We prove that an extreme Kerr initial data set is a unique absolute minimum of the total mass in a (physically relevant) class of vacuum, maximal, asymptotically flat, axisymmetric data for Einstein equations with fixed angular momentum. These data represent non-stationary, axially symmetric black holes. As a consequence, we obtain that any data in this class satisfy the inequality √J ≤ m, where m and J are the total mass and angular momentum of spacetime.

We show that the pre-order defined on the category of contact manifolds by arbitrary symplectic cobordisms is considerably less rigid than its counterparts for exact or Stein cobordisms: in particular, we exhibit large new classes of contact 3-manifolds which are symplectically cobordant to something overtwisted, or to the tight 3-sphere, or which admit symplectic caps containing symplectically embedded spheres with vanishing self-intersection. These constructions imply new and simplified proofs of several recent results involving fillability, planarity, and non-separating contact type embeddings. The cobordisms are built from symplectic handles of the form Σ × D and Σ × [−1, 1] × S1, which have symplectic cores and can be attached to contact 3-manifolds along sufficiently large neighborhoods of transverse links and preLagrangian tori. We also sketch a construction of J-holomorphic foliations in these cobordisms and formulate a conjecture regarding maps induced on Embedded Contact Homology with twisted coefficients.

Topological Quantum Field Theories (TQFTs) pertinent to some emergent low energy phenomena of condensed matter lattice models in 2+1 and 3+1 dimensions are explored. Many of our TQFTs are highly-interacting without free quadratic analogs. Some of our bosonic TQFTs can be regarded as the continuum field theory formulation of Dijkgraaf-Witten twisted discrete gauge theories. Other bosonic TQFTs beyond the Dijkgraaf-Witten description and all fermionic TQFTs (namely the spin TQFTs) are either higher-form gauge theories where particles must have strings attached, or fermionic discrete gauge theories obtained by gauging the fermionic Symmetry-Protected Topological states (SPTs). We analytically calculate both the Abelian and non-Abelian braiding statistics data of anyonic particle and string excitations in these theories, where the statistics data can one-to-one characterize the underlying topological orders of TQFTs. Namely, we derive path integral expectation values of links formed by line and surface operators in these TQFTs. The acquired link invariants include not only the familiar Aharonov-Bohm linking number, but also Milnor triple linking number in 3 dimensions, triple and quadruple linking numbers of surfaces, and intersection number of surfaces in 4 dimensions. We also construct new spin TQFTs with the corresponding knot/link invariants of Arf(-Brown-Kervaire), Sato-Levine and others. We propose a new relation between the fermionic SPT partition function and the Rokhlin invariant. As an example, we can use these invariants and other physical observables, including ground state degeneracy, reduced modular $\mathcal{S}^{xy}$ and $\mathcal{T}^{xy}$ matrices, and the partition function on $\mathbb{RP}^3$ manifold, to identify all $\nu \in \mathbbm{Z}_8$ classes of 2+1 dimensional gauged $\mathbb{Z}_2$-Ising-symmetric $\mathbb{Z}_2^f$-fermionic Topological Superconductors (realized by stacking $\nu$ layers of a pair of chiral and anti-chiral $p$-wave superconductors [$p_x+ip_y$ and $p_x-ip_y$], where boundary supports non-chiral Majorana-Weyl modes) with continuum spin-TQFTs.

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Omicron, the latest SARS-CoV-2 Variant of Concern (VOC), first appeared in Africa in November 2021. At present, the question of whether a new VOC will out-compete the currently predominant variant is important for governments seeking to determine if current surveillance strategies and responses are appropriate and reasonable. Based on both virus genomes and daily-confirmed cases, we compare the additive differences in growth rates and reproductive numbers (R_0) between VOCs and their predominant variants through a Bayesian framework and phylo-dynamics analysis. Faced with different variants, we evaluate the effects of current policies and vaccinations against VOCs and predominant variants. The model also predicts the date on which a VOC may become dominant based on simulation and real data in the early stage. The results suggest that the overall additive difference in growth rates of B.1.617.2 and predominant variants was 0.44 (95% confidence interval, 95% CI: −0.38, 1.25) in February 2021, and that the VOC had a relatively high R_0. The additive difference in the growth rate of BA.1 in the United Kingdom was 6.82 times the difference between Delta and Alpha, and the model successfully predicted the dominating process of Alpha, Delta and Omicron. Current vaccination strategies remain similarly effective against Delta compared to the previous variants. Our model proposes a reliable Bayesian framework to predict the spread trends of VOCs based on early-stage data, and evaluates the effects of public health policies, which may help us better prepare for the upcoming Omicron variant, which is now spreading at an unprecedented speed.