We hypothesize a new and more complete set of anomalies of certain quantum field theories (QFTs) and then give an eclectic proof. First, we propose a set of 't Hooft anomalies of 2d $\mathbb{CP}^{\mathrm{N}-1}$-sigma models at $\theta=\pi$, with N $=2,3,4$ and others, by enlisting all possible 3d cobordism invariants and selecting the matched terms. Second, we propose a set of 't Hooft higher anomalies of 4d time-reversal symmetric SU(N)-Yang-Mills (YM) gauge theory at $\theta=\pi$, via 5d cobordism invariants (higher symmetry-protected topological states) such that compactifying YM theory on a 2-torus matches the constrained 3d cobordism invariants from sigma models. Based on algebraic/geometric topology, QFT analysis, manifold generator dimensional reduction, condensed matter inputs and additional physics criteria, we derive a correspondence between 5d and 3d new invariants, thus broadly prove a more complete anomaly-matching between 4d YM and 2d $\mathbb{CP}^{\mathrm{N}-1}$ models via a twisted 2-torus reduction, done by taking the Poincar\'e dual of specific cohomology class with $\mathbb{Z}_2$ coefficients. We formulate a higher-symmetry analog of "Lieb-Schultz-Mattis theorem" to constrain the low-energy dynamics.

Let 1 p+. We show that the positive part of the closed unit ball of a non-commutative L p-space, as a metric space, is a complete Jordan-invariant for the underlying von Neumann algebra.

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We study the isoperimetric structure of asymptotically flat Riemannian 3-manifolds (M, g) that are C0-asymptotic to Schwarzschild of mass m > 0. Refining an argument due to H. Bray, we obtain an effective volume comparison theorem in Schwarzschild. We use it to show that isoperimetric regions exist in (M, g) for all sufficiently large volumes, and that they are close to centered coordinate spheres. This implies that the volume-preserving stable constant mean curvature spheres constructed by G. Huisken and S.-T. Yau as well as R. Ye as perturbations of large centered coordinate spheres minimize area among all competing surfaces that enclose the same volume. This confirms a conjecture of H. Bray. Our results are consistent with the uniqueness results for volumepreserving stable constant mean curvature surfaces in initial data sets obtained by G. Huisken and S.-T. Yau and strengthened by J. Qing and G. Tian. The additional hypotheses that the surfaces be spherical and far out in the asymptotic region in their results are not necessary in our work.

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