We present an abstract version of Goerss-Hopkins theory in the setting of a prestable @math -category equipped with a suitable periodicity operator. In the case of the @math -category of synthetic spectra, this yields obstructions to realizing a comodule algebra as a homology of a commutative ring spectrum, recovering the results of Goerss and Hopkins.
To give a different example, Goerss-Hopkins towers were also used by the first author to prove that at large primes, there is an equivalence @math of homotopy categories between @math -local spectra and differential @math -comodules @cite_0 . The proof of that result proceeds by comparing topological and algebraic towers and so crucially depends on the additional generality provided in this note.
@cite_0
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We show that if @math is a @math -local Landweber exact homology theory of height @math and @math , then there exists an equivalence @math between homotopy categories of @math -local spectra and differential @math -comodules, generalizing Bousfield's and Franke's results to heights @math . Goerss--Hopkins obstruction theory is a powerful tool for constructing structured ring spectra from purely algebraic data. Using the formalism of model @math -categories, we provide a generalization that applies in an arbitrary presentably symmetric monoidal stable @math -category (such as that of equivariant spectra or of motivic spectra). We prove a homological mirror symmetry equivalence between an @math -brane category for the pair of pants, computed as a wrapped microlocal sheaf category, and a @math -brane category for a mirror LG model, understood as a category of matrix factorizations. The equivalence improves upon prior results in two ways: it intertwines evident affine Weyl group symmetries on both sides, and it exhibits the relation of wrapped microlocal sheaves along different types of Lagrangian skeleta for the same hypersurface. The equivalence proceeds through the construction of a combinatorial realization of the @math -model via arboreal singularities. The constructions here represent the start of a program to generalize to higher dimensions many of the structures which have appeared in topological approaches to Fukaya categories of surfaces. Author(s): Mazel-Gee, Aaron | Advisor(s): Teichner, Peter | Abstract: We develop a theory of model ∞-categories -- that is, of model structures on ∞-categories -- which provides a robust theory of resolutions entirely native to the ∞-categorical context. Using model ∞-categories, we generalize Goerss--Hopkins obstruction theory from spectra to an arbitrary (presentably symmetric monoidal stable) ∞-category. We give a sample application of this generalized obstruction theory in the setting of motivic homotopy theory, where we construct E_∞ structures on the motivic Morava E-theories and compute their automorphism spaces (as E_∞ algebras). We prove a comparison formula for the Donaldson-Thomas curve-counting invariants of two smooth and projective Calabi-Yau threefolds related by a flop. By results of Bridgeland any two such varieties are derived equivalent. Furthermore there exist pairs of categories of perverse coherent sheaves on both sides which are swapped by this equivalence. Using the theory developed by Joyce we construct the motivic Hall algebras of these categories. These algebras provide a bridge relating the invariants on both sides of the flop. The goal of this thesis is to prove that π4(S3) ≃ Z 2Z in homotopy type theory. In particular it is a constructive and purely homotopy-theoretic proof. We first recall the basic concepts of homotopy type theory, and we prove some well-known results about the homotopy groups of spheres: the computation of the homotopy groups of the circle, the triviality of those of the form πk(Sn) with k < n, and the construction of the Hopf fibration. We then move to more advanced tools. In particular, we define the James construction which allows us to prove the Freudenthal suspension theorem and the fact that there exists a natural number n such that π4(S3) ≃ Z nZ. Then we study the smash product of spheres, we construct the cohomology ring of a space, and we introduce the Hopf invariant, allowing us to narrow down the n to either 1 or 2. The Hopf invariant also allows us to prove that all the groups of the form π4n−1(S2n) are infinite. Finally we construct the Gysin exact sequence, allowing us to compute the cohomology of CP2 and to prove that π4(S3) ≃ Z 2Z and that more generally πn+1(Sn) ≃ Z 2Z for every n ≥ 3 This paper tackles the problem of computing topological invariants of geometric objects in a robust manner, using only point cloud data sampled from the object. It is now widely recognised that this kind of topological analysis can give qualitative information about data sets which is not readily available by other means. In particular, it can be an aid to visualisation of high dimensional data. Standard simplicial complexes for approximating the topological type of the underlying space (such as Cech, Rips, or a-shape) produce simplicial complexes whose vertex set has the same size as