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Commutativity versus anti-commutativity is a ubiquitous dichotomy in mathematics, and beyond. Given a “commutative theory” (in a loose sense), one can ask if it admits an odd analogue, that is, a twin “anti-commutative theory”; for instance, a Lie algebra versus a Lie superalgebra. In higher representation theory, odd analogues have started to appear as a means to understand the higher structure of quantum supergroups and their representations. On the other hand, Khovanov homology is a certain link homology known to relate to the higher representation theory of quantum groups; it admits an odd analogue, known as odd Khovanov homology, whose representation theoretic understanding had remained elusive.
This talk will survey the ongoing search for odd analogues, and explain how odd Khovanov homology relates to higher representation theory. If time permits, I may hint on the underlying technical toolbox allowing such construction, given by rewriting theory. This is partially joint with Pedro Vaz.