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Noether Lefschetz Problems For Degeneracy Loci
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Book Synopsis Noether-Lefschetz Problems for Degeneracy Loci by : Jeroen Spandaw
Download or read book Noether-Lefschetz Problems for Degeneracy Loci written by Jeroen Spandaw and published by American Mathematical Soc.. This book was released on 2003 with total page 136 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this monograph we study the cohomology of degeneracy loci of the following type. Let $X$ be a complex projective manifold of dimension $n$, let $E$ and $F$ be holomorphic vector bundles on $X$ of rank $e$ and $f$, respectively, and let $\psi\colon F\to E$ be a holomorphic homomorphism of vector bundles. Consider the degeneracy locus $Z:=D_r(\psi):=\{x\in X\colon \mathrm{rk} (\psi(x))\le r\}.$ We assume without loss of generality that $e\ge f >r\ge 0$. We assume furthermore that $E\otimes F^\vee$ is ample and globally generated, and that $\psi$ is a general homomorphism. Then $Z$ has dimension $d:=n-(e-r)(f-r)$. In order to study the cohomology of $Z$, we consider the Grassmannian bundle $\pi\colon Y:=\mathbb{G}(f-r,F)\to X$ of $(f-r)$-dimensional linear subspaces of the fibres of $F$. In $Y$ one has an analogue $W$ of $Z$: $W$ is smooth and of dimension $d$, the projection $\pi$ maps $W$ onto $Z$ and $W\stackrel{\sim}{\to} Z$ if $n(e-r+1)(f-r+1)$. (If $r=0$ then $W=Z\subseteq X=Y$ is the zero-locus of $\psi\in H^0(X,E\otimes F^\vee)$.) Fulton and Lazarsfeld proved that $ H^q(Y;\mathbb{Z}) \to H^q(W;\mathbb{Z}) $ is an isomorphism for $q
Book Synopsis Hodge Theory, Complex Geometry, and Representation Theory by : Robert S. Doran
Download or read book Hodge Theory, Complex Geometry, and Representation Theory written by Robert S. Doran and published by American Mathematical Soc.. This book was released on 2014 with total page 330 pages. Available in PDF, EPUB and Kindle. Book excerpt: Contains carefully written expository and research articles. Expository papers include discussions of Noether-Lefschetz theory, algebraicity of Hodge loci, and the representation theory of SL2(R). Research articles concern the Hodge conjecture, Harish-Chandra modules, mirror symmetry, Hodge representations of Q-algebraic groups, and compactifications, distributions, and quotients of period domains.
Book Synopsis $\mathcal {R}$-Boundedness, Fourier Multipliers and Problems of Elliptic and Parabolic Type by : Robert Denk
Download or read book $\mathcal {R}$-Boundedness, Fourier Multipliers and Problems of Elliptic and Parabolic Type written by Robert Denk and published by American Mathematical Soc.. This book was released on 2003 with total page 130 pages. Available in PDF, EPUB and Kindle. Book excerpt: The property of maximal $L_p$-regularity for parabolic evolution equations is investigated via the concept of $\mathcal R$-sectorial operators and operator-valued Fourier multipliers. As application, we consider the $L_q$-realization of an elliptic boundary value problem of order $2m$ with operator-valued coefficients subject to general boundary conditions. We show that there is maximal $L_p$-$L_q$-regularity for the solution of the associated Cauchy problem provided the top order coefficients are bounded and uniformly continuous.
Book Synopsis Yang-Mills Measure on Compact Surfaces by : Thierry Lévy
Download or read book Yang-Mills Measure on Compact Surfaces written by Thierry Lévy and published by American Mathematical Soc.. This book was released on 2003 with total page 144 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this memoir we present a new construction and new properties of the Yang-Mills measure in two dimensions. This measure was first introduced for the needs of quantum field theory and can be described informally as a probability measure on the space of connections modulo gauge transformations on a principal bundle. We consider the case of a bundle over a compact orientable surface. Our construction is based on the discrete Yang-Mills theory of which we give a full acount. We are able to take its continuum limit and to define a pathwise multiplicative process of random holonomy indexed by the class of piecewise embedded loops. We study in detail the links between this process and a white noise and prove a result of asymptotic independence in the case of a semi-simple structure group. We also investigate global Markovian properties of the measure related to the surgery of surfaces.