the underlying set of point cloud data. Such constructions are sometimes still tractable, but are wasteful (of computing resources) since the homotopy types of the underlying objects are generally realisable on much smaller vertex sets. We obtain smaller complexes by choosing a set of 'landmark' points from our data set, and then constructing a "witness complex" on this set using ideas motivated by the usual Delaunay complex in Euclidean space. The key idea is that the remaining (non-landmark) data points are used as witnesses to the existence of edges or simplices spanned by combinations of landmark points. Our construction generalises the topology-preserving graphs of Martinetz and Schulten [MS94] in two directions. First, it produces a simplicial complex rather than a graph. Secondly it actually produces a nested family of simplicial complexes, which represent the data at different feature scales, suitable for calculating persistent homology [ELZ00, ZC04]. We find that in addition to the complexes being smaller, they also provide (in a precise sense) a better picture of the homology, with less noise, than the full scale constructions using all the data points. We illustrate the use of these complexes in qualitatively analyzing a data set of 3 × 3 pixel patches studied by David [LPM03]. The spectrum of a first-order sentence is the set of cardinalities of its finite models. We refine the well-known equality between the class of spectra and the class of sets (of positive integers) accepted by nondeterministic Turing machines in polynomial time. Let @math denote the class of spectra of sentences with d universal quantifiers. For any integer @math and each set of positive integers, A, we obtain: [ A NTIME (n^d ) A Sp (d ) A NTIME (n^d ( n)^2 ). ] Further the first implication holds even if we use multidimensional nondeterministic Turing machines. These results hold similarly for generalized spectra. As a consequence, we obtain a simplified proof of a hierarchy result of P. Pudlak about (generalized) spectra. We also prove that the set of primes is the spectrum of a certain sentence with only one variable. The goal of this thesis is to bring together two different theories about critical points of a scalar function and their relation to topology: Discrete Morse theory and Persistent homology. While the goals and fundamental techniques are different, there are certain themes appearing in both theories that closely resemble each other. In certain cases, the two threads can be joined, leading to new insights beyond the classical realm of one particular theory.Discrete Morse theory provides combinatorial equivalents of several core concepts of classical Morse theory, such as discrete Morse functions, discrete gradient vector fields, critical points, and a cancelation theorem for the elimination of critical points of a vector field. Because of its simplicity, it not only maintains the intuition of the classical theory but allows to surpass it in a certain sense by providing explicit and canonical constructions that would become quite complicated in the smooth setting.Persistent homology quantifies topological features of a function. It defines the birth and death of homology classes at critical points, identifies pairs of these (persistence pairs), and provides a quantitative notion of their stability (persistence).Whereas (discrete) Morse theory makes statements about the homotopy type of the sublevel sets of a function, persistence is concerned with their homology. While homology is an invariant of homotopy equivalences, the converse is not true: not every map inducing an isomorphism in homology is a homotopy equivalence. In this thesis we establish a connection between both theories and use this combination to solve problems that are not easily accessibly by any single theory alone. In particular, we solve the problem of minimizing the number of critical points of a function on a surface within a certain tolerance from a given input function. In this paper we consider the localization of a collection of small, three-dimensional bounded homogeneous inclusions via time-harmonic electromagnetic means, typically using arrays of electric or magnetic dipole transmitters and receivers with given polarization(s) at some distance from the collection, possibly also lying in the far field. The inclusions, somewhat apart or closely spaced, are buried within a homogeneous medium, and are of arbitrary contrast of permittivity, conductivity, and permeability vis-ag-vis this embedding medium. The problem is formulated as an inverse scattering problem for the full Maxwell equations and it involves a robust asymptotic modeling of the multistatic response matrix. No specific application is studied at this stage, but characterization of obstacles in subsoils, nondestructive evaluation of man-made structures, and medical imaging are primary fields of application envisaged. The proposed approach uses a MUSIC (multiple signal classification)-type algorithm, and it yields fast numbering, accurate localization, and estimates of the electromagnetic and geometric parameters (polarization tensors) of