We quantized the full Einstein equations in a globally hyperbolic spacetime $N=N^{n+1}$, $n\ge 3$, and found solutions of the resulting hyperbolic equation in a fiber bundle $E$ which can be expressed as a product of spatial eigenfunctions (eigendistributions) and temporal eigenfunctions. The spatial eigenfunctions form a basis in an appropriate Hilbert space while the temporal eigenfunctions are solutions to a second order ordinary differential equation in $\R[]_+$. In case $n\ge 17$ and provided the cosmological constant $\Lam$ is negative the temporal eigenfunctions are eigenfunctions of a self-adjoint operator $\hat H_0$ such that the eigenvalues are countable and the eigenfunctions form an orthonormal basis of a Hilbert space.

In a recent paper we presented a model for a unified quantization of gravity with other fundamental sources of nature. After quantization we had to solve a Wheeler-DeWitt equation which was a hyperbolic equation in a fiber bundle which was equipped with a Lorentzian metric with a time function $t$ ranging from $0$ to infinity. The Lorentzian metric had a big bang singularity in $t=0$ and the coefficients of the hyperbolic operator also inherited this singularity. The solutions of the Wheeler-DeWitt equation were products of spatial and temporal eigenfunctions and in order to prove that these solutions also experience a big bang singularity in $t=0$ the temporal eigenfunctions $w(t)$ had to become unbounded if $t$ tends to $0$. In our former paper this was only proved in special cases where $w$ could be expressed with the help of Bessel functions but not in general. In the present paper we prove that also in the general case the temporal solutions become unbounded near $t=0$, or more precisely: $\lim_{t\ra 0} (\abs w^2+t^2 \abs{\dot w}^2)=\un$ and $\limsup_{t\ra 0}\abs w^2=\un$.

Let O_25 be the vertex algebraic braided tensor category of finite-length modules for the Virasoro Lie algebra at central charge 25 whose composition factors are the irreducible quotients of reducible Verma modules. We show that O_25 is rigid and that its simple objects generate a semisimple tensor subcategory that is braided tensor equivalent to an abelian 3-cocycle twist of the category of finite-dimensional sl_2-modules. We also show that this sl_2-type subcategory is braid-reversed tensor equivalent to a similar category for the Virasoro algebra at central charge 1. As an application, we construct a simple conformal vertex algebra which contains the Virasoro vertex operator algebra of central charge 25 as a PSL_2(C)-orbifold. We also use our results to study Arakawa's chiral universal centralizer algebra of SL_2 at level -1, showing that it has a symmetric tensor category of representations equivalent to Rep PSL_2(C). This algebra is an extension of the tensor product of Virasoro vertex operator algebras of central charges 1 and 25, analogous to the modified regular representations of the Virasoro algebra constructed earlier for generic central charges by I. Frenkel-Styrkas and I. Frenkel-M. Zhu.

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We determine exact exponential asymptotics of eigenfunctions and of corresponding transfer matrices of the almost Mathieu operators for all frequencies in the localization regime. This uncovers a universal structure in their behavior, governed by the continued fraction expansion of the frequency, explaining some predictions in physics literature. In addition it proves the arithmetic version of the frequency transition conjecture. Finally, it leads to an explicit description of several non-regularity phenomena in the corresponding non-uniformly hyperbolic cocycles, which is also of interest as both the first natural example of some of those phenomena and, more generally, the first non-artificial model where non-regularity can be explicitly studied.

The first author constructed a q-parameterized spherical category $\sC$ over C(q) in [Liu15], whose simple objects are labelled by all Young diagrams. In this paper, we compute closed-form expressions for the fusion rule of $\sC$, using Littlewood-Richardson coefficients, as well as the characters (including a generating function), using symmetric functions with infinite variables.

Graph theory is important in information theory. We introduce a quantization process on graphs and apply the quantized graphs in quantum information. The quon language provides a mathematical theory to study such quantized graphs in a general framework. We give a new method to construct graphical quantum error correcting codes on quantized graphs and characterize all optimal ones. We establish a further connection to geometric group theory and construct quantum low-density parity-check stabilizer codes on the Cayley graphs of groups. Their logical qubits can be encoded by the ground states of newly constructed exactly solvable models with translation-invariant local Hamiltonians. Moreover, the Hamiltonian is gapped in the large limit when the underlying group is infinite.

We give a new type of Schur-Weyl duality for the representations of a family of quantum subgroups and their centralizer algebra. We define and classify singly-generated, Yang-Baxter relation planar algebras. We present the skein theoretic construction of a new parameterized planar algebra. We construct infinitely many new subfactors and unitary fusion categories, and compute their trace formula as a closed-form expression, in terms of Young diagrams.

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.

The mathematical picture language project that we began in 2016 has already yielded interesting results. We also point out areas of mathematics and physics where we hope that it will prove useful in the future.