San Vu Ngoc
This article gives a classification, up to symplectic equivalence, of singular Lagrangian foliations given by a completely integrable system of a 4-dimensional symplectic manifold, in a full neighbourhood of a singular leaf of focus-focus (=nodal) type.
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About 6 years ago, semitoric systems were classified by Pelayo & Vu Ngoc by means of five invariants. Standard examples are the coupled spin oscillator on $\mathbb{S}^2 \times \mathbb{R}^2$ and coupled angular momenta on $\mathbb{S}^2 \times \mathbb{S}^2$, both having exactly one focus-focus singularity. But so far there were no explicit examples of systems with more than one focus-focus singularity which are semitoric in the sense of that classification. This paper introduces a 6-parameter family of integrable systems on $\mathbb{S}^2 \times \mathbb{S}^2$ and proves that, for certain ranges o…
Semitoric systems with $2n$ Focus-Focus singularities
该研究构造了含2n个焦点-焦点奇点的半环面系统新显式例子，给出了对应辛流形上的半环面动量映射，丰富了半环面系统的实例库。
We explain how a classical theorem by Arnol'd and Melrose on non-singular functions on a symplectic manifold with boundary can be proved in few lines, and we use the same method to obtain a new result, which is a normal form with functional invariants for the first occurring singularities
The height invariant of a four-parameter semitoric system with two focus-focus singularities
本文针对含两个焦点-焦点奇点的四参数半环面系统，计算了其高度不变量（除多边形不变量外），并证明该不变量的两个分量以复杂方式编码了系统的对称性，推进了多参数半环面系统不变量的计算研究。
We study focus-focus singularities (also known as nodal singularities, or pinched tori) of Lagrangian fibrations on symplectic $4$-manifolds. We show that, in contrast to elliptic and hyperbolic singularities, there exist homeomorphic focus-focus singularities which are not diffeomorphic. Furthermore, we obtain an algebraic description of the moduli space of focus-focus singularities up to smooth equivalence, and show that for double pinched tori this space is one-dimensional. Finally, we apply our construction to disprove Zung's conjecture which says that any non-degenerate singularity can be…
The presence of focus-focus singularities in semi-toric integrables Hamiltonian systems is one of the reasons why there cannot exist global Action-Angle coordinates on such systems. At focus-focus critical points, the Liouville-Arnold-Mineur theorem does not apply. In particular, the affine structure of the image of the moment map around has non-trivial monodromy. In this article, we establish that the singular behaviour and the multi-valuedness of the Action integrals is given by a complex logarithm. This extends a previous result by Vu Ngoc to any dimension. We also calculate the monodromy m…
The paper presents an algorithm for topological classification of nondegenerate saddle-focus singularities of integrable Hamiltonian systems with three degrees of freedom up to semi-local equivalence. In particular, we prove that any singularity of saddle-focus type can be represented as an almost direct product in which the acting group is cyclic. Based on the constructed algorithm, a complete list of singularities of saddle-focus type of complexity $1$, $2$, and $3$, i.e., singularities whose leaf contains one, two, or three singular points of rank $0$, is obtained. Earlier both singularitie…
We construct a 1-parameter family $F_t=(J, H_t)_{0 \leq t \leq 1}$ of integrable systems on a compact $4$-dimensional symplectic manifold $(M, ω)$ that changes smoothly from a toric system $F_0$ with eight elliptic-elliptic singular points via toric type systems to a semitoric system $F_t$ for $ t^- < t < t^+$. These semitoric systems $F_t$ have precisely four elliptic-elliptic and four focus-focus singular points. Moreover, at $t= \frac{1}{2}$, the system has precisely two focus-focus fibres each of which contains exactly two focus-focus points, giving these fibres the shape of double pinched…
We overview our recent work defining and studying normal crossings varieties and subvarieties in symplectic topology. This work answers a question of Gromov on the feasibility of introducing singular (sub)varieties into symplectic topology in the case of normal crossings singularities. It also provides a necessary and sufficient condition for smoothing normal crossings symplectic varieties. In addition, we explain some connections with other areas of mathematics and discuss a few directions for further research.
Classifying affine structures with focus-focus singularities
本文研究可积系统中focus-focus奇异纤维的仿射结构分类，证明其与简单半环系统分类的等价性，并给出多收缩纤维情形下的反例。
We prove, assuming that the Bohr-Sommerfeld rules hold, that the joint spectrum near a focus-focus critical value of a quantum integrable system determines the classical Lagrangian foliation around the full focus-focus leaf. The result applies, for instance, to h-pseudodifferential operators, and to Berezin-Toeplitz operators on prequantizable compact symplectic manifolds.
Let $(M,Ω)$ be a connected symplectic 4-manifold and let $F=(J,H) : M \to \mathbb{R}^2$ be a completely integrable system on $M$ with only non-degenerate singularities and for which $J : M \to \mathbb{R}$ is a proper map. Assume that $F$ does not have singularities with hyperbolic blocks and that $p_1,...,p_n$ are the focus-focus singularities of $F$. For each subset $S=\{i_1,...,i_j\}$ we will show how to modify $F$ locally around any $p_i, i \in S$, in order to create a new integrable system $\tilde{F}=(J, \tilde{H}) : M \to \mathbb{R}^2$ such that its classical spectrum $\tilde{F}(M)$ conta…
In this paper we collect some results on the obstruction spaces for rational surface singularities and minimally elliptic surface singularities. Based on the known results we calculate higher obstruction spaces for such surface singularities. The results imply that in general the higher obstruction spaces of deforming semi-log-canonical surfaces do not vanish. We apply the calculation result to show that there is no Li-Tian and Behrend-Fantechi style virtual fundamental class on such moduli space of semi-log-canonical surfaces.
Symplectic classification for universal unfoldings of $A_n$ singularities in integrable systems
本文研究可积哈密顿系统中$A_n$奇点普适开折的辛分类问题，得到局部、半局部辛正规形及半全局拉格朗日叶状结构分类，明确了不同情形下的辛不变量与结构稳定性。
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