Freshman Seminars

W. Hugh Woodin

2024 Fall (4 Credits)

Schedule: Wednesday, 03:45 PM–05:45 PM

Instructor Permissions: Instructor

Enrollment Cap: 12

Infinity captivates the imagination. A child stands between two mirrors and sees herself reflected over and over again, smaller and smaller, trailing off to infinity. Does it go on forever? … Does anything go on forever? Does life go on forever? Does time go on forever? Does the universe go on forever? Is there anything that we can be certain goes on forever? … It would seem that the counting numbers go on forever, since given any number on can always add one. But is that the extent of forever? Or are there numbers that go beyond that? Are there higher and higher levels of infinity? And, if so, does the totality of all of these levels of infinity itself constitute the highest, most ultimate, level of infinity, the absolutely infinite? In this seminar we will focus on the mathematical infinite. We will start with the so-called paradoxes of the infinite, paradoxes that have led some to the conclusion that the concept of infinity is incoherent. We will see, however, that what these paradoxes ultimately show is that the infinite is just quite different than the finite and that by being very careful we can sharpen the concept of infinity so that these paradoxes are transformed into surprising discoveries. We will follow the historical development, starting with the work of Cantor at the end of the nineteenth century, and proceeding up to the present. The study of the infinite has blossomed into a beautiful branch of mathematics. We will get a glimpse of this subject, and the many levels of infinity, and we will see that the infinite is even more magnificent than one might ever have imagined.

Course open to Freshman Students Only

Additional Course Attributes:

Paul Bamberg

2025 Spring (4 Credits)

Schedule: TBD

Instructor Permissions: Instructor

Enrollment Cap: 12

Projective planes were discovered by Renaissance artists who needed to depict tiled floors on canvas. Quaternions, discovered in the nineteenth century, were used by physicists to represent rotations in three dimensions, which to not commute with one another, In the early 20th century, American mathematicians discovered that quaternions could also be used as coordinates in projective planes where certain theorems of Euclidean geometry fail and the rules of ordinary algebra do not apply to coordinates.This seminar focuses on a single article published at the dawn of the computer era by the great American geometer Marshall Hall, which describes an exhaustive search, with the aid of a primitive computer, for all finite planes of order 9. We will replicate, and perhaps extend, Halls results using the R scripting language, in the process delving into finite geometry, abstract algebra, graph theory, and theory of computation.

Course Notes:
This seminar has no prerequisites. An invitation is extended to all students whether or not they are thinking about studying mathematics.
Course open to Freshman Students Only.