A connectedness property of algebraic moment maps

赋值环空间的连通性定理

A connectedness theorem for spaces of valuation rings

本文将 Zariski 连通性定理提升至赋值环的 Zariski-Riemann 空间，证明子环为局部、剩余代数且整闭当且仅当其是某闭连通子空间中环的交集，并给出该定理的逆命题，且无需 Noetherian 假设。

特征 $p>5$ 下三维簇的连通性原理

Connectedness principle for $3$-folds in characteristic $p>5$

本文在正特征代数几何中研究 Shokurov–Kollár 连通性原理，证明其对特征 $p>5$ 的三维簇成立，并刻画非 klt 轨迹不连通的情形。

$\mathbb{A}^1$-连通性：曲线上的半稳定和抛物半稳定向量丛模堆

$\mathbb{A}^1$--connectedness of moduli stack of semi-stable and parabolic semi-stable vector bundles over a curve

本文研究了曲线上的半稳定向量丛及抛物半稳定向量丛模堆的$\mathbb{A}^1$-连通性，证明了其在固定秩和确定性下的连通性，以及在特定条件下抛物数据下的连通性。

Rational connectedness and Galois covers of the projective line

Let k be a p-adic field. Some time ago, D. Harbater [9] proved that any finite group G may be realized as a regular Galois group over the rational function field in one variable k(t), namely there exists a finite field extension $F/k(t)$, Galois with group G, such that F is a regular extension of k (i.e. k is algebraically closed in F). Moreover, one may arrange that a given k-place of k(t) be totally split in F. Harbater proved this theorem for k an arbitrary complete valued field. Rather formal arguments ([10, §4.5]; \S2 hereafter) then imply that the theorem holds over any `large' field k. …

Families of rationally simply connected varieties over surfaces and torsors for semisimple groups

Under suitable hypotheses, we prove that a form of a projective homogeneous variety $G/P$ defined over the function field of a surface over an algebraically closed field has a rational point. The method uses an algebro-geometric analogue of simple connectedness replacing the unit interval by the projective line. As a consequence, we complete the proof of Serre's Conjecture II in Galois cohomology for function fields over an algebraically closed field.

微分层边界的连通性

Connectedness of the boundaries of the strata of differentials

本文研究微分射影化层边界的连通性，利用亚纯微分层的仿射性证明任意完备代数紧化中边界连通，并借助 Teichmüller 曲线分析全纯微分情形。

On fibers of algebraic invariant moment maps

In this paper we study some properties of fibers of the invariant moment map for a Hamiltonian action of a reductive group on an affine symplectic varieity. We prove that all fibers have equal dimension. Further, under some additional restrictions, we show that the quotients of fibers are irreducible normal schemes. This paper is an expanded version of Section 5 and Subsection 7.2 of AG/0612559v1.

具有有理连通纤维的光滑射影态射的解析与代数 Oka-1 逼近

Analytic and Algebraic Oka-1 Approximation for Smooth Projective Morphisms with Rationally Connected Fibers

本文证明光滑射影态射（纤维有理连通）在开黎曼曲面上的连续提升可被全纯或代数逼近并插值，从而建立代数 Oka-1 性质与有理连通性的等价。

Graphs of Schemes Associated to Group Actions

Let $X$ be a proper algebraic scheme over an algebraically closed field. We assume that a torus $T$ acts on $X$ such that the action has isolated fixed points. The $T$-graph of $X$ can be defined using the fixed points and the one dimensional orbits of the $T$-action. If the upper Borel subgroup of the general linear group with maximal torus $T$ acts on $X$, then we can define a second graph associated to $X$, called the $A$-graph of $X$. We prove that the $A$-graph of $X$ is connected if and only if $X$ is connected. We use this result to give a proof of Hartshorne's theorem on the connectedn…

