Let ( A , m ) be a Gorenstein local ring, and F = { F n } n ∈ Z a Hilbert filtration. In this paper, we give a criterion for Gorensteinness of the associated graded ring of F in terms of the Hilbert coefficients of F in some cases. As a consequence we recover and extend a result proved in [20]. Further, we present ring-theoretic properties of the normal tangent cone of the maximal ideal of A = S / ( f ) where S = K 〚 x , y 1 , … , y m 〛 is a formal power series ring over an algebraically closed field K, and f = x a − g ( y 1 , … , y m ) , where g is a polynomial with g ∈ ( y 1 , … , y m ) b ∖ ( y 1 , … , y m ) b + 1 , and a , b , m are integers. We show that the normal tangent cone G ¯ ( m ) is Cohen-Macaulay if a ≤ b and a ≠ 0 in K. Moreover, we give a criterion of the Gorensteinness of G ¯ ( m ) .

In this article, we compute the minimal free resolutions of the tangent cones of complete intersection monomial curves in affine 4-space under certain conditions. We first show that a complete intersection monomial curve in 3-space has a complete intersection tangent cone if and only if it is obtained by a nice gluing. By using the technique of nice gluing, we construct infinitely many families of complete intersection monomial curves with complete intersection tangent cones. We also construct infinitely many families of the complete intersection monomial curves that do not have complete intersection tangent cones and compute the minimal free resolutions of these families.

We consider an area-minimizing integral current T of codimension higher than 1 in a smooth Riemannian manifold \Sigma . In a previous paper we have subdivided the set of interior singular points with at least one flat tangent cone according to a real parameter, which we refer to as the ‘singularity degree’. In this paper, we show that the set of points for which the singularity degree is strictly larger than 1 is (m-2) -rectifiable. In a subsequent work, we prove that the remaining flat singular points form a \mathcal{H}^{m-2} -null set, thus concluding that the singular set of T is (m-2) -rectifiable.

Abstract We first investigate torus‐equivariant motivic characteristic classes of toric varieties, and then apply them via the equivariant Riemann–Roch formalism to prove very general Euler–Maclaurin‐type formulae for full‐dimensional simple lattice polytopes.We consider ‐equivariant versions and of the motivic Chern and, resp., Hirzebruch characteristic classes of a toric variety (with corresponding torus ), and extend many known results from the non‐equivariant context to the equivariant setting. For example, the equivariant motivic Chern class is computed as the sum of the equivariant Grothendieck classes of the ‐equivariant sheaves of Zariski ‐forms weighted by . Using the motivic, as well as the characteristic class nature of , the corresponding generalized equivariant Hirzebruch ‐genus of a ‐invariant Cartier divisor on is also calculated.Further global formulae for are obtained in the simplicial context based on the Cox construction and the equivariant Lefschetz–Riemann–Roch theorem of Edidin–Graham. Alternative proofs of all these results are given via localization techniques at the torus fixed points in ‐equivariant ‐ and, resp., homology theories of toric varieties, due to Brion–Vergne and, resp., Brylinski–Zhang. These localization results apply to any toric variety with a torus fixed point. In localized ‐equivariant ‐theory, we extend a classical formula of Brion for a full‐dimensional lattice polytope to a weighted version. We also generalize the Molien formula of Brion–Vergne for the localized class of the structure sheaf of a simplicial toric variety to the context of . Similarly, we calculate the localized Hirzebruch class in localized ‐equivariant homology, extending the corresponding results of Brylinski–Zhang for the localized Todd class (fitting with the equivariant Hirzebruch class for ).As main applications of our equivariant characteristic class formulae, we provide a geometric perspective on several weighted Euler–Maclaurin‐type formulae for full‐dimensional simple lattice polytopes (corresponding to simplicial toric varieties), coming from the equivariant toric geometry via the equivariant Hirzebruch–Riemann–Roch (for an ample torus invariant Cartier divisor). Our main results even provide generalizations to arbitrary equivariant coherent sheaf coefficients, including algebraic geometric proofs of (weighted versions of) the Euler–Maclaurin formulae of Cappell–Shaneson, Brion–Vergne, Guillemin, and so forth (all of which correspond to the choice of the structure sheaf), via the equivariant Hirzebruch–Riemann–Roch formalism. In particular, we give a first complete proof of the Euler–Maclaurin formula of Cappell–Shaneson. Our approach, based on motivic characteristic classes, allows us to obtain such Euler–Maclaurin formulae also for (the interior of) a face, as well as for the polytope with several facets (i.e., codimension one faces) removed, for example, for the interior of the polytope (as well as for equivariant characteristic class formulae for locally closed ‐invariant subsets of a toric variety). Moreover, we prove such results also in the weighted context, as well as for ‐Minkowski summands of the given full‐dimensional lattice polytope (corresponding to globally generated torus invariant Cartier divisors in the toric context). Some of these results are extended to local Euler–Maclaurin formulae for the tangent cones at the vertices of the given full‐dimensional lattice polytope (fitting with the localization at the torus fixed points in equivariant ‐theory and equivariant (co)homology). Finally, we also give an application of our abstract Euler–Maclaurin formula to generalized reciprocity for Dedekind sums.

