An algebraic geometry perspective on topological data analysis

Computer Algebra and Algebraic Geometry - Achievements and Perspectives

In this survey I should like to introduce some concepts of algebraic geometry and try to demonstrate the fruitful interaction between algebraic geometry and computer algebra and, more generally, between mathematics and computer science. One of the aims of this article is to show, by means of examples, the usefulness of computer algebra to mathematical research.

Étale 基本群——代数几何中基本群的几何与拓扑方法

Étale Fundamental Groups: a geometric and topological approach to fundamental groups in algebraic geometry

本文通过拓扑覆盖空间、Étale拓扑和Tannakian对偶三种框架，系统构建并比较了代数几何中的基本群概念。

Rational real algebraic models of topological surfaces

Comessatti proved that the set of real points of a rational real algebraic surface is either a nonorientable surface, or the two-sphere, or the torus. Conversely, it is easy to see that all of these surfaces admit a rational real algebraic model. We prove that they admit exactly one rational real algebraic model. This was known earlier only for the two-sphere, torus, projective plane and the Klein bottle.

Overview of topological properties of real algebraic surfaces

These notes reproduce the content of a short, 50-minutes, survey talk given at the Nice University in September, 2004. We added a few topics that have not been touched on in the lecture by lack of time.

Topological noetherianity for algebraic representations of infinite rank classical groups

Draisma recently proved that polynomial representations of $\mathbf{GL}_{\infty}$ are topologically noetherian. We generalize this result to algebraic representations of infinite rank classical groups.

Topological properties of real algebraic varieties: du côté de chez Rokhlin

The survey gives an overview of the achievements in topology of real algebraic varieties in the direction initiated in the early 70th by V.I.Arnold and V.A.Rokhlin. We make an attempt to systematize the principal results in the subject. After an exposition of general tools and results, special attention is paid to surfaces and curves on surfaces.

Families of algebraic varieties parametrized by topological spaces

We study families of algebraic varieties parametrized by topological spaces and generalize some classical results such as Hilbert Nullstellensatz and primary decomposition of commutative rings. We show that there is an equivalence between the category of bivariant coherent sheaves and the category of sheaves of finitely generated modules.

Obstructions to algebraizing topological vector bundles

Suppose $X$ is a smooth complex algebraic variety. A necessary condition for a complex topological vector bundle on $X$ (viewed as a complex manifold) to be algebraic is that all Chern classes must be algebraic cohomology classes, i.e., lie in the image of the cycle class map. We analyze the question of whether algebraicity of Chern classes is sufficient to guarantee algebraizability of complex topological vector bundles. For affine varieties of dimension $\leq 3$, it is known that algebraicity of Chern classes of a vector bundle guarantees algebraizability of the vector bundle. In contrast, w…

Algebraic versus topological entropy for surfaces over finite fields

We show that, as in de Rham cohomology over the complex numbers, the value of the entropy of an automorphism of the surface over a finite field $\F_q $ is taken on the span of the Néron-Severi group inside of $\ell$-adic cohomology. v2: (some) typos removed, exposition (partly) improved.

线性代数、优化与统计中的格拉斯曼簇与旗簇：代数视角

Grassmann and Flag Varieties in Linear Algebra, Optimization, and Statistics: An Algebraic Perspective

本文从代数视角研究格拉斯曼簇与旗簇，将线性代数与统计问题转化为其上的优化问题，以复临界点数量衡量代数复杂度，阐述流形的代数簇实现并计算了多类优化问题的代数复杂度。

Local topological algebraicity of analytic function germs

T. Mostowski showed that every (real or complex) germ of an analytic set is homeomorphic to the germ of an algebraic set. In this paper we show that every (real or complex) analytic function germ, defined on a possibly singular analytic space, is topologically equivalent to a polynomial function germ defined on an affine algebraic variety.

