In this thesis, we apply cochain complexes as an algebraic model of space in a diverse range of mathematical and scientific settings. We begin with an algebraic-discrete Morse theory model of auto-encoding cochain data, connecting the homotopy theory of d ...
We introduce robust principal component analysis from a data matrix in which the entries of its columns have been corrupted by permutations, termed Unlabeled Principal Component Analysis (UPCA). Using algebraic geometry, we establish that UPCA is a well-de ...
We extend the group-theoretic notion of conditional flatness for a localization functor to any pointed category, and investigate it in the context of homological categories and of semi-abelian categories. In the presence of functorial fiberwise localizatio ...
A correspondence functor is a functor from the category of finite sets and correspondences to the category of k-modules, where k is a commutative ring. By means of a suitably defined duality, new correspondence functors are constructed, having remarkable p ...
Visual estimates of stimulus features are systematically biased toward the features of previously encountered stimuli. Such serial dependencies have often been linked to how the brain maintains perceptual continuity. However, serial dependence has mostly b ...
Association for Research in Vision and Ophthalmology (ARVO)2023
Object detection plays a critical role in various computer vision applications, encompassing
domains like autonomous vehicles, object tracking, and scene understanding. These applica-
tions rely on detectors that generate bounding boxes around known object ...
Inductive circuits and devices are ubiquitous and important design elements in many applications, such as magnetic drives, galvanometers, magnetic scanners, applying direct current (DC) magnetic fields to systems, radio frequency coils in nuclear magnetic ...
Shadows for bicategories, defined by Ponto, provide a useful framework that generalizes classical and topological Hochschild homology. In this paper, we define Hochschild-type invariants for monoids in a symmetric monoidal, simplicial model category V, as ...
In this thesis, we study interactions between algebraic and coalgebraic structures in infinity-categories (more precisely, in the quasicategorical model of (infinity, 1)-categories). We define a notion of a Hopf algebra H in an E-2-monoidal infinity-catego ...
The buzzword in industrial policy and innovation policy circles is mission-oriented innovation policy (MOIP) -which means a policy to encourage innovation intended to accomplish a certain missionwhether it be a societal challenge (climate change, global he ...
In this thesis, we give a modern treatment of Dwyer's tame homotopy theory using the language of ∞-categories.
We introduce the notion of tame spectra and show it has a concrete algebraic description.
We then carry out a study of ∞-operads an ...
We prove that four different ways of defining Cartesian fibrations and the Cartesian model structure are all Quillen equivalent: 1.On marked simplicial sets (due to Lurie [31]), 2.On bisimplicial spaces (due to deBrito [12]), 3.On bisimplicial sets, 4.On m ...
In this thesis, we study the homotopical relations of 2-categories, double categories, and their infinity-analogues. For this, we construct homotopy theories for the objects of interest, and show that there are homotopically full embeddings of 2-categories ...
We define filter quotients of -categories and prove that filter quotients preserve the structure of an elementary -topos and in particular lift the filter quotient of the underlying elementary topos. We then specialize to the case of filter products of -ca ...
The cotangent complex of a map of commutative rings is a central object in deformation theory. Since the 1990s, it has been generalized to the homotopical setting of E-infinity-ring spectra in various ways. In this work we first establish, in the context o ...
To achieve conservation objectives for threatened and endangered species, managers must choose among potential recovery actions based on their efficacy. Yet, a lack of standardization in defining how conservation actions support recovery objectives can imp ...
The theory of persistence, which arises from topological data analysis, has been intensively studied in the one-parameter case both theoretically and in its applications. However, its extension to the multi-parameter case raises numerous difficulties, wher ...
In the enriched setting, the notions of injective and projective model structures on a category of enriched diagrams also make sense. In this paper, we prove the existence of these model structures on enriched diagram categories under local presentability, ...
We apply the Acyclicity Theorem of Hess, Kedziorek, Riehl, and Shipley (recently corrected by Garner, Kedziorek, and Riehl) to establishing the existence of model category structure on categories of coalgebras over comonads arising from simplicial adjuncti ...
A correspondence functor is a functor from the category of finite sets and correspondences to the category of k-modules, where k is a commutative ring. A main tool for this study is the construction of a correspondence functor associated to any finite latt ...