##### Department of Mathematics,

University of California San Diego

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### Math 292 - Topology Seminar

## Mingcong Zeng

#### Utrecht

## Real bordism and its friends

##### Abstract:

In the 1960's, Kervaire and Milnor boiled down the problem of counting smooth structure on spheres of dimension greater than 4 to the computation of stable homotopy groups of spheres and the Kervaire invariant one problem. In the following decades, the elements of Kervaire invariant one whose dimension are less or equals to 62 are shown to exist, and finally, Hill, Hopkins and Ravenel in their 2016 paper show that the Kervaire invariant one elements doesn't exist for dimension larger or equals to 254, leaving the 126-dimensional case open. \\ \\ The $C_2$-equivariant Real bordism spectrum and its norms are crucial in HHR's solution, and computation of them is a central topic in equivariant stable homotopy theory. In this talk, I will explore two aspects of Real bordism and its norms: \\ \\ 1. How computation in Real bordism helps us to understand Lubin-Tate E-theories at p = 2. In particular, we can understand almost all actions of finite subgroups of the Morava stablizer groups on E-theories in homotopy. \\ \\ 2. The relation between Real bordism and the Segal conjecture. This relation allows us to bring new tools and perspective into this equivariant computation, and we will show how a spectral sequence based on (Real) topological Hochschild homology can help in understanding Real bordism and its norms. \\ \\ This talk is based on joint work with Beaudry, Hill, Lawson, Meier and Shi.

Host: Zhouli Xu

### February 2, 2021

### 10:30 AM

### Zoom information: Meeting ID: 933 6734 4286 Password: topology

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