Book Synopsis Quasianalytic Monogenic Solutions of a Cohomological Equation by : S. Marmi
Download or read book Quasianalytic Monogenic Solutions of a Cohomological Equation written by S. Marmi and published by American Mathematical Soc.. This book was released on 2003 with total page 83 pages. Available in PDF, EPUB and Kindle. Book excerpt: We prove that the solutions of a cohomological equation of complex dimension one and in the analytic category have a monogenic dependence on the parameter, and we investigate the question of their quasi analyticity. This cohomological equation is the standard linearized conjugacy equation for germs of holomorphic maps in a neighborhood of a fixed point. The parameter is the eigenvalue of the linear part, denoted by $q$. Borel's theory of non-analytic monogenic functions has been first investigated by Arnold and Herman in the related context of the problem of linearization of analytic diffeomorphisms of the circle close to a rotation.Herman raised the question whether the solutions of the cohomological equation had a quasi analytic dependence on the parameter $q$. Indeed they are analytic for $q\in\mathbb{C}\setminus\mathbb{S}^1$, the unit circle $\S^1$ appears as a natural boundary (because of resonances, i.e. roots of unity), but the solutions are still defined at points of $\mathbb{S}^1$ which lie 'far enough from resonances'. We adapt to our case Herman's construction of an increasing sequence of compacts which avoid resonances and prove that the solutions of our equation belong to the associated space of monogenic functions; some general properties of these monogenic functions and particular properties of the solutions are then studied.For instance the solutions are defined and admit asymptotic expansions at the points of $\mathbb{S}^1$ which satisfy some arithmetical condition, and the classical Carleman Theorem allows us to answer negatively to the question of quasi analyticity at these points. But resonances (roots of unity) also lead to asymptotic expansions, for which quasi analyticity is obtained as a particular case of Ecalle's theory of resurgent functions.And at constant-type points, where no quasi analytic Carleman class contains the solutions, one can still recover the solutions from their asymptotic expansions and obtain a special kind of quasi analyticity. Our results are obtained by reducing the problem, by means of Hadamard's product, to the study of a fundamental solution (which turns out to be the so-called $q$-logarithm or 'quantum logarithm'). We deduce as a corollary of our work the proof of a conjecture of Gammel on the monogenic and quasi analytic properties of a certain number-theoretical Borel-Wolff-Denjoy series.
Book Synopsis Radially Symmetric Patterns of Reaction-diffusion Systems by : Arnd Scheel
Download or read book Radially Symmetric Patterns of Reaction-diffusion Systems written by Arnd Scheel and published by American Mathematical Soc.. This book was released on 2003 with total page 86 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this paper, bifurcations of stationary and time-periodic solutions to reaction-diffusion systems are studied. We develop a center-manifold and normal form theory for radial dynamics which allows for a complete description of radially symmetric patterns. In particular, we show the existence of localized pulses near saddle-nodes, critical Gibbs kernels in the cusp, focus patterns in Turing instabilities, and active or passive target patterns in oscillatory instabilities.
