IA Scholar Query: Dualization in lattices given by implicational bases.
https://scholar.archive.org/
Internet Archive Scholar query results feedeninfo@archive.orgTue, 06 Dec 2022 00:00:00 GMTfatcat-scholarhttps://scholar.archive.org/help1440Hilbert-Poincaré series of matroid Chow rings and intersection cohomology
https://scholar.archive.org/work/lawngmuvovftvllwafkbtlt3ei
We study the Hilbert series of four objects arising in the Chow-theoretic and Kazhdan-Lusztig framework of matroids. These are, respectively, the Hilbert series of the Chow ring, the augmented Chow ring, the intersection cohomology module, and its stalk at the empty flat. We develop a parallelism between the Kazhdan-Lusztig polynomial of a matroid and the Hilbert series of its Chow ring. This extends to a parallelism between the Z-polynomial of a matroid and the Hilbert series of its augmented Chow ring. This suggests to bring ideas from one framework to the other. Our two main motivations are the real-rootedness conjecture for all of these polynomials, and the problem of computing them. We provide several intrinsic definitions of these invariants; also, by leveraging that they are valuations under matroid polytope subdivisions, we deduce a fast way for computing them for a large class of matroids. Uniform matroids are a case of combinatorial interest; we link the resulting polynomials with certain real-rooted families such as the (binomial) Eulerian polynomials, and we settle a conjecture of Hameister, Rao, and Simpson. Furthermore, we prove the real-rootedness of the Hilbert series of the augmented Chow rings of uniform matroids via a result of Haglund and Zhang; and in addition, we prove a version of a conjecture of Gedeon in the Chow setting: uniform matroids maximize coefficient-wisely these polynomials for matroids with fixed rank and size. By relying on the nonnegativity of the Kazhdan-Lusztig polynomials and the semi-small decompositions of Braden, Huh, Matherne, Proudfoot, and Wang, we strengthen the unimodality of the Hilbert series of Chow rings, augmented Chow rings, and intersection cohomologies to γ-positivity, a property for palindromic polynomials that lies between unimodality and real-rootedness; this settles a conjecture of Ferroni, Nasr, and Vecchi.Luis Ferroni, Jacob P. Matherne, Matthew Stevens, Lorenzo Vecchiwork_lawngmuvovftvllwafkbtlt3eiTue, 06 Dec 2022 00:00:00 GMTGlobally +-regular varieties and the minimal model program for threefolds in mixed characteristic
https://scholar.archive.org/work/jg5t27fbgbdibllpwknnnrzwfq
We establish the Minimal Model Program for arithmetic threefolds whose residue characteristics are greater than five. In doing this, we generalize the theory of global F-regularity to mixed characteristic and identify certain stable sections of adjoint line bundles. Finally, by passing to graded rings, we generalize a special case of Fujita's conjecture to mixed characteristic.Bhargav Bhatt, Linquan Ma, Zsolt Patakfalvi, Karl Schwede, Kevin Tucker, Joe Waldron, Jakub Witaszekwork_jg5t27fbgbdibllpwknnnrzwfqMon, 05 Dec 2022 00:00:00 GMTEtale and crystalline companions, I
https://scholar.archive.org/work/xgdjxu3uk5eothxlyuaedf634m
Let $X$ be a smooth scheme over a finite field of characteristic $p$. Consider the coefficient objects of locally constant rank on $X$ in $\ell$-adic Weil cohomology: these are lisse Weil sheaves in \'etale cohomology when $\ell \neq p$, and overconvergent $F$-isocrystals in rigid cohomology when $\ell=p$. Using the Langlands correspondence for global function fields in both the \'etale and crystalline settings (work of Lafforgue and Abe, respectively), one sees that on a curve, any coefficient object in one category has "companions" in the other categories with matching characteristic polynomials of Frobenius at closed points. A similar statement is expected for general $X$; building on work of Deligne, Drinfeld showed that any \'etale coefficient object has \'etale companions. We adapt Drinfeld's method to show that any crystalline coefficient object has \'etale companions; this has been shown independently by Abe--Esnault. We also prove some auxiliary results relevant for the construction of crystalline companions of \'etale coefficient objects; this subject will be pursued in a subsequent paper.Kiran S. Kedlayawork_xgdjxu3uk5eothxlyuaedf634mFri, 02 Dec 2022 00:00:00 GMTGeometric Eisenstein Series, Intertwining Operators, and Shin's Averaging Formula
