Every connected sum of lens spaces is a real component of a uniruled algebraic variety

Existence of rational curves on algebraic varieties, minimal rational tangents, and applications

This survey paper discusses some of the recent progress in the study of rational curves on algebraic varieties. It was written for the survey volume of the priority programme "Global Methods in Complex Geometry", supported by the DFG. To start, we discuss existence results for rational curves on foliated manifolds. We construct the tangent morphism, define the variety of minimal rational tangents (VMRT) and name a number of results that show that many of the geometry properies of a uniruled variety are encoded in the projective geometry of the VMRT. The results are then applied in two differen…

On birational involutions of $P^3$

Let $X$ be a rationally connected three-dimensional algebraic variety and let $τ$ be an element of order two in the group of its birational selfmaps. Suppose that there exists a non-uniruled divisorial component of the $τ$-fixed point locus. Using the equivariant minimal model program we give a rough classification of such elements.

Estimating the number of solutions equation of N-point gravitational lens algebraic geometry methods

One of the main problems in the study of system of equations of the gravitational lens, is the computation of coordinates from the known position of the source. In the process of computing finds the solution of equations with two unknowns. The difficulty lies in the fact that, in general, is not known constructive or analytical algorithm for solving systems of polynomial equations In this connection, use numerical methods like the method of tracing. For the N-point gravitational lenses have a system of polynomial equations. Systems Research is advisable to start with an assessment of the numbe…

Real Algebraic Threefolds III: Conic Bundles

This is the third of a series of papers studying real algebraic threefolds, but the methods are mostly independent from the previous two. Let $f:X\to S$ be a map of a smooth projective real algebraic 3-fold to a surface $S$ whose general fibers are rational curves. Assume that the set of real points of $X$ is an orientable 3-manifold $M$. The aim of the paper is to give a topological description of $M$. It is shown that $M$ is either Seifert fibered or a connected sum of lens spaces. Much stronger results hold if $S$ is rational.

The loss surface of deep linear networks viewed through the algebraic geometry lens

By using the viewpoint of modern computational algebraic geometry, we explore properties of the optimization landscapes of the deep linear neural network models. After clarifying on the various definitions of "flat" minima, we show that the geometrically flat minima, which are merely artifacts of residual continuous symmetries of the deep linear networks, can be straightforwardly removed by a generalized $L_2$ regularization. Then, we establish upper bounds on the number of isolated stationary points of these networks with the help of algebraic geometry. Using these upper bounds and utilizing …

The topology of compact Lie group actions through the lens of finite models

Given a compact, connected Lie group $K$, we use principal $K$-bundles to construct manifolds with prescribed finite-dimensional algebraic models. Conversely, let $M$ be a compact, connected, smooth manifold which supports an almost free $K$-action. Under a partial formality assumption on the orbit space and a regularity assumption on the characteristic classes of the action, we describe an algebraic model for $M$ with commensurate finiteness and partial formality properties. The existence of such a model has various implications on the structure of the cohomology jump loci of $M$ and of the r…

透过正几何的视角看 Vandermonde 胞腔

Vandermonde Cells Through the Lens of Positive Geometry

本文研究了 Vandermonde 胞腔的几何与代数结构，推导了其边界分量的显式参数化与代数方程，通过引入允许边界内部奇点的正几何推广，证明了其构成正几何及亏格零对，为其提供了新的代数与几何理解。

On the central path of semidefinite optimization: Degree and worst-case convergence rate

In this paper, we investigate the complexity of the central path of semidefinite optimization through the lens of real algebraic geometry. To that end, we propose an algorithm to compute real univariate representations describing the central path and its limit point, where the limit point is described by taking the limit of central solutions, as bounded points in the field of algebraic Puiseux series. As a result, we derive an upper bound $2^{O(m+n^2)}$ on the degree of the Zariski closure of the central path, when $μ$ is sufficiently small, and for the complexity of describing the limit point…

代数揭开深度学习——神经代数几何学之邀

Algebra Unveils Deep Learning -- An Invitation to Neuroalgebraic Geometry

本文倡导通过代数几何视角研究机器学习模型参数化的函数空间，聚焦多项式激活神经网络等代数模型，建立代数几何不变量与机器学习核心概念的对应关系，奠定神经代数几何学基础。

直线格拉斯曼流形上的最大似然估计

Maximum Likelihood Estimation on the Grassmannian of Lines

研究正格拉斯曼流形，通过代数统计视角给出直线格拉斯曼流形最大似然度封闭公式，推测概率单纯形有唯一局部最大值并提供计算证据。

行列式点过程的似然几何

Likelihood Geometry of Determinantal Point Processes

本文利用代数统计方法研究行列式点过程的似然几何，通过计数并计算小模型的对数似然临界点，反驳了Brunel等人提出的一个猜想。

Theta surfaces

A theta surface in affine 3-space is the zero set of a Riemann theta function in genus 3. This includes surfaces arising from special plane quartics that are singular or reducible. Lie and Poincaré showed that theta surfaces are precisely the surfaces of double translation, i.e. obtained as the Minkowski sum of two space curves in two different ways. These curves are parametrized by abelian integrals, so they are usually not algebraic. This paper offers a new view on this classical topic through the lens of computation. We present practical tools for passing between quartic curves and their th…