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Let $\Sigma$ be a closed genus $g$ surface. Assume that $\mathcal{C}$ is a dg- (or $A_\infty$-) category which admits a faithful action $$ MCG(\Sigma) \to Auteq(\mathcal{C})$$ by the mapping class group by auto-equivalences.

1. What good properties about $\mathcal{C}$ does that imply? (or : why can representation theory tell me about the category or its auto-equivalences)
2. Are there any known cases where such an action arises which do not come from symplectic topology/mirror symmetry? The cases I know of so far come from Seidel-Thomas spherical twists or Ivan's Floer cohomology and pencils of quadrics paper. I assume there must be some examples from geometric representation theory or categorification and alike...
3. What happens if I replace the mapping class group with the Torelli group $\mathcal{I}_g$ (or more generally, level $k$ in the Johnson filtration)?

Let $\Sigma$ be a closed genus $g$ surface. Assume that $\mathcal{C}$ is a smooth and proper dg- (or $A_\infty$-) category which admits a faithful action $$ MCG(\Sigma) \to Auteq(\mathcal{C})$$ by the mapping class group by auto-equivalences.

1. What good properties about $\mathcal{C}$ does that imply? (or : why can representation theory tell me about the category or its auto-equivalences)
2. Are there any known cases where such an action arises which do not come from symplectic topology/mirror symmetry? The cases I know of so far come from Seidel-Thomas spherical twists or Ivan's Floer cohomology and pencils of quadrics paper. I assume there must be some examples from geometric representation theory or categorification and alike...
3. What happens if I replace the mapping class group with the Torelli group $\mathcal{I}_g$ (or more generally, level $k$ in the Johnson filtration)?

Let $\Sigma$ be a closed genus $g$ surface. Assume the $\mathcal{C}$ is a dg- (or $A_\infty$-) category which admits a faithful action $$ MCG(\Sigma) \to Auteq(\mathcal{C})$$ by the mapping class group by auto-equivalences.

1. What good properties about $\mathcal{C}$ does that imply? (or : why can representation theory tell me about the category or its auto-equivalences)
2. Are there any known cases where such an action arises which do not come from symplectic topology/mirror symmetry? The cases I know of so far come from Seidel-Thomas spherical twists or Ivan's Floer cohomology and pencils of quadrics paper. I assume there must be some examples from geometric representation theory or categorification and alike...
3. What happens if I replace the mapping class group with the Torelli group $\mathcal{I}_g$ (or more generally, level $k$ in the Johnson filtration)?

Let $\Sigma$ be a closed genus $g$ surface. Assume that $\mathcal{C}$ is a dg- (or $A_\infty$-) category which admits a faithful action $$ MCG(\Sigma) \to Auteq(\mathcal{C})$$ by the mapping class group by auto-equivalences.

1. What good properties about $\mathcal{C}$ does that imply? (or : why can representation theory tell me about the category or its auto-equivalences)
2. Are there any known cases where such an action arises which do not come from symplectic topology/mirror symmetry? The cases I know of so far come from Seidel-Thomas spherical twists or Ivan's Floer cohomology and pencils of quadrics paper. I assume there must be some examples from geometric representation theory or categorification and alike...
3. What happens if I replace the mapping class group with the Torelli group $\mathcal{I}_g$ (or more generally, level $k$ in the Johnson filtration)?

Let $\Sigma$ be a closed genus $g$ surface. Assume the $\mathcal{C}$ is a dg- (or $A_\infty$-) category which admits a faithful action $$ MCG(\Sigma) \to Auteq(\mathcal{C})$$ by the mapping class group by auto-equivalences.

1. What good properties about $\mathcal{C}$ does that imply? (or why is that useful from a representation theory point of view)
2. Are there any known cases where such an action arises which do not come from symplectic topology/mirror symmetry? The cases I know of so far come from Seidel-Thomas or Ivan's Floer cohomology and pencils of quadrics paper. I assume there must be some examples from geometric representation theory or categorification and alike...
3. What happens if I replace the mapping class group with the Torelli group $\mathcal{I}_g$ (or more generally, level $k$ in the Johnson filtration)?

Let $\Sigma$ be a closed genus $g$ surface. Assume the $\mathcal{C}$ is a dg- (or $A_\infty$-) category which admits a faithful action $$ MCG(\Sigma) \to Auteq(\mathcal{C})$$ by the mapping class group by auto-equivalences.

1. What good properties about $\mathcal{C}$ does that imply? (or : why can representation theory tell me about the category or its auto-equivalences)
2. Are there any known cases where such an action arises which do not come from symplectic topology/mirror symmetry? The cases I know of so far come from Seidel-Thomas spherical twists or Ivan's Floer cohomology and pencils of quadrics paper. I assume there must be some examples from geometric representation theory or categorification and alike...
3. What happens if I replace the mapping class group with the Torelli group $\mathcal{I}_g$ (or more generally, level $k$ in the Johnson filtration)?