拟椭圆上同调和它在几何、拓扑中的应用

An elliptic cohomology theory is an even periodic multiplicative generalized cohomology theory whose associated formal group is the formal completion of an elliptic curve. It is the intersection point of many areas, including algebraic topology, algebraic geometry, mathematical physics, representation theory, number theory. Quasi-elliptic cohomology is both reflecting the geometric nature of elliptic curves and more practicable to study. It is close to elliptic cohomology but has signature difference..It is an old idea of Witten that the elliptic cohomology of a space is related to the circle-equivariant K-theory of its free loop space with the circle acting on it by rotating loops. This relation is usually difficult to interpret. Quasi-elliptic cohomology provides a new idea to interpret it. I constructed it as orbifold K-theory of a constant loop space of bibundles. The idea is being generalized to the construction of elliptic cohomology to finally prove Witten's conjecture..Quasi-elliptic cohomology can be expressed explicitly as equivariant K-theories of fixed point spaces. Other than the good features inherited from equivariant K-theories, many constructions on it, such as its elliptic power operation, its spectrum, has neater forms than those on elliptic cohomology theories and can be generalized to many other cohomology theories. Moreover, quasi-elliptic cohomology is more powerful to study math problems on elliptic curves than elliptic cohomology theories themselves. Applying quasi-elliptic cohomology, I proved the classification of the finite subgroups on the Tate curve, which has the form of Strickland's theorem for Morava E-theories..As equivariant K-theories, quasi-elliptic cohomology has the change-of-group isomorphism, which implies, as equivariant cohomology theories, quasi-elliptic cohomology may be globalized. I gave an explicit construction of an equivariant orthogonal spectrum representing quasi-elliptic cohomology. The tools and techniques that I developed in the process can be applied to the construction of equivariant orthogonal spectrum of a large family of equivariant cohomology theories. Moreover, quasi-elliptic cohomology is a motivating example for a new global homotopy theory that is more suitable to study the globalness of elliptic cohomology theories..Other than its contribution to the study of elliptic cohomology, quasi-elliptic cohomology itself is a distinctive object to study. There are many intriguing and unsual mathematical phenomenon on it. For example, though it is close to elliptic cohomology theories, its chromatic level is 1 whereas the chromatic level of elliptic cohomology theories is 2. In addition, comparing with most elliptic cohomology theories, quasi-elliptic cohomology has more direct relation with Hopkins-Kuhn-Ravenel character theory.

椭圆上同调是以椭圆曲线为形式群的广义上同调理论，是通过代数拓扑研究椭圆曲线的几何性质的重要不变量。它联结了代数拓扑，代数几何，数学物理，群表示论，数论等诸多数学领域。目前椭圆上同调的计算和研究在世界范围内依旧是个难题。拟椭圆上同调是一个接近椭圆上同调但彼此独立的理论。它可以通过不动点空间的等变K理论表示出来。申请人从它入手，通过双丛给出了泰特K理论的闭路空间构造，部分证明了Witten关于椭圆上同调的猜想，并以此为基础在继续证明Witten的猜想。另外，申请人通过拟椭圆上同调证明了泰特曲线上几何结构的分类定理，并在尝试运用类似研究方法给出所有椭圆曲线上几何结构的分类。另外，通过研究拟椭圆上同调申请人给出了构造椭圆上同调等变正交谱的可能方法，并在证实该方法的有效性，同时在着手研究拟椭圆上同调在分色同伦论中的位置以及和其他理论的联系。

郇真研究的主要对象是椭圆上同调。Witten的一个猜想指出，一个空间的椭圆上同调与其自由闭路空间的圆群等变K理论密切相关。郇真在博士论文中，构造了一个闭路轨形（现被称为Huan’s Inertia orbifold），并将拟椭圆上同调定义为它的轨形K理论。该理论非常接近泰特曲线的椭圆上同调，并能反映了椭圆曲线和椭圆上同调理论的关键特征。现在，这个理论的名字，quasi-elliptic cohomology，已经被世界上大多数数学家所理解和认同。他们给出了关于拟椭圆上同调与物理学的关系的猜想，郇真正在参与证明。.在2019-2023年期间，郇真继续研究拟椭圆上同调。她从两个角度推广了这些理论。通过对N个Huan’s Inertia orbifold进行复合，郇真定义了一种新的上同调理论——拟理论，并构建了该理论的等变正交谱。Sati和Schreiber证明了它是某种循环闭路栈的上同调，应该与共形场论有关。此外，郇真从另一角度推广了拟椭圆上同调。她与合作者一起构造了twisted和twisted Real拟椭圆上同调。鉴于共形场论和椭圆上同调之间的关系，人们很自然地期望非定向共形场论与Real椭圆上同调之间存在联系。Twisted Real拟椭圆上同调被构造为Real Huan’s Inertia orbifold的Freed-Moore K-理论，它具有所有我们预期的所有关键性质。此外，郇真构造了Real拟椭圆上同调的幂运算，并在继续研究它和泰特曲线之间的联系。.此外，郇真仍在构造拟椭圆上同调和椭圆上同调的合适代表物。代数拓扑学中的大多数经典理论，包括K-理论和协边理论，都有几何构造。由于椭圆上同调是K-理论的高阶版本，其几何对象是向量丛，因此郇真构建了2向量丛和等变2向量丛的模型。类似于群表示和向量丛之间的关系，2-表示是单点空间上的2-向量丛。郇真正在研究该2-向量丛模型和椭圆上同调之间的关系。.此外，为了证明关于椭圆曲线上几何结构分类的猜想，郇真和合作者研究了generalized Morava-E理论的形式群的水平结构。应用Hopkins-Kuhn-Ravenel特征标理论，他们证明了，对于generalized Morava E-理论，它的形式群上的水平结构可以按照猜想中的形式进行分类。

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