IA Scholar Query: On the Decidability of Some Problems Concerning Morphisms, Regular Sets, and Post Correspondence Problem.
https://scholar.archive.org/
Internet Archive Scholar query results feedeninfo@archive.orgTue, 29 Nov 2022 00:00:00 GMTfatcat-scholarhttps://scholar.archive.org/help1440Topics in the theory of enriched accessible categories
https://scholar.archive.org/work/775tinrmyzgw7o5dzjtff4pcra
The aim of this thesis is to further develop the theory of accessible categories in the enriched context. We study and compare the two notions of accessible and conically accessible 𝒱-categories, both arising as free cocompletions of small 𝒱-categories: the former under flat-weighted colimits and the latter under filtered colimits. These two notions are not the same in general, however we show that they coincide for many significant bases of enrichment such as Cat and SSet, and differ just by Cauchy completeness for many algebraic examples including Ab, R-Mod and GAb. We then provide new characterization theorems for these by considering some notions of virtual orthogonality and virtual reflectivity which generalize the usual reflectivity and orthogonality conditions for locally presentable categories. The word virtual refers to the fact that the reflectivity and orthogonality conditions are given in the free completion of the 𝒱-category involved under small limits, instead of the 𝒱-category itself. We then prove that the 2-category of accessible 𝒱-categories, accessible 𝒱-functors, and 𝒱-natural transformations has all flexible limits. In the final chapters we study, characterize, and provide duality theorems in the setting of accessible 𝒱-categories with limits of a specified class Ψ; in this context, instead of the free completion under small limits, we consider "free completions" under a specific type of colimits ℭ for which, in particular, ℭ-colimits commute in 𝒱 with Ψ-limits. This allows us to capture the theories of weakly locally presentable, locally multipresentable, locally polypresentable, and accessible categories as instances of the same general framework.Giacomo Tendaswork_775tinrmyzgw7o5dzjtff4pcraTue, 29 Nov 2022 00:00:00 GMTProtocorks and monopole Floer homology
https://scholar.archive.org/work/hgvvbavqkrg2lbahqs2bzty2a4
We introduce and study a class of compact 4-manifolds with boundary that we call protocorks. Any exotic pair of simply connected closed 4-manifolds is related by a protocork twist, moreover, any cork is supported by a protocork. We prove a theorem on the relative Seiberg-Witten invariants of a protocork before and after twisting and a splitting theorem on the Floer homology of protocork boundaries. As a corollary we improve a theorem by Morgan and Szabó regarding the variation of Seiberg-Witten invariants with an upper bound which depends only on the topology of the data. Moreover we show that for any cork, only the reduced Floer homology of its boundary contributes to the variation of the Seiberg-Witten invariants after a cork twist.Roberto Laduwork_hgvvbavqkrg2lbahqs2bzty2a4Sun, 27 Nov 2022 00:00:00 GMTRelative elegance and cartesian cubes with one connection
https://scholar.archive.org/work/ixl7soint5eired7rnpmi3ai6i
We establish a Quillen equivalence between the Kan-Quillen model structure and a model structure, derived from a model of a cubical type theory, on the category of cartesian cubical sets with one connection. We thereby identify a second model structure which both constructively models homotopy type theory and presents infinity-groupoids, the first known example being the equivariant cartesian model of Awodey-Cavallo-Coquand-Riehl-Sattler.Evan Cavallo, Christian Sattlerwork_ixl7soint5eired7rnpmi3ai6iSun, 27 Nov 2022 00:00:00 GMTAnother Round of Breaking and Making Quantum Money: How to Not Build It from Lattices, and More
https://scholar.archive.org/work/ghlrkjfs2nef5fwmkalccul75i