Pseudo-reductive and quasi-reductive groups over non-archimedean local fields

Among connected linear algebraic groups, quasi-reductive groups generalize pseudo-reductive groups, which in turn form a useful relaxation of the notion of reductivity. We study quasi-reductive groups over non-archimedean local fields, focusing on aspects involving their locally compact topology. For such groups we construct valuated root data (in the sense of Bruhat--Tits) and we make them act nicely on affine buildings. We prove that they admit Iwasawa and Cartan decompositions, and we construct small compact open subgroups with an Iwahori decomposition. We also initiate the smooth represent…

Real algebraic curves, the moment map and amoebas

In this paper we prove the topological uniqueness of maximal arrangements of a real plane algebraic curve with respect to three lines. More generally, we prove the topological uniqueness of a maximally arranged algebraic curve on a real toric surface. We use the moment map as a tool for studying the topology of real algebraic curves and their complexifications.

$\mathbb{A}^1$-连通叠曲线与椭圆曲线模空间的 Brauer 群

$\mathbb{A}^1$-connected stacky curves and the Brauer group of moduli of elliptic curves

本文提出商叠动机连通分支 Nisnevich 层的计算公式，据此分类 $\mathbb{A}^1$-连通叠曲线并证明椭圆曲线模叠的 $\mathbb{A}^1$-连通性，同时建立光滑代数叠同伦纯性并恢复 Picard 群计算。

Moment Maps, Strict Linear Precision, and Maximum Likelihood Degree One

We study the moment maps of a smooth projective toric variety. In particular, we characterize when the moment map coming from the quotient construction is equal to a weighted Fubini-Study moment map. This leads to an investigation into polytopes with strict linear precision, and in the process we use results from and find remarkable connections between Symplectic Geometry, Geometric Modeling, Algebraic Statistics, and Algebraic Geometry.

Toric ideals, real toric varieties, and the algebraic moment map

This is a tutorial on some aspects of toric varieties related to their potential use in geometric modeling. We discuss projective toric varieties and their ideals, as well as real toric varieties and the algebraic moment map. In particular, we explain the relation between linear precision and the algebraic moment map. This builds on the introduction to toric varieties by David Cox: What is a Toric Variety? at http://www.cs.amherst.edu/~dac/lectures/tutorial.ps

Algebraic polynomials and moments of stochastic integrals

We propose an algebraic method for proving estimates on moments of stochastic integrals. The method uses qualitative properties of roots of algebraic polynomials from certain general classes. As an application, we give a new proof of a variation of the Burkholder-Davis-Gundy inequality for the case of stochastic integrals with respect to real locally square integrable martingales. Further possible applications and extensions of the method are outlined.

Algebraic certificates for the truncated moment problem

The truncated moment problem consists of determining whether a given finitedimensional vector of real numbers y is obtained by integrating a basis of the vector space of polynomials of bounded degree with respect to a non-negative measure on a given set K of a finite-dimensional Euclidean space. This problem has plenty of applications e.g. in optimization, control theory and statistics. When K is a compact semialgebraic set, the duality between the cone of moments of non-negative measures on K and the cone of non-negative polynomials on K yields an alternative: either y is a moment vector, or …

类矩映射与具有指定原像的实代数函数

Moment-like maps and real algebraic functions with prescribed preimages

本文研究构造具有指定原像和单一奇异值的显式实代数函数，推广球面投影与Morse-Bott函数，并首次考虑原像拓扑，推动实代数奇点理论研究。

A relative notion of algebraic Lie group and applications to $n$-stacks

If $S$ is a scheme of finite type over $k=\cc $, let $\Xx /S$ denote the big etale site of schemes over $S$. We introduce {\em presentable group sheaves}, a full subcategory of the category of sheaves of groups on $\Xx /S$ which is closed under kernel, quotient, and extension. Group sheaves which are representable by group schemes of finite type over $S$ are presentable; pullback and finite direct image preserve the notions of presentable group sheaves; over $S=Spec (k)$ then presentable group sheaves are just group schemes of finite type over $Spec(k)$; there is a notion of connectedness exte…

