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.

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We find some necessary conditions for a real Banach space to be an almost CL-space. We also discuss the stability of CL-spaces and almost CL-spaces by$c$_{0}- and$l$_{1}-sums. Finally, we address the question if a space of vector-valued continuous functions can be a CL-space or an almost CL-space.

In the relative trace formula approach to the arithmetic Gan–Gross–Prasad conjecture, we formulate a local conjecture (arithmetic transfer) in the case of an exotic smooth formal moduli space of p-divisible groups, associated to a unitary group relative to a ramified quadratic extension of a p-adic field. We prove our conjecture in the case of a unitary group in three variables.

Given a smooth projective variety X and a smooth divisor D\subset X. We study relative Gromov-Witten invariants of (X,D) and the corresponding orbifold Gromov-Witten invariants of the r-th root stack X_{D,r}. For sufficiently large r, we prove that orbifold Gromov- Witten invariants of X_{D,r} are polynomials in r. Moreover, higher genus relative Gromov-Witten invariants of (X,D) are exactly the constant terms of the corresponding higher genus orbifold Gromov-Witten invari- ants of X_{D,r}. We also provide a new proof for the equality between genus zero relative and orbifold Gromov-Witten invariants, originally proved by Abramovich-Cadman-Wise. When r is sufficiently large and X=C is a curve, we prove that stationary relative invariants of C are equal to the stationary orbifold invariants in all genera.