Proceedings of Symposia in Pure Mathematics最新文献

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A note on knot concordance and involutive knot Floer homology 结的一致性和对合结花的同源性

Proceedings of Symposia in Pure Mathematics

Pub Date : 2017-08-21 DOI: 10.1090/pspum/102/09

Kristen Hendricks, Jennifer Hom

We prove that if two knots are concordant, their involutive knot Floer complexes satisfy a certain type of stable equivalence.

证明了如果两个结是协调的，则它们的对合结花配合物满足一类稳定等价。

引用次数: 6

Weinstein manifolds revisited 重新审视韦恩斯坦流形

Proceedings of Symposia in Pure Mathematics

Pub Date : 2017-07-11 DOI: 10.1090/PSPUM/099/01737

Y. Eliashberg

This is a very biased and incomplete survey of some basic notions, old and new results, as well as open problems concerning Weinstein symplectic manifolds.

这是对温斯坦辛流形的一些基本概念、旧的和新的结果，以及关于温斯坦辛流形的开放问题的一个非常有偏见和不完整的调查。

引用次数: 38

An overview of knot Floer homology 结花同源性综述

Proceedings of Symposia in Pure Mathematics

Pub Date : 2017-06-23 DOI: 10.1090/PSPUM/099/01742

P. Ozsváth, Z. Szabó

Knot Floer homology is an invariant for knots discovered by the authors and, independently, Jacob Rasmussen. The discovery of this invariant grew naturally out of studying how a certain three-manifold invariant, Heegaard Floer homology, changes as the three-manifold undergoes Dehn surgery along a knot. Since its original definition, thanks to the contributions of many researchers, knot Floer homology has emerged as a useful tool for studying knots in its own right. We give here a few selected highlights of this theory, and then move on to some new algebraic developments in the computation of knot Floer homology.

结花同源性是由作者和独立的Jacob Rasmussen发现的结的不变量。这个不变量的发现是在研究一个特定的三流形不变量heeggaard Floer同源性时自然产生的，当三流形沿着一个结进行Dehn手术时，它是如何变化的。由于其最初的定义，由于许多研究人员的贡献，结花同源性已经成为研究结的一个有用的工具。我们在这里给出了这个理论的一些精选的亮点，然后转移到一些新的代数发展在结花同调的计算。

引用次数: 8

Pure 𝑆𝑈(2) gauge theory partition function and generalized Bessel kernel 纯𝑆𝑈(2)规范理论配分函数与广义贝塞尔核

Proceedings of Symposia in Pure Mathematics

Pub Date : 2017-05-04 DOI: 10.1090/PSPUM/098/01727

P. Gavrylenko, O. Lisovyy

We show that the dual partition function of the pure $mathcal N=2$ $SU(2)$ gauge theory in the self-dual $Omega$-background (a) is given by Fredholm determinant of a generalized Bessel kernel and (b) coincides with the tau function associated to the general solution of the Painleve III equation of type $D_8$ (radial sine-Gordon equation). In particular, the principal minor expansion of the Fredholm determinant yields Nekrasov combinatorial sums over pairs of Young diagrams.

我们证明了自对偶$Omega$-背景(a)中纯$mathcal N=2$ $SU(2)$规范理论的对偶配分函数是由广义贝塞尔核的Fredholm行列式给出的，并且(b)与$D_8$型Painleve III方程(径向正弦- gordon方程)通解相关的tau函数一致。特别地，Fredholm行列式的主次展开式产生了Young图对上的Nekrasov组合和。

引用次数: 21

On ELSV-type formulae, Hurwitz numbers and topological recursion 论elv型公式、Hurwitz数与拓扑递归

Proceedings of Symposia in Pure Mathematics

Pub Date : 2017-03-19 DOI: 10.1090/PSPUM/100/01764

D. Lewanski

We present several recent developments on ELSV-type formulae and topological recursion concerning Chiodo classes and several kind of Hurwitz numbers. The main results appeared in D. Lewanski, A. Popolitov, S. Shadrin, D. Zvonkine, "Chiodo formulas for the r-th roots and topological recursion", Lett. Math. Phys. (2016).

本文介绍了关于Chiodo类和几种Hurwitz数的elv型公式和拓扑递推的最新进展。主要结果出现在D. Lewanski, A. Popolitov, S. Shadrin, D. Zvonkine，“关于r- n根和拓扑递归的Chiodo公式”，Lett。数学。理论物理。(2016)。

引用次数: 4

Graded linearisations Graded linearisations

Proceedings of Symposia in Pure Mathematics

Pub Date : 2017-03-15 DOI: 10.1090/pspum/099/01

Gergely B'erczi, B. Doran, F. Kirwan

When the action of a reductive group on a projective variety has a suitable linearisation, Mumford’s geometric invariant theory (GIT) can be used to construct and study an associated quotient variety. In this article we describe how Mumford’s GIT can be extended effectively to suitable actions of linear algebraic groups which are not necessarily reductive, with the extra data of a graded linearisation for the action. Any linearisation in the traditional sense for a reductive group action induces a graded linearisation in a natural way. The classical examples of moduli spaces which can be constructed using Mumford’s GIT are moduli spaces of stable curves and of (semi)stable bundles over a fixed nonsingular curve. This more general construction can be used to construct moduli spaces of unstable objects, such as unstable curves or unstable bundles (with suitable fixed discrete invariants in each case, related to their singularities or Harder–Narasimhan type). In algebraic geometry it is often useful to be able to construct quotients of algebraic varieties by linear algebraic group actions; in particular moduli spaces (or stacks) can be constructed in this way. When the linear algebraic group is reductive, and we have a suitable linearisation for its action on a projective variety, we can use Mumford’s geometric invariant theory (GIT) to construct and study such quotient varieties [32]. The aim of this article is to describe how Mumford’s GIT can be extended effectively to actions of a large family of linear algebraic groups which are not necessarily reductive, with the extra data of a graded linearisation for the action. Any linearisation in the traditional sense for a reductive group action can be regarded as a graded linearisation in a natural way. When a linear algebraic group over an algebraically closed field k of characteristic 0 is a semidirect productH = U ⋊R of its unipotent radical U and a reductive subgroupR ∼= H/U which contains a central one-parameter subgroup λ : Gm → Rwhose adjoint action on the Lie algebra of U has only strictly positive weights, we will see that any linearisation for an action of H on a projective variety X becomes graded if it is twisted by an appropriate (rational) character, and then many of the good properties of Mumford’s GIT hold. Many non-reductive linear algebraic group actions arising in algebraic geometry are actions of groups of this form: for example, any parabolic subgroup of a reductive group has this form, as does the automorphism group of any complete simplicial toric variety [11], and the group of k-jets of germs of biholomorphisms of (C, 0) for any positive integers k and p [6]. Example 0.1. The automorphism group of the weighted projective plane P(1, 1, 2) with weights 1,1 and 2 is Aut(P(1, 1, 2)) ∼= R⋉ U where R ∼= (GL(2)×Gm)/Gm ∼= GL(2) is reductive and U ∼= (k+)3 is unipotent with elements given by (x, y, z) 7→ (x, y, z + λx2 + μxy + νy2) for (λ, μ, ν) ∈ k3. Early work on this project was suppo

当约化群对一个投影变项的作用具有适当的线性化时，Mumford的几何不变理论(GIT)可以用来构造和研究一个相关的商变项。在本文中，我们描述了Mumford的GIT如何有效地扩展到线性代数群的适当动作，这些动作不一定是约化的，并且具有动作的分级线性化的额外数据。任何传统意义上的线性化对于还原性群作用都会以自然的方式引起梯度线性化。可以用Mumford的GIT构造模空间的经典例子是稳定曲线和固定非奇异曲线上的(半)稳定束的模空间。这种更一般的构造可以用来构造不稳定对象的模空间，例如不稳定曲线或不稳定束(每种情况下都有合适的固定离散不变量，与它们的奇点或hard - narasimhan型相关)。在代数几何中，利用线性代数群作用构造代数变量的商是很有用的;特别是模空间(或堆栈)可以用这种方式构造。当线性代数群是约化的，并且我们有一个合适的线性化它对一个射影变量的作用，我们可以使用Mumford的几何不变理论(GIT)来构造和研究这样的商变量[32]。本文的目的是描述Mumford的GIT如何有效地扩展到一大族线性代数群的动作，这些动作不一定是约化的，并且具有动作的分级线性化的额外数据。对于约化群作用，传统意义上的任何线性化都可以看作是自然的梯度线性化。当特征为0的代数闭域k上的线性代数群是其单幂根群U和包含中心单参数子群λ的约化子群pr ~ = H/U的半直积th = U * R时:Gm→r，其在U的李代数上的伴随作用只有严格的正权，我们将看到H在一个投影变量X上的作用的任何线性化，如果它被一个适当的(有理)特征扭曲，那么它就变成了分级，然后Mumford的GIT的许多好的性质成立。代数几何中出现的许多非约化线性代数群作用都是这种形式的群的作用:例如，约化群的任何抛物子群都具有这种形式，任何完全简单环型簇[11]的自同构群也具有这种形式，对于任何正整数k和p[6]的(C, 0)生物纯态的胚的k-射流群也具有这种形式。例0.1。权为1,1和2的加权射影平面P(1,1,2)的自同态群为Aut(P(1,1,2)) ~ = R × U，其中R ~ = (GL(2)×Gm)/Gm ~ = GL(2)是约化的，U ~ = (k+)3是单幂的，对于(λ， μ， ν)∈k3，元素为(x, y, z) 7→(x, y, z + λx2 + μxy + νy2)。该项目的早期工作得到了工程与物理科学研究委员会的支持[批准号GR/T016170/1,EP/G000174/1]。Brent Doran获得瑞士国家科学基金奖200021-138071的部分资助。

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引用次数: 4

Descendents for stable pairs on 3-folds 后代为稳定的3倍配对

Proceedings of Symposia in Pure Mathematics

Pub Date : 2017-03-06 DOI: 10.1090/PSPUM/099/01743

R. Pandharipande

We survey here the construction and the basic properties of descendent invariants in the theory of stable pairs on nonsingular projective 3-folds. The main topics covered are the rationality of the generating series, the functional equation, the Gromov-Witten/Pairs correspondence for descendents, the Virasoro constraints, and the connection to the virtual fundamental class of the stable pairs moduli space in algebraic cobordism. In all of these directions, the proven results constitute only a small part of the conjectural framework. A central goal of the article is to introduce the open questions as simply and directly as possible.

本文研究了非奇异射影3折上稳定对理论中子不变量的构造和基本性质。主要内容包括:生成级数的合理性，泛函方程，后代的Gromov-Witten/Pairs对应，Virasoro约束，以及与代数共坐标中稳定对模空间的虚基类的联系。在所有这些方向上，已证明的结果只构成猜想框架的一小部分。本文的中心目标是尽可能简单直接地介绍开放性问题。

引用次数: 16

Supersymmetric field theories and geometric Langlands: The other side of the coin 超对称场论与几何朗兰兹:硬币的另一面

Proceedings of Symposia in Pure Mathematics

Pub Date : 2017-02-21 DOI: 10.1090/PSPUM/098/01723

A. Balasubramanian, J. Teschner

This note announces results on the relations between the approach of Beilinson and Drinfeld to the geometric Langlands correspondence based on conformal field theory, the approach of Kapustin and Witten based on $N=4$ SYM, and the AGT-correspondence. The geometric Langlands correspondence is described as the Nekrasov-Shatashvili limit of a generalisation of the AGT-correspondence in the presence of surface operators. Following the approaches of Kapustin - Witten and Nekrasov - Witten we interpret some aspects of the resulting picture using an effective description in terms of two-dimensional sigma models having Hitchin's moduli spaces as target-manifold.

本文公布了基于共形场论的Beilinson和Drinfeld的方法与基于$N=4$ SYM的Kapustin和Witten的方法与agt -对应之间关系的结果。几何Langlands对应被描述为表面算子存在下agt对应的推广的Nekrasov-Shatashvili极限。根据Kapustin - Witten和Nekrasov - Witten的方法，我们使用以Hitchin模空间为目标流形的二维sigma模型的有效描述来解释结果图的某些方面。

引用次数: 6

Periods of meromorphic quadratic differentials and Goldman bracket 亚纯二次微分的周期与Goldman括号

Proceedings of Symposia in Pure Mathematics

Pub Date : 2017-02-15 DOI: 10.1090/PSPUM/100/01763

D. Korotkin

We study symplectic properties of monodromy map for second order linear equation with meromorphic potential having only simple poles on a Riemann surface. We show that the canonical symplectic structure on the cotangent bundle $T^*M_{g,n}$ implies the Goldman bracket on the corresponding character variety under the monodromy map, thereby extending the recent results of the paper of M.Bertola, C.Norton and the author from the case of holomorphic to meromorphic potentials with simple poles.

研究了黎曼曲面上具有单极亚纯势的二阶线性方程的单映射的辛性质。我们证明了共切束$T^*M_{g,n}$上的正则交变结构蕴涵了一元映射下对应的特征变化上的Goldman括号，从而将m.b tola, c.n norton等人的论文的最新成果从纯势推广到具有简单极点的亚纯势。

引用次数: 10

Airy structures and symplectic geometry of topological recursion 艾里结构与拓扑递归的辛几何

Proceedings of Symposia in Pure Mathematics

Pub Date : 2017-01-31 DOI: 10.1090/PSPUM/100/01765

M. Kontsevich, Y. Soibelman

We propose a new approach to the topological recursion of Eynard-Orantin based on the notion of Airy structure, which we introduce in the paper. We explain why Airy structure is a more fundamental object than the one of the spectral curve. We explain how the concept of quantization of Airy structure leads naturally to the formulas of topological recursion as well as their generalizations. The notion of spectral curve is also considered in a more general framework of Poisson surfaces endowed with foliation. We explain how the deformation theory of spectral curves is related to Airy structures. Few other topics (e.g. the Holomorphic Anomaly Equation) are also discussed from the general point of view of Airy structures.

本文基于Airy结构的概念，提出了一种求解Eynard-Orantin拓扑递归的新方法。我们解释了为什么艾里结构是一个比光谱曲线更基本的物体。我们解释了艾里结构的量化概念如何自然地导致拓扑递归公式及其推广。谱曲线的概念也在具有叶理的泊松曲面的更一般的框架中被考虑。我们解释了光谱曲线的变形理论与艾里结构的关系。此外，本文还从艾里结构的一般观点出发讨论了其他一些问题(如全纯异常方程)。

引用次数: 49

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