IA Scholar Query: The Stratified Foundations as a Theory Modulo.
https://scholar.archive.org/
Internet Archive Scholar query results feedeninfo@archive.orgThu, 29 Sep 2022 00:00:00 GMTfatcat-scholarhttps://scholar.archive.org/help1440Building Specifications in the Event-B Institution
https://scholar.archive.org/work/defbxururnbjdp4u3ktl5jjwmq
This paper describes a formal semantics for the Event-B specification language using the theory of institutions. We define an institution for Event-B, EVT, and prove that it meets the validity requirements for satisfaction preservation and model amalgamation. We also present a series of functions that show how the constructs of the Event-B specification language can be mapped into our institution. Our semantics sheds new light on the structure of the Event-B language, allowing us to clearly delineate three constituent sub-languages: the superstructure, infrastructure and mathematical languages. One of the principal goals of our semantics is to provide access to the generic modularisation constructs available in institutions, including specification-building operators for parameterisation and refinement. We demonstrate how these features subsume and enhance the corresponding features already present in Event-B through a detailed study of their use in a worked example. We have implemented our approach via a parser and translator for Event-B specifications, EBtoEVT, which also provides a gateway to the Hets toolkit for heterogeneous specification.Marie Farrell, Rosemary Monahan, James F. Powerwork_defbxururnbjdp4u3ktl5jjwmqThu, 29 Sep 2022 00:00:00 GMTDiscrete Microlocal Morse Theory
https://scholar.archive.org/work/o4rhj4kfgvg5bjrk7ok5tril7e
We establish several results combining discrete Morse theory and microlocal sheaf theory in the setting of finite posets and simplicial complexes. Our primary tool is a computationally tractable description of the bounded derived category of sheaves on a poset with the Alexandrov topology. We prove that each bounded complex of sheaves on a finite poset admits a unique (up to isomorphism of complexes) minimal injective resolution, and we provide algorithms for computing minimal injective resolutions, as well as several useful functors between derived categories of sheaves. For the constant sheaf on a simplicial complex, we give asymptotically tight bounds on the complexity of computing the minimal injective resolution with this algorithm. Our main result is a novel definition of the discrete microsupport of a bounded complex of sheaves on a finite poset. We detail several foundational properties of the discrete microsupport, as well as a microlocal generalization of the discrete homological Morse theorem and Morse inequalities.Adam Brown, Ondrej Draganovwork_o4rhj4kfgvg5bjrk7ok5tril7eThu, 29 Sep 2022 00:00:00 GMTFocusing on Liquid Refinement Typing
https://scholar.archive.org/work/dqxe6jc5jvblhjzv7ddobqfs2i
We present a foundation systematizing, in a way that works for any evaluation order, the variety of mechanisms for SMT constraint generation found in index refinement and liquid type systems. Using call-by-push-value, we design a polarized subtyping relation allowing us to prove that our logically focused typing algorithm is sound, complete, and decidable, even in cases seemingly likely to produce constraints with existential variables. We prove type soundness with respect to an elementary domain-theoretic denotational semantics. Soundness implies, relatively simply, our system's totality and logical consistency.Dimitrios J. Economou and Neel Krishnaswami and Jana Dunfieldwork_dqxe6jc5jvblhjzv7ddobqfs2iMon, 26 Sep 2022 00:00:00 GMTOn Supersymmetric Interface Defects, Brane Parallel Transport, Order-Disorder Transition and Homological Mirror Symmetry
https://scholar.archive.org/work/5giksvcv7readlihzldqrfeuwq
