Related papers: Pursuing Stacks
The homotopy continuation method has been widely used in solving parametric systems of nonlinear equations. But it can be very expensive and inefficient due to singularities during the tracking even though both start and end points are…
In complex inferential tasks like question answering, machine learning models must confront two challenges: the need to implement a compositional reasoning process, and, in many applications, the need for this reasoning process to be…
The main aim of this paper is to study chains of model structures arising from cotorsion pairs in extriangulated categories. Starting with a hereditary Hovey triple, we construct further hereditary Hovey triples whose homotopy categories…
Given an associative, not necessarily commutative, ring R with identity, a formal matrix calculus is introduced and developed for pairs of matrices over R. This calculus subsumes the theory of homogeneous systems of linear equations with…
Can we trust the reasoning traces that large reasoning models (LRMs) produce? We investigate whether these traces faithfully reflect what drives model outputs, and whether models will honestly report their influence. We introduce Thought…
Notes on Commutative Alegbra and Algebraic Geometry covering rings, ideals, modules, presheaves, sheaves, schemes, homological algebra, \'etale cohomology and further topics that are more advanced.
We give a homotopy theoretic characterization of stacks on a site $\cC$ as the {\it homotopy sheaves} of groupoids on $\cC$. We use this characterization to construct a model category in which stacks are the fibrant objects. We compare…
An important open question in the use of large language models for knowledge-intensive tasks is how to effectively integrate knowledge from three sources: the model's parametric memory, external structured knowledge, and external…
In this paper, we explore the task of automatically generating natural language descriptions of salient patterns in a time series, such as stock prices of a company over a week. A model for this task should be able to extract high-level…
The aim of this paper is to connect two important and apparently unrelated theories: motivic homotopy theory and ramification theory. We construct motivic homotopy categories over a qcqs base scheme $S$, in which cohomology theories with…
The purpose of these notes is to give an introduction to Deligne-Mumford stacks and their moduli spaces, with emphasis on the moduli problem for curves. The paper has 4 sections. In section 1 we discuss the general problem of constructing a…
Suppose we are given a graph and want to show a property for all its cycles (closed chains). Induction on the length of cycles does not work since sub-chains of a cycle are not necessarily closed. This paper derives a principle reminiscent…
Recollements were introduced originally by Beilinson, Bernstein and Deligne to study the derived categories of perverse sheaves, and nowadays become very powerful in understanding relationship among three algebraic, geometric or topological…
We systematically study the so-called auto-arc spaces. Auto-arc spaces were originally introduced by Schoutens in 2012 and later generalized and studied by the author in his PhD Thesis and subsequent work. In that aforementioned work, only…
The need for tree structure modelling on top of sequence modelling is an open issue in neural dependency parsing. We investigate the impact of adding a tree layer on top of a sequential model by recursively composing subtree representations…
In this note we provide an example of an endomorphism of a short exact sequence of perfect complexes, with the trace of the middle map not equal to the sum of the traces of the two other ones. The point is that the squares involved are…
Covering spaces are a fundamental tool in algebraic topology because of the close relationship they bear with the fundamental groups of spaces. Indeed, they are in correspondence with the subgroups of the fundamental group: this is known as…
The de Rham stack construction of Simpson shows that D-modules are quasicoherent sheaves on a modified geometry. Drinfeld furthermore introduced the ring stack perspective (aka transmutation), which asserts that a coefficient theory is…
Stanley and Odlyzko proposed a method for greedily constructing sets with no 3-term arithmetic progressions. It is conjectured that there is a dichotomy between such sequences: those that have a periodic structure as the sequence satisfies…
While automatically generated polynomial elimination templates have sparked great progress in the field of 3D computer vision, there remain many problems for which the degree of the constraints or the number of unknowns leads to…