Symmetric powers of algebraic and tropical curves: a non-Archimedean perspective

We show that the non-Archimedean skeleton of the $d$-th symmetric power of a smooth projective algebraic curve $X$ is naturally isomorphic to the $d$-th symmetric power of the tropical curve that arises as the non-Archimedean skeleton of $X$. The retraction to the skeleton is precisely the specialization map for divisors. Moreover, we show that the process of tropicalization naturally commutes with the diagonal morphisms and the Abel-Jacobi map and we exhibit a faithful tropicalization for symmetric powers of curves. Finally, we prove a version of the Bieri-Groves Theorem that allows us, under…

The Nash problem from a geometric and topological perspective

We survey the proof of the Nash conjecture for surfaces and show how geometric and topological ideas developed in previous articles by the authors influenced it. Later we summarize the main ideas in the higher dimensional statement and proof by de Fernex and Docampo. We end the paper by explaining later developments on generalized Nash problem and on Kollár and Nemethi's study about holomorphic arcs.

Topological rigidity of algebraic $¶_3$-bundles over curves

A projective algebraic surface which is homeomorphic to a ruled surface over a curve of genus $g\ge 1$ is itself a ruled surface over a curve of genus $g$. In this note, we prove the analogous result for projective algebraic manifolds of dimension 4 in case $g\ge 2$.

Numerical Algebraic Geometry: A New Perspective on String and Gauge Theories

The interplay rich between algebraic geometry and string and gauge theories has recently been immensely aided by advances in computational algebra. However, these symbolic (Gröbner) methods are severely limited by algorithmic issues such as exponential space complexity and being highly sequential. In this paper, we introduce a novel paradigm of numerical algebraic geometry which in a plethora of situations overcomes these short-comings. Its so-called 'embarrassing parallelizability' allows us to solve many problems and extract physical information which elude the symbolic methods. We describe …

实代数与半代数集拓扑类型的有效界

Effective Bounds on Topological Types of Real Algebraic and Semialgebraic Sets

本文基于Hardt定理与模型论框架，证明了由有界次多项式定义的实代数集拓扑类型数量及半代数同胚复杂度的有效界存在性，为显式分类提供了新视角并扩展至半代数集。

Orthogonal and unitary tensor decomposition from an algebraic perspective

While every matrix admits a singular value decomposition, in which the terms are pairwise orthogonal in a strong sense, higher-order tensors typically do not admit such an orthogonal decomposition. Those that do have attracted attention from theoretical computer science and scientific computing. We complement this existing body of literature with an algebro-geometric analysis of the set of orthogonally decomposable tensors. More specifically, we prove that they form a real-algebraic variety defined by polynomials of degree at most four. The exact degrees, and the corresponding polynomials, are…

A Topological Characterization of the Middle Perversity Intersection Complex for Arbitrary Complex Algebraic Varieties

For an arbitrary complex algebraic variety which is not necessarily pure dimensional, the intersection complex can be defined as the direct sum of the Deligne-Goresky-MacPherson intersection complexes of each irreducible component. We give two axiomatic topological characterizations of the middle perversity direct sum intersection complex, one stratification dependent and the other stratification independent. To accomplish this, we show that this direct sum intersection complex can be constructed using Deligne's construction in the more general context of topologically stratified spaces. A con…

幂代数表示的拓扑诺ether性

Topological noetherianity for powers of algebraic representations

研究幂代数表示在对称与线性群作用下的拓扑诺ether性，扩展到正交和辛群，证明其拓扑诺ether性。

The Landscape of Multi-Layer Linear Neural Network From the Perspective of Algebraic Geometry

The clear understanding of the non-convex landscape of neural network is a complex incomplete problem. This paper studies the landscape of linear (residual) network, the simplified version of the nonlinear network. By treating the gradient equations as polynomial equations, we use algebraic geometry tools to solve it over the complex number field, the attained solution can be decomposed into different irreducible complex geometry objects. Then three hypotheses are proposed, involving how to calculate the loss on each irreducible geometry object, the losses of critical points have a certain ran…

Virtual Poincaré polynomial of the link of a real algebraic variety

The Euler characteristic of the link of a real algebraic variety is an interesting topological invariant in order to discuss local topological properties. We prove in the paper that an invariant stronger than the Euler Characteristic is well defined for the link of an algebraic variety: its virtual Poincaré polynomial.