We concentrate on a treatment of a Higgs-Coulomb duality as an absence of manifest phase transition between ordered and disordered phases of 2d 𝒩=(2,2) theories. We consider these examples of QFTs in the Schrödinger picture and identify Hilbert spaces of BPS states with morphisms in triangulated categories of D-brane boundary conditions. As a result of Higgs-Coulomb duality D-brane categories on IR vacuum moduli spaces are equivalent, this resembles an analog of homological mirror symmetry. Following construction ideas behind the Gaiotto-Moore-Witten algebra of the infrared one is able to introduce interface defects in these theories and associate them to D-brane parallel transport functors. We concentrate on surveying simple examples, analytic when possible calculations, numerical estimates and simple physical picture behind curtains of geometric objects. Categorification of hypergeometric series analytic continuation is derived as an Atiyah flop of the conifold. Finally we arrive to an interpretation of the braid group action on the derived category of coherent sheaves on cotangent bundles to flag varieties as a categorification of Berry connection on the Fayet-Illiopolous parameter space of a sigma-model with a quiver variety target space.Dmitry Galakhovwork_5giksvcv7readlihzldqrfeuwqMon, 26 Sep 2022 00:00:00 GMTInvariance of immersed Floer cohomology under Lagrangian surgery
https://scholar.archive.org/work/mmtjxyj6vnck3g4dh5x3g73q2e
We show that cellular Floer cohomology of an immersed Lagrangian brane is invariant under smoothing of a self-intersection point if the quantum valuation of the weakly bounding cochain vanishes and the Lagrangian has dimension at least two. The chain-level map replaces the two orderings of the self-intersection point with meridianal and longitudinal cells on the handle created by the surgery, and uses a bijection between holomorphic disks developed by Fukaya-Oh-Ohta-Ono. Our result generalizes invariance of potentials for certain Lagrangian surfaces in Dimitroglou-Rizell--Ekholm--Tonkonog, and implies the invariance of Floer cohomology under mean curvature flow with this type of surgery, as conjectured by Joyce.Joseph Palmer, Chris Woodwardwork_mmtjxyj6vnck3g4dh5x3g73q2eSun, 25 Sep 2022 00:00:00 GMTThe Axiomatic Approach to Non-Classical Model Theory
https://scholar.archive.org/work/eis66kn4lvfj7ewxu3xnqlxzwy
Institution theory represents the fully axiomatic approach to model theory in which all components of logical systems are treated fully abstractly by reliance on category theory. Here, we survey some developments over the last decade or so concerning the institution theoretic approach to non-classical aspects of model theory. Our focus will be on many-valued truth and on models with states, which are addressed by the two extensions of ordinary institution theory known as L-institutions and stratified institutions, respectively. The discussion will include relevant concepts, techniques, and results from these two areas.Răzvan Diaconescuwork_eis66kn4lvfj7ewxu3xnqlxzwyWed, 21 Sep 2022 00:00:00 GMTAlgebraic Presentations of Dependent Type Theories
https://scholar.archive.org/work/rsnnmooesjhlbcxmqedquui53a
C-systems were defined by Cartmell as models of generalized algebraic theories. B-systems were defined by Voevodsky in his quest to formulate and prove an initiality conjecture for type theories. They play a crucial role in Voevodsky's construction of a syntactic C-system from a term monad. In this work, we construct an equivalence between the category of C-systems and the category of B-systems, thus proving a conjecture by Voevodsky. We construct this equivalence as the restriction of an equivalence between more general structures, called CE-systems and E-systems, respectively. To this end, we identify C-systems and B-systems as "stratified" CE-systems and E-systems, respectively; that is, systems whose contexts are built iteratively via context extension, starting from the empty context.Benedikt Ahrens and Jacopo Emmenegger and Paige Randall North and Egbert Rijkework_rsnnmooesjhlbcxmqedquui53aTue, 20 Sep 2022 00:00:00 GMTIsometric Lie 2-group actions on Riemannian groupoids
https://scholar.archive.org/work/7cnztahgarfd3ewtqjjo73lfuy
We study isometric actions of Lie 2-groups on Riemannian groupoids and exhibit some of their immediate properties. Firstly, we prove an existence result, describe equivariant weakly groupoid linearizations, state versions of both the Slice Theorem and the Equivariant Tubular Neighborhood Theorem, and define orthogonal Lie 2-groups. We also provide a few interesting examples and applications. Secondly, we present a description of the Lie 2-group of strong (weak) groupoid isometries of a Lie groupoid equipped with a 0-metric and determine a model for its Lie 2-algebra of strong (weak) Killing multiplicative vector fields. We show that the Lie 2-algebra of weak Killing multiplicative vector fields is Morita invariant, thus yielding a good notion of geometric Killing vector field on a quotient Riemannian stack.Juan Sebastian Herrera-Carmona, Fabricio Valenciawork_7cnztahgarfd3ewtqjjo73lfuySun, 18 Sep 2022 00:00:00 GMTTopological symmetry in quantum field theory
