Inspired by the quantum McKay correspondence, we consider the classical ADE Lie theory as a quantum theory over sl2. We introduce anti-symmetric characters for representations of quantum groups and investigate the Fourier duality to study the spectral theory. In the ADE Lie theory, there is a correspondence between the eigenvalues of the Coxeter element and the eigenvalues of the adjacency matrix. We formalize related notions and prove such a correspondence for representations of Verlinde algebras of quantum groups: this includes generalized Dynkin diagrams over any simple Lie algebra g at any level k. This answers a recent comment of Terry Gannon on an old question posed by Victor Kac in 1994.

Surface registration between cortical surfaces is crucial in medical imaging for performing systematic comparisons between brains. Landmark-matching registration that matches anatomical features, called the sulcal landmarks, is often required to obtain a meaningful 1-1 correspondence between brain surfaces. This is commonly done by parameterizing the surface onto a simple parameter domain, such as the unit sphere, in which the sulcal landmarks are consistently aligned. Landmark matching surface registration can then be obtained from the landmark aligned parameterizations. For genus-0 closed brain surfaces, the optimized spherical harmonic parameterization, which aligns landmarks to consistent locations on the sphere, has been widely used. This approach is limited by the loss of bijectivity under large deformations and the slow computation. In this paper, we propose FLASH, a fast algorithm to compute the optimized spherical harmonic parameterization with consistent landmark alignment. This is achieved by formulating the optimization problem to ℂ and thereby linearizing the problem. Errors introduced near the pole are corrected using quasiconformal theories. Also, by adjusting the Beltrami differential of the mapping, a diffeomorphic (1-1, onto) spherical parameterization can be effectively obtained. The proposed algorithm has been tested on 38 human brain surfaces. Experimental results demonstrate that the computation of the landmark aligned spherical harmonic parameterization is significantly accelerated using the proposed algorithm.

We prove that there are no stable intersections of regular Cantor sets in the$C$^{1}topology: given any pair ($K$,$K$′) of regular Cantor sets, we can find, arbitrarily close to it in the$C$^{1}topology, pairs $ \left( {\tilde{K},\tilde{K}'} \right) $ of regular Cantor sets with $ \tilde{K} \cap \tilde{K}' = \emptyset $ .

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In this paper we describe a machinery for homological calculations of representations of FI_G, and use it to develop a local cohomology theory over any commutative Noetherian ring. As an application, we show that the depth introduced by the second author coincides with a more classical invariant from commutative algebra, and obtain upper bounds of a few important invariants of FI_G-modules in terms of torsion degrees of their local cohomology groups.