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1. Cohomology of finite tensor categories: Duality and Drinfeld centers

   https://doi.org/10.1090/tran/8548  

   Negron, Cris; Plavnik, Julia (March 2022, Transactions of the American Mathematical Society)

   We consider the finite generation property for cohomology of a finite tensor category C \mathscr {C} , which requires that the self-extension algebra of the unit \operatorname {Ext}^\text {\tiny ∙ }_\mathscr {C}(\mathbf {1},\mathbf {1}) is a finitely generated algebra and that, for each object V V in C \mathscr {C} , the graded extension group \operatorname {Ext}^\text {\tiny ∙ }_\mathscr {C}(\mathbf {1},V) is a finitely generated module over the aforementioned algebra. We prove that this cohomological finiteness property is preserved under duality (with respect to exact module categories) and taking the Drinfeld center, under suitable restrictions on C \mathscr {C} . For example, the stated result holds when C \mathscr {C} is a braided tensor category of odd Frobenius-Perron dimension. By applying our general results, we obtain a number of new examples of finite tensor categories with finitely generated cohomology. In characteristic 0 0 , we show that dynamical quantum groups at roots of unity have finitely generated cohomology. We also provide a new class of examples in finite characteristic which are constructed via infinitesimal group schemes. 

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2. Equivariant quantum differential equation and qKZ equations for a projective space: Stokes bases as exceptional collections, Stokes matrices as Gram matrices, and B-Theorem

   Giordano Cotti, Alexander Varchenko (January 2021, Proceedings of symposia in pure mathematics)

   Novikov, Krichever (Ed.)

   In [TV19a] the equivariant quantum differential equation (qDE) for a projective space was considered and a compatible system of difference qKZ equations was introduced; the space of solutions to the joint system of the qDE and qKZ equations was identified with the space of the equivariant K-theory algebra of the projective space; Stokes bases in the space of solutions were identified with exceptional bases in the equivariant K-theory algebra. This paper is a continuation of [TV19a]. We describe the relation between solutions to the joint system of the qDE and qKZ equations and the topological-enumerative solution to the qDE only, defined as a generating function of equivariant descendant Gromov-Witten invariants. The relation is in terms of the equivariant graded Chern character on the equivariant K-theory algebra, the equivariant Gamma class of the projective space, and the equivariant first Chern class of the tangent bundle of the projective space. We consider a Stokes basis, the associated exceptional basis in the equivariant K-theory algebra, and the associated Stokes matrix. We show that the Stokes matrix equals the Gram matrix of the equivariant Grothendieck-Euler-Poincaré pairing wrt to the basis, which is the left dual to the associated exceptional basis. We identify the Stokes bases in the space of solutions with explicit full exceptional collections in the equivariant derived category of coherent sheaves on the projective space, where the elements of those exceptional collections are just line bundles on the projective space and exterior powers of the tangent bundle of the projective space. These statements are equivariant analogs of results of G. Cotti, B. Dubrovin, D. Guzzetti, and S. Galkin, V. Golyshev, H. Iritani. 

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3. Mapping class group representations from non-semisimple TQFTs

   https://doi.org/10.1142/S0219199721500917  

   De_Renzi, Marco; Gainutdinov, Azat M; Geer, Nathan; Patureau-Mirand, Bertrand; Runkel, Ingo (February 2023, Communications in Contemporary Mathematics)

   In [M. De Renzi, A. Gainutdinov, N. Geer, B. Patureau-Mirand and I. Runkel, 3-dimensional TQFTs from non-semisimple modular categories, preprint (2019), arXiv:1912.02063[math.GT]], we constructed 3-dimensional topological quantum field theories (TQFTs) using not necessarily semisimple modular categories. Here, we study projective representations of mapping class groups of surfaces defined by these TQFTs, and we express the action of a set of generators through the algebraic data of the underlying modular category [Formula: see text]. This allows us to prove that the projective representations induced from the non-semisimple TQFTs of the above reference are equivalent to those obtained by Lyubashenko via generators and relations in [V. Lyubashenko, Invariants of 3-manifolds and projective representations of mapping class groups via quantum groups at roots of unity, Comm. Math. Phys. 172(3) (1995) 467–516, arXiv:hep-th/9405167]. Finally, we show that, when [Formula: see text] is the category of finite-dimensional representations of the small quantum group of [Formula: see text], the action of all Dehn twists for surfaces without marked points has infinite order. 

