Whitney Encounters of the Second Kind

I’m speaking tomorrow in the AMS Current Events Bulletin about the work of Adiprasito, Huh, and Katz on the Rota-Welsh conjecture and Hodge theory for matroids.   See this previous post for an introduction to their work.  My write-up for the Current Events Bulletin is included in this booklet.

Here’s an excerpt from the last section of my slides which I may or may not have time to discuss in tomorrow’s talk.  It concerns this recent paper of June Huh and Botong Wang.  (Note added: As anticipated I did not have time to cover this material!  Here are the slides themselves: ceb_talk)

Continue reading

Hodge Theory in Combinatorics

AHK_picture

From L to R: Karim Adiprasito, June Huh, Eric Katz

In January 2016, my colleague Josephine Yu and I are organizing a workshop called Hodge Theory in Combinatorics. The goal of the workshop is to present the recent proof of a 50-year-old conjecture of Rota by Karim Adiprasito, June Huh, and Eric Katz. In this post, I want to explain what the conjecture says and give a brief outline of its marvelous proof. I will follow rather closely this paper by Adiprasito-Huh-Katz (henceforth referred to as [AHK]) as well as these slides from a talk by June Huh. Continue reading

A Celebration of Independence

Yesterday marked the second anniversary of my blog, and today is the 239th birthday of the U.S. In celebration of Independence Day, I want to explain what matroids are. Matroids were invented by Hassler Whitney (and independently by Takeo Nakasawa) to abstract the notion of linear independence from vector spaces to a much more general setting that includes acyclicity in graphs. Other pioneering early work on matroids was done by Garrett Birkhoff and Saunders MacLane. Matroid theory is a rich subject about which we will only scratch the surface here. In particular, there are many different (“cryptomorphic“) ways to present the matroid axioms which all turn out to be (non-obviously) equivalent to one another. We will focus on just a couple of ways of looking at matroids, emphasizing their connections to tropical geometry. Continue reading