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1. A New Proof of the Atomic Decomposition of Hardy Spaces

   Dekel, S; Kerkyacharian, G; Kyriazis, G; Petrushev, P (April 2018, Constructive Theory of Functions)

   A new proof is given of the atomic decomposition of Hardy spaces H^p, 0< p≤1, in the classical setting on R^n. The new method can be used to establish atomic decomposition of maximal Hardy spaces in general and nonclassical settings. 

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2. Lebesgue Decomposition of Non-Commutative Measures

   https://doi.org/10.1093/imrn/rnaa231  

   Jury, Michael T; Martin, Robert T (September 2020, International Mathematics Research Notices)

   null (Ed.)

   Abstract We extend the Lebesgue decomposition of positive measures with respect to Lebesgue measure on the complex unit circle to the non-commutative (NC) multi-variable setting of (positive) NC measures. These are positive linear functionals on a certain self-adjoint subspace of the Cuntz–Toeplitz $$C^{\ast }-$$algebra, the $$C^{\ast }-$$algebra of the left creation operators on the full Fock space. This theory is fundamentally connected to the representation theory of the Cuntz and Cuntz–Toeplitz $$C^{\ast }-$$algebras; any *−representation of the Cuntz–Toeplitz $$C^{\ast }-$$algebra is obtained (up to unitary equivalence), by applying a Gelfand–Naimark–Segal construction to a positive NC measure. Our approach combines the theory of Lebesgue decomposition of sesquilinear forms in Hilbert space, Lebesgue decomposition of row isometries, free semigroup algebra theory, NC reproducing kernel Hilbert space theory, and NC Hardy space theory. 

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3. Grothendieck Shenanigans: Permutons from pipe dreams via integrable probability

   Morales, Alejandro H; Panova, Greta; Petrov, Leonid; Yeliussizov, Damir (June 2025, FPSAC Proceedings)

   We study random permutations corresponding to pipe dreams. Our main model is motivated by the Grothendieck polynomials with parameter ß = 1 arising in the K-theory of the flag variety. The probability weight of a permutation is proportional to the principal specialization (setting all variables to 1) of its Grothendieck polyno- mial. By mapping this random permutation to a version of TASEP (Totally Asymmetric Simple Exclusion Process, we describe the limiting permuton and fluctuations around it as the order n of the permutation grows to infinity. The fluctuations are of order n$ and have the Tracy-Widom GUE distribution, which places this algebraic (K-theoretic) model into the Kardar-Parisi-Zhang universality class. Inspired by Stanley's question for the maximal value of principal specializations of Schubert polynomials, we resolve the analogous question for B = 1 Grothendieck polynomials, and provide bounds for general B. This analysis uses a correspondence with the free fermion six-vertex model, and the frozen boundary of the Aztec diamond. 

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4. Grothendieck Shenanigans: Permutons from pipe dreams via integrable probability

   Morales, Alejandro H; Panova, Greta; Petrov, Leonid; Yeliussizov, Damir (July 2024, arXiv)

   We study random permutations arising from reduced pipe dreams. Our main model is motivated by Grothendieck polynomials with parameter $$\beta=1$$ arising in K-theory of the flag variety. The probability weight of a permutation is proportional to the principal specialization (setting all variables to 1) of the corresponding Grothendieck polynomial. By mapping this random permutation to a version of TASEP (Totally Asymmetric Simple Exclusion Process), we describe the limiting permuton and fluctuations around it as the order $$n$$ of the permutation grows to infinity. The fluctuations are of order $$n^{\frac13}$$ and have the Tracy-Widom GUE distribution, which places this algebraic (K-theoretic) model into the Kardar-Parisi-Zhang universality class. We also investigate non-reduced pipe dreams and make progress on a recent open problem on the asymptotic number of inversions of the resulting permutation. Inspired by Stanley's question for the maximal value of principal specializations of Schubert polynomials, we resolve the analogous question for $$\beta=1$$ Grothendieck polynomials, and provide bounds for general $$\beta$$. 

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5. Repurposing Mu-Mimo Downlink For Joint Wireless Communications And Imaging Via Virtual Users

   Li, Kris; Ramirez, David; Mishra, Kumar Vijay; