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1. Non-vanishing of geometric Whittaker coefficients for reductive groups

   https://doi.org/10.1090/jams/1051  

   Færgeman, Joakim; Raskin, Sam (April 2025, Journal of the American Mathematical Society)

   We prove that cuspidal automorphic $D$-modules have non-vanishing Whittaker coefficients, generalizing known results in the geometric Langlands program from $G{L}_n$to general reductive groups. The key tool is a microlocal interpretation of Whittaker coefficients. We establish various exactness properties in the geometric Langlands context that may be of independent interest. Specifically, we show Hecke functors are $t$-exact on the category of tempered $D$-modules, strengthening a classical result of Gaitsgory (with different hypotheses) for $G{L}_n$. We also show that Whittaker coefficient functors are $t$-exact for sheaves with nilpotent singular support. An additional consequence of our results is that the tempered, restricted geometric Langlands conjecture must be $t$-exact. We apply our results to show that for suitably irreducible local systems, Whittaker-normalized Hecke eigensheaves are perverse sheaves that are irreducible on each connected component of ${\mathrm{Bun}}_G$. 

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2. Co-𝑡-structures on derived categories of coherent sheaves and the cohomology of tilting modules

   https://doi.org/10.1090/ert/655  

   Achar, Pramod; Hardesty, William (February 2024, Representation Theory of the American Mathematical Society)

   We construct a co- $t$-structure on the derived category of coherent sheaves on the nilpotent cone $\mathcal{N}$of a reductive group, as well as on the derived category of coherent sheaves on any parabolic Springer resolution. These structures are employed to show that the push-forwards of the “exotic parity objects” (considered by Achar, Hardesty, and Riche [Transform. Groups 24 (2019), pp. 597–657]), along the (classical) Springer resolution, give indecomposable objects inside the coheart of the co- $t$-structure on $\mathcal{N}$. We also demonstrate how the various parabolic co- $t$-structures can be related by introducing an analogue to the usual translation functors. As an application, we give a proof of a scheme-theoretic formulation of the relative Humphreys conjecture on support varieties of tilting modules in type $A$for $p>h$. 

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3. Regularity, singularities and ℎ-vector of graded algebras

   https://doi.org/10.1090/tran/9089  

   Dao, Hailong; Ma, Linquan; Varbaro, Matteo (January 2024, Transactions of the American Mathematical Society)

   Let $R$be a standard graded algebra over a field. We investigate how the singularities of $\mathrm{Spec}<\#comment/>R$or $\mathrm{Proj}<\#comment/>R$affect the $h$-vector of $R$, which is the coefficient of the numerator of its Hilbert series. The most concrete consequence of our work asserts that if $R$satisfies Serre’s condition $\left(S_r\right)$and has reasonable singularities (Du Bois on the punctured spectrum or $F$-pure), then $h_0$, …, $h_r\ge <\#comment/>0$. Furthermore the multiplicity of $R$is at least $h_0+h_1+\cdots <\#comment/>+h_{r-<\#comment/>1}$. We also prove that equality in many cases forces $R$to be Cohen-Macaulay. The main technical tools are sharp bounds on regularity of certain $\mathrm{Ext}$modules, which can be viewed as Kodaira-type vanishing statements for Du Bois and $F$-pure singularities. Many corollaries are deduced, for instance that nice singularities of small codimension must be Cohen-Macaulay. Our results build on and extend previous work by de Fernex-Ein, Eisenbud-Goto, Huneke-Smith, Murai-Terai and others. 

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4. An optimal approximation problem for free polynomials

   https://doi.org/10.1090/proc/16474  

   Arora, Palak; Augat, Meric; Jury, Michael; Sargent, Meredith (November 2023, Proceedings of the American Mathematical Society)

   Motivated by recent work on optimal approximation by polynomials in the unit disk, we consider the following noncommutative approximation problem: for a polynomial $f$in $d$freely noncommuting arguments, find a free polynomial $p_n$, of degree at most $n$, to minimize $c_n≔\Vert <\#comment/>p_{n}f-<\#comment/>1{\Vert <\#comment/>}^2$. (Here the norm is the ${\mathrm{\ell <\#comment/>}}^2$norm on coefficients.) We show that $c_n\to <\#comment/>0$if and only if $f$is nonsingular in a certain nc domain (the row ball), and prove quantitative bounds. As an application, we obtain a new proof of the characterization of polynomials cyclic for the $d$-shift. 

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5. Rigidity of nonpositively curved manifolds with convex boundary

   https://doi.org/10.1090/proc/16475  

   Ghomi, Mohammad; Spruck, Joel (November 2023, Proceedings of the American Mathematical Society)

   We show that a compact Riemannian $3$-manifold $M$with strictly convex simply connected boundary and sectional curvature $K\le <\#comment/>a\le <\#comment/>0$is isometric to a convex domain in a complete simply connected space of constant curvature $a$, provided that $K\equiv <\#comment/>a$on planes tangent to the boundary of $M$. This yields a characterization of strictly convex surfaces with minimal total curvature in Cartan-Hadamard $3$-manifolds, and extends some rigidity results of Greene-Wu, Gromov, and Schroeder-Strake. Our proof is based on a recent comparison formula for total curvature of Riemannian hypersurfaces, which also yields some dual results for $K\ge <\#comment/>a\ge <\#comment/>0$. 

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