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Definable Additive Categories Purity And Model Theory
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Book Synopsis Definable Additive Categories: Purity and Model Theory by : Mike Prest
Download or read book Definable Additive Categories: Purity and Model Theory written by Mike Prest and published by American Mathematical Soc.. This book was released on 2011-02-07 with total page 122 pages. Available in PDF, EPUB and Kindle. Book excerpt: Most of the model theory of modules works, with only minor modifications, in much more general additive contexts (such as functor categories, categories of comodules, categories of sheaves). Furthermore, even within a given category of modules, many subcategories form a ``self-sufficient'' context in which the model theory may be developed without reference to the larger category of modules. The notion of a definable additive category covers all these contexts. The (imaginaries) language which one uses for model theory in a definable additive category can be obtained from the category (of structures and homomorphisms) itself, namely, as the category of those functors to the category of abelian groups which commute with products and direct limits. Dually, the objects of the definable category--the modules (or functors, or comodules, or sheaves)--to which that model theory applies may be recovered as the exact functors from the, small abelian, category (the category of pp-imaginaries) which underlies that language.
Book Synopsis Functor Categories, Model Theory, Algebraic Analysis and Constructive Methods by : Alexander Martsinkovsky
Download or read book Functor Categories, Model Theory, Algebraic Analysis and Constructive Methods written by Alexander Martsinkovsky and published by Springer Nature. This book was released on with total page 256 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Models, Logics, and Higher-dimensional Categories by : Bradd T. Hart
Download or read book Models, Logics, and Higher-dimensional Categories written by Bradd T. Hart and published by American Mathematical Soc.. This book was released on with total page 440 pages. Available in PDF, EPUB and Kindle. Book excerpt: Proceedings of a conference held at Centre de recherches mathematiques of the Universite de Montreal, June 18-20, 2009.
Book Synopsis Logic and Its Applications by : Md. Aquil Khan
Download or read book Logic and Its Applications written by Md. Aquil Khan and published by Springer. This book was released on 2019-02-13 with total page 210 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book collects the refereed proceedings of the 8th Indian Conference on Logic and Its Applications, ICLA 2019, held in Delhi, India, in March 2019. The volume contains 13 full revised papers along with 6 invited talks presented at the conference. The aim of this conference series is to bring together researchers from a wide variety of fields in which formal logic plays a significant role. Areas of interest include mathematical and philosophical logic, computer science logic, foundations and philosophy of mathematics and the sciences, use of formal logic in areas of theoretical computer science and artificial intelligence, logic and linguistics, and the relationship between logic and other branches of knowledge. Of special interest are studies in systems of logic in the Indian tradition, and historical research on logic.
Book Synopsis Purity, Spectra and Localisation by : Mike Prest
Download or read book Purity, Spectra and Localisation written by Mike Prest and published by Cambridge University Press. This book was released on 2009-06-04 with total page 798 pages. Available in PDF, EPUB and Kindle. Book excerpt: A unified, coherent account of the algebraic aspects and uses of the Ziegler spectrum. It may be used as an introductory graduate-level text, providing relevant background material and a wealth of illustrated examples. An extensive index and thorough referencing also make this book an ideal reference.
Book Synopsis Categorical, Homological and Combinatorial Methods in Algebra by : Ashish K. Srivastava
Download or read book Categorical, Homological and Combinatorial Methods in Algebra written by Ashish K. Srivastava and published by American Mathematical Soc.. This book was released on 2020-06-23 with total page 357 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book contains the proceedings of the AMS Special Session, in honor of S. K. Jain's 80th birthday, on Categorical, Homological and Combinatorial Methods in Algebra held from March 16–18, 2018, at Ohio State University, Columbus, Ohio. The articles contained in this volume aim to showcase the current state of art in categorical, homological and combinatorial aspects of algebra.
Book Synopsis Surveys in Representation Theory of Algebras by : Alex Martsinkovsky
Download or read book Surveys in Representation Theory of Algebras written by Alex Martsinkovsky and published by American Mathematical Soc.. This book was released on 2018-09-12 with total page 203 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume contains selected expository lectures delivered at the annual Maurice Auslander Distinguished Lectures and International Conference over the last several years. Reflecting the diverse landscape of modern representation theory of algebras, the selected articles include: a quick introduction to silting modules; a survey on the first decade of co-t-structures in triangulated categories; a functorial approach to the notion of module; a representation-theoretic approach to recollements in abelian categories; new examples of applications of relative homological algebra; connections between Coxeter groups and quiver representations; and recent progress on limits of approximation theory.