https://scholar.archive.org/work/zykzvhif3fcufjmyv32ics7ake
We introduce a framework for internal topological symmetries in quantum field theory, including "noninvertible symmetries" and "categorical symmetries". This leads to a calculus of topological defects which takes full advantage of well-developed theorems and techniques in topological field theory. Our discussion focuses on finite symmetries, and we give indications for a generalization to other symmetries. We treat quotients and quotient defects (often called "gauging" and "condensation defects"), finite electromagnetic duality, and duality defects, among other topics. We include an appendix on finite homotopy theories, which are often used to encode finite symmetries and for which computations can be carried out using methods of algebraic topology. Throughout we emphasize exposition and examples over a detailed technical treatment.Daniel S. Freed, Gregory W. Moore, Constantin Telemanwork_zykzvhif3fcufjmyv32ics7akeThu, 15 Sep 2022 00:00:00 GMTA case for DOT: Theoretical Foundations for Objects With Pattern Matching and GADT-style Reasoning
https://scholar.archive.org/work/ll2tdshywzdkdkizvfbpyxqmey
Many programming languages in the OO tradition now support pattern matching in some form. Historical examples include Scala and Ceylon, with the more recent additions of Java, Kotlin, TypeScript, and Flow. But pattern matching on generic class hierarchies currently results in puzzling type errors in most of these languages. Yet this combination of features occurs naturally in many scenarios, such as when manipulating typed ASTs. To support it properly, compilers needs to implement a form of subtyping reconstruction: the ability to reconstruct subtyping information uncovered at runtime during pattern matching. We introduce cDOT, a new calculus in the family of Dependent Object Types (DOT) intended to serve as a formal foundation for subtyping reconstruction. Being descended from pDOT, itself a formal foundation for Scala, cDOT can be used to encode advanced object-oriented features such as generic inheritance, type constructor variance, F-bounded polymorphism, and first-class recursive modules. We demonstrate that subtyping reconstruction subsumes GADTs by encoding λ_2,Gμ, a classical constraint-based GADT calculus, into cDOT.Aleksander Boruch-Gruszecki, Radosław Waśko, Yichen Xu, Lionel Parreauxwork_ll2tdshywzdkdkizvfbpyxqmeyThu, 15 Sep 2022 00:00:00 GMTOn locally analytic vectors of the completed cohomology of modular curves II
https://scholar.archive.org/work/c5m5chfxlvbuzly5z4dvr3a2bm
This is a continuation of our previous work on the locally analytic vectors of the completed cohomology of modular curves. We construct differential operators on modular curves with infinite level at p in both "holomorphic" and "anti-holomorphic" directions. As applications, we reprove a classicality result of Emerton which says that every absolutely irreducible two dimensional Galois representation which is regular de Rham at p and appears in the completed cohomology of modular curves comes from an eigenform. Moreover we give a geometric description of the locally analytic representations of GL_2(ℚ_p) attached to such a Galois representation in the completed cohomology.Lue Panwork_c5m5chfxlvbuzly5z4dvr3a2bmWed, 14 Sep 2022 00:00:00 GMTA Bunch of Sessions: A Propositions-as-Sessions Interpretation of Bunched Implications in Channel-Based Concurrency
https://scholar.archive.org/work/klbrcgex3jdpjns3bguucxeqv4