https://scholar.archive.org/work/pw6luff4pzgrvbmzchn3s7plle
In the geometric Langlands program over function fields, Braverman-Gaitsgory and Laumon constructed geometric Eisenstein functors which geometrize the classical construction of Eisenstein series. Fargues and Scholze very recently constructed a general candidate for the local Langlands correspondence, via a geometric Langlands correspondence occurring over the Fargues-Fontaine curve. We carry some of the theory of geometric Eisenstein series over to the Fargues-Fontaine setting. Namely, given a quasi-split connected reductive group G/ℚ_p with simply connected derived group and maximal torus T, we construct an Eisenstein functor nEis(-), which takes sheaves on Bun_T to sheaves on Bun_G. We show that, given a sufficiently nice L-parameter ϕ_T: W_ℚ_p→^LT, there is a Hecke eigensheaf on Bun_G with eigenvalue ϕ, given by applying nEis(-) to the Hecke eigensheaf 𝒮_ϕ_T on Bun_T attached to ϕ_T by Fargues and Zou. We show that nEis(𝒮_ϕ_T) interacts well with Verdier duality, and, assuming compatibility of the Fargues-Scholze correspondence with a suitably nice form of the local Langlands correspondence, provide an explicit formula for the stalks of the eigensheaf in terms of parabolic inductions of the character χ attached to ϕ_T. This has several surprising consequences. First, it recovers special cases of an averaging formula of Shin for the cohomology of local Shimura varieties with rational coefficients, and generalizes it to the non-minuscule case. Second, it refines the averaging formula in the cases where the parameter ϕ is sufficiently nice, giving an explicit formula for the degrees of cohomology that certain parabolic inductions sit in, and this refined formula holds even with torsion coefficients.Linus Hamannwork_pw6luff4pzgrvbmzchn3s7plleFri, 02 Dec 2022 00:00:00 GMTCompatibility of the Fargues-Scholze and Gan-Takeda Local Langlands
https://scholar.archive.org/work/4uvzyeix3ffwbb7thftqmu4rwu
Given a prime p, a finite extension L/ℚ_p, a connected p-adic reductive group G/L, and a smooth irreducible representation π of G(L), Fargues-Scholze recently attached a semisimple Weil parameter to such π, giving a general candidate for the local Langlands correspondence. It is natural to ask whether this construction is compatible with known instances of the correspondence after semisimplification. For G = GL_n and its inner forms, Fargues-Scholze and Hansen-Kaletha-Weinstein showed that the correspondence is compatible with the correspondence of Harris-Taylor/Henniart. We verify a similar compatibility for G = GSp_4 and its unique non-split inner form G = GU_2(D), where D is the quaternion division algebra over L, assuming that L/ℚ_p is unramified and p > 2. In this case, the local Langlands correspondence has been constructed by Gan-Takeda and Gan-Tantono. Analogous to the case of GL_n and its inner forms, this compatibility is proven by describing the Weil group action on the cohomology of a local Shimura variety associated to GSp_4, using basic uniformization of abelian type Shimura varieties due to Shen, combined with various global results of Kret-Shin and Sorensen on Galois representations in the cohomology of global Shimura varieties associated to inner forms of GSp_4 over a totally real field. After showing the parameters are the same, we apply some ideas from the geometry of the Fargues-Scholze construction explored recently by Hansen, to give a more precise description of the cohomology of this local Shimura variety, verifying a strong form of the Kottwitz conjecture in the process.Linus Hamannwork_4uvzyeix3ffwbb7thftqmu4rwuFri, 02 Dec 2022 00:00:00 GMTEtale and crystalline companions, I
https://scholar.archive.org/work/fz3gcgj3vrcfrpu7omvqdpp734