the inclusions. The mathematical machinery is detailed first, some specific attention being given to triaxial ellipsoidal inclusions and degenerate spherical shapes (for the latter known results are retrieved). Then, the viability of this algorithm—which would be easily extended to planarly layered environments by introduction of their Green’s functions—is documented by a variety of numerical results from synthetic, noiseless, and severely noisy field data. Given a knot K in the 3-sphere, let QK be its fundamental quan- dle as introduced by D.Joyce. Its first homology group is easily seen to be H1(QK) = Z. We prove that H2(QK) = 0 if and only if K is trivial, and H2(QK) = Z whenever K is non-trivial. An analogous result holds for links, thus characterizing trivial components. More detailed information can be derived from the conjugation quandle: let Q K be the conjugacy class of a meridian in the knot group 1(S 3 rK). We show that H2(Q K) = Z p , where p is the number of prime summands in a connected sum decomposition of K. Let @math be a smooth scheme over an algebraically closed field @math of characteristic zero and @math a regular function, and write @math Crit @math , as a closed subscheme of @math . The motivic vanishing cycle @math is an element of the @math -equivariant motivic Grothendieck ring @math defined by Denef and Loeser math.AG 0006050 and Looijenga math.AG 0006220, and used in Kontsevich and Soibelman's theory of motivic Donaldson-Thomas invariants, arXiv:0811.2435. We prove three main results: (a) @math depends only on the third-order thickenings @math of @math . (b) If @math is another smooth scheme, @math is regular, @math Crit @math , and @math is an embedding with @math and @math an isomorphism, then @math equals @math "twisted" by a motive associated to a principal @math -bundle defined using @math , where now we work in a quotient ring @math of @math . (c) If @math is an "oriented algebraic d-critical locus" in the sense of Joyce arXiv:1304.4508, there is a natural motive @math , such that if @math is locally modelled on Crit @math , then @math is locally modelled on @math . Using results from arXiv:1305.6302, these imply the existence of natural motives on moduli schemes of coherent sheaves on a Calabi-Yau 3-fold equipped with "orientation data", as required in Kontsevich and Soibelman's motivic Donaldson-Thomas theory arXiv:0811.2435, and on intersections of oriented Lagrangians in an algebraic symplectic manifold. This paper is an analogue for motives of results on perverse sheaves of vanishing cycles proved in arXiv:1211.3259. We extend this paper to Artin stacks in arXiv:1312.0090. We study the expected topological properties of Cech and Vietoris–Rips complexes built on random points in ℝd . We find higher-dimensional analogues of known results for connectivity and component counts for random geometric graphs. However, higher homology H k is not monotone when k>0. In particular, for every k>0, we exhibit two thresholds, one where homology passes from vanishing to nonvanishing, and another where it passes back to vanishing. We give asymptotic formulas for the expectation of the Betti numbers in the sparser regimes, and bounds in the denser regimes. We prove the motivic version of Joyce-Song formula for the Behrend function identities proposed in Jiang2 . The main method we use is Nicaise's motivic integration for formal schemes and Cluckers-Loeser's motivic constructible functions. As an application we prove that there is a Poisson algebra homomorphism from the motivic Hall algebra of the abelian category of coherent sheaves on a Calabi-Yau threefold @math to the motivic quantum torus of @math , thus generalizing the integration map of Joyce-Song in JS and Bridgeland in Bridgeland10 to the motivic level. Such an integration map has applications in the wall crossing of motivic Donaldson-Thomas invariants. We study quantitatively the effective large-scale behavior of discrete elliptic equations on the lattice ( Z^d ) with random coefficients. The theory of stochastic homogenization relates the random, stationary, and ergodic field of coefficients with a deterministic matrix of effective coefficients. This is done via the corrector problem, which can be viewed as a highly degenerate elliptic equation on the infinite-dimensional space of admissible coefficient fields. In this contribution we develop new quantitative methods for the corrector problem based on the assumption that ergodicity holds in the quantitative form of a Spectral Gap Estimate w.r.t. a Glauber dynamics on coefficient fields—as it is the case for independent and identically distributed coefficients. As a main result we prove an optimal decay in time of the semigroup associated with the corrector problem (i.e. of the generator of the process called “random environment as seen from the particle”). As a corollary we recover existence of stationary correctors (in dimensions (d>2 )) and prove new optimal estimates for regularized versions of the corrector (in dimensions (d 2 )). We also give a self-contained proof of a new estimate on the gradient of the parabolic, variable-coefficient