Speaker: Zhen Huan (Center for Mathematical Sciences, Huazhong University of Science and Technology)

Time: 15:00-16:00 pm, Februray 24, 2023, GMT+8

Venue: Room 1114, Sciences Building No. 1

Abstract:

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 at the intersection of a variety of areas in mathematics, including algebraic topology, algebraic geometry, mathematical physics, representation theory and number theory. From different perspectives we have different interpretations of elliptic cohomology, which gives us different ways to study it. In the talk I will present two approaches to study elliptic cohomology.

One is an idea indicated by Witten that the elliptic cohomology of a space is related to the circle-equivariant K-theory of the free loop space of it. Motivated by this, I constructed quasi-elliptic cohomology during my PhD. It is closely related to Tate K-theory. I formulate the stringy power operation of this theory. Applying that I prove the finite subgroups of Tate curve can be classified by the Tate K-theory of symmetric groups modulo a certain transfer ideal. Recently, together with Young, we construct twisted Real quasi-elliptic cohomology as the twisted KR-theory of loop groupoids. The theory systematically incorporates loop rotation and reflection. After establishing basic properties of the theory, we construct Real analogues of the stringy power operation of quasi-elliptic cohomology as well as its twisted elliptic Pontryagin character, to further study its relation with field theories.

The other approach is via a representing object of elliptic cohomology. Other than elliptic spectrum, a good choice is its geometric object. For example, the geometric object of K-theory is vector bundle. As a higher version of K-theory, the geometric object of elliptic cohomology should be “2-vector bundle”. Analogous to the relation between vector bundles and group representations, a 2-representation of a 2-group is a 2-vector bundle at a point. We glue the local (equivariant) 2-vector bundles together by higher sheafification and obtain the 2-stack of (equivariant) 2-vector bundles. Currently I’m exploring further the relation between this model of 2-vector bundles and elliptic cohomology.

Biography:

郇真的主要研究方向是代数拓扑，代数几何和数学物理。2006年本科毕业于北京大学数学学院，2017年博士毕业于伊利诺伊大学厄巴纳-香槟分校，导师是Charles Rezk。郇真的主要研究对象是椭圆上同调。博士期间她构造了一个闭路空间（现被称为Huan’s Inertia orbifold），在高阶几何领域具有启发意义。该闭路空间的K-理论和椭圆上同调关系紧密，被称为拟椭圆上同调。另外，除了通过代数拓扑领域的正交等变谱，郇真也在尝试从高阶几何的角度构造椭圆上同调的表示物，即2-向量丛。该项工作仍在进一步发展完善中。

Source: School of Mathematical Sciences