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1. Stabilization of the cohomology of thickenings

   Bhatt, Bhargav; Blickle, Manuel; Lyubeznik, Gennady; Singh, Anurag; Zhang, Wenliang. (April 2019, American journal of mathematics)

   For a local complete intersection subvariety $X = V (I)$ in $P^n$ over a field of characteristic zero, we show that, in cohomological degrees smaller than the codimension of the singular locus of $$X$$, the cohomology of vector bundles on the formal completion of $P^n$ along $$X$$ can be effectively computed as the cohomology on any sufficiently high thickening $$X_t = V (I^t)$$; the main ingredient here is a positivity result for the normal bundle of $$X$$. Furthermore, we show that the Kodaira vanishing theorem holds for all thickenings $$X_t$$ in the same range of cohomological degrees; this extends the known version of Kodaira vanishing on $$X$$, and the main new ingredient is a version of the Kodaira- Akizuki-Nakano vanishing theorem for $$X$$, formulated in terms of the cotangent complex. 

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2. Eulerianity of Fourier coefficients of automorphic forms

   https://doi.org/10.1090/ert/565  

   Gourevitch, Dmitry; Gustafsson, Henrik; Kleinschmidt, Axel; Persson, Daniel; Sahi, Siddhartha (June 2021, Representation Theory of the American Mathematical Society)

   We study the question of Eulerianity (factorizability) for Fourier coefficients of automorphic forms, and we prove a general transfer theorem that allows one to deduce the Eulerianity of certain coefficients from that of another coefficient. We also establish a ‘hidden’ invariance property of Fourier coefficients. We apply these results to minimal and next-to-minimal automorphic representations, and deduce Eulerianity for a large class of Fourier and Fourier–Jacobi coefficients. In particular, we prove Eulerianity for parabolic Fourier coefficients with characters of maximal rank for a class of Eisenstein series in minimal and next-to-minimal representations of groups of ADE-type that are of interest in string theory. 

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3. Torsion Invariants of complexes of groups

   https://doi.org/10.1215/00127094-2023-0024  

   Okun, Boris; Schreve, K (February 2024, Duke)

   Hain, Richard (Ed.)

   Suppose a residually finite group G acts cocompactly on a contractible complex with strict fundamental domain Q, where the stabilizers are either trivial or have normal Z-subgroups. Let dQ be the subcomplex of Q with nontrivial stabilizers. Our main result is a computation of the homology torsion growth of a chain of finite index normal subgroups of G. We show that independent of the chain, the normalized torsion limits to the torsion of dQ shifted a degree. Under milder assumptions of acyclicity of nontrivial stabilizers, we show similar formulas for the mod p-homology growth. We also obtain formulas for the universal and the usual L^2-torsion of G in terms of the torsion of stabilizers and topology of dQ. In particular, we get complete answers for right-angled Artin groups, which shows that they satisfy a torsion analogue of Lück’s approximation theorem. 

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4. Equivalence of codes for countable sets of reals

   https://doi.org/10.4153/S0008439520000661  

   Chan, William (September 2021, Canadian Mathematical Bulletin)

   null (Ed.)

   Abstract A set $$U \subseteq {\mathbb {R}} \times {\mathbb {R}}$$ is universal for countable subsets of $${\mathbb {R}}$$ if and only if for all $$x \in {\mathbb {R}}$$ , the section $$U_x = \{y \in {\mathbb {R}} : U(x,y)\}$$ is countable and for all countable sets $$A \subseteq {\mathbb {R}}$$ , there is an $$x \in {\mathbb {R}}$$ so that $$U_x = A$$ . Define the equivalence relation $$E_U$$ on $${\mathbb {R}}$$ by $$x_0 \ E_U \ x_1$$ if and only if $$U_{x_0} = U_{x_1}$$ , which is the equivalence of codes for countable sets of reals according to U . The Friedman–Stanley jump, $=^+$ , of the equality relation takes the form $$E_{U^*}$$ where $U^*$ is the most natural Borel set that is universal for countable sets. The main result is that $=^+$ and $$E_U$$ for any U that is Borel and universal for countable sets are equivalent up to Borel bireducibility. For all U that are Borel and universal for countable sets, $$E_U$$ is Borel bireducible to $=^+$ . If one assumes a particular instance of $$\mathbf {\Sigma }_3^1$$ -generic absoluteness, then for all $$U \subseteq {\mathbb {R}} \times {\mathbb {R}}$$ that are $$\mathbf {\Sigma }_1^1$$ (continuous images of Borel sets) and universal for countable sets, there is a Borel reduction of $=^+$ into $$E_U$$ . 

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5. Structural basis of higher order oligomerization of KSHV inhibitor of cGAS

   https://doi.org/10.1073/pnas.2200285119  

   Bhowmik, Debipreeta; Tian, Yuan; Wang, Bing; Zhu, Fanxiu; Yin, Qian (August 2022, Proceedings of the National Academy of Sciences)

   Kaposi's sarcoma–associated herpesvirus (KSHV) inhibitor of cyclic GMP–AMP synthase (cGAS) (KicGAS) encoded by ORF52 is a conserved major tegument protein of KSHV and the first reported viral inhibitor of cGAS. In our previous study, we found that KicGAS is highly oligomerized in solution and that oligomerization is required for its cooperative DNA binding and for inhibiting DNA-induced phase separation and activation of cGAS. However, how KicGAS oligomerizes remained unclear. Here, we present the crystal structure of KicGAS at 2.5 Å resolution, which reveals an “L”-shaped molecule with each arm of the L essentially formed by a single α helix (α1 and α2). Antiparallel dimerization of α2 helices from two KicGAS molecules leads to a unique “Z”-shaped dimer. Surprisingly, α1 is also a dimerization domain. It forms a parallel dimeric leucine zipper with the α1 from a neighboring dimer, leading to the formation of an infinite chain of KicGAS dimers. Residues involved in leucine zipper dimer formation are among the most conserved residues across ORF52 homologs of gammaherpesviruses. The self-oligomerization increases the valence and cooperativity of interaction with DNA. The resultant multivalent interaction is critical for the formation of liquid condensates with DNA and consequent sequestration of DNA from being sensed by cGAS, explaining its role in restricting cGAS activation. The structure presented here not only provides a mechanistic understanding of the function of KicGAS but also informs a molecular target for rational design of antivirals against KSHV and related viruses. 

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