Preprint: "A quantum analogue of the dihedral action on Grassmannians" (arXiv:1102.0422)

A preprint of the paper "A quantum analogue of the dihedral action on Grassmannians" by Justin Allman and myself is now available on the arXiv via the above link. The abstract for this paper is as follows:

In recent work, Launois and Lenagan have shown how to construct a cocycle twisting of the quantum Grassmannian and an isomorphism of the twisted and untwisted algebras that sends a given quantum minor to the minor whose index set is permuted according to the n-cycle c=(1 2 ... n). This twisting is needed because c does not induce an automorphism of the quantum Grassmannian, as it does classically and semi-classically.

We extend this construction to give a quantum analogue of the action on the Grassmannian of the dihedral subgroup of S_n generated by c and w₀, the longest element, and show that this induces an action of this subgroup on the torus-invariant prime ideals of the quantum Grassmannian. We also show that this subgroup acts on the totally nonnegative and totally positive Grassmannians. Then we see that this dihedral subgroup action exists classically, semi-classically (by Poisson automorphisms and anti-automorphisms, a result of Yakimov) and in the quantum and nonnegative settings.

2010 Abel Prize awarded to John T. Tate

From the Abel Prize website:

The Norwegian Academy of Science and Letters has decided to award the Abel Prize for 2010 to John Torrence Tate, University of Texas at Austin, for his vast and lasting impact on the theory of numbers. The President of the Norwegian Academy of Science and Letters, Nils Christian Stenseth, announced the name of the 2010 Abel Laureate at the Academy in Oslo today, 24. March. John Tate will receive the Abel Prize from His Majesty King Harald at an award ceremony in Oslo, Norway, May 25.

The Abel Prize recognizes contributions of extraordinary depth and influence to the mathematical sciences and has been awarded annually since 2003. It carries a cash award of NOK 6,000,000 (close to € 730,000 or US$ 1 mill.)

The theory of numbers stretches from the mysteries of prime numbers to the ways in which we store, transmit, and secure information in modern computers. Over the past century it has developed into one of the most elaborate and sophisticated branches of mathematics, interacting profoundly with other key areas. John Tate is a prime architect of this development.

John Tate's scientific accomplishments span six decades. A wealth of essential mathematical ideas and constructions were initiated by Tate and later named after him, such as the Tate module, Tate curve, Tate cycle, Hodge-Tate decompositions, Tate cohomology, Serre-Tate parameter, Lubin-Tate group, Tate trace, Shafarevich-Tate group, Néron-Tate height, to mention just a few.

According to the Abel committee, "Many of the major lines of research in algebraic number theory and arithmetic geometry are only possible because of the incisive contributions and illuminating insights of John Tate. He has truly left a conspicuous imprint on modern mathematics".

First Clay Mathematics Institute Millennium Prize Announced

From the Clay Mathematics Institute website:

The Clay Mathematics Institute hereby awards the Millennium Prize for resolution of the Poincaré conjecture to Grigoriy Perelman.

The Poincaré conjecture is one of the seven Millennium Prize Problems established by CMI in 2000. The Prizes were conceived to record some of the most difficult problems with which mathematicians were grappling at the turn of the second millennium; to elevate in the consciousness of the general public the fact that in mathematics, the frontier is still open and abounds in important unsolved problems; to emphasize the importance of working towards a solution of the deepest, most difficult problems; and to recognize achievement in mathematics of historical magnitude.

For more information, see the page linked from this post's title.

Preprint:"Quantum cluster algebra structures on quantum Grassmannians and their quantum Schubert cells: the finite-type cases" (arXiv:0912.4397)

A preprint of the paper "Quantum cluster algebra structures on quantum Grassmannians and their quantum Schubert cells: the finite-type cases", by myself and Stéphane Launois, is now available on the arXiv via the above link. The abstract for this paper is as follows:

We exhibit quantum cluster algebra structures on quantum Grassmannians K_q[Gr(2,n)] and their quantum Schubert cells, as well as on K_q[Gr(3,6)], K_q[Gr(3,7)] and K_q[Gr(3,8)]. These cases are precisely those where the quantum cluster algebra is of finite type and the structures we describe quantize those found by Scott for the classical situation.

Election of new Warden at Keble College, Oxford

The new Warden of Keble College, Oxford, who will succeed Professor Dame Averil Cameron in Michaelmas Term 2010 is Sir Jonathan Phillips KCB, currently Permanent Secretary to the Northern Ireland Executive.