Hausdorff Stability of Persistence Spaces

Multidimensional persistence modules do not admit a concise representation analogous to that provided by persistence diagrams for real-valued functions. However, there is no obstruction for multidimensional persistent Betti numbers to admit one. Therefore, it is reasonable to look for a generalization of persistence diagrams concerning those properties that are related only to persistent Betti numbers. In this paper, the persistence space of a vector-valued continuous function is introduced to generalize the concept of persistence diagram in this sense. The main result is its stability under function perturbations: Any change in vector-valued functions implies a not greater change in the Hausdorff distance between their persistence spaces.     

Betti numbers of Springer fibers in type A

We determine the Betti numbers of the Springer fibers in type A. To do this, we construct a cell decomposition of the Springer fibers. The codimension of the cells is given by an analogue of the Coxeter length. This makes our cell decomposition well suited for the calculation of Betti numbers.     

Reduced Gorenstein codimension three subschemes of projective space

It is known, from work of Diesel, which graded Betti numbers are possible for Artinian Gorenstein height three ideals. In this paper we show that any such set of graded Betti numbers in fact occurs for a reduced set of points in , a stick figure in , or more generally, a good linear configuration in . Consequently, any Gorenstein codimension three scheme specializes to such a ``nice' configuration, preserving the graded Betti numbers in the process. This is the codimension three Gorenstein analog of a classical result of arithmetically Cohen-Macaulay codimension two schemes.

     

Integral curvature and topological obstructions for submanifolds

We provide integral curvature bounds for compact Riemannian manifolds that allow isometric immersions into a Euclidean space with low codimension in terms of the Betti numbers.     

The hilbert function and the minimal free resolution of some gorenstein ideals of codimension 4

We give a construction for Gorenstein ideals of codimension 4 which we believe have maximal graded Betti numbers for their Hilbert function.     

关于高维Willmore问题

本文考虑高维欧氏空间中子流形Ｍ的一组有较好意义的共形不变的泛函．给出这些泛函通过Ｍ的Ｂｅｔｔｉ数的下界估计;给出对于管状超曲面的下界和对于双球环的下界以及达到这些下界的相应的子流形,并且证明对于管状超曲面所得的有关Ｂｅｔｔｉ数的下界是不精确的,方法是不适当的．给出类似Ｗｉｌｌｍｏｒｅ猜测的一些猜测．     

Unimodal betti numbers for a class of nilpotent lie algebras

In this paper we prove that all finite dimens~onal complex nilpotent Lie algebras containing an abelian ideal of codimension 1 have unirnodal Betti numbers.     

Poincaré polynomial of the moduli spaces of parabolic bundles

In this paper we use Weil conjectures (Deligne’s theorem) to calculate the Betti numbers of the moduli spaces of semi-stable parabolic bundles on a curve. The quasi parabolic analogue of the Siegel formula, together with the method of HarderNarasimhan filtration gives us a recursive formula for the Poincaré polynomials of the moduli. We solve the recursive formula by the method of Zagier, to give the Poincaré polynomial in a closed form. We also give explicit tables of Betti numbers in small rank, and genera.     

Resolutions of letterplace ideals of posets

We investigate resolutions of letterplace ideals of posets. We develop topological results to compute their multigraded Betti numbers, and to give structural results on these Betti numbers. If the poset is a union of no more than c chains, we show that the Betti numbers may be computed from simplicial complexes of no more than c vertices. We also give a recursive procedure to compute the Betti diagrams when the Hasse diagram of P has tree structure.     

拟遗传代数的诱导模与广义Betti数

本文引入了模的广义Betti数,给出了经典Betti数与广义Betti数之间的关系,证明了具有纯粹强正合Borel子代数的零关系拟遗传代数的诱导模的极小投射分解可通过其正合Borel子代数的相应模的极小投射分解的诱导给出,从而两者具有相同的广义Betti数.     

Betti numbers in multidimensional persistent homology are stable functions

Multidimensional persistence mostly studies topological features of shapes by analyzing the lower level sets of vector‐valued functions, called filtering functions. As is well known, in the case of scalar‐valued filtering functions, persistent homology groups can be studied through their persistent Betti numbers, that is, the dimensions of the images of the homomorphisms induced by the inclusions of lower level sets into each other. Whenever such inclusions exist for lower level sets of vector‐valued filtering functions, we can consider the multidimensional analog of persistent Betti numbers. Varying the lower level sets, we obtain that persistent Betti numbers can be seen as functions taking pairs of vectors to the set of non‐negative integers. In this paper, we prove stability of multidimensional persistent Betti numbers. More precisely, we prove that small changes of the vector‐valued filtering functions imply only small changes of persistent Betti numbers functions. This result can be obtained by assuming the filtering functions to be just continuous. Multidimensional stability opens the way to a stable shape comparison methodology based on multidimensional persistence. In order to obtain our stability theorem, some other new results are proved for continuous filtering functions. They concern the finiteness of persistent Betti numbers for vector‐valued filtering functions and the representation via persistence diagrams of persistent Betti numbers, as well as their stability, in the case of scalar‐valued filtering functions. Finally, from the stability of multidimensional persistent Betti numbers, we obtain a lower bound for the natural pseudo‐distance. Copyright © 2013 John Wiley & Sons, Ltd.     

