arXiv subjects

Ezra Miller

Publications and source records attributed to Ezra Miller.

At least 19 recordsLinked to original sources

arXiv2025

Quasipolynomial behavior via constructibility in multigraded algebra

Piecewise quasipolynomial growth of Presburger counting functions combines with tame persistent homology module theory to conclude piecewise quasipolynomial behavior of constructible families of finely graded modules over constructible commutative semigroup rings. Functorial preservation of constructibility for families under local cohomology, $\operatorname{Tor}$, and $\operatorname{Ext}$ yield piecewise quasipolynomial, quasilinear, or quasiconstant growth statements for length of local cohomology, $a$-invariants, regularity, depth; length of $\operatorname{Tor}$ and Betti numbers; length of $\operatorname{Ext}$ and Bass numbers; associated primes via $v$-invariants; and extended degrees, including the usual degree, Hilbert-Samuel multiplicity, arithmetic degree, and homological degree.

math.AC↗

arXiv2025

Essential graded algebra over polynomial rings with real exponents

The geometric and algebraic theory of monomial ideals and multigraded modules is initiated over real-exponent polynomial rings and, more generally, monoid algebras for real polyhedral cones. The main results include the generalization of Nakayama's lemma; complete theories of minimal and dense primary, secondary, and irreducible decomposition, including associated and attached faces; socles and tops; minimality and density for downset hulls, upset covers, and fringe presentations; Matlis duality; and geometric analysis of staircases. Modules that are semialgebraic or piecewise-linear (PL) have the relevant property preserved by functorial constructions as well as by minimal primary and secondary decompositions. And when the modules in question are subquotients of the group itself, such as monomial ideals and quotients modulo them, minimal primary and secondary decompositions are canonical, as are irreducible decompositions up to the new real-exponent notion of density.

math.AC↗

arXiv2025

A central limit theorem for random tangent fields on stratified spaces

Variation of empirical Fréchet means on a metric space with curvature bounded above is encoded via random fields indexed by unit tangent vectors. A central limit theorem shows these random tangent fields converge to a Gaussian such field and lays the foundation for more traditionally formulated central limit theorems in subsequent work.

math.PR↗

arXiv2024

Fibers of maps to totally nonnegative spaces

This paper undertakes a study of the structure of the fibers of the Chevalley exponentiation maps $f_{(i_1,\dots ,i_d)}$. The fibers of these maps $f_{(i_1,\dots ,i_d)}$ encode the nonnegative real relations amongst exponentiated Chevalley generators. Our main theorems show that the fibers admit cell stratifications, that these cell stratifications have the same face posets as interior dual block complexes of subword complexes, and that these posets are contractible. We conjecture that each such fiber is a regular CW complex homeomorphic to the interior dual block complex of a subword complex. This conjecture is shown to have as a corollary a new proof of the Fomin-Shapiro Conjecture by way of general topological results regarding approximating maps by homeomorphisms.

math.CO↗

arXiv2024

Minimal resolutions of lattice ideals

A canonical minimal free resolution of an arbitrary co-artinian lattice ideal over the polynomial ring is constructed over any field whose characteristic is 0 or any but finitely many positive primes. The differential has a closed-form combinatorial description as a sum over lattice paths in $\mathbb{Z}^n$ of weights that come from sequences of faces in simplicial complexes indexed by lattice points. Over a field of any characteristic, a non-canonical but simpler resolution is constructed by selecting choices of higher-dimensional analogues of spanning trees along lattice paths. These constructions generalize sylvan resolutions for monomial ideals by lifting them equivariantly to lattice modules.

math.AC↗

arXiv2024

Minimal free resolutions of numerical semigroup algebras via Apéry specialization

Numerical semigroups with multiplicity $m$ are parameterized by integer points in a polyhedral cone $C_m$, according to Kunz. For the toric ideal of any such semigroup, the main result here constructs a free resolution whose overall structure is identical for all semigroups parametrized by the relative interior of a fixed face of $C_m$. The matrix entries of this resolution are monomials whose exponents are parametrized by the coordinates of the corresponding point in $C_m$, and minimality of the resolution is achieved when the semigroup is maximal embedding dimension, which is the case parametrized by the interior of $C_m$ itself.

math.AC↗

arXiv2023

Shadow geometry at singular points of CAT(k) spaces

In any CAT(k) space M, the "shadow" of a tangent vector Z at a point p is the set vectors that form an angle of πor more with Z. Taking logarithm maps at points approaching p along a fixed geodesic ray from p with tangent Z collapses the shadow to a single ray while leaving isometrically intact every convex cone that avoids the shadow of Z.

math.MG↗

arXiv2023

Geometry of measures on smoothly stratified metric spaces

Any measure $μ$ on a CAT(k) space M that is stratified as a finite union of manifolds and has local exponential maps near the Fréchet mean $\barμ$ yields a continuous "tangential collapse" from the tangent cone of M at $\barμ$ to a vector space that preserves the Fréchet mean, restricts to an isometry on the "fluctuating cone" of directions in which the Fréchet mean can vary under perturbation of $μ$, and preserves angles between arbitrary and fluctuating tangent vectors at the Fréchet mean.

