Real algebraic structures

Signatures of real algebraic curves via plumbing diagrams

We define and calculate signature and nullity invariants for complex schemes for curves in the real projective plane. We use an analog of the Murasugi-Tristram inequality to prohibit certain schemes from being realized by real algebraic curves. We give new formulas for Casson-Gordon invariants of graph manifolds, and signatures of graph links.

Symmetric plane curves of degree 7 : pseudo-holomorphic and algebraic classifications

This paper is motivated by the real symplectic isotopy problem : does there exists a nonsingular real pseudoholomorphic curve not isotopic in the projective plane to any real algebraic curve of the same degree? Here, we focus our study on symmetric real curves on the projective plane. We give a classification of real schemes (resp. complex schemes) realizable by symmetric real curves of degree 7 with respect to the type of the curve (resp. M-symmetric real curves of degree 7). In particular, we exhibit two real schemes which are realizable by real symmetric dividing pseudoholomorphic curves of…

Real algebraic knots of low degree

In this paper we study rational real algebraic knots in $\R P^3$. We show that two real algebraic knots of degree $\leq5$ are rigidly isotopic if and only if their degrees and encomplexed writhes are equal. We also show that any irreducible smooth knot which admits a plane projection with less than or equal to four crossings has a rational parametrization of degree $\leq 6$. Furthermore an explicit construction of rational knots of a given degree with arbitrary encomplexed writhe (subject to natural restrictions) is presented.

显式光滑实代数函数在光滑实代数流形上可能同时具有紧致与非紧致原像

Explicit smooth real algebraic functions which may have both compact and non-compact preimages on smooth real algebraic manifolds

本文提出一种变体，在满足原像紧致性显式条件下于光滑实代数流形上构造非真光滑实代数函数，推进了可微映射奇点理论与实代数几何的发展。

关于 codimension $-1$ 的 round fold 映射的实代数实现

On real algebraic realization of round fold maps of codimension $-1$

本文研究了 codimension $-1$ 的 round fold 映射的实代数实现，探讨其在实代数几何中的基础和重要性。

关于3维球面中与混合多项式相关的实代数链环

On real algebraic links in the 3-sphere associated with mixed polynomials

针对3维球面中实代数链环的分类与刻画问题，本文构造新的混合奇点类以实现这类链环，并为Benedetti-Shiota猜想提供线索。

三维球面上的实代数过度扭转接触结构

Real algebraic overtwisted contact structures on 3-spheres

本文证明了三维球面上所有具有正三维不变量$d_3$的过度扭转接触结构（除可能9个例外）均为实代数结构，且相关开书的页是平面的。

Moduli spaces and algebraic cycles in real algebraic geometry

This thesis intends to make a contribution to the theories of algebraic cycles and moduli spaces over the real numbers. In the study of the subvarieties of a projective algebraic variety, smooth over the field of real numbers, the cycle class map between the Chow ring and the equivariant cohomology ring plays an important role. The image of the cycle class map remains difficult to describe in general; we study this group in detail in the case of real abelian varieties. To do so, we construct integral Fourier transforms on Chow rings of abelian varieties over any field. They allow us to prove t…

广义仿射建筑用于实闭域上半单代数群的构造

Generalized affine buildings for semisimple algebraic groups over real closed fields

本文利用实代数几何构造了半单代数群在实闭域上的仿射建筑，并研究了其无穷远处的球形建筑和基点处的局部建筑，同时得到了群分解结果。

The structure set of an arbitrary space, the algebraic surgery exact sequence and the total surgery obstruction

An introduction to the applications of algebraic surgery to the structure theory of high-dimensional topological manifolds.

Algebraically-Informed Deep Networks (AIDN): A Deep Learning Approach to Represent Algebraic Structures

One of the central problems in the interface of deep learning and mathematics is that of building learning systems that can automatically uncover underlying mathematical laws from observed data. In this work, we make one step towards building a bridge between algebraic structures and deep learning, and introduce \textbf{AIDN}, \textit{Algebraically-Informed Deep Networks}. \textbf{AIDN} is a deep learning algorithm to represent any finitely-presented algebraic object with a set of deep neural networks. The deep networks obtained via \textbf{AIDN} are \textit{algebraically-informed} in the sens…