会议摘要（Abstract）

The interaction between geometry (Riemannian, algebraic, Kaehler, metric,...), quantum field theory and analysis (elliptic and parabolic PDEs, calculus of variations) has brought spectacular advances and generated deep insight in all those disciplines.
Minima of variational integrals may provide optimal solutions to geometric problems, and parabolic PDEs may deform initial geometrical structures into optimal ones, thereby solving important geometric problems. Functionals from QFT contain rich structures that can be exploited for the construction of subtle geometric invariants. In turn, such problems typically lead to very difficult analytical challenges. The resulting PDEs are not only highly non-linear, but from a variational perspective usually are not contained in the range of the Palais-Smale condition, and therefore, standard methods usually break down. This challenge gave an important impetus to the theory of non-linear PDEs. One line of research exploited convexity properties, typically arising from non-positive curvature, another one depended on an extremely careful study of the formation of singularities, which in turn had to use geometric features or algebraic properties from QFT.
During this workshop, we want to bring people together to explore various current research questions in this field, including
-- Analytic methods for studying moduli spaces in algebraic geometry
-- A general mathematical theory of the action functionals of QFT and the resulting challenges  for PDE theory to construct minimizers or other critical points and to understand their regularity properties
-- Relations between QFT functionals and geometric constructions, like Kapustin-Witten and Higgs fields
-- Bernstein and Dirichlet problems for minimal submanifolds of Euclidean spaces and spheres
-- The role of PDEs in metric space geometry
-- The geometry of positive sectional curvature
-- The approximation of geometric objects by discrete ones

我们希望通过本次研讨会组织与会者一起探讨几何、量子场论和分析相互作用领域的前沿问题，包括：

-- 代数几何中模空间的分析方法；

-- 量子场论中作用泛函的一般数学理论及产生的偏微分方程理论挑战，如构造极小解或其它临界点，研究它们的正则性；

--  量子场论中作用泛函与几何构造的关系，如Kapustin-Witten 和 Higgs 场；

--  欧氏空间和球面中极小子流形的Bernstein 和 Dirichlet 问题；

--  偏微分方程在度量空间几何中的作用；

--  正曲率几何学；

--  几何对象的离散逼近。

举办意义(Description of the aim)

几何，量子场论和非线性分析之间的相互作用为这些领域中带来了巨大的推动和深刻的洞察。变分问题的极小点为几何问题提供了最优解，抛物方程把初始几何结构形变为最优解，从而解决重要的几何问题。量子场论中的作用量泛函包含丰富的结构，可以用于构造精妙的几何不变量。反之，这些问题也常常带来困难的分析挑战，其产生的偏微分方程不仅是高度非线性的，而且从变分观点看通常不能包含在满足Palais-Smale条件的范围中，因此，标准的方法往往失效。这种挑战为我们提供了发展非线性分析理论的重要动力，一条途径是运用凸性性质，常见于非正曲率情形，另一条途径是依赖于对奇性形成的精细研究，这反过来需要利用量子场论的几何特性或代数性质。在本次研讨会期间，我们将探讨有关几何，量子场论和非线性分析之间相互作用领域的前沿研究问题。