Jumping Numbers of a Simple Complete Ideal in a Two-Dimensional Regular Local Ring PDF Download
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Author: Tarmo Järvilehto
Publisher: American Mathematical Soc.
ISBN: 0821848119
Category : Curves, Plane
Languages : en
Pages : 93
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Book Description
The multiplier ideals of an ideal in a regular local ring form a family of ideals parameterized by non-negative rational numbers. As the rational number increases the corresponding multiplier ideal remains unchanged until at some point it gets strictly smaller. A rational number where this kind of diminishing occurs is called a jumping number of the ideal. In this manuscript the author gives an explicit formula for the jumping numbers of a simple complete ideal in a two-dimensional regular local ring. In particular, he obtains a formula for the jumping numbers of an analytically irreducible plane curve. He then shows that the jumping numbers determine the equisingularity class of the curve.
Author: Tarmo Järvilehto
Publisher: American Mathematical Soc.
ISBN: 0821848119
Category : Curves, Plane
Languages : en
Pages : 93
Get Book
Book Description
The multiplier ideals of an ideal in a regular local ring form a family of ideals parameterized by non-negative rational numbers. As the rational number increases the corresponding multiplier ideal remains unchanged until at some point it gets strictly smaller. A rational number where this kind of diminishing occurs is called a jumping number of the ideal. In this manuscript the author gives an explicit formula for the jumping numbers of a simple complete ideal in a two-dimensional regular local ring. In particular, he obtains a formula for the jumping numbers of an analytically irreducible plane curve. He then shows that the jumping numbers determine the equisingularity class of the curve.
Author: Antonio Campillo
Publisher: American Mathematical Soc.
ISBN: 0821869000
Category : Mathematics
Languages : en
Pages : 362
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Book Description
Contains the proceedings of the Second International Workshop on Zeta Functions in Algebra and Geometry held May 3-7, 2010 at the Universitat de les Illes Balears, Palma de Mallorca, Spain. The conference focused on the following topics: arithmetic and geometric aspects of local, topological, and motivic zeta functions, Poincare series of valuations, zeta functions of groups, rings, and representations, prehomogeneous vector spaces and their zeta functions, and height zeta functions.
Author: Maria Alberich-Carramiñana
Publisher: Springer Nature
ISBN: 3030848000
Category : Mathematics
Languages : en
Pages : 150
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Book Description
This book comprises an overview of twelve months of intense activity of the research group Geometry, Topology, Algebra, and Applications (GEOMVAP) at the Universitat Politècnica de Catalunya (UPC). Namely, it contains extended abstracts of the group meeting in Cardona and of the international Workshop of Women in Geometry and Topology aligned with a series of workshops in the topic. As such, it includes a panoramic view of the main research interests of the group which focus on varieties and manifolds from the algebraic, topological and differential perspective with a view towards applications. The GEOMVAP group has a long tradition working on various interfaces of algebra, geometry and topology. In the last decade, the group has become active contributor in interdisciplinary science and it is now focused on both a theoretical point of view and the transversal applications to several disciplines including Robotics, Machine Learning, Phylogenetics, Physics and Celestial Mechanics. The increasing interdisciplinarity of modern research and the fact that the boundaries between different areas of mathematics are vanishing, with a constant transfer of problems and techniques between them, makes it difficult to progress without a multidisciplinary approach. GEOMVAP gathers together experts in Algebraic, Symplectic and Arithmetic Geometry to stimulate the interaction between them and to allow the study of each object from different points of view. The book aims at established researchers, as well as at PhD and postdoctoral students who want to learn more about the latest advances in pure and applied Geometry and Topology.
Author: John C. Baez
Publisher: American Mathematical Soc.
ISBN: 0821872842
Category : Mathematics
Languages : en
Pages : 120
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Book Description
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.
Author: Yorck Sommerhäuser
Publisher: American Mathematical Soc.
ISBN: 0821869132
Category : Mathematics
Languages : en
Pages : 134
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Book Description
The authors prove that the kernel of the action of the modular group on the center of a semisimple factorizable Hopf algebra is a congruence subgroup whenever this action is linear. If the action is only projective, they show that the projective kernel is a congruence subgroup. To do this, they introduce a class of generalized Frobenius-Schur indicators and endow it with an action of the modular group that is compatible with the original one.
Author: Mikhail Khovanov
Publisher: American Mathematical Soc.
ISBN: 082188977X
Category : Mathematics
Languages : en
Pages : 87
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Book Description
A categorification of the Beilinson-Lusztig-MacPherson form of the quantum sl(2) was constructed in a paper (arXiv:0803.3652) by Aaron D. Lauda. Here the authors enhance the graphical calculus introduced and developed in that paper to include two-morphisms between divided powers one-morphisms and their compositions. They obtain explicit diagrammatical formulas for the decomposition of products of divided powers one-morphisms as direct sums of indecomposable one-morphisms; the latter are in a bijection with the Lusztig canonical basis elements.
