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1. Products of normal subsets

   https://doi.org/10.1090/tran/8960  

   Larsen, Michael; Shalev, Aner; Tiep, Pham (February 2024, Transactions of the American Mathematical Society)

   In this paper we consider which families of finite simple groups $G$have the property that for each $\mathrm{ϵ<\#comment/>}>0$there exists $N>0$such that, if $|G|\ge <\#comment/>N$and $S,T$are normal subsets of $G$with at least $\mathrm{ϵ<\#comment/>}|G|$elements each, then every non-trivial element of $G$is the product of an element of $S$and an element of $T$. We show that this holds in a strong and effective sense for finite simple groups of Lie type of bounded rank, while it does not hold for alternating groups or groups of the form ${\mathrm{P}\mathrm{S}\mathrm{L}}_n\left(q\right)$where $q$is fixed and $n\to <\#comment/>\mathrm{\infty <\#comment/>}$. However, in the case $S=T$and $G$alternating this holds with an explicit bound on $N$in terms of $\mathrm{ϵ<\#comment/>}$. Related problems and applications are also discussed. In particular we show that, if $w_1,w_2$are non-trivial words, $G$is a finite simple group of Lie type of bounded rank, and for $g\in <\#comment/>G$, $P_{w_1\left(G\right),w_2\left(G\right)}\left(g\right)$denotes the probability that $g_1{g}_2=g$where $g_i\in <\#comment/>w_i\left(G\right)$are chosen uniformly and independently, then, as $|G|\to <\#comment/>\mathrm{\infty <\#comment/>}$, the distribution $P_{w_1\left(G\right),w_2\left(G\right)}$tends to the uniform distribution on $G$with respect to the $L^{\mathrm{\infty <\#comment/>}}$norm. 

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2. A partial converse ghost lemma for the derived category of a commutative Noetherian ring

   https://doi.org/10.1090/proc/16294  

   Liu, Jian; Pollitz, Josh (January 2023, Proceedings of the American Mathematical Society)

   In this article a condition is given to detect the containment among thick subcategories of the bounded derived category of a commutative noetherian ring. More precisely, for a commutative noetherian ring $R$and complexes of $R$-modules with finitely generated homology $M$and $N$, we show $N$is in the thick subcategory generated by $M$if and only if the ghost index of $N_{\mathfrak{p}}$with respect to $M_{\mathfrak{p}}$is finite for each prime $\mathfrak{p}$of $R$. To do so, we establish a “converse coghost lemma” for the bounded derived category of a non-negatively graded DG algebra with noetherian homology. 

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3. On the Boundary Behavior of Mass-Minimizing Integral Currents

   https://doi.org/10.1090/memo/1446  

   De_Lellis, Camillo; De_Philippis, Guido; Hirsch, Jonas; Massaccesi, Annalisa (November 2023, Memoirs of the American Mathematical Society)

   Let $\mathrm{\Sigma <\#comment/>}$be a smooth Riemannian manifold, $\mathrm{\Gamma <\#comment/>}\subset <\#comment/>\mathrm{\Sigma <\#comment/>}$a smooth closed oriented submanifold of codimension higher than $2$and $T$an integral area-minimizing current in $\mathrm{\Sigma <\#comment/>}$which bounds $\mathrm{\Gamma <\#comment/>}$. We prove that the set of regular points of $T$at the boundary is dense in $\mathrm{\Gamma <\#comment/>}$. Prior to our theorem the existence of any regular point was not known, except for some special choice of $\mathrm{\Sigma <\#comment/>}$and $\mathrm{\Gamma <\#comment/>}$. As a corollary of our theorem we answer to a question in Almgren’sAlmgren’s big regularity paperfrom 2000 showing that, if $\mathrm{\Gamma <\#comment/>}$is connected, then $T$has at least one point $p$of multiplicity $\frac{1}{2}$, namely there is a neighborhood of the point $p$where $T$is a classical submanifold with boundary $\mathrm{\Gamma <\#comment/>}$; we generalize Almgren’s connectivity theorem showing that the support of $T$is always connected if $\mathrm{\Gamma <\#comment/>}$is connected; we conclude a structural result on $T$when $\mathrm{\Gamma <\#comment/>}$consists of more than one connected component, generalizing a previous theorem proved by Hardt and Simon in 1979 when $\mathrm{\Sigma <\#comment/>}={\mathbb{R}}^{m+1}$and $T$is $m$-dimensional. 

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4. 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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5. 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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