Public verification of quantum money has been one of the central objects in quantum cryptography ever since Wiesner's pioneering idea of using quantum mechanics to construct banknotes against counterfeiting. So far, we do not know any publicly-verifiable quantum money scheme that is provably secure from standard assumptions. In this work, we provide both negative and positive results for publicly verifiable quantum money. **In the first part, we give a general theorem, showing that a certain natural class of quantum money schemes from lattices cannot be secure. We use this theorem to break the recent quantum money scheme of Khesin, Lu, and Shor. **In the second part, we propose a framework for building quantum money and quantum lightning we call invariant money which abstracts some of the ideas of quantum money from knots by Farhi et al.(ITCS'12). In addition to formalizing this framework, we provide concrete hard computational problems loosely inspired by classical knowledge-of-exponent assumptions, whose hardness would imply the security of quantum lightning, a strengthening of quantum money where not even the bank can duplicate banknotes. **We discuss potential instantiations of our framework, including an oracle construction using cryptographic group actions and instantiations from rerandomizable functional encryption, isogenies over elliptic curves, and knots.Hart Montgomery, Jiahui Liu, Mark Zhandrywork_ghlrkjfs2nef5fwmkalccul75iTue, 22 Nov 2022 00:00:00 GMTAn axiomatic approach to virtual chains
https://scholar.archive.org/work/q5abx64ypjcx7hle5xbma73h4i
We introduce a category of Kuranishi presentations, whose objects are a variant of the Kuranishi structures introduced by Fukaya and Ono, and which can be seen as a refinement of the version studied by Pardon. We then formulate the notion of virtual chains categorically as a natural transformation between two functors from this category to the category of chain complexes; we call such a datum 'a theory of virtual counts'. To show that this definition carries non-trivial content, we then construct a multicategory whose objects are Kuranishi flow categories, and show that a theory of virtual counts determines a multifunctor to the multicategory of chain complexes. We then implement this construction in the setting of Hamiltonian Floer theory, borrowing from some joint work with Groman and Varolgunes, yielding a construction of Hamiltonian Floer groups (and operations on them) as an output of this machine. We plan to provide a similar account for Lagrangian Floer theory in subsequent joint work.Mohammed Abouzaidwork_q5abx64ypjcx7hle5xbma73h4iTue, 22 Nov 2022 00:00:00 GMTA Foundation for Archival Engineering
https://scholar.archive.org/work/mdigtcmzmvd2fh4xoznb23niha
Archives comprise information that individuals and organizations use in their activities. Archival theory is the intellectual framework for organizing, managing, preserving and access to archives both while they serve the needs of those who produce them and later when researchers consult them for other purposes. Archival theory is sometimes called archival science, but it does not constitute a modern science in the sense of a coherent body of knowledge formulated in a way that is appropriate for empirical testing and validation. Both archival theory and practice are seriously challenged by the spread and continuing changes in information technology and its increasing and increasingly diverse use in human activities. This article describes problems with and controversies in archival theory and advocates for a reformulation of concepts to address the digital challenge and to make the field more robust, both by addressing the problems and by enriching its capabilities by adopting concepts from other fields such as taxonomy, semiotics and systemic functional linguistics. The objective of this reformulation is to transform the discipline on the model of modern scientific method in a way that engenders a new discipline of archival engineering that is robust enough to guide the development of automated methods even in the face of continuing and unpredictable change in IT.Kenneth Thibodeauwork_mdigtcmzmvd2fh4xoznb23nihaFri, 18 Nov 2022 00:00:00 GMTMapping analytic surgery to homology, higher rho numbers and metrics of positive scalar curvature
https://scholar.archive.org/work/vsyqxs3drvgarkvnqjlhbfokya
