On the representability of ${\Cal Hilb}^nk[x]_{(x)}$

Representability of Hilbert schemes and Hilbert stacks of points

We show that the Hilbert functor of points on an arbitrary separated algebraic stack is an algebraic space. We also show the algebraicity of the Hilbert stack of points on an algebraic stack and the algebraicity of the Weil restriction of an algebraic stack along a finite flat morphism. For the latter two results, no separation assumptions are necessary.

对偏微分方程解的导出模栈表示性的Bornological视角

A Bornological Perspective on the Representability of Derived Moduli Stacks of Solutions to PDEs

本文通过Bornological理论展示导出模栈的表示性无需复杂分析工具，提出新的微分几何模型以嵌入已有的Bornological几何理论。

On semidefinite representations of non-closed sets

Spectrahedra are sets defined by linear matrix inequalities. Projections of spectrahedra are called semidefinitely representable sets. Both kinds of sets are of practical use in polynomial optimization, since they occur as feasible sets in semidefinite programming. There are several recent results on the question which sets are semidefinite representable. So far, all results focus on the case of closed sets. In this work we develop a new method to prove semidefinite representability of sets which are not closed. For example, the interior of a semidefinite representable set is shown to be semid…