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1. Odd-Dimensional GKM-Manifolds of Non-Negative Curvature

   https://doi.org/10.1093/imrn/rnab137  

   Escher, Christine; Goertsches, Oliver; Searle, Catherine (October 2021, International Mathematics Research Notices)

   Abstract Let $$M$$ be a closed, odd GKM$$_3$$ manifold of non-negative sectional curvature. We show that in this situation one can associate an ordinary abstract GKM$$_3$$ graph to $$M$$ and prove that if this graph is orientable, then both the equivariant and the ordinary rational cohomology of $$M$$ split off the cohomology of an odd-dimensional sphere. 

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2. Stability of Tschirnhausen Bundles

   https://doi.org/10.1093/imrn/rnad075  

   Coskun, Izzet; Larson, Eric; Vogt, Isabel (April 2023, International Mathematics Research Notices)

   Abstract Let $$\alpha \colon X \to Y$$ be a general degree $$r$$ primitive map of nonsingular, irreducible, projective curves over an algebraically closed field of characteristic zero or larger than $$r$$. We prove that the Tschirnhausen bundle of $$\alpha $$ is semistable if $$g(Y) \geq 1$$ and stable if $$g(Y) \geq 2$$. 

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3. Quantitative stability for minimizing Yamabe metrics

   https://doi.org/10.1090/btran/111  

   Engelstein, Max; Neumayer, Robin; Spolaror, Luca (May 2022, Transactions of the American Mathematical Society)

   On any closed Riemannian manifold of dimension , we prove that if a function nearly minimizes the Yamabe energy, then the corresponding conformal metric is close, in a quantitative sense, to a minimizing Yamabe metric in the conformal class. Generically, this distance is controlled quadratically by the Yamabe energy deficit. Finally, we produce an example for which this quadratic estimate is false. 

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4. On the K-theory of division algebras over local fields

   https://doi.org/10.1007/s00222-019-00909-x  

   Hesselholt, Lars; Larsen, Michael; Lindenstrauss, Ayelet (July 2019, Inventiones mathematicae)

   Let K be a complete discrete valuation field with finite residue field of characteristic p, and let D be a central division algebra over K of finite index d. Thirty years ago, Suslin and Yufryakov showed that for all prime numbers ℓ different from p and integers j≥1 , there exists a "reduced norm" isomorphism of ℓ-adic K-groups Nrd_{D/K}:K_j(D,Z_ℓ)→K_j(K,Z_ℓ) such that d⋅Nrd_{D/K} is equal to the norm homomorphism N_{D/K}. The purpose of this paper is to prove the analogous result for the p-adic K-groups. To do so, we employ the cyclotomic trace map to topological cyclic homology and show that there exists a "reduced trace" equivalence Trd_{A/S}:THH(A|D,Z_p)→THH(S|K,Z_p) between two p-complete cyclotomic spectra associated with D and K, respectively. Interestingly, we show that if p divides d, then it is not possible to choose said equivalence such that, as maps of cyclotomic spectra, d⋅Trd_{A/S} agrees with the trace Tr_{A/S}, although this is possible as maps of spectra with T-action 

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5. Lower bounds for Maass forms on semisimple groups

   https://doi.org/10.1112/S0010437X20007125  

   Brumley, Farrell; Marshall, Simon (May 2020, Compositio Mathematica)

   Let $$G$$ be an anisotropic semisimple group over a totally real number field $$F$$ . Suppose that $$G$$ is compact at all but one infinite place $$v_{0}$$ . In addition, suppose that $$G_{v_{0}}$$ is $$\mathbb{R}$$ -almost simple, not split, and has a Cartan involution defined over $$F$$ . If $$Y$$ is a congruence arithmetic manifold of non-positive curvature associated with $$G$$ , we prove that there exists a sequence of Laplace eigenfunctions on $$Y$$ whose sup norms grow like a power of the eigenvalue. 

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