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1. Automorphic cohomology, motivic cohomology, and the adjoint L-function

   Prasanna, Kartik A.; Venkatesh, Akshay (July 2021, Asterisque)

   We propose a relationship between the cohomology of arithmetic groups, and the motivic cohomology of certain (Langlands-)attached motives. The motivic cohomology group in question is that related, by Beilinson’s conjecture, to the adjoint L-function at s=1. We present evidence for the conjecture using the theory of periods of automorphic forms, and using analytic torsion. 

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2. On the Kottwitz conjecture for local shtuka spaces

   https://doi.org/10.1017/fmp.2022.7  

   Hansen, David; Kaletha, Tasho; Weinstein, Jared (May 2022, Forum of mathematics)

   Kottwitz’s conjecture describes the contribution of a supercuspidal representation to the cohomology of a local Shimura variety in terms of the local Langlands correspondence. A natural extension of this conjecture concerns Scholze’s more general spaces of local shtukas. Using a new Lefschetz–Verdier trace formula for v-stacks, we prove the extended conjecture, disregarding the action of the Weil group, and modulo a virtual representation whose character vanishes on the locus of elliptic elements. As an application, we show that, for an irreducible smooth representation of an inner form of GLn, the L-parameter constructed by Fargues–Scholze agrees with the usual semisimplified parameter arising from local Langlands. 

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3. Upper Triangular Linear Relations on Mmultiplicities and the Stanley-Stembridge Conjecture

   https://doi.org/10.37236/10489  

   Harada, Megumi; Precup, Martha (July 2022, The Electronic Journal of Combinatorics)

   In 2015, Brosnan and Chow, and independently Guay-Paquet, proved the Shareshian-Wachs conjecture, which links the Stanley-Stembridge conjecture in combinatorics to the geometry of Hessenberg varieties through Tymoczko's permutation group action on the cohomology ring of regular semisimple Hessenberg varieties. In previous work, the authors exploited this connection to prove a graded version of the Stanley-Stembridge conjecture in a special case. In this manuscript, we derive a new set of linear relations satisfied by the multiplicities of certain permutation representations in Tymoczko's representation. We also show that these relations are upper-triangular in an appropriate sense, and in particular, they uniquely determine the multiplicities. As an application of these results, we prove an inductive formula for the multiplicity coefficients corresponding to partitions with a maximal number of parts. 

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4. Positively curved Riemannian metrics with logarithmic symmetry rank bounds

   https://doi.org/10.4171/CMH/340  

   Kennard, Lee (November 2014, Commentarii Mathematici Helvetici)

   We prove an obstruction at the level of rational cohomology to the existence of positively curved metrics with large symmetry rank. The symmetry rank bound is logarithmic in the dimension of the manifold. As one application, we provide evidence for a generalized conjecture of H. Hopf, which states that no symmetric space of rank at least two admits a metric with positive curvature. Other applications concern product manifolds, connected sums, and manifolds with nontrivial fundamental group. 

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5. $$L^{2}$$-Cohomology of quasi-fibered boundary metrics

   https://doi.org/10.1007/s00222-024-01253-5  

   Kottke, Chris; Rochon, Frédéric (June 2024, Inventiones mathematicae)

   We develop new techniques to compute the weighted L2-cohomology of quasi- fibered boundary metrics (QFB-metrics). Combined with the decay of L2-harmonic forms obtained in a companion paper, this allows us to compute the reduced L2- cohomology for various classes of QFB-metrics. Our results applies in particular to the Nakajima metric on the Hilbert scheme of n points on C2, for which we can show that the Vafa-Witten conjecture holds. Using the compactification of the monopole moduli space announced by Fritzsch, the first author and Singer, we can also give a proof of the Sen conjecture for the monopole moduli space of magnetic charge 3. 

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