Book Synopsis Model Theory of Modules, Algebras and Categories by : Alberto Facchini
Download or read book Model Theory of Modules, Algebras and Categories written by Alberto Facchini and published by American Mathematical Soc.. This book was released on 2019-05-31 with total page 237 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume contains the proceedings of the international conference Model Theory of Modules, Algebras and Categories, held from July 28–August 2, 2017, at the Ettore Majorana Foundation and Centre for Scientific Culture in Erice, Italy. Papers contained in this volume cover recent developments in model theory, module theory and category theory, and their intersection.
Book Synopsis Ring Theory 2019 - Proceedings Of The Eighth China-japan-korea International Symposium On Ring Theory by : Hideto Asashiba
Download or read book Ring Theory 2019 - Proceedings Of The Eighth China-japan-korea International Symposium On Ring Theory written by Hideto Asashiba and published by World Scientific. This book was released on 2021-01-04 with total page 256 pages. Available in PDF, EPUB and Kindle. Book excerpt: Since 1991, the group of ring theorists from China and Japan, joined by Korea from 1995 onwards, took turns to hold the quadrennial international conferences (sometimes also referred to as symposiums). As the proceedings of the eighth conference held in Nagoya, Japan in 2019, this volume consists of a collection of articles by invited speakers (survey) and general speakers (survey and original), all of which were refereed by world experts.The survey articles show the trends of current research and offer clear, thorough explanations that are ideal for researchers also in other specialized areas of ring theory. The original articles display new results, ideas and tools for research investigations in ring theory.The articles cover major areas in ring theory, such as: structures of rings, module theory, homological algebra, groups, Hopf algebras, Lie theory, representation theory of rings, (non-commutative) algebraic geometry, commutative rings (structures, representations), amongst others.This volume is a useful resource for researchers — both beginners and advanced experts — in ring theory.
Book Synopsis The Schrodinger Model for the Minimal Representation of the Indefinite Orthogonal Group $O(p,q)$ by : Toshiyuki Kobayashi
Download or read book The Schrodinger Model for the Minimal Representation of the Indefinite Orthogonal Group $O(p,q)$ written by Toshiyuki Kobayashi and published by American Mathematical Soc.. This book was released on 2011 with total page 145 pages. Available in PDF, EPUB and Kindle. Book excerpt: The authors introduce a generalization of the Fourier transform, denoted by $\mathcal{F}_C$, on the isotropic cone $C$ associated to an indefinite quadratic form of signature $(n_1,n_2)$ on $\mathbb{R}^n$ ($n=n_1+n_2$: even). This transform is in some sense the unique and natural unitary operator on $L^2(C)$, as is the case with the Euclidean Fourier transform $\mathcal{F}_{\mathbb{R}^n}$ on $L^2(\mathbb{R}^n)$. Inspired by recent developments of algebraic representation theory of reductive groups, the authors shed new light on classical analysis on the one hand, and give the global formulas for the $L^2$-model of the minimal representation of the simple Lie group $G=O(n_1+1,n_2+1)$ on the other hand.
Book Synopsis The Hermitian Two Matrix Model with an Even Quartic Potential by : Maurice Duits
Download or read book The Hermitian Two Matrix Model with an Even Quartic Potential written by Maurice Duits and published by American Mathematical Soc.. This book was released on 2012 with total page 105 pages. Available in PDF, EPUB and Kindle. Book excerpt: The authors consider the two matrix model with an even quartic potential $W(y)=y^4/4+\alpha y^2/2$ and an even polynomial potential $V(x)$. The main result of the paper is the formulation of a vector equilibrium problem for the limiting mean density for the eigenvalues of one of the matrices $M_1$. The vector equilibrium problem is defined for three measures, with external fields on the first and third measures and an upper constraint on the second measure. The proof is based on a steepest descent analysis of a $4\times4$ matrix valued Riemann-Hilbert problem that characterizes the correlation kernel for the eigenvalues of $M_1$. The authors' results generalize earlier results for the case $\alpha=0$, where the external field on the third measure was not present.
