Graduate/Undergraduate Reading groups

Spring 2011

During the past couple years there have been several graduate/undergraduate student reading groups in the math department. The idea of these reading groups is to give undergraduates an opportunity to learn about a topic they wouldn't usually see in their classes and a chance to work together more informally. They also have the opportunity to meet and learn from graduate students and postdoctoral fellows.

In these reading groups a graduate student meets with two to five undergraduates once a week for about an hour to work through a book on a chosen topic. The structure of the meetings is flexible. In the past students have taken turns presenting material or all members of the group have discussed what they read the week before.

To participate in a reading group you need to be willing to spend an hour or two a week as a group and time by yourself to prepare for your meeting.

This semesters topics are: To participate in a reading group, contact the graduate student or postdoctoral fellow who will be leading the group you are interested in by Monday, Jan. 24. Send them an email with your name, year, and a list of the math classes you have taken. Please check that your background matches the prerequisites for the group you are interested in.

If you have any questions please email me at arlo "dot" caine "at" nd "dot" edu.

Dr. Caine


Information for Fall 2008, Spring 2009, Fall 2009, Spring 2010, and Fall 2010.
Information for graduate students.


Introductory Set Theory

Consider the following famous question, often called the `Barber paradox.' In a town, there is a barber who shaves exactly those men who do not shave themselves. Does the barber shave himself? After a bit of contemplation, you will see that you cannot answer this question with either `yes' or `no' without causing some contradiction. In mathematics, we often use the notion of a `collection,' `set,' or `class' of objects. The use of these notions in math can be traced as far back as Aristotle, but were first studied as objects in their own right by Georg Cantor in the 1870's. Near the end of the 19th century, paradoxes similar to the Barber paradox were seen to exist in the theory of sets. To rid set theory of these foundational problems, an axiomatic approach had to be taken. In this course, we will discuss the intuitive theory of sets (also called naive set theory), then observe how the standard axioms of set theory (ZFC) are developed in order to avoid paradoxes and inconsistencies. Once we have this set theory as a foundation, we will see how much of mathematics can be built up from ZFC. This will lead to the formulation of numbers as sets. Our goal will be to see how different set theoretic definitions of `number' (ordinals and cardinals) match in the finite case, but diverge in the infinite case. Finally, if time permits, we will discuss the continuum hypothesis in a non-rigorous fashion. This course would be great for you if you are interested in mathematical logic and the foundations of mathematics, or if you just want to know what the heck people mean when they talk about `different infinities.'

Text: The Joy of Sets by Keith Devlin

Graduate student: Steven vanDenDriessche (Email: svandend "at" nd "dot" edu)

Complex Semi-simple Lie Algebras

This reading group will focus on Lie algebras which are vector spaces with additional structure. The study of Lie algebras uses ideas from and has an impact on theoretical physics and many areas of mathematics. In particular, we will learn how to associate Lie algebras to vectors in Euclidean spaces which will lead to a beautiful theorem using graph theory to classify Lie algebras. The prerequisites for this reading group is a course in linear algebra and a desire to learn some amazing mathematics!

Text: Complex Semi-simple Lie Algebras by Jean-Pierre Serre

Graduate Student: Nicole Kroeger (Email: nkroeger "at" nd "dot" edu)

Metric Space Topology

The purpose of this seminar is to introduce undergraduates to a topic in higher mathematics. General Topology is a subject accessible to able first and second year mathematics students in a good mathematics program. Students generally find it interesting and valuable. Knowledge of Topology will prove quite useful in upper level mathematics courses and in graduate work in the mathematical sciences. Students will find that the course requires some work, but not as much work as a regular Mathematics course. Initially the lectures will be given by Professor Connolly. Later, students will be asked to present some of the material. Initially the seminar will focus on metric topology, a subject with a strong intuitive underpinning. Any student in Mathematics 10860 or 20860 is eligible to join this seminar. The seminar is limited to 12 students. This will be done on a "first come, first served" basis. The first twelve students who pay Prof. Connolly for the book, or show him an order they have placed to buy the book, or show that they own the book, will be accepted in the seminar. The text (Gamelin and Greene) costs about $10.00, with shipping, in paperback. Prof. Connolly will submit an order for this text on Thursday, January 20 at 9:00 am. He will order you a copy if you go to his office, Hurley 169, with $10.00 before then. Or you can purchase it (from Amazon.com or other sites, or possibly Notre Dame’s bookstore). If you would like to join this reading group, please:

Text: Introduction to Topology, 2nd Ed. by Gamelin and Greene

Professor: Dr. Connolly