Related papers: Mass problems and intuitionistic higher-order logi…
We show how a novel construction of the sheaf of Cherednik algebras on a quotient orbifold Y=X/G by virtue of formal geometry in author's prior work leads to results for the sheaf of Cherednik algebra which until recently were viewed as…
The so-called topos approach provides a radical reformulation of quantum theory. Structurally, quantum theory in the topos formulation is very similar to classical physics. There is a state object, analogous to the state space of a…
We propose a semantic foundation for logics for reasoning in settings that possess a distinction between equality of variables, a coarser equivalence of variables, and a notion of conditional independence between variables. We show that…
Qualitative spatial models based on Goodman-style mereology and pseudo-topology often pose problems for advanced geometric reasoning, as they lack true Euclidean geometry and fully developed topological spaces. We address this issue by…
In a 1985 commentary to his collected works, Kolmogorov remarked that his 1932 paper "was written in hope that with time, the logic of solution of problems [i.e., intuitionistic logic] will become a permanent part of a [standard] course of…
The overwhelming majority of the attempts in exploring the problems related to quantum logical structures and their interpretation have been based on an underlying set-theoretic syntactic language. We propose a transition in the involved…
This paper studies the combinatorics of ideals which recently appeared in ergodicity results for analytic equivalence relations. The ideals have the following topological representation. There is a separable metrizable space $X$, a…
We introduce new foundations for relative topos theory based on stacks. One of the central results in our theory is an adjunction between the category of toposes over the topos of sheaves on a given site $({\mathcal{C}}, J)$ and that of…
In this survey, we provide an overview of category theory-derived machine learning from four mainstream perspectives: gradient-based learning, probability-based learning, invariance and equivalence-based learning, and topos-based learning.…
In the series of papers Motivic GUT Part I: Grand Unified Theory of Topological Order, Motivic GUT Part II: Grand Unified Theory of Symmetry-Protected Topological Order, and Motivic GUT Part III: Grand Unified Theory of Symmetry-Enriched…
We introduce a hierarchy of degree structures between the Medvedev and Muchnik lattices which allow varying amounts of non-uniformity. We use these structures to introduce the notion of the uniformity of a Muchnik reduction, which expresses…
A key problem in the application of first-order probabilistic methods is the enormous size of graphical models they imply. The size results from the possible worlds that can be generated by a domain of objects and relations. One of the…
Possibilistic logic, an extension of first-order logic, deals with uncertainty that can be estimated in terms of possibility and necessity measures. Syntactically, this means that a first-order formula is equipped with a possibility degree…
It is known that infinitely many Medvedev degrees exist inside the Muchnik degree of any nontrivial $\Pi^0_1$ subset of Cantor space. We shed light on the fine structures inside these Muchnik degrees related to learnability and piecewise…
The Mukhin-Tarasov-Varchenko Theorem (previously the Shapiro Conjecture) asserts that a Schubert problem has all solutions distinct and real if the Schubert varieties involved osculate a rational normal curve at real points. This sparked…
Higher order set theory has been a topic of interest for some time, with recent efforts focused on the strength of second order set theories [KW16]. In this paper we strive to present one 'theory of collections' that allows for a formal…
We give natural examples of factors of the Muchnik lattice which capture intuitionistic propositional logic (IPC), arising from the concepts of lowness, 1-genericity, hyperimmune-freeness and computable traceability. This provides a purely…
This paper proposes an interpretation of Grothendieck's geometric universes as a foundational framework for \emph{information networks}. We argue that Grothendieck topologies, sheaves, and topoi provide a sheaf-theoretic semantics in which…
We generalise sheaf models of intuitionistic logic to univalent type theory over a small category with a Grothendieck topology. We use in a crucial way that we have constructive models of univalence, that can then be relativized to any…
In this paper, I develop a novel version of the multiverse theory of sets called hierarchical pluralism by introducing the notion of `degrees of intentionality' of theories. The presented view is articulated for the purpose of reconciling…