Related papers: Combinatorial presentation of multidimensional per…

Multimodal normal incestual systems are investigated in terms of multiple categories. The different sorted composition of operators are exhibited as 2-cells in multiple categories built up from 2-categories giving rise to different axioms.…

Multigraded linear series generalize the classical morphism to the linear series of a basepoint-free line bundle on a scheme. We investigate the collection of the natural cornering morphisms into elementary bigraded linear series obtained…

Using the category of finite sets and injections, we construct a new model for the multilinearization of multifunctors between spaces that appears in the derivatives of Goodwillie calculus. We show that this model yields a lax monoidal…

Let $k$ be a commutative $\mathbb{Q}$-algebra. We study families of functors between categories of finitely generated $R$-modules which are defined for all commutative $k$-algebras $R$ simultaneously and are compatible with base changes.…

In this note, we study the general form of a multiplicative bijection on several families of functions defined on manifolds, both real or complex valued. In the real case, we prove that it is essentially defined by a composition with a…

To each finitely presented module $M$ over a commutative ring $R$ one can associate an $R$-ideal $\mathrm{Fitt}_{R}(M)$, which is called the (zeroth) Fitting ideal of $M$ over $R$. This is of interest because it is always contained in the…

We investigate the correspondence between generalized persistence modules and graded modules in the case the indexing set has a monoid action. We introduce the notion of an action category over a monoid graded ring. We show that the…

For any additive functor from modules (or, more generally, from an abelian category with enough projectives or injectives), we construct long sequences tying up together the derived functors, the satellites, and the stabilizations of the…

While persistent homology has taken strides towards becoming a wide-spread tool for data analysis, multidimensional persistence has proven more difficult to apply. One reason is the serious drawback of no longer having a concise and…

26 different concrete representations of the space of vector valued distributions on a smooth manifold of dimension n are presented systematically, most of them new. In the particular case of representations as module homomorphisms acting…

When studying families in the moduli space of dynamical systems, choosing an appropriate representative function for a conjugacy class can be a delicate task. The most delicate questions surround rationality of the conjugacy class compared…

This thesis deals with the enumerative study of combinatorial maps, and its application to the enumeration of other combinatorial objects. Combinatorial maps, or simply maps, form a rich combinatorial model. They have an intuitive and…

We consider a pivotal monoidal functor whose domain is a modular tensor category (MTC). We show that the trace of such a functor naturally extends to a representation of the corresponding tube category. As irreducible representations of the…

The main object of study of the present paper is the group $\au_n$ of \emph{unimodular automorphisms} of $\com^n$. Taking $\au_n$ as a working example, our intention was to develop an approach (or rather an edifice) which allows one to…

Let $(R, \mathfrak m)$ be a commutative noetherian local ring and $I$ an ideal of $R$. For every $R$-module $M$, $\gamma_I(M) = \sum\{ \operatorname{Bi} f \,|\, f \in \operatorname{Hom}_R(I,M)\}$ is called the trace of $I$ in $M$. It is…

A connection between moduli spaces of algebro-geometric objects and moduli spaces of polyhedral objects has been under investigation in recent years. Loosely speaking, the skeleton of an algebro-geometric moduli space is expressed as the…

Let $(A,\m)$ be a Noetherian local ring, let $M$ be a finitely generated \CM $A$-module of dimension $r \geq 2$ and let $I$ be an ideal of definition for $M$. Set $L^I(M) = \bigoplus_{n\geq 0}M/I^{n+1}M$. In part one of this paper we showed…

A multicomplex structure is defined from an ordered lattice of multigraphs. This structure will help us to observe the features of Persistent Homology in this context, its interaction with the ordering and the repercussions of the process…

We give a new method for the evaluation of a class of integrals of rational symmetric functions in N pairs of variables {x_a, y_a}_{a=1,... N} arising in coupled matrix models, valid for a broad class of two-variable measures. The result is…

Let $M$ be a finitely generated module over a Noetherian ring $R$ and $N$ a submodule. The index of reducibility ir$_M(N)$ is the number of irreducible submodules that appear in an irredundant irreducible decomposition of $N$ (this number…