Let Γ be a f.g. discrete group and let M̃ be a Galois Γ-covering of a smooth closed manifold M. Let S_*^Γ(M̃) be the analytic structure group, appearing in the Higson-Roe analytic surgery sequence → S_*^Γ(M̃)→ K_*(M)→ K_*(C_r^*Γ)→. We prove that for an arbitrary discrete group Γ it is possible to map the whole Higson-Roe sequence to the long exact sequence of even/odd-graded noncommutative de Rham homology → H_[*-1](𝒜Γ)→ H^del_[*-1](𝒜Γ)→ H^e_[*](𝒜Γ)→, with 𝒜Γ a dense homomorphically closed subalgebra of C^*_rΓ. Here, H_*^del(𝒜Γ) is the delocalized homology and H_*^e(𝒜Γ) is the homology localized at the identity element. Then, under additional assumptions on Γ, we prove the existence of a pairing between HC^*_del(ℂΓ), the delocalized part of the cyclic cohomology of ℂΓ, and H^del_*-1(𝒜Γ). This, in particular, gives a pairing between S^Γ_*(M̃) and HC^*-1_del(ℂΓ). We also prove the existence of a pairing between S^Γ_*(M̃) and the relative cohomology H^[*-1](M→ BΓ). Both these parings are compatible with known pairings associated with the other terms in the Higson-Roe sequence. In particular, we define higher rho numbers associated to the rho class ρ(D̃)∈ S_*^Γ(M̃) of an invertible Γ-equivariant Dirac type operator on M̃. Finally, we provide a precise study for the behavior of all previous K-theoretic and homological objects and of the higher rho numbers under the action of the diffeomorphism group of M. Then, we establish new results on the moduli space of metrics of positive scalar curvature when M is spin.Paolo Piazza, Thomas Schick, Vito Felice Zenobiwork_vsyqxs3drvgarkvnqjlhbfokyaMon, 14 Nov 2022 00:00:00 GMTPrinciples of operator algebras
https://scholar.archive.org/work/cikqosndpje7vmolarptt6mjxu
This is an introduction to the algebras A⊂ B(H) that the bounded linear operators T:H→ H can form, once a complex Hilbert space H is given. Motivated by quantum mechanics, we are mostly interested in the von Neumann algebras, which are stable under taking adjoints, T→ T^*, and weakly closed. When the algebra has a trace tr:A→ℂ, we can think of it as being of the form A=L^∞(X), with X being a quantum measured space, and of particular interest is the free case, where the center of the algebra is Z(A)=ℂ. Following Murray, von Neumann, Connes, Jones, Voiculescu, Woronowicz, we discuss here the basic properties of such algebras A, and how to do algebra, geometry, analysis and probability on the underlying quantum spaces X.Teo Banicawork_cikqosndpje7vmolarptt6mjxuFri, 11 Nov 2022 00:00:00 GMTCoalgebra Encoding for Efficient Minimization
https://scholar.archive.org/work/kicdblvvjzcrpoleh742kh7g7i
Recently, we have developed an efficient generic partition refinement algorithm, which computes behavioural equivalence on a state-based system given as an encoded coalgebra, and implemented it in the tool CoPaR. Here we extend this to a fully fledged minimization algorithm and tool by integrating two new aspects: (1) the computation of the transition structure on the minimized state set, and (2) the computation of the reachable part of the given system. In our generic coalgebraic setting these two aspects turn out to be surprisingly non-trivial requiring us to extend the previous theory. In particular, we identify a sufficient condition on encodings of coalgebras, and we show how to augment the existing interface, which encapsulates computations that are specific for the coalgebraic type functor, to make the above extensions possible. Both extensions have linear run time.Hans-Peter Deifel, Stefan Milius, Thorsten Wißmannwork_kicdblvvjzcrpoleh742kh7g7iWed, 09 Nov 2022 00:00:00 GMTAbstract evolution systems
https://scholar.archive.org/work/lmkko3uqz5hqzoh7bhwjo6twn4
