Mathematics Program Presents
A Taste of Algebraic Combinatorics
Wednesday, April 7, 2021
Online Event
3:00 pm – 4:00 pm EDT/GMT-4
3:00 pm – 4:00 pm EDT/GMT-4
Galen Dorpalen-Barry '15, University of Minnesota
In 1943, J. L. Woodbridge of Philadelphia submitted the following problem to American Mathematical Monthly: “Show that n cuts can divide a cheese into as many as $(n+1)(n^2 - n + 6)/6$ pieces.”This question and its solution are deeply connected to the study of collections of lines in $mathbb{R}^2$, planes in $mathbb{R}^3$, and more generally hyperplanes in $mathbb{R}^n$. We will explore the solution and a more general version: given n (hyper)planes in a real, d-dimensional vector space, how can we figure out the number of chambers of an arrangement of hyperplanes, without necessarily being able to see and count them?
There are many wonderful solutions to this question. We present one provided by the Varchenko-Gel’fand ring, which is the ring of functions from the chambers of the arrangement to the integers with pointwise addition and multiplication. Varchenko and Gel’fand gave a simple presentation for this ring, which can be computed using simple facts about linear algebra.
We will assume very little background but expect that the audience is familiar with linear independence and dependence. We will give a ring-theoretic solution to this problem, so it may be helpful (but not necessary) to be familiar with quotient rings.
Zoom Info: https://bard.zoom.us/j/86398169686?pwd=M0pvT25ETmFhbUhkb1FUc2FuaGl0QT09
Meeting ID: 863 9816 9686
Passcode: 742619
For more information, call 845-758-6822, e-mail [email protected],
or visit https://bard.zoom.us/j/86398169686?pwd=M0pvT25ETmFhbUhkb1FUc2FuaGl0QT09.
Time: 3:00 pm – 4:00 pm EDT/GMT-4
Location: Online Event