The emergence of propositions-as-sessions, a Curry-Howard correspondence between propositions of Linear Logic and session types for concurrent processes, has settled the logical foundations of message-passing concurrency. Central to this approach is the resource consumption paradigm heralded by Linear Logic. In this paper, we investigate a new point in the design space of session type systems for message-passing concurrent programs. We identify O'Hearn and Pym's Logic of Bunched Implications (BI) as a fruitful basis for an interpretation of the logic as a concurrent programming language. This leads to a treatment of non-linear resources that is radically different from existing approaches based on Linear Logic. We introduce a new π-calculus with sessions, called πBI; its most salient feature is a construct called spawn, which expresses new forms of sharing that are induced by structural principles in BI. We illustrate the expressiveness of πBI and lay out its fundamental theory: type preservation, deadlock-freedom, and weak normalization results for well-typed processes; an operationally sound and complete typed encoding of an affine λ-calculus; and a non-interference result for access of resources.Dan Frumin, Emanuele D'Osualdo, Bas van den Heuvel, Jorge A. Pérezwork_klbrcgex3jdpjns3bguucxeqv4Mon, 12 Sep 2022 00:00:00 GMTStratifying systems and Jordan-Hölder extriangulated categories
https://scholar.archive.org/work/dths5g3b7rfwhodcpfyeveg4ky
Stratifying systems, which have been defined for module, triangulated and exact categories previously, were developed to produce examples of standardly stratified algebras. A stratifying system Φ is a finite set of objects satisfying some orthogonality conditions. One very interesting property is that the subcategory ℱ(Φ) of objects admitting a composition series-like filtration with factors in Φ has the Jordan-Hölder property on these filtrations. This article has two main aims. First, we introduce notions of subobjects, simple objects and composition series relative to the extriangulated structure in order to define a Jordan-Hölder extriangulated category. Moreover, we characterise these categories in terms of the associated Grothendieck monoid and Grothendieck group. Second, we develop a theory of stratifying systems in extriangulated categories. We define projective stratifying systems and show that every stratifying system Φ in an extriangulated category is part of a projective one (Φ, Q). We prove that ℱ(Φ) is a Jordan-Hölder extriangulated category when (Φ,Q) satisfies some extra conditions.Thomas Brüstle, Souheila Hassoun, Amit Shah, Aran Tattarwork_dths5g3b7rfwhodcpfyeveg4kyTue, 06 Sep 2022 00:00:00 GMTConstrainahedra
https://scholar.archive.org/work/2pkt4jm6ebhupp2ytde227wkle
We define a family of convex polytopes called constrainahedra, which index collisions of horizontal and vertical lines. Our construction proceeds by first defining a poset C(m,n) of good rectangular preorders, then proving that C(m,n) is a lattice, and finally constructing a polytopal realization by taking the convex hull of a certain explicitly-defined collection of points. The constrainahedra will form the combinatorial backbone of the second author's construction of strong homotopy duoids. We indicate how constrainahedra could be realized as Gromov-compactified configuration spaces of horizontal and vertical lines; viewed from this perspective, the constrainahedra include naturally into the first author's notion of 2-associahedra.Nathaniel Bottman, Daria Poliakovawork_2pkt4jm6ebhupp2ytde227wkleTue, 30 Aug 2022 00:00:00 GMTEquivariant Oka theory: survey of recent progress
https://scholar.archive.org/work/njxrsirkm5aaneh6w4pzhkztq4
We survey recent work, published since 2015, on equivariant Oka theory. The main results described in the survey are as follows. Homotopy principles for equivariant isomorphisms of Stein manifolds on which a reductive complex Lie group G acts. Applications to the linearisation problem. A parametric Oka principle for sections of a bundle E of homogeneous spaces for a group bundle , all over a reduced Stein space X with compatible actions of a reductive complex group on E, , and X. Application to the classification of generalised principal bundles with a group action. Finally, an equivariant version of Gromov's Oka principle based on a notion of a G-manifold being G-Oka.Frank Kutzschebauch, Finnur Lárusson, Gerald W Schwarzwork_njxrsirkm5aaneh6w4pzhkztq4Wed, 24 Aug 2022 00:00:00 GMTUniformization of some weight 3 variations of Hodge structure, Anosov representations, and Lyapunov exponents
https://scholar.archive.org/work/ftnmmghjjnghlaoagnqrx6vusy
