Related papers: K-Theory of Multi-parameter Persistence Modules: A…
We provide a universal characterization of the construction taking a scheme $X$ to its stable $\infty$-category $\text{Mot}(X)$ of noncommutative motives, patterned after the universal characterization of algebraic K-theory due to…
Using sheaf theory, I introduce a continuous theory of persistence for mappings between compact manifolds. In the case both manifolds are orientable, the theory holds for integer coefficients. The sheaf introduced here is stable to…
We introduce persistence matching diagrams induced by set mappings of metric spaces, based on 0-persistent homology of Vietoris-Rips filtrations. Also, we present a geometric definition of the persistence matching diagram that is more…
These notes develop the foundations of Milnor-Witt K-theory for fields of arbitrary characteristic, without any perfectness assumptions. Extending the work of Morel and Feld, we establish all functorial properties of Milnor-Witt K-theory…
The theory of minimal K-types for p-adic reductive groups was developed in part to classify irreducible admissible representations with wild ramification. An important observation was that minimal K-types associated to such representations…
The natural occurrence of singular spaces in applications has led to recent investigations on performing topological data analysis (TDA) in a stratified framework. In many applications, there is no a priori information on what points should…
We show that a pointwise finite-dimensional persistence module indexed over a small category decomposes into a direct sum of indecomposables with local endomorphism rings. As an application of this result we give new, short proofs of…
In the recent years, the generalization of the Erd\H{o}s-Ko-Rado (EKR) theorem to permutation groups has been of much interest. A transitive group is said to satisfy the EKR-module property if the characteristic vector of every maximum…
In this paper we study a model structure on a category of schemes with a group action and the resulting unstable and stable equivariant motivic homotopy theories. The new model structure introduced here samples a comparison to the one by…
A rigidity theory is developed for frameworks in a metric space with two types of distance constraints. Mixed sparsity graph characterisations are obtained for the infinitesimal and continuous rigidity of completely regular bar-joint…
We construct moduli spaces of linear self-maps of projective space with marked points, up to projective equivalence. That is, we let the special linear group act simultaneously by conjugation on projective linear maps and diagonally on…
We investigate the optimization of neural networks on symmetric data, and compare the strategy of constraining the architecture to be equivariant to that of using data augmentation. Our analysis reveals that that the relative geometry of…
One-dimensional persistent homology is arguably the most important and heavily used computational tool in topological data analysis. Additional information can be extracted from datasets by studying multi-dimensional persistence modules and…
This paper applies the multiplicity polar theorem to the study of hypersurfaces with non-isolated singularities. The multiplicity polar theorem controls the multiplicity of a pair of modules in a family by relating the multiplicity at the…
In this paper we explore the representation property over sets. This property generalizes constructibility, however is weak enough to enable us to prove that the class of theories $T$ whose models are representable is exactly the class of…
In this paper, by introducing a wider class of one-parameter group actions for test configurations, we have a stronger form of the definition of K-stability. This allows us to obtain some key step of my preceding work in proving that…
When a category $\mathcal{C}$ satisfies certain conditions, we define the notion of rank invariant for arbitrary poset-indexed functors $F:\mathbf{P} \rightarrow \mathcal{C}$ from a category theory perspective. This generalizes the standard…
The relationship between overparameterization, stability, and generalization remains incompletely understood in the setting of discontinuous classifiers. We address this gap by establishing a generalization bound for finite function classes…
This paper develops the idea of homology for 1-parameter families of topological spaces. We express parametrized homology as a collection of real intervals with each corresponding to a homological feature supported over that interval or,…
Over-extended Kac-Moody algebras contain so-called gradient structures - a gl(d)-covariant level decomposition of the algebra contains strings of modules at different levels that can be interpreted as spatial gradients. We present an…