Regular Functions Transversal at Infinity

Fukaya categories of two-tori revisited

We construct an A-infinity structure of the Fukaya category explicitly for any flat symplectic two-torus. The structure constants of the non-transversal A-infinity products are obtained as derivatives of those of transversal A-infinity products.

On irregular links at infinity of algebraic plane curves

We give two proofs of a conjecture of the first author (Inv. Math. 98, 1989) that a reduced algebraic plane curve is regular at infinity if and only if its link at infinity is a regular toral link. This conjecture has also been proved by Ha H.~V. using Lojasiewicz numbers at infinity. Our first proof uses the polar invariant and the second proof uses linear systems of plane curve singularities. The second approach (based on a paper in preparation) also proves a stronger conjecture (loc. cit.) describing topologically the regular link at infinity associated with an irregular link at infinity.

On Affine Real Cubic Surfaces

We prove that the space of affine, transversal at infinity, non-singular real cubic surfaces has 15 connected components. We give a topological criterion to distinguish them and show also how these 15 components are adjacent to each other via wall-crossing.

On the set of points at infinity of a polynomial image of ${\mathbb R}^n$

In this work we prove that the set of points at infinity $S_\infty:={\rm Cl}_{{\mathbb R}{\mathbb P}^m}(S)\cap\mathsf{H}_\infty$ of a semialgebraic set $S\subset{\mathbb R}^m$ which is the image of a polynomial map $f:{\mathbb R}^n\to{\mathbb R}^m$ is connected. This result is no further true in general if $f$ is a regular map, although it still works for a large family of regular maps that we call quasi-polynomial maps.

Semicontinuity of the spectrum at infinity

We prove that, for an analytic family of ``weakly tame'' regular functions on an affine manifold, the spectrum at infinity of each function of the family is semicontinuous in the sense of Varchenko.

关于仿射空间正则自同态的DML(1)性质：无穷远点的重数

On the DML(1) property for regular endomorphisms of affine spaces: multiplicities of points at infinity

本文研究仿射空间正则自同态的动力学Mordell-Lang猜想，在无穷远点重数几何平均小于代数度的条件下，证明该猜想对曲线成立。