Book Synopsis The $RO(G)$-Graded Equivariant Ordinary Homology of $G$-Cell Complexes with Even-Dimensional Cells for $G=\mathbb {Z}/p$ by :
Download or read book The $RO(G)$-Graded Equivariant Ordinary Homology of $G$-Cell Complexes with Even-Dimensional Cells for $G=\mathbb {Z}/p$ written by and published by American Mathematical Soc.. This book was released on with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Moduli Spaces of Polynomials in Two Variables by : Javier Fernández de Bobadilla Olarzábal
Download or read book Moduli Spaces of Polynomials in Two Variables written by Javier Fernández de Bobadilla Olarzábal and published by American Mathematical Soc.. This book was released on 2005 with total page 136 pages. Available in PDF, EPUB and Kindle. Book excerpt: In the space of polynomials in two variables $\mathbb{C}[x,y]$ with complex coefficients we let the group of automorphisms of the affine plane $\mathbb{A}^2_{\mathbb{C}}$ act by composition on the right. In this paper we investigate the geometry of the orbit space. We associate a graph with each polynomial in two variables that encodes part of its geometric properties at infinity; we define a partition of $\mathbb{C}[x,y]$ imposing that the polynomials in the same stratum are the polynomials with a fixed associated graph. The graphs associated with polynomials belong to certain class of graphs (called behaviour graphs), that has a purely combinatorial definition.We show that any behaviour graph is actually a graph associated with a polynomial. Using this we manage to give a quite precise geometric description of the subsets of the partition. We associate a moduli functor with each behaviour graph of the class, which assigns to each scheme $T$ the set of families of polynomials with the given graph parametrized over $T$. Later, using the language of groupoids, we prove that there exists a geometric quotient of the subsets of the partition associated with the given graph by the equivalence relation induced by the action of Aut$(\mathbb{C}^2)$. This geometric quotient is a coarse moduli space for the moduli functor associated with the graph. We also give a geometric description of it based on the combinatorics of the associated graph. The results presented in this memoir need the development of a certain combinatorial formalism. Using it we are also able to reprove certain known theorems in the subject.
Book Synopsis Dynamics of Topologically Generic Homeomorphisms by : Ethan Akin
Download or read book Dynamics of Topologically Generic Homeomorphisms written by Ethan Akin and published by American Mathematical Soc.. This book was released on 2003 with total page 146 pages. Available in PDF, EPUB and Kindle. Book excerpt: The goal of this work is to describe the dynamics of generic homeomorphisms of certain compact metric spaces $X$. Here ``generic'' is used in the topological sense -- a property of homeomorphisms on $X$ is generic if the set of homeomorphisms with the property contains a residual subset (in the sense of Baire category) of the space of all homeomorphisms on $X$. The spaces $X$ we consider are those with enough local homogeneity to allow certain localized perturbations of homeomorphisms; for example, any compact manifold is such a space. We show that the dynamics of a generic homeomorphism is quite complicated, with a number of distinct dynamical behaviors coexisting (some resemble subshifts of finite type, others, which we call `generalized adding machines', appear strictly periodic when viewed to any finite precision, but are not actually periodic). Such a homeomorphism has infinitely many, intricately nested attractors and repellors, and uncountably many distinct dynamically-connected components of the chain recurrent set. We single out several types of these ``chain components'', and show that each type occurs densely (in an appropriate sense) in the chain recurrent set. We also identify one type that occurs generically in the chain recurrent set. We also show that, at least for $X$ a manifold, the chain recurrent set of a generic homeomorphism is a Cantor set, so its complement is open and dense. Somewhat surprisingly, there is a residual subset of $X$ consisting of points whose limit sets are chain components of a type other than the type of chain components that are residual in the space of all chain components. In fact, for each generic homeomorphism on $X$ there is a residual subset of points of $X$ satisfying a stability condition stronger than Lyapunov stability.
Book Synopsis Hilbert Modular Forms: mod $p$ and $p$-Adic Aspects by : Fabrizio Andreatta
Download or read book Hilbert Modular Forms: mod $p$ and $p$-Adic Aspects written by Fabrizio Andreatta and published by American Mathematical Soc.. This book was released on 2005 with total page 100 pages. Available in PDF, EPUB and Kindle. Book excerpt: We study Hilbert modular forms in characteristic $p$ and over $p$-adic rings. In the characteristic $p$ theory we describe the kernel and image of the $q$-expansion map and prove the existence of filtration for Hilbert modular forms; we define operators $U$, $V$ and $\Theta_\chi$ and study the variation of the filtration under these operators. Our methods are geometric - comparing holomorphic Hilbert modular forms with rational functions on a moduli scheme with level-$p$ structure, whose poles are supported on the non-ordinary locus.In the $p$-adic theory we study congruences between Hilbert modular forms. This applies to the study of congruences between special values of zeta functions of totally real fields. It also allows us to define $p$-adic Hilbert modular forms 'a la Serre' as $p$-adic uniform limit of classical modular forms, and compare them with $p$-adic modular forms 'a la Katz' that are regular functions on a certain formal moduli scheme. We show that the two notions agree for cusp forms and for a suitable class of weights containing all the classical ones. We extend the operators $V$ and $\Theta_\chi$ to the $p$-adic setting.