Abstract We consider a CMC hypersurface with an isolated singular point at which the tangent cone is regular, and such that, in a neighbourhood of said point, the hypersurface is the boundary of a Caccioppoli set that minimises the standard prescribed-mean-curvature functional. We prove that in a ball centred at the singularity there exists a sequence of smooth CMC hypersurfaces, with the same prescribed mean curvature, that converge to the given one. Moreover, these hypersurfaces arise as boundaries of minimisers. In ambient dimension 8 the condition on the cone is redundant. (When the mean curvature vanishes identically, the result is the well-known Hardt–Simon approximation theorem.

Abstract This paper is concerned with the space of harmonic functions vanishing on a given subset of Euclidean space. In dimensions three and higher, we show that for the cone cut out by a generically chosen harmonic quadratic polynomial this space is exactly two-dimensional. This phenomenon allows the following generalization to arbitrary elliptic differential operators of second order: Consider the level set of a solution at a nondegenerate critical value. As long as the tangent cone to the level set at a critical point satisfies a certain genericity condition, the space of solutions vanishing on the level set is at most two-dimensional.

Abstract In this article, we develop and present a novel regularization scheme for ill-posed inverse problems governed by nonlinear time-dependent partial differential equations (PDEs). In our recent work, we introduced a bi-level regularization framework. This study significantly improves upon the bi-level algorithm by sequentially initializing the lower-level problem, yielding accelerated convergence and demonstrable multi-scale effect, while retaining regularizing effect and allows for the usage of inexact PDE solvers. Moreover, by collecting the lower-level trajectory, we uncover an interesting connection to the incremental load method. The sequential bi-level approach illustrates its universality through several reaction-diffusion applications, in which the nonlinear reaction law needs to be determined. We moreover prove that the proposed tangential cone condition is satisfied.

Let E \subset \Omega be a local almost-minimizer of the relative perimeter in the open set \Omega\subset \R^{n} . We prove a free-boundary monotonicity inequality for E at a point x\in \delta\Omega , under a geometric property called “visibility”, that \Omega is required to satisfy in a neighborhood of x . Incidentally, the visibility property is satisfied by a considerably large class of Lipschitz and possibly non-smooth domains. Then, we prove the existence of the density of the relative perimeter of E at x , as well as the fact that any blow-up of E at x is necessarily a perimeter-minimizing cone within the tangent cone to \Omega at x .

We present recent advances in the analysis of nonlinear problems involving singular (degenerate) operators. The results are obtained within the framework of p-regularity theory, which has been successfully developed over the past four decades. We illustrate the theory with applications to degenerate problems in various areas of mathematics, including optimization and differential equations. In particular, we address the problem of describing the tangent cone to the solution set of nonlinear equations in singular cases. The structure of p-factor operators is used to propose optimality conditions and to construct novel numerical methods for solving degenerate nonlinear equations and optimization problems. The numerical methods presented in this paper represent the first approaches targeting solutions to degenerate problems such as the Van der Pol differential equation, boundary-value problems with small parameters, and partial differential equations where Poincaré's method of small parameters fails. Additionally, these methods may be extended to nonlinear degenerate dynamical systems and other related problems.

