Cup and cap products in intersection (co)homology

We construct cup and cap products in intersection (co)homology with field coefficients. The existence of the cap product allows us to give a new proof of Poincaré duality in intersection (co)homology which is similar in spirit to the usual proof for ordinary (co)homology of manifolds.     

Duality methods for a class of quasilinear systems

Duality methods are used to generate explicit solutions to nonlinear Hodge systems, demonstrate the well-posedness of boundary value problems, and reveal, via the Hodge–Bäcklund transformation, underlying symmetries among superficially different forms of the equations.     

HL-2M 真空室保温层设计及安装

基于 HL-2M 真空室烘烤保温要求，通过有限元分析和原型件实验确定采用陶瓷纤维与纳米级微孔材 料组合作为 HL-2M 真空室保温材料。在 30℃时，保温层的导热系数小于 0.027W⋅m⁻¹·℃⁻¹；300℃时，导热系数 小于 0.038W⋅m⁻¹·℃⁻¹。在保温层厚度 25mm、热面温度 300℃且达到稳态时，冷面可控制在 85℃以下，线圈侧的 温度低于 60℃，整体热损失小于 12kW，满足 HL-2M 真空室烘烤需求。      

Errata and addendum to “On the prime geodesic theorem for hyperbolic ‐manifolds” Math. Nachr. 291 (2018), no. 14–15, 2160–2167

We correct the exponent in the error term of the prime geodesic theorem for hyperbolic 3‐manifolds 1 and in Park's theorem for higher dimensions [ 3 , 2 ].     

The projective dimension of three cubics is at most 5

Let R be a polynomial ring over a field and I an ideal generated by three forms of degree three. Motivated by Stillman's question, Engheta proved that the projective dimension $\mathrm{pd}(R/I)$ of $R/I$ is at most 36, although the example with largest projective dimension he constructed has $\mathrm{pd}(R/I)=5$. Based on computational evidence, it had been conjectured that $\mathrm{pd}(R/I)\le 5$. In the present paper we prove this conjectured sharp bound.     

Gaps in the number of generators of monomial Togliatti systems

Let $I_{d,n}?k[x_0,?,x_n]$ be a minimal monomial Togliatti system of forms of degree d. In [4], Mezzetti and Miró-Roig proved that the minimal number of generators $\mu (I_{d,n})$ of $I_{d,n}$ lies in the interval $[2n+1,\left(\begin{array}{c}n+d?1\\ n?1\end{array}\right)]$. In this paper, we prove that for $n\ge 4$ and $d\ge 3$, the integer values in $[2n+3,3n?1]$ cannot be realized as the number of minimal generators of a minimal monomial Togliatti system. We classify minimal monomial Togliatti systems $I_{d,n}?k[x_0,?,x_n]$ of forms of degree d with $\mu (I_{d,n})=2n+2$ or 3n (i.e. with the minimal number of generators reaching the border of the non-existence interval). Finally, we prove that for $n=4$, $d\ge 3$ and $\mu \in [9,\left(\begin{array}{c}d+3\\ 3\end{array}\right)]?\{11\}$ there exists a minimal monomial Togliatti system $I_{d,n}?k[x_0,?,x_n]$ of forms of degree d with $\mu (I_{n,d})=\mu$.     

On the variety of triangles for a hyper-Kähler fourfold constructed by Debarre and Voisin

We study the similarities between the Fano varieties of lines on a cubic fourfold, a hyper-Kähler fourfold studied by Beauville and Donagi, and the hyper-Kähler fourfold constructed by Debarre and Voisin in [3]. We exhibit an analog of the notion of “triangle” for these varieties and prove that the 6-dimensional variety of “triangles” is a Lagrangian subvariety in the cube of the constructed hyper-Kähler fourfold.