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Abstract: We inspect Vietoris–Rips complexes VRt(X) of certain metric spaces X using a new generalization of Bestvina–Brady discrete Morse theory. Our main result is a pair of metric criteria on X, called the Morse Criterion and Link Criterion, that allow us to deduce information about the homotopy types of certain VRt(X). One application is to topological data analysis, specifically persistence of homotopy type for certain Vietoris–Rips complexes. For example we recover some results of Adamaszek–Adams and Hausmann regarding homotopy types of VRt(Sn). Another application is to geometric group theory; we prove that any group acting geometrically on a metric space satisfying a version of the Link Criterion admits a geometric ... Read More PubDate: 2022-10-02T00:00:00-05:00

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Abstract: Following ideas of Iriyeh and Shibata we give a short proof of the three-dimensional Mahler conjecture for symmetric convex bodies. Our contributions include, in particular, simple self-contained proofs of their two key statements. The first of these is an equipartition (ham sandwich type) theorem which refines a celebrated result of Hadwiger and, as usual, can be proved using ideas from equivariant topology. The second is an inequality relating the product volume to areas of certain sections and their duals. Finally we give an alternative proof of the characterization of convex bodies that achieve the equality case and establish a new stability ... Read More PubDate: 2022-10-02T00:00:00-05:00

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Abstract: In this paper we construct families of bounded domains Ωε and solutions uε of{−Δuε=1inΩεuε=0on∂Ωεsuch that, for any integer k ≥ 2, uε admits at least k maximum points for small enough. The domain Ωε is "not far" to be convex in the sense that it is starshaped, the curvature of ∂Ωε vanishes at exactly two points and the minimum of the curvature of ∂Ωε goes to 0 as ε → ... Read More PubDate: 2022-10-02T00:00:00-05:00

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Abstract: In this paper, we prove an averaged version of an algebraicity conjecture of Gross and Zagier concerning the values of higher Green's function at CM points. Furthermore, we give the factorization of the ideal generated by such algebraic value in the spirit of the famous work of Gross and Zagier on singular ... Read More PubDate: 2022-10-02T00:00:00-05:00

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Abstract: We consider the following variational problem: among all curves in ℝn of fixed length with prescribed end points and prescribed tangents at the end points, minimise the L∞-norm of the curvature. We show that the solutions of this problem, and of a generalised version, are characterised by a system of differential equations. Furthermore, we have a lot of information about the structure of solutions, which allows a ... Read More PubDate: 2022-10-02T00:00:00-05:00

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Abstract: This article is concerned with the interplay between the theory of smooth valuation on manifolds and Riemannian geometry. We confirm the angularity conjecture formulated by A. Bernig, J. H. G. Fu, and G. Solanes which sheds new light on the geometric meaning of the Lipschitz-Killing valuations. The proof relies on a complete classification of translation-invariant angular curvature measures on ℝn, a result of independent ... Read More PubDate: 2022-10-02T00:00:00-05:00

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Abstract: An analogue of the Riesz-Sobolev convolution inequality is formulated and proved for arbitrary compact connected Abelian groups. Maximizers are characterized, and a quantitative stability theorem is proved, under natural hypotheses. A corresponding stability theorem for sets whose sumset has nearly minimal measure is also proved, sharpening recent results of other authors. For the special case of the group ℝ/ℤ, a continuous deformation of sets is developed, under which a scaled Riesz-Sobolev functional is shown to be ... Read More PubDate: 2022-10-02T00:00:00-05:00

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Abstract: Given an integer matrix A ∈ ℤd×n, we study the natural mixed Hodge module structure in the sense of Saito on the Gauß–Manin system attached to the monomial map h: (ℂ*)d → ℂn induced by A. We completely determine in the normal case the associated graded object to the weight filtration, by computing the intersection complexes with respective multiplicities that form its constituents. Our results show that these data are purely combinatorial, and not arithmetic, in the sense that they only depend on the polyhedral structure of the cone of A, but not on the semigroup itself. In particular, we extend results of de Cataldo, Migliorini and Musta\c{t}\v{a} to the setting of torus embeddings and give a closed form for the ... Read More PubDate: 2022-10-02T00:00:00-05:00

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Abstract: In this paper we determine the automorphism groups of the profinite braid groups with four or more strings in terms of the profinite Grothendieck-Teichmüller ... Read More PubDate: 2022-10-02T00:00:00-05:00