The present article surmmarizes our recent results about Calabi–Yau threefolds with infinite fundamental group. This class of Calabi–Yau manifolds is relatively simple yet rich enough to display the essential complexities of Calabi–Yau geometries, and thus it provides good testing-grounds for general theories and conjectures.
The statistical structure on a manifold M is predicated upon a special kind of coupling between the Riemannian metric g and a torsion-free affine connection ∇ on the TM, such that ∇g is totally symmetric, forming, by definition, a “Codazzi pair” t∇, gu. In this paper, we first investigate various transformations of affine connections, including additive translation (by an arbitrary (1,2)-tensor K), multiplicative perturbation (through an arbitrary invertible operator L on TM), and conjugation (through a non-degenerate two-form h). We then study the Codazzi coupling of ∇ with h and its coupling with L, and the link between these two couplings. We introduce, as special cases of K-translations, various transformations that generalize traditional projective and dual-projective transformations, and study their commutativity with L-perturbation and h-conjugation transformations. Our derivations allow affine connections to carry torsion, and we investigate conditions under which torsions are preserved by the various transformations mentioned above. While reproducing some known results regarding Codazzi transform, conformal-projective transformation, etc., we extend much of these geometric relations, and hence obtain new geometric insights, for the general case of a non-degenerate two-form h (instead of the symmetric g) and an affine connection with possibly non-vanishing torsion. Our systematic approach establishes a general setting for the study of Information Geometry based on transformations and coupling relations of affine connections.
We provide a unified approach for constructing Wick words in mixed q-Gaussian algebras, which are generated by sj = aj +a ∗ j , j = 1, · · · , N, where aia ∗ j −qija ∗ j ai = δij . Here we also allow equality in −1 ≤ qij = qji ≤ 1. This approach relies on Speicher’s central limit theorem and the ultraproduct of von Neumann algebras. We also use the unified argument to show that the Ornstein–Uhlenbeck semigroup is hypercontractive, the Riesz transform associated to the number operator is bounded, and the number operator satisfies the Lp Poincar´e inequalities with constants C √p. Finally we prove that the mixed q-Gaussian algebra is weakly amenable and strongly solid in the sense of Ozawa and Popa. Our approach is mainly combinatorial and probabilistic. The results in this paper can be regarded as generalizations of previous results due to Speicher, Biane, Lust-Piquard, Avsec, et al.
Benjamin AllenHarvard University, Emmanuel CollegeChristine SampleEmmanuel CollegeYulia DementievaEmmanuel CollegeRuben C. MedeirosEmmanuel CollegeChristopher PaolettiEmmanuel CollegeMartin A. NowakHarvard University
Publications of CMSA of Harvardmathscidoc:1702.38002
Over time, a population acquires neutral genetic substitutions as a consequence of random
drift. A famous result in population genetics asserts that the rate, K, at which these substitutions
accumulate in the population coincides with the mutation rate, u, at which they arise in
individuals: K = u. This identity enables genetic sequence data to be used as a “molecular
clock” to estimate the timing of evolutionary events. While the molecular clock is known to
be perturbed by selection, it is thought that K = u holds very generally for neutral evolution.
Here we show that asymmetric spatial population structure can alter the molecular clock
rate for neutral mutations, leading to either K<u or K>u. Our results apply to a general class
of haploid, asexually reproducing, spatially structured populations. Deviations from K = u
occur because mutations arise unequally at different sites and have different probabilities of
fixation depending on where they arise. If birth rates are uniform across sites, then K u. In
general, K can take any value between 0 and Nu. Our model can be applied to a variety of
population structures. In one example, we investigate the accumulation of genetic mutations
in the small intestine. In another application, we analyze over 900 Twitter networks to
study the effect of network topology on the fixation of neutral innovations in social evolution.
We study the heat kernel expansion of the Laplacian on n-forms defined on a subgraph of a directed complete graph. We derive two expressions for the subgraph heat kernel on 0-forms and compute the coefficients of the expansion. We also obtain the subgraph heat kernel of the Laplacian on 1-forms.