These formulas have integral coefficients and imply that one of the main results of Lauda's paper--identification of the Grothendieck ring of his 2-category with the idempotented quantum sl(2)--also holds when the 2-category is defined over the ring of integers rather than over a field. A new diagrammatic description of Schur functions is also given and it is shown that the the Jacobi-Trudy formulas for the decomposition of Schur functions into elementary or complete symmetric functions follows from the diagrammatic relations for categorified quantum sl(2).
Author: Aleksandr Vladimirovich Sobolev
Publisher: American Mathematical Soc.
ISBN: 0821884875
Category : Mathematics
Languages : en
Pages : 104
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Book Description
Relying on the known two-term quasiclassical asymptotic formula for the trace of the function $f(A)$ of a Wiener-Hopf type operator $A$ in dimension one, in 1982 H. Widom conjectured a multi-dimensional generalization of that formula for a pseudo-differential operator $A$ with a symbol $a(\mathbf{x}, \boldsymbol{\xi})$ having jump discontinuities in both variables. In 1990 he proved the conjecture for the special case when the jump in any of the two variables occurs on a hyperplane. The present paper provides a proof of Widom's Conjecture under the assumption that the symbol has jumps in both variables on arbitrary smooth bounded surfaces.
Author: Paul Mezo
Publisher: American Mathematical Soc.
ISBN: 0821875655
Category : Mathematics
Languages : en
Pages : 94
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Book Description
Suppose $G$ is a real reductive algebraic group, $\theta$ is an automorphism of $G$, and $\omega$ is a quasicharacter of the group of real points $G(\mathbf{R})$. Under some additional assumptions, the theory of twisted endoscopy associates to this triple real reductive groups $H$. The Local Langlands Correspondence partitions the admissible representations of $H(\mathbf{R})$ and $G(\mathbf{R})$ into $L$-packets. The author proves twisted character identities between $L$-packets of $H(\mathbf{R})$ and $G(\mathbf{R})$ comprised of essential discrete series or limits of discrete series.
Author: David A. Cox
Publisher: American Mathematical Soc.
ISBN: 0821887432
Category : Mathematics
Languages : en
Pages : 116
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Book Description
Consider a rational projective curve $\mathcal{C}$ of degree $d$ over an algebraically closed field $\pmb k$. There are $n$ homogeneous forms $g_{1},\dots, g_{n}$ of degree $d$ in $B=\pmb k[x, y]$ which parameterize $\mathcal{C}$ in a birational, base point free, manner. The authors study the singularities of $\mathcal{C}$ by studying a Hilbert-Burch matrix $\varphi$ for the row vector $[g_{1},\dots, g_{n}]$. In the ``General Lemma'' the authors use the generalized row ideals of $\varphi$ to identify the singular points on $\mathcal{C}$, their multiplicities, the number of branches at each singular point, and the multiplicity of each branch. Let $p$ be a singular point on the parameterized planar curve $\mathcal{C}$ which corresponds to a generalized zero of $\varphi$. In the `'triple Lemma'' the authors give a matrix $\varphi'$ whose maximal minors parameterize the closure, in $\mathbb{P}^{2}$, of the blow-up at $p$ of $\mathcal{C}$ in a neighborhood of $p$. The authors apply the General Lemma to $\varphi'$ in order to learn about the singularities of $\mathcal{C}$ in the first neighborhood of $p$. If $\mathcal{C}$ has even degree $d=2c$ and the multiplicity of $\mathcal{C}$ at $p$ is equal to $c$, then he applies the Triple Lemma again to learn about the singularities of $\mathcal{C}$ in the second neighborhood of $p$. Consider rational plane curves $\mathcal{C}$ of even degree $d=2c$. The authors classify curves according to the configuration of multiplicity $c$ singularities on or infinitely near $\mathcal{C}$. There are $7$ possible configurations of such singularities. They classify the Hilbert-Burch matrix which corresponds to each configuration. The study of multiplicity $c$ singularities on, or infinitely near, a fixed rational plane curve $\mathcal{C}$ of degree $2c$ is equivalent to the study of the scheme of generalized zeros of the fixed balanced Hilbert-Burch matrix $\varphi$ for a parameterization of $\mathcal{C}$.
Author: Matthias Lesch
Publisher: American Mathematical Soc.
ISBN: 0821872966
Category : Mathematics
Languages : en
Pages : 92
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Book Description
"November 2012, volume 220, number (end of volume)."
