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1. Rigidity properties of the cotangent complex

   https://doi.org/10.1090/jams/1000  

   Briggs, Benjamin; Iyengar, Srikanth (January 2023, Journal of the American Mathematical Society)

   This work concerns a map φ : R → S \varphi \colon R\to S of commutative noetherian rings, locally of finite flat dimension. It is proved that the André-Quillen homology functors are rigid, namely, if D n ( S / R ; − ) = 0 \mathrm {D}_n(S/R;-)=0 for some n ≥ 1 n\ge 1 , then D i ( S / R ; − ) = 0 \mathrm {D}_i(S/R;-)=0 for all i ≥ 2 i\ge 2 and φ {\varphi } is locally complete intersection. This extends Avramov’s theorem that draws the same conclusion assuming D n ( S / R ; − ) \mathrm {D}_n(S/R;-) vanishes for all n ≫ 0 n\gg 0 , confirming a conjecture of Quillen. The rigidity of André-Quillen functors is deduced from a more general result about the higher cotangent modules which answers a question raised by Avramov and Herzog, and subsumes a conjecture of Vasconcelos that was proved recently by the first author. The new insight leading to these results concerns the equivariance of a map from André-Quillen cohomology to Hochschild cohomology defined using the universal Atiyah class of φ \varphi . 

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2. Constructing nonproxy small test modules for the complete intersection property

   https://doi.org/10.1017/nmj.2021.7  

   Briggs, Benjamin; Grifo, Eloísa; Pollitz, Josh (June 2021, Nagoya Mathematical Journal)

   Abstract A local ring R is regular if and only if every finitely generated R -module has finite projective dimension. Moreover, the residue field k is a test module: R is regular if and only if k has finite projective dimension. This characterization can be extended to the bounded derived category $$\mathsf {D}^{\mathsf f}(R)$$ , which contains only small objects if and only if R is regular. Recent results of Pollitz, completing work initiated by Dwyer–Greenlees–Iyengar, yield an analogous characterization for complete intersections: R is a complete intersection if and only if every object in $$\mathsf {D}^{\mathsf f}(R)$$ is proxy small. In this paper, we study a return to the world of R -modules, and search for finitely generated R -modules that are not proxy small whenever R is not a complete intersection. We give an algorithm to construct such modules in certain settings, including over equipresented rings and Stanley–Reisner rings. 

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3. Decomposing multitwists

   https://doi.org/10.1007/s11854-023-0301-4  

   Fletcher, Alastair N; Vellis, Vyron (April 2024, Journal d'Analyse Mathématique)

   The Decomposition Problem in the class $$LIP(\S^2)$$ is to decompose any bi-Lipschitz map $$f:\S^2 \to \S^2$$ as a composition of finitely many maps of arbitrarily small isometric distortion. In this paper, we construct a decomposition for certain bi-Lipschitz maps which spiral around every point of a Cantor set $$X$$ of Assouad dimension strictly smaller than one. These maps are constructed by considering a collection of Dehn twists on the Riemann surface $$\S^2 \setminus X$$. The decomposition is then obtained via a bi-Lipschitz path which simultaneously unwinds these Dehn twists. As part of our construction, we also show that $$X \subset \S^2$$ is uniformly disconnected if and only if the Riemann surface $$\S^2 \setminus X$$ has a pants decomposition whose cuffs have hyperbolic length uniformly bounded above, which may be of independent interest. 

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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. Doubly-Affine Extractors, and Their Applications

   Dodis, Yevgeniy and (July 2021, 2nd Conference on Information-Theoretic Cryptography (ITC 2021))

   In this work we challenge the common misconception that information-theoretic (IT) privacy is too impractical to be used in the real-world: we propose to build simple and reusable IT-encryption solutions whose only efficiency penalty (compared to computationally-secure schemes) comes from a large secret key size, which is often a rather minor inconvenience, as storage is cheap. In particular, our solutions are stateless and locally computable at the optimal rate, meaning that honest parties do not maintain state and read only (optimally) small portions of their large keys with every use. Moreover, we also propose a novel architecture for outsourcing the storage of these long keys to a network of semi-trusted servers, trading the need to store large secrets with the assumption that it is hard to simultaneously compromise too many publicly accessible ad-hoc servers. Our architecture supports everlasting privacy and post-application security of the derived one-time keys, resolving two major limitations of a related model for outsourcing key storage, called bounded storage model. Both of these results come from nearly optimal constructions of so called doubly-affine extractors: locally-computable, seeded extractors Ext(X,S) which are linear functions of X (for any fixed seed S), and protect against bounded affine leakage on X. This holds unconditionally, even if (a) affine leakage may adaptively depend on the extracted key R = Ext(X,S); and (b) the seed S is only computationally secure. Neither of these properties are possible with general-leakage extractors. 

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