Book Synopsis On the Classification of Polish Metric Spaces Up to Isometry by : Su Gao
Download or read book On the Classification of Polish Metric Spaces Up to Isometry written by Su Gao and published by American Mathematical Soc.. This book was released on 2003 with total page 78 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Invariants of Boundary Link Cobordism by : Desmond Sheiham
Download or read book Invariants of Boundary Link Cobordism written by Desmond Sheiham and published by American Mathematical Soc.. This book was released on 2003 with total page 110 pages. Available in PDF, EPUB and Kindle. Book excerpt: An $n$-dimensional $\mu$-component boundary link is a codimension $2$ embedding of spheres $L=\sqcup_{\mu}S^n \subset S^{n+2}$ such that there exist $\mu$ disjoint oriented embedded $(n+1)$-manifolds which span the components of $L$. An $F_\mu$-link is a boundary link together with a cobordism class of such spanning manifolds. The $F_\mu$-link cobordism group $C_n(F_\mu)$ is known to be trivial when $n$ is even but not finitely generated when $n$ is odd. Our main result is an algorithm to decide whether two odd-dimensional $F_\mu$-links represent the same cobordism class in $C_{2q-1}(F_\mu)$ assuming $q>1$. We proceed to compute the isomorphism class of $C_{2q-1}(F_\mu)$, generalizing Levine's computation of the knot cobordism group $C_{2q-1}(F_1)$.Our starting point is the algebraic formulation of Levine, Ko and Mio who identify $C_{2q-1}(F_\mu)$ with a surgery obstruction group, the Witt group $G^{(-1)^q,\mu}(\Z)$ of $\mu$-component Seifert matrices. We obtain a complete set of torsion-free invariants by passing from integer coefficients to complex coefficients and by applying the algebraic machinery of Quebbemann, Scharlau and Schulte. Signatures correspond to 'algebraically integral' simple self-dual representations of a certain quiver (directed graph with loops). These representations, in turn, correspond to algebraic integers on an infinite disjoint union of real affine varieties. To distinguish torsion classes, we consider rational coefficients in place of complex coefficients, expressing $G^{(-1)^q,\mu}(\mathbb{Q})$ as an infinite direct sum of Witt groups of finite-dimensional division $\mathbb{Q}$-algebras with involution.The Witt group of every such algebra appears as a summand infinitely often. The theory of symmetric and hermitian forms over these division algebras is well-developed. There are five classes of algebras to be considered; complete Witt invariants are available for four classes, those for which the local-global principle applies. An algebra in the fifth class, namely a quaternion algebra with non-standard involution, requires an additional Witt invariant which is defined if all the local invariants vanish.
Book Synopsis Necessary Conditions in Dynamic Optimization by : Francis Clarke
Download or read book Necessary Conditions in Dynamic Optimization written by Francis Clarke and published by American Mathematical Soc.. This book was released on 2005 with total page 113 pages. Available in PDF, EPUB and Kindle. Book excerpt: This monograph derives necessary conditions of optimality for a general control problem formulated in terms of a differential inclusion. These conditions constitute a new state of the art, subsuming, unifying, and substantially extending the results in the literature. The Euler, Weierstrass and transversality conditions are expressed in their sharpest known forms. No assumptions of boundedness or convexity are made, no constraint qualifications imposed, and only weak pseudo-Lipschitz behavior is postulated on the underlying multifunction. The conditions also incorporate a `stratified' feature of a novel nature, in which both the hypotheses and the conclusion are formulated relative to a given radius function. When specialized to the calculus of variations, the results yield necessary conditions and regularity theorems that go significantly beyond the previous standard. They also apply to parametrized control systems, giving rise to new and stronger maximum principles of Pontryagin type. The final chapter is devoted to a different issue, that of the Hamiltonian necessary condition. It is obtained here, for the first time, in the case of nonconvex values and in the absence of any constraint qualification; this has been a longstanding open question in the subject. Apart from the final chapter, the treatment is self-contained, and calls upon only standard results in functional and nonsmooth analysis.