Recent Progress in the Study of Representations of Integers as Sums of Squares

In this article, the authors collect the recent results concerningthe representations of integers as sums of an even number ofsquares that are inspired by conjectures of Kac and Wakimoto.They start with a sketch of Milne's proof of two of these conjectures,and they also show an alternative route to deduce these twoconjectures from Milne's determinant formulas for sums of, respectively,4s² or 4s(s+1) triangular numbers. This approach is inspiredby Zagier's proof of the Kac–Wakimoto formulas via modularforms. The survey ends with recent conjectures of the firstauthor and Chua. 2000 Mathematics Subject Classification 11E25,11F11.     

On the Weak-Lefschetz property for Artinian Gorenstein algebras

We deal with the weak Lefschetz property (WLP) for Artinian standard graded Gorenstein algebras of codimension 3. We prove that many Gorenstein sequences force the WLP for such algebras. Moreover for every Gorenstein sequence \(H\) of codimension 3 we found several Gorenstein Betti sequences compatible with \(H\) which again force the WLP. Finally we show that for every Gorenstein Betti sequence the general Artinian standard graded Gorenstein algebra with such Betti sequence has the WLP.     

Topological obstructions for submanifolds in low codimension

We prove integral curvature bounds in terms of the Betti numbers for compact submanifolds of the Euclidean space with low codimension. As an application, we obtain topological obstructions for \(\delta \)-pinched immersions. Furthermore, we obtain intrinsic obstructions for minimal submanifolds in spheres with pinched second fundamental form.     

An Improved Multiplicity Conjecture for Codimension 3 Gorenstein Algebras

The Multiplicity Conjecture is a deep problem relating the multiplicity (or degree) of a Cohen–Macaulay standard graded algebra with certain extremal graded Betti numbers in its minimal free resolution. In the case of level algebras of codimension three, Zanello has proposed a stronger conjecture. We prove this conjecture in the Gorenstein case.     

Generic initial ideals of Artinian ideals having Lefschetz properties or the strong Stanley property

For a standard Artinian k-algebra A=R/I, we give equivalent conditions for A to have the weak (or strong) Lefschetz property or the strong Stanley property in terms of the minimal system of generators of gin(I). Using the equivalent condition for the weak Lefschetz property, we show that some graded Betti numbers of gin(I) are determined just by the Hilbert function of I if A has the weak Lefschetz property. Furthermore, for the case that A is a standard Artinian k-algebra of codimension 3, we show that every graded Betti number of gin(I) is determined by the graded Betti numbers of I if A has the weak Lefschetz property. And if A has the strong Lefschetz (respectively Stanley) property, then we show that the minimal system of generators of gin(I) is determined by the graded Betti numbers (respectively by the Hilbert function) of I.     

Organizing center for the bifurcation analysis of a generalized Gause model with prey harvesting and Holling response function of type III

The present note is an addendum to the paper of Etoua-Rousseau (2010) [1] which presented a study of a generalized Gause model with prey harvesting and a generalized Holling response function of type III: . Complete bifurcation diagrams were proposed, but some parts were conjectural. An organizing center for the bifurcation diagram was given by a nilpotent point of saddle type lying on an invariant line and of codimension greater than or equal to 3. This point was of codimension 3 when b≠0, and was conjectured to be of infinite codimension when b=0. This conjecture was in line with a second conjecture that the Hopf bifurcation of order 2 degenerates to a Hopf bifurcation of infinite codimension when b=0. In this note we prove these two conjectures.     

Betti numbers and lifting of Gorenstein codimension three ideals

In this paper we consider homogeneous Gorenstein ideals of codimension three in a polynomial ring and determine their graded Betti numbers in terms of their Hilbert function. For such ideals we prove also a lifting theorem in the vein of a classical result of Hartshorne concerning monomial ideals.     

Resolutions and Betti Numbers of Polynomial Modules via Involutive Bases

We describe a novel approach to the computation of free resolutions and of Betti numbers of polynomial modules based on a combination of the theory of involutive bases with algebraic discrete Morse theory. This approach allows for the first time to compute Betti numbers (even single ones) without determining a whole resolution which in many cases drastically reduces the computation time. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Minimal Resolutions and the Homology of Matching and Chessboard Complexes

We generalize work of Lascoux and Józefiak-Pragacz-Weyman on Betti numbers for minimal free resolutions of ideals generated by 2 × 2 minors of generic matrices and generic symmetric matrices, respectively. Quotients of polynomial rings by these ideals are the classical Segre and quadratic Veronese subalgebras, and we compute the analogous Betti numbers for some natural modules over these Segre and quadratic Veronese subalgebras. Our motivation is two-fold: We immediately deduce from these results the irreducible decomposition for the symmetric group action on the rational homology of all chessboard complexes and complete graph matching complexes as studied by Björner, Lovasz, Vreica and ivaljevi. This follows from an old observation on Betti numbers of semigroup modules over semigroup rings described in terms of simplicial complexes. The class of modules over the Segre rings and quadratic Veronese rings which we consider is closed under the operation of taking canonical modules, and hence exposes a pleasant symmetry inherent in these Betti numbers.