Green’s function, which is a crucial analytic ingredient in our approach. As an application of these results, we prove the first (and optimal) estimates for the approximation of the homogenized coefficients by the popular periodization method in case of independent and identically distributed coefficients. This note is a sequel to our earlier paper of the same title [4] and describes invariants of rational homology 3-spheres associated to acyclic orthogonal local systems. Our work is in the spirit of the Axelrod–Singer papers [1], generalizes some of their results, and furnishes a new setting for the purely topological implications of their work. Comparing graphs to determine the level of underlying structural similarity between them is a widely encountered problem in computer science. It is particularly relevant to the study of Internet topologies, such as the generation of synthetic topologies to represent the Internet's AS topology. We derive a new metric that enables exactly such a structural comparison: the weighted spectral distribution. We then apply this metric to three aspects of the study of the Internet's AS topology. i) We use it to quantify the effect of changing the mixing properties of a simple synthetic network generator. ii) We use this quantitative understanding to examine the evolution of the Internet's AS topology over approximately seven years, finding that the distinction between the Internet core and periphery has blurred over time. iii) We use the metric to derive optimal parameterizations of several widely used AS topology generators with respect to a large-scale measurement of the real AS topology. Prologue Point-set topology, change functors, and proper actions: Introduction to Part I The point-set topology of parametrized spaces Change functors and compatibility relations Proper actions, equivariant bundles and fibrations Model categories and parametrized spaces: Introduction to Part II Topologically bicomplete model categories Well-grounded topological model categories The @math -model structure on @math Equivariant @math -type model structures Ex-fibrations and ex-quasifibrations The equivalence between Ho @math and @math Parametrized equivariant stable homotopy theory: Introduction to Part III Enriched categories and @math -categories The category of orthogonal @math -spectra over @math Model structures for parametrized @math -spectra Adjunctions and compatibility relations Module categories, change of universe, and change of groups Parametrized duality theory: Introduction to Part IV Fiberwise duality and transfer maps Closed symmetric bicategories The closed symmetric bicategory of parametrized spectra Costenoble-Waner duality Fiberwise Costenoble-Waner duality Homology and cohomology, Thom spectra, and addenda: Introduction to Part V Parametrized homology and cohomology theories Equivariant parametrized homology and cohomology Twisted theories and spectral sequences Parametrized FSP's and generalized Thom spectra Epilogue: Cellular philosophy and alternative approaches Bibliography Index Index of notation. Abstract The homotopy method is used to find all eigenpairs of symmetric matrices. A special homotopy is constructed for Jacobi matrices. It is shown that there are exactly n distinct smooth curves connecting trivial solutions to desired eigenpairs. These curves are solutions of a certain ordinary differential equation with different initial values. Hence, they can be followed numerically. Incorporated with sparse matrix techniques, this method might be used to solve eigenvalue problems for large scale matrices. Given a simplicial complex with weights on its simplices, and a nontrivial cycle on it, we are interested in finding the cycle with minimal weight which is homologous to the given one. Assuming that the homology is defined with integer ( @math ) coefficients, we show the following (Theorem 5.2): For a finite simplicial complex @math of dimension greater than @math , the boundary matrix @math is totally unimodular if and only if @math is torsion-free for all pure subcomplexes @math in @math of dimensions @math and @math , respectively, where @math . Because of the total unimodularity of the boundary matrix, we can solve the optimization problem, which is inherently an integer programming problem, as a linear program and obtain an integer solution. Thus, the problem of finding optimal cycles in a given homology class can be solved in polynomial time. This result is surprising in the backdrop of a recent result which says that the problem is NP-hard under @math coefficients which, being a field, is in general easier to deal with. Our result implies, among other things, that one can compute in polynomial time an optimal @math -cycle in a given homology class for any triangulation of an orientable compact @math -manifold or for any finite simplicial complex embedded in @math . Our optimization approach can also be used for various related problems, such as finding an optimal chain homologous to a given one when these are not cycles. Our result can also be viewed as providing a topological characterization of total unimodularity.