Hölder continuity of tangent cones in RCD(K,N) spaces and applications to nonbranching

In compact Riemannian manifolds of dimension 3 or higher with positive Ricci curvature, we prove that every constant mean curvature hypersurface produced by the Allen–Cahn min–max procedure in Bellettini and Wickramasekera (arXiv:2010.05847, 2020) (with constant prescribing function) is a local minimiser of the natural area-type functional around each isolated singularity. In particular, every tangent cone at each isolated singularity of the resulting hypersurface is area-minimising. As a consequence, for any real λ we show, through a surgery procedure, that for a generic 8-dimensional compact Riemannian manifold with positive Ricci curvature there exists a closed embedded smooth hypersurface of constant mean curvature λ; the minimal case (λ=0) of this result was obtained in Chodosh et al. (Ars Inveniendi Analytica, 2022) .

The Plateau problem asks: which are the surfaces of least m-dimensional area spanning a given (m-1)-dimensional boundary? To guarantee existence of minimizers and desirable compactness properties for sequences of surfaces, one must consider a weak notion of surface, thus allowing for area-minimizing “surfaces” to have singularities. A particularly natural framework for this problem is via integral currents, allowing for surfaces to have integer multiplicities. The Plateau problem has been studied in great depth in this setting since the 1950s, pioneered by works of De Giorgi, Federer & Fleming, Almgren, White and built upon by many others. We present the history of the problem and some recent breakthroughs in the regularity theory, together with the uniqueness of blow-ups, for area-minimizing surfaces in this framework. We additionally demonstrate that semicalibrated integral currents, which are a natural subclass of almost area-minimizers in this framework, exhibit the same regularity and structural properties as area-minimizers. This is based on a series of joint works with Camillo De Lellis and Paul Minter, and a joint work with Paul Minter, Davide Parise and Luca Spolaor.

Abstract We prove uniqueness of tangent cones for forced mean curvature flow, at both closed self-shrinkers and round cylindrical self-shrinkers, in any codimension. The corresponding results for mean curvature flow in Euclidean space were proven by Schulze and Colding–Minicozzi, respectively. We adapt their methods to handle the presence of the forcing term, which vanishes in the blow-up limit but complicates the analysis along the rescaled flow. Our results naturally include the case of mean curvature flows in Riemannian manifolds.

Abstract The object of this paper is to further investigate the notion of shape and topological derivatives in the light of the general notion of Hadamard semidifferential for a function defined on a subset of a topological vector space. The use of semitrajectories and the characterization of the adjacent tangent cone provide simple tools for defining Hadamard semi-differentials and differentials without a priori introduction of geometric structures such as, for instance, a differential manifold. Such a simple notion retains all the operations of the classical differential calculus, including the chain rule, for a large class of nondifferentiable functions, in particular, the norms and the convex functions. It also provides a direct access to functions defined on a lousy set or a manifold with boundary. This direct approach is first illustrated in the context of the classical matrix subgroups of the general linear group GL(n) of invertible n×n matrices, which are the prototypes of Lie groups. For the shape derivative we have groups of diffeomorphisms of the Euclidean space ℝ n with the composition operation, and the adjacent tangent cone is a linear space; for the topological derivative we have the group of characteristic functions with the symmetric difference operation and the adjacent tangent cone is only a cone at some points.

Abstract The object of this paper is to further investigate the notion of shape and topological derivatives in the light of the general notion of Hadamard semidifferential for a function defined on a subset of a topological vector space. The use of semitrajectories and the characterization of the adjacent tangent cone provide simple tools for defining Hadamard semi-differentials and differentials without a priori introduction of geometric structures such as, for instance, a differential manifold. Such a simple notion retains all the operations of the classical differential calculus, including the chain rule, for a large class of nondifferentiable functions, in particular, the norms and the convex functions. It also provides a direct access to functions defined on a lousy set or a manifold with boundary. This direct approach is first illustrated in the context of the classical matrix subgroups of the general linear group GL(n) of invertible n×n matrices, which are the prototypes of Lie groups. For the shape derivative we have groups of diffeomorphisms of the Euclidean space ℝ n with the composition operation, and the adjacent tangent cone is a linear space; for the topological derivative we have the group of characteristic functions with the symmetric difference operation and the adjacent tangent cone is only a cone at some points.

