Orbifold subfactors from Hecke algebras II --- Quantum doubles and braiding ---

A. Ocneanu has observed that a mysterious orbifold phenomenon occurs in the system of the M_infinity-M_infinity bimodules of the asymptotic inclusion, a subfactor analogue of the quantum double, of the Jones subfactor of type A_2n+1. We show that this is a general phenomenon and identify some of his orbifolds with the ones in our sense as subfactors given as simultaneous fixed point algebras by working on the Hecke algebra subfactors of type A of Wenzl. That is, we work on their asymptotic inclusions and show that the M_infinity-M_infinity bimodules are described by certain orbifolds (with ghosts) for SU(3)_3k. We actually compute several examples of the (dual) principal graphs of the asymptotic inclusions. As a corollary of the identification of Ocneanu's orbifolds with ours, we show that a non-degenerate braiding exists on the even vertices of D_2n, n>2.Comment: 37 pages, Late

If Archimedes would have known functions

These are notes and slides from a Pecha-Kucha talk given on March 6, 2013. The presentation tinkered with the question whether calculus on graphs could have emerged by the time of Archimedes, if the concept of a function would have been available 2300 years ago. The text first attempts to boil down discrete single and multivariable calculus to one page each, then presents the slides with additional remarks and finally includes 40 "calculus problems" in a discrete or so-called 'quantum calculus' setting. We also added some sample Mathematica code, gave a short overview over the emergence of the function concept in calculus and included comments on the development of calculus textbooks over time.Comment: 31 pages, 36 figure

Dense packing on uniform lattices

We study the Hard Core Model on the graphs ${\rm {\bf \scriptstyle G}}$ obtained from Archimedean tilings i.e. configurations in $\scriptstyle \{0,1\}^{{\rm {\bf G}}}$ with the nearest neighbor 1's forbidden. Our particular aim in choosing these graphs is to obtain insight to the geometry of the densest packings in a uniform discrete set-up. We establish density bounds, optimal configurations reaching them in all cases, and introduce a probabilistic cellular automaton that generates the legal configurations. Its rule involves a parameter which can be naturally characterized as packing pressure. It can have a critical value but from packing point of view just as interesting are the noncritical cases. These phenomena are related to the exponential size of the set of densest packings and more specifically whether these packings are maximally symmetric, simple laminated or essentially random packings.Comment: 18 page

Applying parabolic Peterson: affine algebras and the quantum cohomology of the Grassmannian

The Peterson isomorphism relates the homology of the affine Grassmannian to the quantum cohomology of any flag variety. In the case of a partial flag, Peterson’s map is only a surjection, and one needs to quotient by a suitable ideal on the affine side to map isomorphically onto the quantum cohomology. We provide a detailed exposition of this parabolic Peterson isomorphism in the case of the Grassmannian of m-planes in complex n-space, including an explicit recipe for doing quantum Schubert calculus in terms of the appropriate subset of non-commutative k-Schur functions. As an application, we recast Postnikov’s affine approach to the quantum cohomology of the Grassmannian as a consequence of parabolic Peterson by showing that the affine nilTemperley–Lieb algebra arises naturally when forming the requisite quotient of the homology of the affine Grassmannian

Regularization of Discontinuous Foliations: Blowing up and Sliding Conditions via Fenichel Theory

We study the regularization of an oriented 1-foliation $\mathcal{F}$ on $M \setminus \Sigma$ where $M$ is a smooth manifold and $\Sigma \subset M$ is a closed subset, which can be interpreted as the discontinuity locus of $\mathcal{F}$. In the spirit of Filippov's work, we define a sliding and sewing dynamics on the discontinuity locus $\Sigma$ as some sort of limit of the dynamics of a nearby smooth 1-foliation and obtain conditions to identify whether a point belongs to the sliding or sewing regions.Comment: 32 page

Non-involutory Hopf algebras and 3-manifold invariants

We present a definition of an invariant #(M,H), defined for every finite-dimensional Hopf algebra (or Hopf superalgebra or Hopf object) H and for every closed, framed 3-manifold M. When H is a quantized universal enveloping algebra, #(M,H) is closely related to well-known quantum link invariants such as the HOMFLY polynomial, but it is not a topological quantum field theory.Comment: 36 page