://www.heldermann.de/SSPM/sspmcovr.htm

The grades will determined by attendance and activity of students in the classroom.

View the Course Descriptions from the University Catalog