CAG@ICH#49
C*-trek: the next generalization
Please excuse the unpardonable but irresistible title.
Apparently, had i read more closely the Vakil paper i'm preparing to present next semester, i would have realized that one natural next generalization of Bott-Samelson varieties is that which Vakil uses, throughout, to play his checker game. They seem to go like this: Picture a poset like the one described before, except connect every vertex to every other vertex "below" it, and on each edge specify the desired dimension of the intersection of the corresponding subspaces. In the previous post i put conditions on all this indexing data that made the correspondence a true correspondence (that is, bijective). This time i'll just say, "exclude nonsense and mod out by repetition" without specifying what combinatorial conditions accomplish this (because i don't know, and they'd take too long to write anyway). The point is that we are able to enforce intermediate conditions on the various subspaces comprising the lattice, rather than just the containment indicator. This makes for a much richer world of closed ("precisely this-dimensional intersection" in each case) and open ("at most this-dimensional intersection" in at least one case) "varieties" - that is, a much finer "Bruhat" decomposition (certain unions of which comprise the "usual" (if one may call it that) decomposition).
One of the great things about blogging, as Arianna Huffington posited on The Daily Show last week, is that initial impressions and hasty, often virulent opinions are worth something to the world. I'm therefore at least precedented [Someone tell my why that's not in my dictionary while "unprecedented" is.] in my intuitive, nonrigorous, and very likely misguided and flat-wrong posts about mathematics i don't understand in the first place.
visualizing complex functions
If you've ever met a complex function, you know that it can feel like an online acquaintanceship: You have a pretty keen feel for how ey behaves and some idea of how ey looks, but you may be at a loss to explain em to someone else. Then, when you finally get a good look at em, all the fragmented understanding comes together to form a new, solid, but familiar impression. Take a look here and see if you recognize anyone.
funereal unease
It occurred to me several weeks ago to wonder why i occasionally think forward to whose funerals i'll decide to attend. These people i choose tend to be discernible only by the magnitude of the (positive) impacts they've had on my life, rather than this factor being convoluted by other, a priori expected factors like the particular aspect of my life (professional, social, educational, moral), our closeness (former friend, long-term friend, whether or not we're friends when they die), or ever their knowledge of me. (I may choose to attend Barack Obama's funeral.) These are the factors, more than the one that seems to directly influence my choice, that will determine how many people i know at the event and how advantageous it ends up being for me. So, why does this magnitude of impact matter so much?
I've come to the tentative conclusion that this is a (rare) example of humility and decency on my part, and something to reflect positively on people in general: We attend someone's funeral to demonstrate to other people our respect and admiration for the person. (I've of course modded out "obligatory" appearances at family members' and family friends' funerals, nonattendance at which would shade one rather negatively.) What do you think?