We introduce the concept of an abstract evolution system, which provides a convenient framework for studying generic mathematical structures and their properties. Roughly speaking, an evolution system is a category endowed with a selected class of morphisms called transitions, and with a selected object called the origin. We illustrate it by a series of examples from several areas of mathematics. We formulate sufficient conditions for the existence of the unique "most complicated" evolution. In case the evolution system "lives" in model theory and nontrivial transitions are one-point extensions, the limit of the most complicated evolution is known under the name Fraisse limit, a unique countable universal homogeneous model determined by a fixed class of finitely generated models satisfying some obvious axioms. Evolution systems can also be viewed as a generalization of abstract rewriting systems, where the partially ordered set is replaced by a category. In our setting, the process of rewriting plays a nontrivial role, whereas in rewriting systems only the result of a rewriting procedure is relevant. An analogue of Newman's Lemma holds in our setting, although the proof is a bit more delicate, nevertheless, still based on Huet's idea using well founded induction.Wiesław Kubiś, Paulina Radeckawork_lmkko3uqz5hqzoh7bhwjo6twn4Sun, 06 Nov 2022 00:00:00 GMTRemarks on classification theory for abstract elementary classes with applications to abelian group theory and ring theory
https://scholar.archive.org/work/ezootjr2pffmxpwvzs2sy3qbdu
This thesis has two parts. The first part deals with the classification theory of abstract elementary classes and the second part deals with links and applications of this theory to algebra. Part I: Remarks on classification theory for abstract elementary classes This part of the thesis is made up of three chapters based on the corresponding papers: [Ch. 2], [Ch. 3] (a joint work with S. Vasey), and [Ch. 4] (a joint work with R. Grossberg). Chapter 2, Non-forking w-good frames. We introduce and study the notion of a w-good λ-frame which is a weakening of Shelah's notion of a good λ-frame. W-good λ-frames are useful as they imply the existence of larger models. We show that if K has a w-good λ-frame, then K has a model of size λ ++. This result extends [Sh:h, §II.4.13.3], [JaSh13, 3.1.9], and [Vas16a, 8.9]. Chapter 3, Universal classes near ℵ1 (a joint work with S. Vasey). Shelah has provided sufficient conditions for an Lω1,-ωsentence ψ to have arbitrarily large models and for a Morley-like theorem to hold of ψ. These conditions involve structural and set-theoretic assumptions on all the ℵn's. Using tools of Boney, Shelah, and Vasey, we give assumptions on ℵ0 and ℵ1 which suffice when ψ is restricted to be universal. Chapter 4, Simple-like independence relations in abstract elementary classes (a joint work with R. Grossberg). We introduce and study simple and supersimple independence relations in the context of AECs with a monster model. We show that if K has a simple independence relation with the (< ℵ0)-witness property for singletons, then K does not have the tree property. We characterize supersimple independence relations by finiteness of the Lascar rank under locality assumptions on the independence relation. Part II: Applications to abelian group theory and ring theory [...]Marcos Mazari Armidawork_ezootjr2pffmxpwvzs2sy3qbduMon, 31 Oct 2022 00:00:00 GMTDistortion element in the automorphism group of a full shift
https://scholar.archive.org/work/rzg3vkkqjvegnhdeqvittrbf6y
We show that there is a distortion element in a finitely-generated subgroup G of the automorphism group of the full shift, namely an element of infinite order whose word norm grows polylogarithmically. As a corollary, we obtain a lower bound on the entropy dimension of any subshift containing a copy of G, and that a sofic shift's automorphism group contains a distortion element if and only if the sofic shift is uncountable. We obtain also that groups of Turing machines and the higher-dimensional Brin-Thompson groups mV admit distortion elements; in particular, 2V (unlike V) does not admit a proper action on a CAT(0) cube complex. The distortion element is essentially the SMART machine.Antonin Callard, Ville Salowork_rzg3vkkqjvegnhdeqvittrbf6yMon, 31 Oct 2022 00:00:00 GMTThe Geometry of Rings of Components of Hurwitz Spaces
https://scholar.archive.org/work/rqu2e4bqdrd23blzgkmfddfrxq
