Five connected research areas五条彼此连接的研究主线
I work across restriction theory, local smoothing, projections, distance problems, Furstenberg sets, and Kakeya.
我的研究涵盖限制性理论、局部光滑、投影、距离问题、Furstenberg 集与 Kakeya。
How tightly can a wave concentrate as it travels?
一束波在传播时可以集中到什么程度？
Work on wave equations, Strichartz estimates, cones, and square functions uses wave packets, decoupling, and multiscale arguments to measure concentration and dispersion.
围绕波方程、Strichartz 估计、锥面与平方函数的工作，以波包、解耦和多尺度论证量化波的集中与扩散。
What spatial structure is forced when frequency lives near a curved surface?
当频率被限制在弯曲曲面附近，空间中会被迫出现什么结构？
Restriction theory links curved frequency surfaces to wave concentration. Wang's work brings brooms, weighted estimates, decoupling, tube incidences, and two-ends inequalities into this still-open landscape.
限制性理论把弯曲频率曲面与波的集中联系起来。王虹的工作将扫帚结构、加权估计、解耦、细管相交和 two-ends 不等式带入这一仍然开放的领域。
What remains visible when a fractal set is measured from many viewpoints?
从不同角度测量一个分形集合时，什么信息不会消失？
Falconer distance sets, radial projections, and restricted projections ask how dimension controls the distances and shadows produced by a set.
Falconer 距离集、径向投影与受限投影研究集合的维数如何控制它产生的距离与影子。
How small can a set be if it is rich along many directions?
如果一个集合沿许多方向都很丰富，它最小能有多小？
Dimension estimates for directional fractals connect incidence geometry, projection theorems, and multiscale decompositions. These results form an independent research arc as well as a bridge to restriction and Kakeya.
方向性分形的维数估计连接相交几何、投影定理与多尺度分解。这既是一条独立研究主线，也通向限制性问题与 Kakeya。
How much space is forced by line segments pointing in every direction?
指向所有方向的线段会迫使集合占据多少空间？
A sequence of joint works with Joshua Zahl—from sticky structure and Assouad dimension to volume estimates for unions of convex sets—culminated in a proof of the three-dimensional Kakeya set conjecture.
王虹与 Joshua Zahl 从 sticky 结构、Assouad 维数推进到凸集并集的体积估计，最终给出三维 Kakeya 集合猜想的证明。