Volume 2022, Issue 788

This paper presents some results concerning the size of magnetic fields that support zero modes for the three-dimensional Dirac equation and related problems for spinor equations. It is a well-known fact that for the Schrödinger equation in three dimensions to have a negative energy bound state, the 3/2{3/2} norm of the potential has to be greater than the Sobolev constant. We prove an analogous result for the existence of zero modes, namely that the 3/2{3/2} norm of the magnetic field has to greater than twice the Sobolev constant. The novel point here is that the spinorial nature of the wave function is crucial. It leads to an improved diamagnetic inequality from which the bound is derived. While the results are probably not sharp, other equations are analyzed where the results are indeed optimal.

Morrison, Walker, and Wedrich used the blob complex to construct a generalization of Khovanov–Rozansky homology to links in the boundary of a 4-manifold. The degree zero part of their theory, called the skein lasagna module, admits an elementary definition in terms of certain diagrams in the 4-manifold. We give a description of the skein lasagna module for 4-manifolds without 1- and 3-handles, and present some explicit calculations for disk bundles over S2{S^{2}}.

The Weyl principle is extended from the Riemannian to the pseudo-Riemannian setting, and subsequently to manifolds equipped with generic symmetric (0,2){(0,2)}-tensors. More precisely, we construct a family of generalized curvature measures attached to such manifolds, extending the Riemannian Lipschitz–Killing curvature measures introduced by Federer. We then show that they behave naturally under isometric immersions, in particular they do not depend on the ambient signature. Consequently, we extend Theorema Egregium to surfaces equipped with a generic metric of changing signature, and more generally, establish the existence as distributions of intrinsically defined Lipschitz–Killing curvatures for such manifolds of arbitrary dimension. This includes in particular the scalar curvature and the Chern–Gauss–Bonnet integrand. Finally, we deduce a Chern–Gauss–Bonnet theorem for pseudo-Riemannian manifolds with generic boundary.

We investigate the variety of commuting matrices. We classify its components for any number of matrices of size at most 7. We prove that starting from quadruples of 8×8{8\times 8} matrices, this scheme has generically nonreduced components, while up to degree 7 it is generically reduced. Our approach is to recast the problem as deformations of modules and generalize an array of methods: apolarity, duality and Białynicki–Birula decompositions to this setup. We include a thorough review of our methods to make the paper self-contained and accessible to both algebraic and linear-algebraic communities. Our results give the corresponding statements for the Quot schemes of points, in particular we classify the components of Quotd⁡(𝒪𝔸n⊕r){\operatorname{Quot}_{d}(\mathcal{O}_{\mathbb{A}^{n}}^{\oplus r})} for d≤7{d\leq 7} and all r , n .

We show that every convex ancient solution of mean curvature flow with Type I curvature growth is either spherical, cylindrical, or planar. We then prove the corresponding statement for flows by a natural class of curvature functions which are convex or concave in the second fundamental form. Neither of these results assumes interior noncollapsing.

Let ℓ{\ell} be a prime, k a finitely generated field of characteristic different from ℓ{\ell}, and X a smooth geometrically connected curve over k . Say a semisimple representation of π1ét⁢(Xk¯){\pi_{1}^{{\text{\'{e}t}}}(X_{\bar{k}})} is arithmetic if it extends to a finite index subgroup of π1ét⁢(X){\pi_{1}^{{\text{\'{e}t}}}(X)}. We show that there exists an effective constant N=N⁢(X,ℓ){N=N(X,\ell)} such that any semisimple arithmetic representation of π1ét⁢(Xk¯){\pi_{1}^{{\text{\'{e}t}}}(X_{\bar{k}})} into GLn⁡(ℤℓ¯){\operatorname{GL}_{n}(\overline{\mathbb{Z}_{\ell}})}, which is trivial mod ℓN{\ell^{N}}, is in fact trivial. This extends a previous result of the second author from characteristic zero to all characteristics. The proof relies on a new noncommutative version of Siegel’s linearization theorem and the ℓ{\ell}-adic form of Baker’s theorem on linear forms in logarithms.

Let G be a unimodular locally compact group. We define a property of irreducible unitary G -representations V which we call c-temperedness, and which for the trivial V boils down to Følner’s condition (equivalent to the trivial V being tempered, i.e. to G being amenable). The property of c-temperedness is a-priori stronger than the property of temperedness. We conjecture that for semisimple groups over local fields temperedness implies c-temperedness. We check the conjecture for a special class of tempered V ’s, as well as for all tempered V ’s in the cases of G:=SL2⁢(ℝ){G:=\mathrm{SL}_{2}({\mathbb{R}})} and of G=PGL2⁢(Ω){G=\mathrm{PGL}_{2}(\Omega)} for a non-Archimedean local field Ω of characteristic 0 and residual characteristic not 2. We also establish a weaker form of the conjecture, involving only K -finite vectors. In the non-Archimedean case, we give a formula expressing the character of a tempered V as an appropriately-weighted conjugation-average of a matrix coefficient of V , generalising a formula of Harish-Chandra from the case when V is square-integrable.