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The classical matching pipeline used for visual localization typically involves three steps: (i) local feature detection and description, (ii) feature matching, and (iii) outlier rejection. Recently emerged correspondence networks propose…
We classify affine operators on a unitary or Euclidean space U up to topological conjugacy. An affine operator is a map f: U-->U of the form f(x)=Ax+b, in which A: U-->U is a linear operator and b in U. Two affine operators f and g are said…
The thesis presents the subject of synthetic topology, especially with relation to metric spaces. A model of synthetic topology is a categorical model in which objects possess an intrinsic topology in a suitable sense, and all morphisms are…
We compute rationally the topological (complex) K-theory of the classifying space BG of a discrete group provided that G has a cocompact G-CW-model for its classifying space for proper G-actions. For instance word-hyperbolic groups and…
In many areas of applied geometric/numeric computational mathematics, including geo-mapping, computer vision, computer graphics, finite element analysis, medical imaging, geometric design, and solid modeling, one has to compute incidences,…
We use the notion of topological data analysis to compare metrics on data sets. We provide two different motivating examples for this. The first of these is a point cloud data set that has $\mathbb{R}^2$ as its ambient space, and is…
Let W be an associative PI algebra over a field F of characteristic zero, graded by a finite group G. Let id_{G}(W) denote the T-ideal of G-graded identities of W. We prove: 1. {[G-graded PI equivalence]} There exists a field extension K of…
I briefly discuss a method of obtaining distinct classes of topologically equivalent knots by developing appropriate computer programs.
For every finite closure space $X$ one can define a finite topological space $\operatorname{Top} X$ together with a natural projection $\operatorname{Top} X\longrightarrow X$. This could allow to apply the techniques of topological…
A ringed finite space is a ringed space whose underlying topological space is finite. The category of ringed finite spaces contains, fully faithfully, the category of finite topological spaces and the category of affine schemes. Any ringed…
We derive a one-parameter deformation of the refined topological vertex that, when used to compute non-periodic web diagrams, reproduces the six-dimensional topological string partition functions that are computed using the refined vertex…
We prove a characterization of profinite algebras, i.e., topological algebras that are isomorphic to a projective limit of finite discrete algebras. In general profiniteness concerns both the topological and algebraic characteristics of a…
Various kinds of data are routinely represented as discrete probability distributions. Examples include text documents summarized by histograms of word occurrences and images represented as histograms of oriented gradients. Viewing a…
In this paper, we consider a simple class of stratified spaces -- 2-complexes. We present an algorithm that learns the abstract structure of an embedded 2-complex from a point cloud sampled from it. We use tools and inspiration from…
We introduce a topology on the space of all isomorphism types represented in a given class of countable models, and use this topology as an aid in classifying the isomorphism types. This mixes ideas from effective descriptive set theory and…
We investigate which topological spaces can be constructed as topological realisations of higher-rank graphs. We describe equivalence relations on higher-rank graphs for which the quotient is again a higher-rank graph, and show that…
We use topological methods to study complexity of deep computations and limit computations. We use topology of function spaces, specifically, the classification Rosenthal compacta, to identify new complexity classes. We use the language of…
The sets of closed and closed-normal subgroups of a profinite group carry a natural profinite topology. Through a combination of algebraic and topological methods the size of these subgroup spaces is calculated, and the spaces partially…
The concept of $typed$ $topology$ is introduced. In a typed topological space, some open sets are assigned "types", and topological concepts such as closure, connectedness can be defined using types. A finite data set in $R^2$ is a…
Software and hardware architectures are prone to modifications. We demonstrate how a mathematically founded powerful refinement calculus for a class of architectures, namely pipe and filter architectures, can be used to modify a system in a…