Definable Topological Dynamics for Trigonalizable Algebraic Groups over Qp

We study the flow (G(Qp); SG(Qp)) of trigonalizable algebraic group acting on its type space, focusing on the problem raised in [17] of whether weakly generic types coincide with almost periodic types if the group has global definable f-generic types, equivalently whether the union of minimal subflows of a suitable type space is closed. We will give a description of of f-generic types of trigonalizable algebraic groups, and prove that every f-generic type is almost periodic.

Topological versions of Abel-Jacobi, the height pairing, and the Poincaré bundle

We extend to the topological setting the classical constructions of the Abel-Jacobi mapping on homologically trivial algebraic cycles and the height pairing between two such cycles. We further interpret the height pairing between homologically trivial topological cycles (with disjoint support) as giving a lifting of their Abel-Jacobi images to the fiber of the Poincaré bundle, extending work of R. Hain in the algebraic setting. Part II of the current revision further explores the relationship between the topological height pairing and the classical height pairing in the case of algebraic cycle…

关于全纯叶层代数重数的拓扑不变性

On the topological invariance of the algebraic multiplicity of holomorphic foliations

本文研究全纯叶层代数重数的拓扑不变性，证明二维下重数为1是拓扑不变量并推广至高维，验证多类叶层的不变性，还修正了Rosas-Bazan的相关结论。

Equivariant Semi-topological Invariants, Atiyah's KR-theory, and Real Algebraic Cycles

We establish an Atiyah-Hirzebruch type spectral sequence relating real morphic cohomology and real semi-topological K-theory and prove it to be compatible with the Atiyah-Hirzebruch spectral sequence relating Bredon cohomology and Atiyah's KR-theory constructed by Dugger. An equivariant and a real version of Suslin's conjecture on morphic cohomology are formulated, proved to come from the complex version of Suslin conjecture and verified for certain real varieties. In conjunction with the spectral sequences constructed here this allows the computation of the real semi-topological K-theory of s…

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.

对拓扑数据分析进行实代数簇采样

Sampling real algebraic varieties for topological data analysis

本文提出了一种新的自适应算法，用于在给定定义多项式的情况下，保证密度地生成实代数簇上的点集，同时通过数值代数几何方法提供正式保证，并利用几何启发式方法减少样本大小，从而更有效地应用拓扑数据分析方法。

Stratified-algebraic vector bundles

We investigate stratified-algebraic vector bundles on a real algebraic variety X. A stratification of X is a finite collection of pairwise disjoint, Zariski locally closed subvarieties whose union is X. A topological vector bundle on X is called a stratified-algebraic vector bundle if, roughly speaking, there exists a stratification of X such that the restriction of the bundle to each stratum is an algebraic vector bundle. In particular, every algebraic vector bundle on X is stratified-algebraic. It turns out that stratified-algebraic vector bundles have many surprising properties, which disti…

Weight zero part of the first cohomology of complex algebraic varieties

We show that the weight 0 part of the first cohomology of a complex algebraic variety $X$ is a topological invariant, and give an explicit description of its dimension using a topological construction of the normalization of $X$, where $X$ can be reducible, but must be equidimensional. The first assertion is known in the $X$ compact case by A. Weber, where intersection cohomology is used. Note that the weight 1 or 2 part of the first cohomology is not a topological (or even analytic) invariant in the non-compact case by Serre's example.

The Ceresa class: tropical, topological, and algebraic

The Ceresa cycle is an algebraic cycle attached to a smooth algebraic curve with a marked point, which is trivial when the curve is hyperelliptic with a marked Weierstrass point. The image of the Ceresa cycle under a certain cycle class map provides a class in étale cohomology called the Ceresa class. Describing the Ceresa class explicitly for non-hyperelliptic curves is in general not easy. We present a "combinatorialization" of this problem, explaining how to define a Ceresa class for a tropical algebraic curve, and also for a topological surface endowed with a multiset of commuting Dehn twi…

Rational cuspidal curves in projective surfaces. Topological versus algebraic obstructions

We study rational cuspidal curves in projective surfaces. We specify two criteria obstructing possible configurations of singular points that may occur on such curves. One criterion generalizes the result of Fernandez de Bobadilla, Luengo, Melle--Hernandez and Nemethi and is based on the Bezout theorem. The other one is a generalization of the result obtained by Livingston and the author and relies on Ozsvath--Szabo inequalities for $d$-invariants in Heegaard Floer homology. We show by means of explicit calculations that the two approaches give very similar obstructions.