math.MG↗

arXiv2023

Central limit theorems for Fréchet means on stratified spaces

Fréchet means of samples from a probability measure $μ$ on any smoothly stratified metric space M with curvature bounded above are shown to satisfy a central limit theorem (CLT). The methods and results proceed by introducing and proving analytic properties of the "escape vector" of any finitely supported measure $δ$ in M, which records infinitesimal variation of the Fréchet mean $\barμ$ of $μ$ in response to perturbation of $μ$ by adding the mass $tδ$ for $t \to 0$. The CLT limiting distribution $N$ on the tangent cone $T$ at the Fréchet mean is characterized in four ways. The first uses tangential collapse $L$ to compare $T$ with a linear space and then applies a distortion map to the usual linear CLT to transfer back to $T$. Distortion is defined by applying escape after taking preimages under $L$. The second characterization constructs singular analogues of Gaussian measures on smoothly stratified spaces and expresses $N$ as the escape vector of any such "Gaussian mass". The third characterization expresses $N$ as the directional derivative, in the space of measures on $M$, of the barycenter map at $μ$ in the (random) direction given by any Gaussian mass. The final characterization expresses $N$ as the directional derivative, in the space $C$ of continuous real-valued functions on $T$, of a minimizer map, with the derivative taken at the Fréchet function $F \in C$ along the (random) direction given by the negative of the Gaussian tangent field induced by $μ$. Precise mild hypotheses on the measure $μ$ guarantee these CLTs, whose convergence is proved via the second characterization of $N$ by formulating a duality between Gaussian masses and Gaussian tangent fields.

math.PR↗

arXiv2023

Stratifications of real vector spaces from constructible sheaves with conical microsupport

Interpreting the syzygy theorem for tame modules over posets in the setting of derived categories of subanalytically constructible sheaves proves two conjectures due to Kashiwara and Schapira concerning the existence of stratifications of real vector spaces that play well with sheaves having microsupport in a given cone or, equivalently, sheaves in the corresponding conic topology.

math.AT↗

arXiv2022

Global dimension of real-exponent polynomial rings

The ring R of real-exponent polynomials in n variables over any field has global dimension n+1 and flat dimension n. In particular, the residue field k = R/m of R modulo its maximal graded ideal m has flat dimension n via a Koszul-like resolution. Projective and flat resolutions of all R-modules are constructed from this resolution of k. The same results hold when R is replaced by the monoid algebra for the positive cone of any subgroup of $\mathbb{R}^n$ satisfying a mild density condition.

math.AC↗

arXiv2020

Data structures for real multiparameter persistence modules

A theory of modules over posets is developed to define computationally feasible, topologically interpretable data structures, in terms of birth and death of homology classes, for persistent homology with multiple real parameters. To replace the noetherian hypothesis in the general setting of modules over posets, a "finitely encoded" condition is defined combinatorially and developed algebraically. It captures topological tameness of persistent homology. Poset-modules satisfying it can be specified by "fringe presentations" that reflect birth-and-death descriptions of persistence. A syzygy theorem characterizes finitely encoded modules as admitting appropriately finite presentations and resolutions. The geometric and algebraic theory focuses on modules over real polyhedral groups (real vector spaces with polyhedral positive cones) and a parallel theory over discrete polyhedral groups (abelian groups with finitely generated positive cones). Existence of primary decomposition is proved over arbitrary polyhedral partially ordered abelian groups, but the real and discrete cases carry enough geometry and, crucially in the real case, topology to induce complete theories of minimal primary and secondary decomposition, associated and attached faces, minimal generators and cogenerators, socles and tops, minimal upset covers and downset hulls, Matlis duality, and minimal fringe presentation. Real semialgebraic properties of data are preserved by functorial constructions. Tops and socles become functorial birth and death spaces for multiparameter persistence modules. They yield functorial "QR codes" and "elder morphisms" for modules over real and discrete polyhedral groups that generalize and categorify the bar code and elder rule for persistent homology in one parameter. The disparate ways that QR codes and elder morphisms model bar codes coalesce, in one parameter, to functorial bar codes.

math.AT↗

arXiv2020

Modules over posets: commutative and homological algebra

The commutative and homological algebra of modules over posets is developed, as closely parallel as possible to the algebra of finitely generated modules over noetherian commutative rings, in the direction of finite presentations, primary decompositions, and resolutions. Interpreting this finiteness in the language of derived categories of subanalytically constructible sheaves proves two conjectures due to Kashiwara and Schapira concerning sheaves with microsupport in a given cone. The motivating case is persistent homology of arbitrary filtered topological spaces, especially the case of multiple real parameters. The algebraic theory yields computationally feasible, topologically interpretable data structures, in terms of birth and death of homology classes, for persistent homology indexed by arbitrary posets. The exposition focuses on the nature and ramifications of a suitable finiteness condition to replace the noetherian hypothesis. The tameness condition introduced for this purpose captures finiteness for variation in families of vector spaces indexed by posets in a way that is characterized equivalently by distinct topological, algebraic, combinatorial, and homological manifestations. Tameness serves both the theoretical and computational purposes: it guarantees finite primary decompositions, as well as various finite presentations and resolutions all related by a syzygy theorem, and the data structures thus produced are computable in addition to being interpretable. The tameness condition and its resulting theory are new even in the finitely generated discrete setting, where being tame is materially weaker than being noetherian.

