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Talks at Kansas State University
April 7, 2009
Talk I Talk II
Talk I: (u, v, and w knots) x (topology, combinatorics, low algebra,
and high algebra)
Abstract. My subject is a Cartesian product. It runs in
three parallel columns - the u column, for usual knots,
the v column, for virtual knots, and the w column,
for welded, or weakly virtual, or warmup knots. Each
class of knots has a topological meaning and a "finite type" theory,
which leads to some combinatorics, somewhat different combinatorics
in each case. In each column the resulting combinatorics ends up
describing tensors within a different "low algebra" universe - the
universe of metrized Lie algebras for u, the richer universe of
Lie bialgebras for v, and for w, the wider and therefore
less refined universe of general Lie algebras. In each column there is
a "fundamental theorem" to be proven (or conjectured), and the means,
in each column, is a different piece of "high algebra": associators and
quasi-Hopf algebras in one, deformation quantization à la Etingof and
Kazhdan in the second, and in the third, the Kashiwara-Vergne theory
of convolutions on Lie groups. Finally, u maps to v
and v maps to w at topology level, and the relationship
persists and deepens the further down the columns one goes.
The 12 boxes in this product each deserves its own talk, and the few that
are not yet fully understood deserve a few further years of research. Thus my
talk will only give the flavour of a few of the boxes that I understand, and
only hint at my expectations for the contents the (2,4) box, the one I
understand the least and the one I wish to understand the most.
Appetizer Handout. Homomorphic_Expansions.pdf.
Main Handout. 3x4.html, 3x4.pdf, 3x4.png (source files:
My scratch work. Pensieve:
Talk and Pensieve:
1st Kansas Talk.
Talk II: The Hardest Math I've Ever Really Used
Abstract. What's the hardest math I've ever used in real
life? Me, myself, directly - not by using a cellphone or a GPS device
that somebody else designed. And in "real life" - not while studying or
I use addition and subtraction daily, adding up bills or calculating
change. I use percentages often, though mostly it is just "add 15
percents". I seldom use multiplication and division: when I buy in
bulk, or when I need to know how many tiles I need to replace my kitchen
floor. I've used powers twice in my life, doing calculations related to
mortgages. I've used a tiny bit of 2x2 linear algebra for a tiny bit of
non-math-related computer graphics I've played with. And for a long
time, that was all. In my talk I will tell you how recently a math
topic discovered only in the 1800s made a brief and modest appearance
in my non-mathematical life. There are many books devoted to that topic
and a lot of active research. Yet for all I know, nobody ever needed
the actual gory formulas for such a simple reason before.
Handout. hardest.html, hardest.pdf, hardest.png (source files:
My scratch work. Pensieve:
2nd Kansas Talk.
"God created the knots, all else in topology is the work of mortals."