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4. Classification of Frobenius Forms in Five Variables

   https://doi.org/10.1007/978-3-030-91986-3_15  

   Kadyrsizova, Zhibek; Kenkel, Jennifer; Page, Janet; Singh, Jyoti; Smith, Karen E.; Vraciu, Adela; Witt, Emily E. (January 2021, Women in Commutative Algebra: Proceedings of the 2019 WICA Workshop)

   Miller, Claudia; Striuli, Janet; Witt, Emily E. (Ed.)

   Cubic surfaces in characteristic two are investigated from the point of view of prime characteristic commutative algebra. In particular, we prove that the non-Frobenius split cubic surfaces form a linear subspace of codimension four in the 19-dimensional space of all cubics, and that up to projective equivalence, there are finitely many non-Frobenius split cubic surfaces. We explicitly describe defining equations for each and characterize them as extremal in terms of configurations of lines on them. In particular, a (possibly singular) cubic surface in characteristic two fails to be Frobenius split if and only if no three lines on it form a “triangle”. 

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5. Edge Expansion and Spectral Gap of Nonnegative Matrices

   https://doi.org/10.1137/1.9781611975994.73  

   Mehta, Jenish C.; Schulman, Leonard J. (January 2020, Proceedings of the 2020 ACM-SIAM Symposium on Discrete Algorithms)

   The classic graphical Cheeger inequalities state that if $$M$$ is an $$n\times n$$ \emph{symmetric} doubly stochastic matrix, then \[ \frac{1-\lambda_{2}(M)}{2}\leq\phi(M)\leq\sqrt{2\cdot(1-\lambda_{2}(M))} \] where $$\phi(M)=\min_{S\subseteq[n],|S|\leq n/2}\left(\frac{1}{|S|}\sum_{i\in S,j\not\in S}M_{i,j}\right)$$ is the edge expansion of $$M$$, and $$\lambda_{2}(M)$$ is the second largest eigenvalue of $$M$$. We study the relationship between $$\phi(A)$$ and the spectral gap $$1-\re\lambda_{2}(A)$$ for \emph{any} doubly stochastic matrix $$A$$ (not necessarily symmetric), where $$\lambda_{2}(A)$$ is a nontrivial eigenvalue of $$A$$ with maximum real part. Fiedler showed that the upper bound on $$\phi(A)$$ is unaffected, i.e., $$\phi(A)\leq\sqrt{2\cdot(1-\re\lambda_{2}(A))}$$. With regards to the lower bound on $$\phi(A)$$, there are known constructions with \[ \phi(A)\in\Theta\left(\frac{1-\re\lambda_{2}(A)}{\log n}\right), \] indicating that at least a mild dependence on $$n$$ is necessary to lower bound $$\phi(A)$$. In our first result, we provide an \emph{exponentially} better construction of $$n\times n$$ doubly stochastic matrices $$A_{n}$$, for which \[ \phi(A_{n})\leq\frac{1-\re\lambda_{2}(A_{n})}{\sqrt{n}}. \] In fact, \emph{all} nontrivial eigenvalues of our matrices are $$0$$, even though the matrices are highly \emph{nonexpanding}. We further show that this bound is in the correct range (up to the exponent of $$n$$), by showing that for any doubly stochastic matrix $$A$$, \[ \phi(A)\geq\frac{1-\re\lambda_{2}(A)}{35\cdot n}. \] As a consequence, unlike the symmetric case, there is a (necessary) loss of a factor of $$n^{\alpha}$ for $$\frac{1}{2}\leq\alpha\leq1$$ in lower bounding $$\phi$$ by the spectral gap in the nonsymmetric setting. Our second result extends these bounds to general matrices $$R$$ with nonnegative entries, to obtain a two-sided \emph{gapped} refinement of the Perron-Frobenius theorem. Recall from the Perron-Frobenius theorem that for such $$R$$, there is a nonnegative eigenvalue $$r$$ such that all eigenvalues of $$R$$ lie within the closed disk of radius $$r$$ about $$0$$. Further, if $$R$$ is irreducible, which means $$\phi(R)>0$$ (for suitably defined $$\phi$$), then $$r$$ is positive and all other eigenvalues lie within the \textit{open} disk, so (with eigenvalues sorted by real part), $$\re\lambda_{2}(R) 

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