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. Pr
ojective 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.
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 decompositi
ons, 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.
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 com
bined 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.
This paper undertakes a study of the structure of the fibers of a family of maps $f_{(i_1,dots ,i_d)}$ arising from representation theory, motivated both by connections to Lusztigs theory of canonical bases and also by the fact that these fibers enco
de the nonnegative real relations amongst exponentiated Chevalley generators. In particular, we prove that the fibers of these maps $f_{(i_1,dots ,i_d)}$ (restricted to the standard simplex $Delta_{d-1}$ in a way that still captures the full structure) admit cell decompositions induced by the decomposition of $Delta_{d-1}$ into open simplices of various dimensions. We also prove that these cell decompositions have the same face posets as interior dual block complexes of subword complexes and that these interior dual block complexes are contractible. We conjecture that each such fiber is a regular CW complex homeomorphic to the interior dual block complex of a subword complex. We show how this conjecture would yield as a corollary a new proof of the Fomin-Shapiro Conjecture by way of general topological results regarding approximating maps by homeomorphisms.
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 noeth
erian 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.
Possibilities for using geometry and topology to analyze statistical problems in biology raise a host of novel questions in geometry, probability, algebra, and combinatorics that demonstrate the power of biology to influence the future of pure mathem
atics. This expository article is a tour through some biological explorations and their mathematical ramifications. The article starts with evolution of novel topological features in wing veins of fruit flies, which are quantified using the algebraic structure of multiparameter persistent homology. The statistical issues involved highlight mathematical implications of sampling from moduli spaces. These lead to geometric probability on stratified spaces, including the sticky phenomenon for Frechet means and the origin of this mathematical area in the reconstruction of phylogenetic trees.
Building on coprincipal mesoprimary decomposition [Kahle and Miller, 2014], we combinatorially construct an irreducible decomposition of any given binomial ideal. In a parallel manner, for congruences in commutative monoids we construct decomposition
s that are direct combinatorial analogues of binomial irreducible decompositions, and for binomial ideals we construct decompositions into ideals that are as irreducible as possible while remaining binomial. We provide an example of a binomial ideal that is not an intersection of binomial irreducible ideals, thus answering a question of Eisenbud and Sturmfels [1996].
New representations of tree-structured data objects, using ideas from topological data analysis, enable improved statistical analyses of a population of brain artery trees. A number of representations of each data tree arise from persistence diagrams
that quantify branching and looping of vessels at multiple scales. Novel approaches to the statistical analysis, through various summaries of the persistence diagrams, lead to heightened correlations with covariates such as age and sex, relative to earlier analyses of this data set. The correlation with age continues to be significant even after controlling for correlations from earlier significant summaries
Scattered over the past few years have been several occurrences of simplicial complexes whose topological behavior characterize the Cohen-Macaulay property for quotients of polynomial rings by arbitrary (not necessarily squarefree) monomial ideals. T
he purpose of this survey is to gather the developments into one location, with self-contained proofs, including direct combinatorial topological connections between them.
Weighted enumeration of reduced pipe dreams (or rc-graphs) results in a combinatorial expression for Schubert polynomials. The duality between the set of reduced pipe dreams and certain antidiagonals has important geometric implications [A. Knutson a
nd E. Miller, Grobner geometry of Schubert polynomials, Ann. Math. 161, 1245-1318]. The original proof of the duality was roundabout, relying on the algebra of certain monomial ideals and a recursive characterization of reduced pipe dreams. This paper provides a direct combinatorial proof.