Let X be a smooth scheme over a finite field of characteristic p. Consider the coefficient objects of locally constant rank on X in ℓ-adic Weil cohomology: these are lisse Weil sheaves in étale cohomology when ℓ≠ p, and overconvergent F-isocrystals in rigid cohomology when ℓ=p. Using the Langlands correspondence for global function fields in both the étale and crystalline settings (work of Lafforgue and Abe, respectively), one sees that on a curve, any coefficient object in one category has "companions" in the other categories with matching characteristic polynomials of Frobenius at closed points. A similar statement is expected for general X; building on work of Deligne, Drinfeld showed that any étale coefficient object has étale companions. We adapt Drinfeld's method to show that any crystalline coefficient object has étale companions; this has been shown independently by Abe–Esnault. We also prove some auxiliary results relevant for the construction of crystalline companions of étale coefficient objects; this subject will be pursued in a subsequent paper.Kiran S. Kedlayawork_fz3gcgj3vrcfrpu7omvqdpp734Thu, 01 Dec 2022 00:00:00 GMTExtrapolation in general quasi-Banach function spaces
https://scholar.archive.org/work/qnr5untepvbb5kc3ejemswxlqy
In this paper we prove off-diagonal, limited range, multilinear, vector-valued, and two-weight versions of the Rubio de Francia extrapolation theorem in general quasi-Banach function spaces. We prove mapping properties of the generalization of the Hardy-Littlewood maximal operator to very general bases that includes a method to obtain self-improvement results that are sharp with respect to its operator norm. Furthermore, we prove bounds for the Hardy-Littlewood maximal operator in weighted Lorentz, variable Lebesgue, and Morrey spaces, and recover and extend several extrapolation theorems in the literature. Finally, we provide an application of our results to the Riesz potential and the Bilinear Hilbert transform.Zoe Nieraethwork_qnr5untepvbb5kc3ejemswxlqyTue, 29 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 GMTBi-intermediate logics of trees and co-trees
https://scholar.archive.org/work/nf2zi4giizbnfg5wlmv5sp54tm
A bi-Heyting algebra validates the Gödel-Dummett axiom (p→ q)∨ (q→ p) iff the poset of its prime filters is a disjoint union of co-trees (i.e., order duals of trees). Bi-Heyting algebras of this kind are called bi-Gödel algebras and form a variety that algebraizes the extension 𝖻𝗂-𝖫𝖢 of bi-intuitionistic logic axiomatized by the Gödel-Dummett axiom. In this paper we initiate the study of the lattice Λ(𝖻𝗂-𝖫𝖢) of extensions of 𝖻𝗂-𝖫𝖢. We develop the methods of Jankov-style formulas for bi-Gödel algebras and use them to prove that there are exactly continuum many extensions of 𝖻𝗂-𝖫𝖢. We also show that all these extensions can be uniformly axiomatized by canonical formulas. Our main result is a characterization of the locally tabular extensions of 𝖻𝗂-𝖫𝖢. We introduce a sequence of co-trees, called the finite combs, and show that a logic in 𝖻𝗂-𝖫𝖢 is locally tabular iff it contains at least one of the Jankov formulas associated with the finite combs. It follows that there exists the greatest non-locally tabular extension of 𝖻𝗂-𝖫𝖢 and consequently, a unique pre-locally tabular extension of 𝖻𝗂-𝖫𝖢. These results contrast with the case of the intermediate logic axiomatized by the Gödel-Dummett axiom, which is known to have only countably many extensions, all of which are locally tabular.N. Bezhanishvili, M. Martins, T. Moraschiniwork_nf2zi4giizbnfg5wlmv5sp54tmSun, 27 Nov 2022 00:00:00 GMTAlgebraic dependence and Milnor K-theory
https://scholar.archive.org/work/bihjss7vj5dbvpsrbigg7mdv3u
This paper shows that algebraic (in)dependence is encoded in Milnor K-theory of fields. As an application, we show that the isomorphism type of a field is determined by its Milnor K-theory, up to purely inseparable extensions, in most situations.Adam Topazwork_bihjss7vj5dbvpsrbigg7mdv3uSat, 26 Nov 2022 00:00:00 GMTChain-order polytopes: toric degenerations, Young tableaux and monomial bases
https://scholar.archive.org/work/ws3gjhtku5fgziqkkhkntsgm2i
Our first result realizes the toric variety of every marked chain-order polytope (MCOP) of the Gelfand--Tsetlin poset as an explicit Gr\"obner (sagbi) degeneration of the flag variety. This generalizes the Sturmfels/Gonciulea--Lakshmibai/Kogan--Miller construction for the Gelfand--Tsetlin degeneration to the MCOP setting. The key idea of our approach is to use pipe dreams to define realizations of toric varieties in Pl\"ucker coordinates. We then use this approach to generalize two more well-known constructions to arbitrary MCOPs: standard monomial theories such as those given by semistandard Young tableaux and PBW-monomial bases in irreducible representations such as the FFLV bases. In an addendum we introduce the notion of semi-infinite pipe dreams and use it to obtain an infinite family of poset polytopes each providing a toric degeneration of the semi-infinite Grassmannian.Igor Makhlinwork_ws3gjhtku5fgziqkkhkntsgm2iThu, 24 Nov 2022 00:00:00 GMTOn M-Theory Dual of Large-N Thermal QCD-Like Theories up to O(R^4) and G-Structure Classification of Underlying Non-Supersymmetric Geometries