We develop a class of uniformizations for certain weight 3 variations of Hodge structure (VHS). The analytic properties of the VHS are used to establish a conjecture of Eskin, Kontsevich, M\"oller, and Zorich on Lyapunov exponents. Additionally, we prove that the monodromy representations are log-Anosov, a dynamical property that has a number of global consequences for the VHS. We establish a strong Torelli theorem for the VHS and describe appropriate domains of discontinuity. Additionally, we classify the hypergeometric differential equations that satisfy our assumptions. We obtain several multi-parameter families of equations, which include the mirror quintic as well as the six other thin cases of Doran--Morgan and Brav--Thomas.Simion Filipwork_ftnmmghjjnghlaoagnqrx6vusyTue, 23 Aug 2022 00:00:00 GMTSubsets of rectifiable curves in Banach spaces II: universal estimates for almost flat arcs
https://scholar.archive.org/work/sjwvdwjiffcu5lhyyungobq5w4
We prove that in any Banach space the set of windows in which a rectifiable curve resembles two or more straight line segments is quantitatively small with constants that are independent of the curve, the dimension of the space, and the choice of norm. Together with Part I, this completes the proof of the necessary half of the Analyst's Traveling Salesman theorem with sharp exponent in uniformly convex spaces.Matthew Badger, Sean McCurdywork_sjwvdwjiffcu5lhyyungobq5w4Mon, 22 Aug 2022 00:00:00 GMTDeformed graphical zonotopal algebras
https://scholar.archive.org/work/jfyrn5qobbfifgcnumpxgkhuxu
We study certain filtered deformations of the external zonotopal algebra of a given graph parametrized by univariate polynomials. We establish some general properties of these algebras, compute their Hilbert series for a number of graphs using Macaulay2, and formulate several conjectures.Boris Shapiro, Ilya Smirnov, Arkady Vaintrobwork_jfyrn5qobbfifgcnumpxgkhuxuMon, 22 Aug 2022 00:00:00 GMTStability theorems for H-type Carnot groups
https://scholar.archive.org/work/5div5ped4jbx3h46ak22zgkjgi
We introduce the H-type deviation δ(𝔾) of a step two Carnot group 𝔾, which measures the deviation of the group from the class of Heisenberg-type groups. We show that δ(𝔾)=0 if and only if 𝔾 carries a vertical metric which endows it with the structure of an H-type group. We compute the H-type deviation for several naturally occurring families of step two groups. In addition, we provide analytic expressions which are comparable to the H-type deviation. As a consequence, we establish new analytic characterizations for the class of H-type groups. For instance, denoting by N(g)=(||x||_h^4+16||t||_v^2)^1/4, g=exp(x+t), the canonical Kaplan-type quasi-norm in a step two group 𝔾 with taming Riemannian metric g_h⊕ g_v, we show that 𝔾 is H-type if and only if ||∇_0 N(g)||_h^2=||x||_h^2/N(g)^2 for all g 0. Similarly, we show that 𝔾 is H-type if and only if N^2-Q is ℒ-harmonic in 𝔾∖{0}. Here ∇_0 denotes the horizontal differential operator, ℒ the canonical sub-Laplacian, and Q = 𝔳_1+2𝔳_2 the homogeneous dimension of 𝔾, where 𝔳_1⊕𝔳_2 is the stratification of the Lie algebra. It is well-known that H-type groups satisfy both of these analytic conclusions. The new content of these results lies in the converse directions. Motivation for this work comes from a longstanding conjecture regarding polarizable Carnot groups. We formulate a quantitative stability conjecture regarding the fundamental solution for the sub-Laplacian on step two Carnot groups. Its validity would imply that all step two polarizable groups admit an H-type group structure. We confirm this conjecture for a sequence of anisotropic Heisenberg groups.Jeremy T. Tysonwork_5div5ped4jbx3h46ak22zgkjgiTue, 09 Aug 2022 00:00:00 GMTThe Calogero–Moser Derivative Nonlinear Schrödinger Equation
https://scholar.archive.org/work/3thlxwcydvdc5lo5dtrhdwvf7i
We study the Calogero–Moser derivative NLS equation i ∂_t u +∂_xx u + (D+|D|)(|u|^2) u =0 posed on the Hardy-Sobolev space H^s_+(ℝ) for s large enough. We display a Lax pair structure for this equation, from which we prove global well-posedness in the energy space for every data with mass not bigger than 2π. We also classify all traveling solitary waves and we study in detail the class of multi-soliton solutions u(t), which exhibit energy cascades as t tends to infinity in the following strong sense such that u(t)_H^s≃ |t|^2s for every s > 0.Patrick Gérard, Enno Lenzmannwork_3thlxwcydvdc5lo5dtrhdwvf7iMon, 08 Aug 2022 00:00:00 GMT