math.AC↗

arXiv2020

Homological algebra of modules over posets

Homological algebra of modules over posets is developed, as closely parallel as possible to that of finitely generated modules over noetherian commutative rings, in the direction of finite presentations and resolutions. Centrally at issue is how to define finiteness to replace the noetherian hypothesis which fails. The tameness condition introduced for this purpose captures finiteness for variation in families of vector spaces indexed by posets in a way that is characterized equivalently by distinct topological, algebraic, combinatorial, and homological manifestations. Tameness serves both theoretical and computational purposes: it guarantees finite presentations and resolutions of various sorts, all related by a syzygy theorem, amenable to algorithmic manipulation. Tameness and its homological theory are new even in the finitely generated discrete setting of $\mathbb{N}^n$-gradings, where tame is materially weaker than noetherian. In the context of persistent homology of filtered topological spaces, especially with multiple real parameters, the algebraic theory of tameness yields topologically interpretable data structures in terms of birth and death of homology classes.

math.AT↗

arXiv2020

Primary decomposition over partially ordered groups

Over any partially ordered abelian group whose positive cone is closed in an appropriate sense and has finitely many faces, modules that satisfy a weak finiteness condition admit finite primary decompositions. This conclusion rests on the introduction of basic notions in the relevant generality, such as closedness of partially ordered abelian groups, faces and their coprimary modules, and finiteness conditions as well local and global support functors for modules over partially ordered groups.

math.AC↗

arXiv2020

Minimal resolutions of monomial ideals

An explicit combinatorial minimal free resolution of an arbitrary monomial ideal $I$ in a polynomial ring in $n$ variables over a field of characteristic $0$ is defined canonically, without any choices, using higher-dimensional generalizations of combined spanning trees for cycles and cocycles ("hedges") in the upper Koszul simplicial complexes of $I$ at lattice points in $\mathbb{Z}^n$. The differentials in these "sylvan resolutions" are expressed as matrices whose entries are sums over lattice paths of weights determined combinatorially by sequences of hedges ("hedgerows") along each lattice path. This combinatorics enters via an explicit matroidal expression for the Moore-Penrose pseudoinverses of the differentials in any CW complex as weighted averages of splittings defined by hedges. This "Hedge Formula" also yields a projection formula from CW chains to boundaries. The translation from Moore-Penrose combinatorics to free resolutions relies on Wall complexes, which construct minimal free resolutions of graded ideals from vertical splittings of Koszul bicomplexes. The algebra of Wall complexes applied to individual hedgerows yields explicit but noncanonical combinatorial minimal free resolutions of arbitrary monomial ideals in any characteristic.

math.AC↗

arXiv2017

When is a polynomial ideal binomial after an ambient automorphism?

Can an ideal I in a polynomial ring k[x] over a field be moved by a change of coordinates into a position where it is generated by binomials $x^a - cx^b$ with c in k, or by unital binomials (i.e., with c = 0 or 1)? Can a variety be moved into a position where it is toric? By fibering the G-translates of I over an algebraic group G acting on affine space, these problems are special cases of questions about a family F of ideals over an arbitrary base B. The main results in this general setting are algorithms to find the locus of points in B over which the fiber of F - is contained in the fiber of a second family F' of ideals over B; - defines a variety of dimension at least d; - is generated by binomials; or - is generated by unital binomials. A faster containment algorithm is also presented when the fibers of F are prime. The big-fiber algorithm is probabilistic but likely faster than known deterministic ones. Applications include the setting where a second group T acts on affine space, in addition to G, in which case algorithms compute the set of G-translates of I - whose stabilizer subgroups in T have maximal dimension; or - that admit a faithful multigrading by $Z^r$ of maximal rank r. Even with no ambient group action given, the final application is an algorithm to - decide whether a normal projective variety is abstractly toric. All of these loci in B and subsets of G are constructible; in some cases they are closed.

math.AC↗

arXiv2017

Positivity and Kleiman transversality in equivariant K-theory of homogeneous spaces

We prove the conjectures of Graham-Kumar and Griffeth-Ram concerning the alternation of signs in the structure constants for torus-equivariant K-theory of generalized flag varieties G/P. These results are immediate consequences of an equivariant homological Kleiman transversality principle for the Borel mixing spaces of homogeneous spaces, and their subvarieties, under a natural group action with finitely many orbits. The computation of the coefficients in the expansion of the equivariant K-class of a subvariety in terms of Schubert classes is reduced to an Euler characteristic using the homological transversality theorem for non-transitive group actions due to S. Sierra. A vanishing theorem, when the subvariety has rational singularities, shows that the Euler characteristic is a sum of at most one term--the top one--with a well-defined sign. The vanishing is proved by suitably modifying a geometric argument due to M. Brion in ordinary K-theory that brings Kawamata-Viehweg vanishing to bear.

math.AG↗