https://scholar.archive.org/work/ritq2a2l65hgnfldwrztqpdz2u
Construction of a top-down holographic dual of thermal QCD-like theories (equivalence class of theories which are UV-conformal, IR-confining and have fundamental quarks) at intermediate 't Hooft coupling and the G-structure (torsion classes) classification of the underlying geometries (in the Infra Red (IR)/non-conformal sector in particular) of the non-supersymmetric string/M-theory duals, have been missing in the literature. We take the first important steps in this direction by studying the M theory dual of large-N thermal QCD-like theories at intermediate gauge and 't Hooft couplings and obtaining the O(l_p^6) corrections arising from the O(R^4) terms to the "MQGP" background (M-theory dual of large-N thermal QCD-like theories at intermediate gauge/string coupling, but large 't Hooft coupling) of . The main Physics lesson learnt is that there is a competition between non-conformal IR enhancement and Planckian and large-N suppression and going to orders beyond the O(l_p^6) is necessitated if the IR enhancement wins out. The main lesson learnt in Math is in the context of the differential geometry (G-structure classification) of the internal manifolds relevant to the string/M-theory duals of large-N thermal QCD-like theories, wherein we obtain for the first time inclusive of the O(R^4) corrections in the Infra-Red (IR), the SU(3)-structure torsion classes of the type IIA mirror of (making contact en route with Siegel theta functions related to appropriate hyperelliptic curves, as well as the Kiepert's algorithm of solving quintics), and the G_2/SU(4)/Spin(7)-structure torsion classes of the seven- and eight-folds associated with its M theory uplift.Vikas Yadav, Aalok Misrawork_ritq2a2l65hgnfldwrztqpdz2uWed, 23 Nov 2022 00:00:00 GMTSheaves and symplectic geometry of cotangent bundles
https://scholar.archive.org/work/txrrdh74kbdhlg5b2xwp324m7i
This paper is essentially made of the three preprints arXiv:1212.5818, arXiv:1311.0187, arXiv:1603.07876 gathered in a single text, with simplified proofs. We recall several results of the microlocal theory of sheaves of Kashiwara-Schapira and apply them to study the symplectic geometry of cotangent bundles. We explain how we can recover the Gromov nonsqueezing theorem, the Gromov-Eliashberg rigidity theorem, the existence of graph selectors, we prove a three cusps conjecture about curves on the sphere and we recover more recent results on the topology of exact Lagrangian submanifolds of cotangent bundles.Stéphane Guillermouwork_txrrdh74kbdhlg5b2xwp324m7iTue, 22 Nov 2022 00:00:00 GMTMean string field theory: Landau-Ginzburg theory for 1-form symmetries
https://scholar.archive.org/work/ug4zuimjvrgrdcr4pfqz22c7cm
By analogy with the Landau-Ginzburg theory of ordinary zero-form symmetries, we introduce and develop a Landau-Ginzburg theory of one-form global symmetries, which we call mean string field theory. The basic dynamical variable is a string field – defined on the space of closed loops – that can be used to describe the creation, annihilation, and condensation of effective strings. Like its zero-form cousin, the mean string field theory provides a useful picture of the phase diagram of broken and unbroken phases. We provide a transparent derivation of the area law for charged line operators in the unbroken phase and describe the dynamics of gapless Goldstone modes in the broken phase. The framework also provides a theory of topological defects of the broken phase and a description of the phase transition that should be valid above an upper critical dimension, which we discuss. We also discuss general consequences of emergent one-form symmetries at zero and finite temperature.Nabil Iqbal, John McGreevywork_ug4zuimjvrgrdcr4pfqz22c7cmTue, 22 Nov 2022 00:00:00 GMTOn Bruhat-Tits theory over a higher dimensional base
https://scholar.archive.org/work/dgp6gnmcprb3vdaguey3zel4iy