Book Synopsis On the Shape of a Pure O-sequence by : Mats Boij
Download or read book On the Shape of a Pure O-sequence written by Mats Boij and published by American Mathematical Soc.. This book was released on 2012 with total page 78 pages. Available in PDF, EPUB and Kindle. Book excerpt: A monomial order ideal is a finite collection $X$ of (monic) monomials such that, whenever $M\in X$ and $N$ divides $M$, then $N\in X$. Hence $X$ is a poset, where the partial order is given by divisibility. If all, say $t$, maximal monomials of $X$ have the same degree, then $X$ is pure (of type $t$). A pure $O$-sequence is the vector, $\underline{h}=(h_0=1,h_1,...,h_e)$, counting the monomials of $X$ in each degree. Equivalently, pure $O$-sequences can be characterized as the $f$-vectors of pure multicomplexes, or, in the language of commutative algebra, as the $h$-vectors of monomial Artinian level algebras. Pure $O$-sequences had their origin in one of the early works of Stanley's in this area, and have since played a significant role in at least three different disciplines: the study of simplicial complexes and their $f$-vectors, the theory of level algebras, and the theory of matroids. This monograph is intended to be the first systematic study of the theory of pure $O$-sequences.
Book Synopsis Finite Order Automorphisms and Real Forms of Affine Kac-Moody Algebras in the Smooth and Algebraic Category by : Ernst Heintze
Download or read book Finite Order Automorphisms and Real Forms of Affine Kac-Moody Algebras in the Smooth and Algebraic Category written by Ernst Heintze and published by American Mathematical Soc.. This book was released on 2012 with total page 66 pages. Available in PDF, EPUB and Kindle. Book excerpt: Let $\mathfrak{g}$ be a real or complex (finite dimensional) simple Lie algebra and $\sigma\in\mathrm{Aut}\mathfrak{g}$. The authors study automorphisms of the twisted loop algebra $L(\mathfrak{g},\sigma)$ of smooth $\sigma$-periodic maps from $\mathbb{R}$ to $\mathfrak{g}$ as well as of the ``smooth'' affine Kac-Moody algebra $\hat L(\mathfrak{g},\sigma)$, which is a $2$-dimensional extension of $L(\mathfrak{g},\sigma)$. It turns out that these automorphisms which either preserve or reverse the orientation of loops, and are correspondingly called to be of first and second kind, can be described essentially by curves of automorphisms of $\mathfrak{g}$. If the order of the automorphisms is finite, then the corresponding curves in $\mathrm{Aut}\mathfrak{g}$ allow us to define certain invariants and these turn out to parametrize the conjugacy classes of the automorphisms. If their order is $2$ the authors carry this out in detail and deduce a complete classification of involutions and real forms (which correspond to conjugate linear involutions) of smooth affine Kac-Moody algebras.
The resulting classification can be seen as an extension of Cartan's classification of symmetric spaces, i.e. of involutions on $\mathfrak{g}$. If $\mathfrak{g}$ is compact, then conjugate linear extensions of involutions from $\hat L(\mathfrak{g},\sigma)$ to conjugate linear involutions on $\hat L(\mathfrak{g}_{\mathbb{C}},\sigma_{\mathbb{C}})$ yield a bijection between their conjugacy classes and this gives existence and uniqueness of Cartan decompositions of real forms of complex smooth affine Kac-Moody algebras.
The authors show that their methods work equally well also in the algebraic case where the loops are assumed to have finite Fourier expansions.
Book Synopsis Infinite-dimensional Representations of 2-groups by : John C. Baez
Download or read book Infinite-dimensional Representations of 2-groups written by John C. Baez and published by American Mathematical Soc.. This book was released on 2012 with total page 120 pages. Available in PDF, EPUB and Kindle. Book excerpt: A “$2$-group'' is a category equipped with a multiplication satisfying laws like those of a group. Just as groups have representations on vector spaces, $2$-groups have representations on “$2$-vector spaces'', which are categories analogous to vector spaces. Unfortunately, Lie $2$-groups typically have few representations on the finite-dimensional $2$-vector spaces introduced by Kapranov and Voevodsky. For this reason, Crane, Sheppeard and Yetter introduced certain infinite-dimensional $2$-vector spaces called ``measurable categories'' (since they are closely related to measurable fields of Hilbert spaces), and used these to study infinite-dimensional representations of certain Lie $2$-groups. Here they continue this work.
