### February 12 3:00pm, BRH 105

Joshua Wiscons (Sacramento State)

• Title:
Investigating groups via the geometry of their involutions
• Abstract:
The set of involutions (elements of order two) of a group is often highly structured, and investigating this set can be an effective way to better understand the group. This talk will explore the involutions of two familiar groups: $\operatorname{SO}_3(\mathbb{R})$ (the group of rotations the sphere), and $\operatorname{PSL}_2(\mathbb{C})$ (the group of Möbius transformations of $\mathbb{C}\cup\{\infty\}$). In the former case, the group structure endows the set of involutions with the structure of a projective plane. In the latter case, the involutions form a projective plane with a missing quadric. The talk will conclude by indicating how the tension between these two examples (genuine plane vs. almost plane) is driving progress on a 15 year-old problem related to the classification of the (model-theoretically important) simple groups of finite Morley rank. Familiarity with basic group theory and linear algebra will be assumed---group actions will also appear. But sadly, no model theory will be needed.

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