We study a variant of the ring of components of Hurwitz moduli spaces for covers, introduced by Ellenberg, Venkatesh and Westerland in 2016 in their proof of the Cohen-Lenstra conjecture for function fields. For G-covers of the projective line, we show that the ring of components is a commutative algebra of finite type. We therefore study it using tools from algebraic geometry. We obtain a description of the spectrum, relating its geometry to group-theoretical properties and combinatorial aspects of Galois covers. When G is a symmetric group, we are able to fully describe the geometric points of the spectrum.Béranger Seguinwork_rqu2e4bqdrd23blzgkmfddfrxqSun, 23 Oct 2022 00:00:00 GMTIntroduction to quantum groups
https://scholar.archive.org/work/p3qxhufzhvguzpzc7crlx7ygha
This is an introduction to the quantum groups, or rather to the simplest quantum groups. The idea is that the unitary group U_N has a free analogue U_N^+, whose standard coordinates u_ij∈ C(U_N^+) are allowed to be free, and the closed subgroups G⊂ U_N^+ can be thought of as being the compact quantum Lie groups. There are many interesting examples of such quantum groups, for the most designed in order to help with questions in quantum mechanics and statistical mechanics, and some general theory available as well, including Peter-Weyl theory, Tannakian duality, Brauer theorems and Weingarten integration. We discuss here the basic aspects of all this.Teo Banicawork_p3qxhufzhvguzpzc7crlx7yghaSat, 22 Oct 2022 00:00:00 GMTClassical Set Theory: Theory of Sets and Classes
https://scholar.archive.org/work/3sn3ycbmg5acdidye3hpq3w7dm
This is a short introductory course to Set Theory, based on axioms of von Neumann--Bernays--G\"odel (briefly NBG). The text can be used as a base for a lecture course in Foundations of Mathematics, and contains a reasonable minimum which a good (post-graduate) student in Mathematics should know about foundations of this science.Taras Banakhwork_3sn3ycbmg5acdidye3hpq3w7dmThu, 20 Oct 2022 00:00:00 GMTRich groups, weak second order logic, and applications
https://scholar.archive.org/work/c2mrnkfxcrfqtluhqip22m5574
In this paper we initiate a study of first-order rich groups, i.e., groups where the first-order logic has the same power as the weak second order logic. Surprisingly, there are quite a lot of finitely generated rich groups, they are somewhere in between hyperbolic and nilpotent groups (these ones are not rich). We provide some methods to prove that groups (and other structures) are rich and describe some of their properties. As corollaries we look at Malcev's problems in various groups.Olga Kharlampovich, Alexei Myasnikov, Mahmood Sohrabiwork_c2mrnkfxcrfqtluhqip22m5574Sat, 15 Oct 2022 00:00:00 GMTEstimation under group actions: recovering orbits from invariants
https://scholar.archive.org/work/fqfsejx2cre33febqbuhxfjd5y
We study a class of orbit recovery problems in which we observe independent copies of an unknown element of ℝ^p, each linearly acted upon by a random element of some group (such as ℤ/p or SO(3)) and then corrupted by additive Gaussian noise. We prove matching upper and lower bounds on the number of samples required to approximately recover the group orbit of this unknown element with high probability. These bounds, based on quantitative techniques in invariant theory, give a precise correspondence between the statistical difficulty of the estimation problem and algebraic properties of the group. Furthermore, we give computer-assisted procedures to certify these properties that are computationally efficient in many cases of interest. The model is motivated by geometric problems in signal processing, computer vision, and structural biology, and applies to the reconstruction problem in cryo-electron microscopy (cryo-EM), a problem of significant practical interest. Our results allow us to verify (for a given problem size) that if cryo-EM images are corrupted by noise with variance σ^2, the number of images required to recover the molecule structure scales as σ^6. We match this bound with a novel (albeit computationally expensive) algorithm for ab initio reconstruction in cryo-EM, based on invariant features of degree at most 3. We further discuss how to recover multiple molecular structures from mixed (or heterogeneous) cryo-EM samples.Afonso S. Bandeira, Ben Blum-Smith, Joe Kileel, Amelia Perry, Jonathan Weed, Alexander S. Weinwork_fqfsejx2cre33febqbuhxfjd5yThu, 13 Oct 2022 00:00:00 GMTFrom Samples to Persistent Stratified Homotopy Types
