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New chiral boundary conditions are found for quantum gravity with matter on AdS3. The associated asymptotic symmetry group is generated by a single right-moving U(1) Kac-Moody-Virasoro algebra with c_R = 3l/2G. The Kac-Moody zero mode generates global left-moving translations and equals, for a BTZ black hole, the sum of the total mass and spin. The level is positive about the global vacuum and negative in the black hole sector, corresponding to ergosphere formation. Realizations arising in Chern-Simons gravity and string theory are analyzed. The new boundary conditions are shown to naturally arise for warped AdS3 in the limit that the warp parameter is taken to zero.

It is well known that the spectral measure of eigenvalues of a rescaled square non-Hermitian random matrix with independent entries satises the circular law. We consider the product $T X$, where $T$ is a deterministic $N\times M$ matrix and $X$ is a random $M\times  N$ matrix with independent entries having zero mean and variance $(N \wedge M) ^{-1}$. We prove a general local circular law for the empirical spectral distribution (ESD) of $TX$ at any point $z$ away from the unit circle under the assumptions that$ N \sim M$, and the matrix entries $X_{ij} have suciently high moments. More precisely, if $z$ satisfes $| |z|
- 1| \ge \tau$  for arbitrarily small $\tau > 0, the ESD of $TX$ converges to $\tilde{X}_D (z) d A(z)$, where $\tilde{X}_D$ is a rotation-invariant function determined by the singular valuau of $T$ and $dA$ denotes the Lebesgue measure on $\mathbb{C}$. The local circular law is valid around $z$ up to scale $(N \wedge M) ^{-1/4+\epsilon}$ for any  $\epsilon >0$. Moreover, if $|z| >1$ or the matrix entries of $X$ have vanishing third moments, the local circular law is valid around $z$ up to scale $(N \wedge M) ^{-1/2+\epsilon}$ for any  $\epsilon >0$.

We obtain a growth estimate for the number of lattice points inside any Q-Gorenstein cone. Our proof uses the result of Futaki-Ono-Wang on Sasaki-Einstein metric for the toric Sasakian manifold associated to the cone, a Yau’s inequality, and the Kawasaki-Riemann-Roch formula for orbifolds.

For a sub-Riemannian manifold provided with a smooth volume, we relate the small-time asymptotics of the heat kernel at a point y of the cut locus from x with roughly “how much” y is conjugate to x. This is done under the hypothesis that all minimizers connecting x to y are strongly normal, i.e. all pieces of the trajectory are not abnormal. Our result is a refinement of the one of Leandre 4t log pt(x, y) ! −d2(x, y) for t ! 0, in which only the leading exponential term is detected. Our results are obtained by extending an idea of Molchanov from the Riemannian to the sub-Riemannian case, and some details we get appear to be new even in the Riemannian context. These results permit us to obtain properties of the sub-Riemannian distance starting from those of the heat kernel and vice versa. For the Grushin plane endowed with the Euclidean volume, we get the expansion pt(x, y)  t−5/4 exp(−d2(x, y)/4t) where y is reached from a - Riemannian point x by a minimizing geodesic which is conjugate at y.