Let 𝒪__n := k z__1, ..., z__n over an algebraically closed residue field k of characteristic zero. Set K__n := Fract __n. Let G be an almost-simple, simply-connected affine algebraic group over k with a maximal torus T and a Borel subgroup B. Given a n-tuple f = (f__1, ..., f__n) of concave functions on the root system of G as in Bruhat-Tits , , we define n-bounded subgroups P__ f(k)⊂ G(K__n) as a direct generalization of Bruhat-Tits groups for the case n=1. We show that these groups are schematic, i.e. they are valued points of smooth quasi-affine group schemes with connected fibres and adapted to the divisor with normal crossing z_1 ⋯ z_n =0 in the sense that the restriction to the generic point of the divisor z_i=0 is given by f_i. This provides a higher-dimensional analogue of the Bruhat-Tits group schemes with natural specialization properties. In 10, under suitable tameness assumptions, we extend all these results for a n+1-tuple f = (f__0, ..., f__n) of concave functions on the root system of G replacing 𝒪__n by x__1,⋯,x__n where is a complete discrete valuation ring with residue field of characteristic p. In particular, if x_0 is the uniformizer of 𝒪, then the group scheme is adapted to the divisor x_0 ⋯ x_n=0. In the last part of the paper we give applications to constructing certain natural group schemes on wonderful embeddings of groups and also certain families of 2-parahoric group schemes on minimal resolutions of surface singularities.Vikraman Balaji, Yashonidhi Pandeywork_dgp6gnmcprb3vdaguey3zel4iyMon, 21 Nov 2022 00:00:00 GMTHigher-group symmetry in finite gauge theory and stabilizer codes
https://scholar.archive.org/work/hi2arvakvfanfabmuqbgd72t6a
A large class of gapped phases of matter can be described by topological finite group gauge theories. In this paper, we derive the d-group global symmetry and its 't Hooft anomaly for topological finite group gauge theories in (d+1) space-time dimensions, including non-Abelian gauge groups and Dijkgraaf-Witten twists. We focus on the 1-form symmetry generated by invertible (Abelian) magnetic defects and the higher-form symmetries generated by invertible topological defects decorated with lower dimensional gauged symmetry-protected topological (SPT) phases. We show that due to a generalization of the Witten effect and charge-flux attachment, the 1-form symmetry generated by the magnetic defects mixes with other symmetries into a higher group. We describe such higher-group symmetry in various lattice model examples. We discuss several applications, including the classification of fermionic SPT phases in (3+1)D for general fermionic symmetry groups, where we also derive a simpler formula for the [O_5] ∈ H^5(BG, U(1)) obstruction than has appeared in previous work. We also show how the d-group symmetry is related to fault-tolerant non-Pauli logical gates and a refined Clifford hierarchy in stabilizer codes. We construct new logical gates in stabilizer codes using the d-group symmetry, such as the control-Z gate in (3+1)D ℤ_2 toric code.Maissam Barkeshli, Yu-An Chen, Po-Shen Hsin, Ryohei Kobayashiwork_hi2arvakvfanfabmuqbgd72t6aMon, 21 Nov 2022 00:00:00 GMTOn complete classes of valuated matroids
https://scholar.archive.org/work/zdyzn5oq2rejjfppqwoqurxhny
We characterize a rich class of valuated matroids, called R-minor valuated matroids that includes the indicator functions of matroids, and is closed under operations such as taking minors, duality, and induction by network. We refute the refinement of a 2003 conjecture by Frank, exhibiting valuated matroids that are not R-minor. The family of counterexamples is based on sparse paving matroids. Valuated matroids are inherently related to gross substitute valuations in mathematical economics. By the same token we refute the Matroid Based Valuation Conjecture by Ostrovsky and Paes Leme (Theoretical Economics 2015) asserting that every gross substitute valuation arises from weighted matroid rank functions by repeated applications of merge and endowment operations. Our result also has implications in the context of Lorentzian polynomials: it reveals the limitations of known construction operations.Edin Husić, Georg Loho, Ben Smith, László A. Véghwork_zdyzn5oq2rejjfppqwoqurxhnyMon, 21 Nov 2022 00:00:00 GMTTopological Holography: Towards a Unification of Landau and Beyond-Landau Physics
https://scholar.archive.org/work/p6ncec5ppjao5eqex2pjxnxkzy
