Preprint: "Examples of quantum cluster algebras associated to partial flag varieties" (arXiv:0907.4922)

A preprint of my latest paper is now available on the arXiv via the above link. The abstract for this paper is as follows:

We give several explicit examples of quantum cluster algebra structures, as introduced by Berenstein and Zelevinsky, on quantized coordinate rings of partial flag varieties and their associated unipotent radicals. These structures are shown to be quantizations of the cluster algebra structures found on the corresponding classical objects by Geiß, Leclerc and Schröer, whose work generalizes that of several other authors. We also exhibit quantum cluster algebra structures on the quantized enveloping algebras of the Lie algebras of the unipotent radicals.

Lifetime Achievement prize for George Lusztig

From the American Mathematical Society web pages:

George Lusztig of the Massachusetts Institute of Technology has received the 2008 AMS Leroy P. Steele Prize for Lifetime Achievement. Presented annually by the American Mathematical Society, the Steele Prize is one of the highest distinctions in mathematics. The prize was awarded today [7th January 2008] at the Joint Mathematics Meetings in San Diego, California.

"The work of George Lusztig has entirely reshaped representation theory, and in the process changed much of mathematics," the prize citation states. "His work has touched widely separated parts of mathematics, reshaping them and knitting them together. He has built new bridges to combinatorics and algebraic geometry, solving classical problems in those disciplines and creating exciting new ones. This is a remarkable career, and as exciting to watch today as it was at the beginning more than thirty years ago."

The full citation may be found in the prize booklet (PDF), from page 46 onwards.

Many congratulations to Prof. Lusztig. I have personally been inspired by his work, especially on quantum groups. Indeed, my whole research career would likely not exist without him.

Preprint: "On the inductive construction of quantized enveloping algebras" (arXiv:0706.0455)

A preprint of my latest paper is now available on the arXiv (the mathematical preprint repository) via the above link. The subject of this paper is a construction for a class of Hopf algebras called quantized enveloping algebras. The construction is a way of encoding in the algebraic structure the idea of adding nodes to the associated Dynkin diagrams.

A new mathematical celebrity: E8

Towards the end of last month, the popular media across the world covered a story about pure mathematics research. While this is not unheard of, it was rather surprising - though not in a bad way! The story in question was about a calculation carried out by a group of mathematicians, investigating the structure of an object called "E₈". (Actually, many different objects can be reasonably labelled E₈ but the one they were interested in is the split real form of the Lie group. Don't worry if you don't know what that is! The links below will help explain.)

For those interested in knowing more, I suggest starting with the original information put out by the group or the BBC's report of it. These also include some "pretty pictures" (they're mathematical, so they do mean something, rather than simply being decorative!) There's a more in-depth discussion in comments posted by John Baez and others on the n-Category Cafe blog. This is aimed at the semi-knowledgeable and could be considered a warm-up for the full technical details.

This story highlights several features of modern pure mathematical research:

• even the explicit calculations often take substantial computing power;
• the calculations provide glimpses of more general theory (they're not the end of the story, just the beginning);
• it's hard to convey in layperson's language what's actually being done (if you're not a mathematician, please don't take it personally when we say it's hard to explain!);
• sometimes you get very pretty pictures: it's all about symmetry! And finally
  
• representation theory is cool*
  

(* OK, I'm biased. But really, it is!)