Book Synopsis Mutually Catalytic Super Branching Random Walks: Large Finite Systems and Renormalization Analysis by : J. T. Cox
Download or read book Mutually Catalytic Super Branching Random Walks: Large Finite Systems and Renormalization Analysis written by J. T. Cox and published by American Mathematical Soc.. This book was released on 2004 with total page 97 pages. Available in PDF, EPUB and Kindle. Book excerpt: We study features of the longtime behavior and the spatial continuum limit for the diffusion limit of the following particle model. Consider populations consisting of two types of particles located on sites labeled by a countable group. The populations of each of the types evolve as follows: each particle performs a random walk and dies or splits in two with probability $\frac{1} {2}$ and the branching rates of a particle of each type at a site $x$ at time $t$ is proportional to the size of the population at $x$ at time $t$ of the other type. The diffusion limit of ''small mass, large number of initial particles'' is a pair of two coupled countable collections of interacting diffusions, the mutually catalytic super branching random walk.Consider now increasing sequences of finite subsets of sites and define the corresponding finite versions of the process. We study the evolution of these large finite spatial systems in size-dependent time scales and compare them with the behavior of the infinite systems, which amounts to establishing the so-called finite system scheme. A dichotomy is known between transient and recurrent symmetrized migrations for the infinite system, namely, between convergence to equilibria allowing for coexistence in the first case and concentration on monotype configurations in the second case.Correspondingly we show in the recurrent case both large finite and infinite systems behave similar in all time scales, in the transient case we see for small time scales a behavior resembling the one of the infinite system, whereas for large time scales the system behaves as in the finite case with fixed size and finally in intermediate scales interesting behavior is exhibited, the system diffuses through the equilibria of the infinite system which are indexed by the pair of intensities and this diffusion process can be described as mutually catalytic diffusion on $(\R^ )^2$. At the same time, the above finite system asymptotics can be applied to mean-field systems of $N$ exchangeable mutually catalytic diffusions. This is the building block for a renormalization analysis of the spatially infinite hierarchical model and leads to an association of this system with the so-called interaction chain, which reflects the behavior of the process on large space-time scales.Similarly we introduce the concept of a continuum limit in the hierarchical mean field limit and show that this limit always exists and that the small-scale properties are described by another Markov chain called small scale characteristics. Both chains are analyzed in detail and exhibit the following interesting effects. The small scale properties of the continuum limit exhibit the dichotomy, overlap or segregation of densities of the two populations, as a function of the underlying random walk kernel. A corresponding concept to study hot spots is presented. Next we look in the transient regime for global equilibria and their equilibrium fluctuations and in the recurrent regime on the formation of monotype regions.For particular migration kernels in the recurrent regime we exhibit diffusive clustering, which means that the sizes (suitable defined) of monotype regions have a random order of magnitude as time proceeds and its distribution is explicitly identifiable. On the other hand in the regime of very large clusters we identify the deterministic order of magnitude of monotype regions and determine the law of the random size. These two regimes occur for different migration kernels than for the cases of ordinary branching or Fisher-Wright diffusion. Finally we find a third regime of very rapid deterministic spatial cluster growth which is not present in other models just mentioned. A further consequence of the analysis is that mutually catalytic branching has a fixed point property under renormalization and gives a natural example different from the trivial case of multitype models consisting of two independent versions of the fixed points for the one type case.