We prove three theorems about the asymptotic behavior of solutions u to the homogeneous Dirichlet problem for the Laplace equation at boundary points with tangent cones. First, under very mild hypotheses, we show that the doubling index of u either has a unique finite limit, or goes to infinity; in other words, there is a well-defined order of vanishing. Second, under more quantitative hypotheses, we prove that if the order of vanishing of u is finite at a boundary point 0 , then locally u(x) = |x|^{m} \psi(x/|x|) + o(|x|^{m}) , where |x|^{m} \psi(x/|x|) is a homogeneous harmonic function on the tangent cone. Finally, we construct a convex domain in three dimensions where such an expansion fails at a boundary point, showing that some quantitative hypotheses are necessary in general. The assumptions in all of the results only involve regularity at a single point, and in particular are much weaker than what is necessary for unique continuation, monotonicity of Almgren’s frequency, Carleman estimates, or other related techniques.

ABSTRACTA new Riemannian rank adaptive method (RRAM) is proposed for the low‐rank tensor completion problem (LRTCP). This problem is formulated as a least‐squares optimization problem on the algebraic variety of tensors of bounded tensor‐train (TT) rank. The RRAM iteratively optimizes over fixed‐rank smooth manifolds using a Riemannian conjugate gradient algorithm from Steinlechner. In between, the rank is increased by computing a descent direction selected in the tangent cone to the variety. A numerical method to estimate the rank increase is proposed. This numerical method is based on a new theoretical result for the low‐rank tensor approximation problem and a definition of an estimated TT‐rank. When the iterate comes close to a lower‐rank set, the RRAM decreases the rank based on the TT‐rounding algorithm from Oseledets and a definition of a numerical rank. It is shown that the TT‐rounding algorithm can be considered an approximate projection onto the lower‐rank set, which satisfies a certain angle condition to ensure that the image is sufficiently close to that of an exact projection. Several numerical experiments illustrate the use of the RRAM and its subroutines in Matlab. In all experiments, the proposed RRAM significantly outperforms the state‐of‐the‐art RRAM for tensor completion in the TT format from Steinlechner in terms of computation time.

We establish uniqueness and regularity results for tangent cones (at a point or at infinity), with isolated singularities arising from a given immersed stable minimal hypersurface with suitably small (non-immersed) singular set. In particular, our results allow the tangent cone to occur with any integer multiplicity.

Abstract In this article, we prove that the fundamental group π 1 ⁢ ( M ) \pi_{1}(M) of a complete open manifold 𝑀 with nonnegative Ricci curvature is finitely generated, under the condition that the Riemannian universal cover M ̃ \tilde{M} satisfies an “almost 𝑘-polar at infinity” condition. Additionally, such π 1 ⁢ ( M ) \pi_{1}(M) is virtually abelian. Furthermore, we demonstrate that the base point of any tangent cone at infinity of such a manifold is nearly a pole. In the case where M ̃ \tilde{M} exhibits almost maximal Euclidean volume growth, we prove that 𝑀 deformation retracts to a closed submanifold 𝐹 which is diffeomorphic to a flat manifold, provided 𝑀 is not simply connected.

Abstract We continue our study, started in [A. Češík, G. Gravina and M. Kampschulte, Inertial evolution of non-linear viscoelastic solids in the face of (self-)collision, Calc. Var. Partial Differential Equations 63 2024, 2, Paper No. 55], of (self-)collisions of viscoelastic solids in an inertial regime. We show existence of weak solutions with a corresponding contact force measure in the case of solids with only Lipschitz-regular boundaries. This necessitates a careful study of different concepts of tangent and normal cones and the role these play both in the proofs and in the formulation of the problem itself. Consistent with our previous approach, we study contact without resorting to penalization, i.e., by only relying on a strict non-interpenetration condition. Additionally, we improve the strategies of our previous proof, eliminating the need for regularization terms across all levels of approximation.