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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)

Twenty-Seventh Annual Friends of Mathematics Lecture

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 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: 3x4.zip).

My scratch work. Pensieve: PSU Talk and Pensieve: 1st Kansas Talk.

Talk II: The Hardest Math I've Ever Really Used

Twenty-Seventh Annual Friends of Mathematics Award Banquet

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 teaching mathematics?

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: hardest.zip).

My scratch work. Pensieve: 2nd Kansas Talk.

Some propaganda...

"God created the knots, all else in topology is the work of mortals."

Leopold Kronecker (modified)