Martingales arising from minimal submanifolds and other geometric contexts

We consider a class of martingales on Cartan-Hadamard manifolds that includes Brownian motion on a minimal submanifold. We give sufficient conditions for such martingales to be transient, extending previous results on the transience of minimal submanifolds. We also give conditions for the almost sure convergence of the angular component (in polar coordinates) of a martingale in this class, including both the negatively pinched case (using earlier results on martingales of bounded dilation), and the radially symmetric case with quadratic decay of the upper curvature bound. Applied to minimal submanifolds, this gives curvature conditions on the ambient Cartan-Hadamard manifold under which any minimal submanifold admits a non-constant, bounded, harmonic function. Though our discussion is primarily motivated by minimal submanifolds, this class of martingales includes diffusions naturally associated to ancient solutions of mean curvature flow and to certain sub-Riemannian structures, and we briefly discuss these contexts as well. Our techniques are elementary, consisting mainly of comparison geometry and Ito's rule.Comment: Accepted version (some mistakes corrected from the previous), to appear in Illinois Journal of Mathematic

Heat kernel asymptotics on sub-Riemannian manifolds with symmetries and applications to the bi-Heisenberg group

By adapting a technique of Molchanov, we obtain the heat kernel asymptotics at the sub-Riemannian cut locus, when the cut points are reached by an $r$-dimensional parametric family of optimal geodesics. We apply these results to the bi-Heisenberg group, that is, a nilpotent left-invariant sub-Rieman\-nian structure on $\mathbb{R}^{5}$ depending on two real parameters $\alpha_{1}$ and $\alpha_{2}$. We develop some results about its geodesics and heat kernel associated to its sub-Laplacian and we illuminate some interesting geometric and analytic features appearing when one compares the isotropic ($\alpha_{1}=\alpha_{2}$) and the non-isotropic cases ($\alpha_{1}\neq \alpha_{2}$). In particular, we give the exact structure of the cut locus, and we get the complete small-time asymptotics for its heat kernel.Comment: 17 pages, 1 figur

Small time heat kernel asymptotics at the sub-Riemannian cut locus

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 $4t\log p_t(x,y)\to -d^2(x,y)$ for $t\to 0$, 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 $p_t(x,y)\sim t^{-5/4}\exp(-d^2(x,y)/4t)$ where $y$ is reached from a Riemannian point $x$ by a minimizing geodesic which is conjugate at $y$

Development of molecular approaches to estimating germinal mutation rates I. Detection of insertion/deletion/rearrangement variants in the human genome

DNA from 130 individuals was studied with up to 18 (primarily cDNA) probes for the frequency of variants in this initial experiment to determine the feasibility of this approach to screening for germinal gene mutations. This approach, a modification of the usual restriction enzyme mapping strategy, focuses on the detection of insertion/deletion/rearrangement (I/D/R) variants, because the DNA is digested with only two restriction enzymes before transfer to membranes and hybridization with an extensive series of unrelated probes. Some 4000 noncontiguous, independent DNA fragments ("loci"), functional loci, pseudogenes or anonymous fragments, (a total of ~ 77 400 kb) were screened. 19 different classes and 31 copies of presumably I/D/R variants were detected while 4 different classes and 24 individuals exhibiting base substitution variants were observed. 18 of the 19 I/D/R classes were rare variants, that is, each were observed at a frequency, within this population, of less than 0.01; 3 of the base substitution classes existed at polymorphic frequencies and only 1 was a rare variant. 10 of the I/D/R classes, occurring in a total of 18 individuals, were detected with probes which are not known to be associated with repetitive elements. This is a variant frequency for I/D/R variants without known repetitive elements of 0.15 classes and 0.23 copies for each 1000 kb screened; this would extrapolate to 1600 such variant sites in the genome of each individual. Within the context of a mutation screening program, the rare variants, either with or without repetitive elements, would have a higher probability of being de novo mutations than would polymorphic variants; this former group would be the focus of family studies to test for the heritability of the allele (fragment pattern). Sufficient DNA probes are available to screen a significant portion of the human genome for genetic variation and de novo mutations of this type.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/27913/1/0000334.pd

Observing the QuantumBehavior of Light in an Undergraduate Laboratory

While the classical, wavelike behavior of light (interference and diffraction) has been easily observed in undergraduate laboratories for many years, explicit observation of the quantum nature of light (i.e., photons) is much more difficult. For example, while well-known phenomena such as the photoelectric effect and Compton scattering strongly suggest the existence of photons, they are not definitive proof of their existence. Here we present an experiment, suitable for an undergraduate laboratory, that unequivocally demonstrates the quantum nature of light. Spontaneously downconverted light is incident on a beamsplitter and the outputs are monitored with single-photon counting detectors. We observe a near absence of coincidence counts between the two detectors—a result inconsistent with a classical wave model of light, but consistent with a quantum description in which individual photons are incident on the beamsplitter. More explicitly, we measured the degree of second-order coherence between the outputs to be g(2)(0)=0.0177±0.0026,g(2)(0)=0.0177±0.0026, which violates the classical inequality g(2)(0)⩾1g(2)(0)⩾1 by 377 standard deviations