They begin with a detailed study of measurable categories. Then they give a geometrical description of the measurable representations, intertwiners and $2$-intertwiners for any skeletal measurable $2$-group. They study tensor products and direct sums for representations, and various concepts of subrepresentation. They describe direct sums of intertwiners, and sub-intertwiners--features not seen in ordinary group representation theory and study irreducible and indecomposable representations and intertwiners. They also study “irretractable'' representations--another feature not seen in ordinary group representation theory. Finally, they argue that measurable categories equipped with some extra structure deserve to be considered “separable $2$-Hilbert spaces'', and compare this idea to a tentative definition of $2$-Hilbert spaces as representation categories of commutative von Neumann algebras.
Book Synopsis General Relativistic Self-similar Waves that Induce an Anomalous Acceleration Into the Standard Model of Cosmology by : Joel Smoller
Download or read book General Relativistic Self-similar Waves that Induce an Anomalous Acceleration Into the Standard Model of Cosmology written by Joel Smoller and published by American Mathematical Soc.. This book was released on 2012 with total page 69 pages. Available in PDF, EPUB and Kindle. Book excerpt: The authors prove that the Einstein equations for a spherically symmetric spacetime in Standard Schwarzschild Coordinates (SSC) close to form a system of three ordinary differential equations for a family of self-similar expansion waves, and the critical ($k=0$) Friedmann universe associated with the pure radiation phase of the Standard Model of Cosmology is embedded as a single point in this family. Removing a scaling law and imposing regularity at the center, they prove that the family reduces to an implicitly defined one-parameter family of distinct spacetimes determined by the value of a new acceleration parameter $a$, such that $a=1$ corresponds to the Standard Model. The authors prove that all of the self-similar spacetimes in the family are distinct from the non-critical $k\neq0$ Friedmann spacetimes, thereby characterizing the critical $k=0$ Friedmann universe as the unique spacetime lying at the intersection of these two one-parameter families. They then present a mathematically rigorous analysis of solutions near the singular point at the center, deriving the expansion of solutions up to fourth order in the fractional distance to the Hubble Length. Finally, they use these rigorous estimates to calculate the exact leading order quadratic and cubic corrections to the redshift vs luminosity relation for an observer at the center.
Book Synopsis Iwasawa Theory, Projective Modules, and Modular Representations by : Ralph Greenberg
Download or read book Iwasawa Theory, Projective Modules, and Modular Representations written by Ralph Greenberg and published by American Mathematical Soc.. This book was released on 2010 with total page 198 pages. Available in PDF, EPUB and Kindle. Book excerpt: This paper shows that properties of projective modules over a group ring $\mathbf{Z}_p[\Delta]$, where $\Delta$ is a finite Galois group, can be used to study the behavior of certain invariants which occur naturally in Iwasawa theory for an elliptic curve $E$. Modular representation theory for the group $\Delta$ plays a crucial role in this study. It is necessary to make a certain assumption about the vanishing of a $\mu$-invariant. The author then studies $\lambda$-invariants $\lambda_E(\sigma)$, where $\sigma$ varies over the absolutely irreducible representations of $\Delta$. He shows that there are non-trivial relationships between these invariants under certain hypotheses.
Book Synopsis A Theory of Generalized Donaldson-Thomas Invariants by : Dominic D. Joyce
Download or read book A Theory of Generalized Donaldson-Thomas Invariants written by Dominic D. Joyce and published by American Mathematical Soc.. This book was released on 2012 with total page 199 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book studies generalized Donaldson-Thomas invariants $\bar{DT}{}^\alpha(\tau)$. They are rational numbers which `count' both $\tau$-stable and $\tau$-semistable coherent sheaves with Chern character $\alpha$ on $X$; strictly $\tau$-semistable sheaves must be counted with complicated rational weights. The $\bar{DT}{}^\alpha(\tau)$ are defined for all classes $\alpha$, and are equal to $DT^\alpha(\tau)$ when it is defined. They are unchanged under deformations of $X$, and transform by a wall-crossing formula under change of stability condition $\tau$. To prove all this, the authors study the local structure of the moduli stack $\mathfrak M$ of coherent sheaves on $X$. They show that an atlas for $\mathfrak M$ may be written locally as $\mathrm{Crit}(f)$ for $f:U\to{\mathbb C}$ holomorphic and $U$ smooth, and use this to deduce identities on the Behrend function $\nu_\mathfrak M$. They compute the invariants $\bar{DT}{}^\alpha(\tau)$ in examples, and make a conjecture about their integrality properties. They also extend the theory to abelian categories $\mathrm{mod}$-$\mathbb{C}Q\backslash I$ of representations of a quiver $Q$ with relations $I$ coming from a superpotential $W$ on $Q$.
