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**MATH 4510 – Abstract Algebra II****Instructor: Elizabeth Drellich****MW 2:00 - 3:20 PM****Text:** A First Course in Abstract Algebra, Fraleigh, 7th ed.

So you have taken or are taking Abstract Algebra I (3510) and know what a group is, for example: ℤ_{17}.

- You can add, for example 10 + 14 = 24 = 7 in your group.

- You have inverses, -(10) = 7 since 10 + 7 = 17 = 0 in your group.

- You have an identity element, 0 + *x* = *x*.

- You might even know that the only subgroup of ℤ_{17} is the trivial group, since the order of any subgroup must divide the order, 17 of your group.

But wait! There's more! Your group has other stuff happening!

- Can you multiply in ℤ_{17}? After all, these are numbers, does 5 x 5 = 25 = 8?

- Do you have a multiplicative identity? Does 1 x *y* = *y* for all elements *y*?

- Does every non-zero element have a multiplicative inverse? For example 3 x 6 = 18 = 1.

The answers to all these questions, for ℤ_{17}, are YES! This means your group is a ring, with unity, and in fact it is a field.

But you know what else? Our friend ℤ_{17} can multiply with elements in ℤ_{17} ⊕ ℤ_{17}: 4 x (5,9) = (20,36) = (3,2)

We say ℤ_{17} ⊕ ℤ_{17} is a ℤ_{17}-module.

If you have taken Abstract Algebra (Math 3510) and want to learn more about groups, subgroups, rings, fields, ideals, and modules, then Abstract Algebra II (Math 4510) is for you!

We will be using the same textbook, and picking up where the 3510 classes leave off this semester.

**Math 4980 - Financial MathInstructor: Huguette Tran**

Prerequisites:

This course covers fundamental concepts of financial mathematics and their applications in calculating present and future values, annuities and variable cash flows, yield rates, and valuation of stocks, bonds and other securities. The course will also provide an introduction to financial instruments, including options, contracts, and hedging, and the concept of no–arbitrage as it relates to financial mathematics.

Students who meet the prerequisites and have an interest in financial mathematics are welcome to enroll (whether they plan to complete the actuarial certificate). Students interested in a career in actuarial science are encouraged to take this course to be familiar with the topics covered in Exam FM/2.

This course may be applied toward a Math Major or Math Minor as an upper-level math elective. Non-math majors may also apply the course toward the Actuarial Science Certificate with the approval of Dr. H. Tran or of another faculty advisor in the Department of Mathematics.

**Math 5270 - Mathematical Theory of ComputationInstructor: Su GaoTR 9:30 - 10:50amText: **Computability Theory, S. Barry Cooper, 1st ed (Chapman & Hall / CRC)

The subtitle of this course is: What “The Imitation Game” didn’t tell you about Alan Turing.

The drama thriller film focused on Turing’s life, with plenty of twists and fictitious reconstructions of the history. This course, however, looks into the mathematical contributions of Turing and gives a truthful story of Computability Theory, the subject which Turing was one of the founding fathers of.

In this course we introduce the basic notion and results of Computability Theory. We will discuss different models of computation (including Turing machines), the Church-Turing Thesis, recursive enumerability, algorithmic decidability, definability in arithmetic, and relative computability. As applications of the theory we will discuss Turing’s solution of Hilbert’s Entscheidungsproblem, Goedel’s First Incompleteness Theorem, and the basics of the Computational Complexity Theory. We will introduce the complexity classes P and NP, and talk about the million-dollar P vs. NP question.

The textbook of the course will be S. Barry Cooper, *Computability Theory*, Chapman & Hall/CRC, 2004. A renowned computability theorist, Professor Cooper was the only student of Mike Yates, who was a student of Robin Gandy, who was the only student of Alan Turing.

**Math 6010 - Topics in Logic & FoundationsInstructor: Stephen JacksonTR 11:00 - 12:20 PM**

This course will focus on the theory of definable equivalence relations, and the associated descriptive set theory. Conventional descriptive set theory deals with the theory of definable subsets of Polish spaces. In the past couple of decades there has been much activity into developing the theory of definable equivalence relations. This can be viewed as extending the conventional theory from the context of Polish/standard Borel spaces to more general definable cardinalities. A simple example is the Vitali equivalence relation R/Q. A standard argument shows there is no Borel (or in the right context, definable)

selector for this equivalence relation. That is, we cannot view the quotient space R/Q as a subset of a Polish space, it is a different kind of object. This turns out to be a basic example of a hyperfinite equivalence relation. Perhaps surprisingly, there are several deep and important questions about hyperfinite relations that are still open. What kind of objects do we get in this way, and how are they related? This is the basic motivation for the subject.

Although the basic tools from descriptive set theory are heavily used, not a large background is required. We will go over the basic tools from descriptive set theory needed along the way, so no formal background is required.

The theory largely splits into the case of countable equivalence relations, and the theory of more general equivalence relations, including the theory of Polish group actions. Thus, the theory intersects with the theory of Polish groups. We will, as time permits, cover topics from both the countable as well more general case. Aside from descriptive set theory, the theory also interacts with many different areas including dynamics, ergodic theory, group theory (the groups SL_n(Z), for example, play a role in the countable theory).

Some references for the course:

- Invariant descriptive set theory by S. Gao.

- Topics in orbit equivalence by A. Kechris and B. Miller.

- The structure of hyperfinite Borel equivalence relations by Dougherty, Jackson, and Kechris (Transactions of the AMS)

- Countable Borel Equivalence relations by Jackson, Kechris, Louveau (Journal of Mathematical Logic)

- Rigidity theorems for actions of countable groups and countable Borel equivalence relations by G. Hjorth and A. Kechris (Memoirs of the AMS vol 833).

**Math 6510 - Methods in permutation groupsInstructor: Doug BrozovicMW 2:00 - 3:20 PM**

The goal of this class will be the development of methods in the theory of finite permutation groups. The goal will be to work towards some classification problems while providing a cursory but sufficient introduction to a collection of tools, both theoretical and computational. In particular, I am going to invest time regularly over the course of the semester demonstrating the use of GAP4 for various computations in group theory.

**MATH 6810 – Probability and Fractals****Instructor: Pieter Allaart****MW 12:30 - 1:50 PM****Prerequisites:** Math 5320 (measure theory). Some knowledge of undergraduate-level probability is helpful, but not required.

The purpose of this course is to explore the interface between probability theory and fractal geometry. The first four weeks or so will be spent on a rigorous development of probability based on measure theory. Topics will include the law of large numbers, conditional expectation and martingales, both in discrete and continuous time. In the middle part of the course we will introduce Hausdorff dimension, iterated function systems, self-similar sets (including the ternary Cantor set) and their generalizations. We will then learn some important probabilistic techniques for computing the Hausdorff dimension of these sets. From there, we will proceed to study random fractals, with randomized versions of the classical Weierstrass function as one of the main examples. The last third of the course will be devoted to Brownian motion and, if there is sufficient time, its generalization fractional Brownian motion. The sample paths of these stochastic processes exhibit "statistical self-similarity", and we will compute the Hausdorff dimension of the graphs and the level sets of these processes.

There is no required text for this course, but recommended reading includes:

1. P. Billingsley, Probability and Measure (3rd edition), Wiley

2. K. Falconer, Fractal geometry. Mathematical foundations and applications (2nd edition), Wiley

Your grade will be based on attendance, active class participation and one 30 minute presentation.

This course satisfies the Analysis breadth requirement.

View the Course Descriptions from the University Catalog