通过拓扑量子场论对代数不变量的量子化

Quantization of algebraic invariants through Topological Quantum Field Theories

本文研究用拓扑量子场论量子化代数不变量，给出基于欧拉示性数的必要条件及曲面情形的部分充分条件，并证明正维数代数群的表示簇庞加莱多项式不可幺半TQFT量子化。

The AKSZ Construction in Derived Algebraic Geometry as an Extended Topological Field Theory

We construct a family of oriented extended topological field theories using the AKSZ construction in derived algebraic geometry, which can be viewed as an algebraic and topological version of the classical AKSZ field theories that occur in physics. These have as their targets higher categories of symplectic derived stacks, with higher morphisms given by iterated Lagrangian correspondences. We define these, as well as analogous higher categories of oriented derived stacks and iterated oriented cospans, and prove that all objects are fully dualizable. Then we set up a functorial version of the A…

Non-commutative extensions of two-dimensional topological field theories and Hurwitz numbers for real algebraic curves

It is well-known that classical two-dimensional topological field theories are in one-to-one correspondence with commutative Frobenius algebras. An important extension of classical two-dimensional topological field theories is provided by open-closed two-dimensional topological field theories. In this paper we extend open-closed two-dimensional topological field theories to nonorientable surfaces. We call them Klein topological field theories(KTFT). We prove that KTFTs bijectively correspond to algebras with certain additional structures, called structure algebras. Semisimple structure algebra…

Elements of spin Hurwitz theory: closed algebraic formulas, blobbed topological recursion, and a proof of the Giacchetto-Kramer-Lewanski conjecture

In this paper, we discuss the properties of the generating functions of spin Hurwitz numbers. In particular, for spin Hurwitz numbers with arbitrary ramification profiles, we construct the weighed sums which are given by Orlov's hypergeometric solutions of the 2-component BKP hierarchy. We derive the closed algebraic formulas for the correlation functions associated with these tau-functions, and under reasonable analytical assumptions we prove the loop equations (the blobbed topological recursion). Finally, we prove a version of topological recursion for the spin Hurwitz numbers with the spin …

从代数统计角度看离散时间、离散状态多状态马尔可夫模型

Discrete-time, discrete-state multistate Markov models from the perspective of algebraic statistics

本文从代数统计角度研究离散时间、离散状态多状态马尔可夫模型，通过研究联合路径概率间多项式关系，建立与事件历史分析的联系，推导非齐次和齐次模型相关结论，从两角度研究最大似然估计并给出数据应用。

Three-dimensional topological field theory and symplectic algebraic geometry II

Motivated by the path integral analysis of boundary conditions in a 3-dimensional topological sigma-model, we suggest a definition of the 2-category associated with a holomorphic symplectic manifold X and study its properties. The simplest objects of this 2-category are holomorphic lagrangian submanifolds of X. We pay special attention to the case when X is the total space of the cotangent bundle of a complex manifold U or a deformation thereof. In the latter case the endomorphism category of the zero section is a monoidal category which is an A-infinity deformation of the 2-periodic derived c…

Orthogonal tensor decomposition and orbit closures from a linear algebraic perspective

We study orthogonal decompositions of symmetric and ordinary tensors using methods from linear algebra. For the field of real numbers we show that the sets of decomposable tensors can be defined be equations of degree 2. This gives a new proof of some of the results of Robeva and Boralevi et al. Orthogonal decompositions over the field of complex numbers had not been studied previously; we give an explicit description of the set of decomposable tensors using polynomial equalities and inequalities, and we begin a study of their closures. The main open problem that arises from this work is to ob…

$q$-变形的拓扑递归、权重向量与代数结构

$q$-Deformed Topological Recursion, Weight Vectors and Algebraic Structures

本文通过研究$q$-变形的$\mathcal{W}$-代数表示中的最高权重向量，推导出$q$-变形的拓扑递归，并证明其产生$q$-变形的量子曲线，统一了量子可积性的多种方法。