https://scholar.archive.org/work/mmvmewcchzgjbdfow63ou5tvr4
The natural occurrence of singular spaces in application has led to recent investigations on performing topological data analysis (TDA) in a stratified framework. Non-stratified applications in TDA rely on a series of convenient properties of (persistent) homotopy types of sufficiently regular spaces. Classical notions of stratified homotopy equivalence generally turn out to be too rigid to display similar behavior. At the same time, recent developments in abstract stratified homotopy theory employ weaker notions of stratified equivalences. These exhibit behavior more suitable for the purpose of TDA while still retaining the same homotopy theoretical information as classical stratified homotopy equivalences for common examples such as Whitney stratified spaces. In this work, we establish a notion of persistent stratified homotopy type obtained from a sample with two strata. We show that it behaves much like its non-stratified counterpart and exhibits many properties (such as stability) necessary for an application in TDA. Using local approximations of tangent cones, we describe a pipeline to construct a persistent stratified homotopy type from a non-stratified sample without prior knowledge of the location of the singularity. Our main result is a sampling theorem which guarantees that for a class of Whitney stratified spaces with two strata, our method produces arbitrarily close approximations of the persistent stratified homotopy type of the original space for sufficiently good samples.Tim Mäder, Lukas Waaswork_mmvmewcchzgjbdfow63ou5tvr4Tue, 11 Oct 2022 00:00:00 GMTCharacter varieties of a transitioning Coxeter 4-orbifold
https://scholar.archive.org/work/7f7ca7k2gvh4nj7y34udwxqmue
In 2010, Kerckhoff and Storm discovered a path of hyperbolic 4-polytopes eventually collapsing to an ideal right-angled cuboctahedron. This is expressed by a deformation of the inclusion of a discrete reflection group (a right-angled Coxeter group) in the isometry group of hyperbolic 4-space. More recently, we have shown that the path of polytopes can be extended to Anti-de Sitter geometry so as to have geometric transition on a naturally associated 4-orbifold, via a transitional half-pipe structure. In this paper, we study the hyperbolic, Anti-de Sitter, and half-pipe character varieties of Kerckhoff and Storm's right-angled Coxeter group near each of the found holonomy representations, including a description of the singularity that appears at the collapse. An essential tool is the study of some rigidity properties of right-angled cusp groups in dimension four.Stefano Riolo, Andrea Seppiwork_7f7ca7k2gvh4nj7y34udwxqmueTue, 11 Oct 2022 00:00:00 GMTA flag version of Beilinson-Drinfeld Grassmannian for surfaces
https://scholar.archive.org/work/lspu4yq7mndj5iyrttvdnslknq
In this paper we define and study a generalization of the Belinson-Drinfeld Grassmannian to the case where the curve is replaced by a smooth projective surface X, and the trivialization data are given with respect to a flag of closed subschemes. In order to do this, we first establish some general formal gluing results for moduli of almost perfect complexes, perfect complexes and torsors. We then construct a simplicial object of flags of closed subschemes of a smooth projective surface X, naturally associated to the operation of taking union of flags. We prove that this simplicial object has the 2-Segal property. For an affine complex algebraic group G, we finally define a flag analog 𝒢r_X of the Beilinson-Drinfeld Grassmannian of G-bundles on the surface X, and show that most of the properties of the Beilinson-Drinfeld Grassmannian for curves can be extended to our flag generalization. In particular, we prove a factorization formula, the existence of a canonical flat connection, we construct actions of the loop group and of the positive loop group on 𝒢r_X, and define a fusion product on sheaves on 𝒢r_X.Benjamin Hennion, Valerio Melani, Gabriele Vezzosiwork_lspu4yq7mndj5iyrttvdnslknqMon, 10 Oct 2022 00:00:00 GMT