Compact Kaehler quotients of algebraic varieties and Geometric Invariant Theory

Given an action of a complex reductive Lie group G on a normal variety X, we show that every analytically Zariski-open subset of X admitting an analytic Hilbert quotient with projective quotient space is given as the set of semistable points with respect to some G-linearised Weil divisor on X. Applying this result to Hamiltonian actions on algebraic varieties we prove that semistability with respect to a momentum map is equivalent to GIT-semistability in the sense of Mumford and Hausen. It follows that the number of compact momentum map quotients of a given algebraic Hamiltonian G-variety is f…

第12章：紧半代数集上的矩问题（修订版）

Chapter 12: The moment problem on compact semi-algebraic sets (revised version)

本文是著作第12章的修订版，为二次模和半环的Archimedean Positivstellensätze提出统一方法，并补充了半环Positivstellensätze与矩问题的新结果。

局部像矩映射那样重构具有指定像、且与到1维实仿射空间的典范投射复合的实代数映射

Reconstructing real algebraic maps locally like moment-maps with prescribed images and compositions with the canonical projections to the $1$-dimensional real affine space

本研究提出一类局部类似矩映射、原像为单点集或球面乘积的非正余维数显式实代数映射，补充了Nash-Tognoli理论下显式实代数映射实例不足的问题，并给出其与典范投射复合的显式函数族。

非负多项式与代数曲线上的矩问题

Nonnegative Polynomials and Moment Problems on Algebraic Curves

本文研究了代数曲线上的非负多项式锥的面结构及其对偶矩锥的Carathéodory数，揭示了拓扑对四次规则复杂性的影响，并应用于截断矩问题。

霍奇闭链的谱分析：一种通过广义矩量法研究霍奇猜想的新途径

Spectral Analysis of Hodge Cycles: A Novel Approach to the Hodge Conjecture via Generalized Moments

本研究提出研究霍奇猜想的新方法，通过推广Zernike矩定义霍奇闭链的特征形式与谱展开系数，经玩具模型验证后构建适用于K3曲面的框架，为霍奇猜想研究提供新方向。

半代数集的横截性定理及其在二阶矩信号恢复与冷冻电镜中的应用

A transversality theorem for semi-algebraic sets with application to signal recovery from the second moment and cryo-EM

本文证明了半代数集在群表示中的横截性定理，并将其应用于从二阶矩恢复信号的问题，特别针对冷冻电镜推导出分子结构恢复的显式界，同时给出Gram矩阵分解、多参考对齐、相位恢复及置换不变分离器设计的信息论界。

Algebraic Identifiability of Gaussian Mixtures

We prove that all moment varieties of univariate Gaussian mixtures have the expected dimension. Our approach rests on intersection theory and Terracini's classification of defective surfaces. The analogous identifiability result is shown to be false for mixtures of Gaussians in dimension three and higher. Their moments up to third order define projective varieties that are defective. Our geometric study suggests an extension of the Alexander-Hirschowitz Theorem for Veronese varieties to the Gaussian setting.

An Algebraic Geometry Method for Calculating DOS for 2D tight binding models

An algebraic geometry method is used to calculate the moments of the electron density of states as a function of the energy for lattices in the tight binding approximation. Interpreting the moments as the Mellin transform of the density allows writing down a formula for the density as an inverse Mellin transform. The method is illustrated by working out the density function for the two-dimensional square and honeycomb lattices.

Some algebraic and arithmetic properties of Feynman diagrams

This article reports on some recent progresses in Bessel moments, which represent a class of Feynman diagrams in 2-dimensional quantum field theory. Many challenging mathematical problems on these Bessel moments have been formulated as a vast set of conjectures, by David Broadhurst and collaborators, who work at the intersection of high energy physics, number theory and algebraic geometry. We present the main ideas behind our verifications of several such conjectures, which revolve around linear and non-linear sum rules of Bessel moments, as well as relations between individual Feynman diagram…