Book Synopsis Conformal and Harmonic Measures on Laminations Associated with Rational Maps by : Vadim A. Kaimanovich
Download or read book Conformal and Harmonic Measures on Laminations Associated with Rational Maps written by Vadim A. Kaimanovich and published by American Mathematical Soc.. This book was released on 2005 with total page 134 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is dedicated to Dennis Sullivan on the occasion of his 60th birthday. The framework of affine and hyperbolic laminations provides a unifying foundation for many aspects of conformal dynamics and hyperbolic geometry. The central objects of this approach are an affine Riemann surface lamination $\mathcal A$ and the associated hyperbolic 3-lamination $\mathcal H$ endowed with an action of a discrete group of isomorphisms. This action is properly discontinuous on $\mathcal H$, which allows one to pass to the quotient hyperbolic lamination $\mathcal M$. Our work explores natural ``geometric'' measures on these laminations. We begin with a brief self-contained introduction to the measure theory on laminations by discussing the relationship between leafwise, transverse and global measures. The central themes of our study are: leafwise and transverse ``conformal streams'' on an affine lamination $\mathcal A$ (analogues of the Patterson-Sullivan conformal measures for Kleinian groups), harmonic and invariant measures on the corresponding hyperbolic lamination $\mathcal H$, the ``Anosov--Sinai cocycle'', the corresponding ``basic cohomology class'' on $\mathcal A$ (which provides an obstruction to flatness), and the Busemann cocycle on $\mathcal H$. A number of related geometric objects on laminations -- in particular, the backward and forward Poincare series and the associated critical exponents, the curvature forms and the Euler class, currents and transverse invariant measures, $\lambda$-harmonic functions and the leafwise Brownian motion -- are discussed along the lines. The main examples are provided by the laminations arising from the Kleinian and the rational dynamics. In the former case, $\mathcal M$ is a sublamination of the unit tangent bundle of a hyperbolic 3-manifold, its transversals can be identified with the limit set of the Kleinian group, and we show how the classical theory of Patterson-Sullivan measures can be recast in terms of our general approach. In the latter case, the laminations were recently constructed by Lyubich and Minsky in [LM97]. Assuming that they are locally compact, we construct a transverse $\delta$-conformal stream on $\mathcal A$ and the corresponding $\lambda$-harmonic measure on $\mathcal M$, where $\lambda=\delta(\delta-2)$. We prove that the exponent $\delta$ of the stream does not exceed 2 and that the affine laminations are never flat except for several explicit special cases (rational functions with parabolic Thurston orbifold).
Book Synopsis Infinite Dimensional Complex Symplectic Spaces by : William Norrie Everitt
Download or read book Infinite Dimensional Complex Symplectic Spaces written by William Norrie Everitt and published by American Mathematical Soc.. This book was released on 2004 with total page 76 pages. Available in PDF, EPUB and Kindle. Book excerpt: Complex symplectic spaces, defined earlier by the authors in their ""AMS Monograph"", are non-trivial generalizations of the real symplectic spaces of classical analytical dynamics. These spaces can also be viewed as non-degenerate indefinite inner product spaces, although the authors here follow the lesser known exposition within complex symplectic algebra and geometry, as is appropriate for their prior development of boundary value theory. In the case of finite dimensional complex symplectic spaces it was shown that the corresponding symplectic algebra is important for the description and classification of all self-adjoint boundary value problems for (linear) ordinary differential equations on a real interval.In later ""AMS Memoirs"" infinite dimensional complex symplectic spaces were introduced for the analysis of multi-interval systems and elliptic partial differential operators. In this current Memoir the authors present a self-contained, systematic investigation of general complex symplectic spaces, and their Lagrangian subspaces, regardless of the finite or infinite dimensionality - starting with axiomatic definitions and leading towards general Glazman-Krein-Naimark (GKN) theorems.In particular, the appropriate relevant topologies on such a symplectic space $\mathsf{S}$ are compared and contrasted, demonstrating that $\mathsf{S}$ is a locally convex linear topological space in terms of the symplectic weak topology. Also the symplectic invariants are defined (as cardinal numbers) characterizing $\mathsf{S}$, in terms of suitable Hilbert structures on $\mathsf{S}$. The penultimate section is devoted to a review of the applications of symplectic algebra to the motivating of boundary value problems for ordinary and partial differential operators. The final section, the Aftermath, is a review and summary of the relevant literature on the theory and application of complex symplectic spaces. The Memoir is completed by symbol and subject indexes.
Book Synopsis $v_1$-Periodic Homotopy Groups of $SO(n)$ by : Martin Bendersky
Download or read book $v_1$-Periodic Homotopy Groups of $SO(n)$ written by Martin Bendersky and published by American Mathematical Soc.. This book was released on 2004 with total page 106 pages. Available in PDF, EPUB and Kindle. Book excerpt: Computes the 2-primary $v_1$-periodic homotopy groups of the special orthogonal groups $SO(n)$; the method is to calculate the Bendersky-Thompson spectral sequence, a $K_*$-based unstable homotopy spectral sequence, of $\operatorname{Spin}(n)$.
