##### Department of Mathematics,

University of California San Diego

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### Math 295 - Mathematics Colloquium

## Jianfeng Lin

#### MIT

## The Pin(2)-equivariant Borsukâ€“Ulam theorem and the Geography Problem of 4-manifolds

##### Abstract:

The classical Borsuk-Ulam theorem states that a continuous map from a n-dimensional sphere to m-dimensional sphere which preserves the antipodal Z/2-actions only exists when m is greater than or equal to n. One can ask a similar question, by replacing the antipodal Z/2-action with an action of the Lie group Pin(2). On a seemingly unrelated side, the Geography Problem of 4-manifolds asks which simply connected topological 4-manifolds admits a smooth structure. By the celebrated works of Kirby-Siebenmann, Freedman, Donaldson, Seiberg-Witten and Furuta, there is a surprising connection between these two questions. In this talk, I will: 1. Explain this beautiful connection between the two problems. 2. Present a solution to the Pin(2)-equivariant Borsukâ€“Ulam problem. 3. State its application to the Geography Problem. In particular, a partial result on the famous 11/8-conjecture. 4. Describe the ideas of our proof, which uses Pin(2)-equivariant stable homotopy theory. This talk is based on a joint work with Mike Hopkins, XiaoLin Danny Shi and Zhouli Xu. No familiarity of homotopy theory or 4-dimensional topology will be assumed.

Host: James McKernan

### November 28, 2018

### 2:00 PM

### AP&M 6402

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