Mathematical Sciences: Logarithmic Sobolev Inequalities and Knot Theory

格罗斯教授和金教授将在谐波分析和几何量子场论领域进行研究。格罗斯教授的研究将集中在对数索博列夫不等式这一蓬勃发展的领域，他将尝试在各种未经尝试的情况下证明这一不等式，包括一些无限维的情况。金教授的研究将涉及量子场论及其与几何学和拓扑学的关系。特别是，他将使用量子场论的思想来研究欧几里得三维空间中的结、链和辫的理论。对数索博列夫不等式在许多数学领域都很有用，从光滑表面的理论和它们如何振动到各种系统达到平衡的速度。结理论是数学分支中被称为拓扑学的一个领域，最近被证明与物理学和数学中的其他形式化理论有关，特别是量子场理论。

Professors Gross and King will conduct research in the areas of harmonic analysis and geometrical quantum field theory. Professor Gross' research will be in the flourishing field of logarithmic Sobolev inequalities which he will attempt to prove in various untried circumstances including some infinite dimensional situations. Professor King's research will involve quantum field theory and its relation to geometry and topology. In particular he will use ideas of quantum field theory to investigate the theory of knots, links, and braids in Euclidean three space. Logarithmic Sobolev inequalities have been useful in many areas of mathematics ranging from the theory of smooth surfaces and how they vibrate to how fast systems of various kinds reach equilibrium. The theory of knots, an area within the branch of mathematics called topology, was recently shown to be connected to other formalisms in physics and mathematics, in particular the theory of quantum fields.