We outline a holographic framework that attempts to unify Landau and beyond-Landau paradigms of quantum phases and phase transitions. Leveraging a modern understanding of symmetries as topological defects/operators, the framework uses a topological order to organize the space of quantum systems with a global symmetry in one lower dimension. The global symmetry naturally serves as an input for the topological order. In particular, we holographically construct a String Operator Algebra (SOA) which is the building block of symmetric quantum systems with a given symmetry G in one lower dimension. This exposes a vast web of dualities which act on the space of G-symmetric quantum systems. The SOA facilitates the classification of gapped phases as well as their corresponding order parameters and fundamental excitations, while dualities help to navigate and predict various corners of phase diagrams and analytically compute universality classes of phase transitions. A novelty of the approach is that it treats conventional Landau and unconventional topological phase transitions on an equal footing, thereby providing a holographic unification of these seemingly-disparate domains of understanding. We uncover a new feature of gapped phases and their multi-critical points, which we dub fusion structure, that encodes information about which phases and transitions can be dual to each other. Furthermore, we discover that self-dual systems typically posses emergent non-invertible, i.e., beyond group-like symmetries. We apply these ideas to 1+1d quantum spin chains with finite Abelian group symmetry, using topologically-ordered systems in 2+1d. We predict the phase diagrams of various concrete spin models, and analytically compute the full conformal spectra of non-trivial quantum phase transitions, which we then verify numerically.Heidar Moradi, Seyed Faroogh Moosavian, Apoorv Tiwariwork_p6ncec5ppjao5eqex2pjxnxkzyMon, 21 Nov 2022 00:00:00 GMTLarge U(1) charges from flux breaking in 4D F-theory models
https://scholar.archive.org/work/jdcqh5j2xffozofemms6iyzip4
We study the massless charged spectrum of U(1) gauge fields in F-theory that arise from flux breaking of a nonabelian group. The U(1) charges that arise in this way can be very large. In particular, using vertical flux breaking, we construct an explicit 4D F-theory model with a U(1) decoupled from other gauge sectors, in which the massless/light fields have charges as large as 657. This result greatly exceeds prior results in the literature. We argue heuristically that this result may provide an upper bound on charges for light fields under decoupled U(1) factors in the F-theory landscape. We also show that the charges can be even larger when the U(1) is coupled to other gauge groups.Shing Yan Li, Washington Taylorwork_jdcqh5j2xffozofemms6iyzip4Mon, 21 Nov 2022 00:00:00 GMTMetrizable uniform spaces
https://scholar.archive.org/work/h5vzwcuaurfrrjfree4vmotpky
Three themes of general topology: quotient spaces; absolute retracts; and inverse limits - are reapproached here in the setting of metrizable uniform spaces, with an eye to applications in geometric and algebraic topology. The results include: 1) If f: A -> Y is a uniformly continuous map, where X and Y are metric spaces and A is a closed subset of X, we show that the adjunction space X\cup_f Y with the quotient uniformity (hence also with the topology thereof) is metrizable, by an explicit metric. This yields natural constructions of cone, join and mapping cylinder in the category of metrizable uniform spaces, which we show to coincide with those based on subspace (of a normed linear space); on product (with a cone); and on the isotropy of the l_2 metric. 2) We revisit Isbell's theory of uniform ANRs, as refined by Garg and Nhu in the metrizable case. The iterated loop spaces \Omega^n P of a pointed compact polyhedron P are shown to be uniform ANRs. Four characterizations of uniform ANRs among metrizable uniform spaces X are given: (i) the completion of X is a uniform ANR, and the remainder is uniformly a Z-set in the completion; (ii) X is uniformly locally contractible and satisfies the Hahn approximation property; (iii) X is uniformly \epsilon-homotopy dominated by a uniform ANR for each \epsilon>0; (iv) X is an inverse limit of uniform ANRs with "nearly splitting" bonding maps. Several chapters are devoted primarily to exposition: (I) an introduction to uniform spaces, with a focus on the metrizable case; (V) the space of measurable functions; (VI) the space of probability measures and other measure spaces.Sergey A. Melikhovwork_h5vzwcuaurfrrjfree4vmotpkyThu, 17 Nov 2022 00:00:00 GMT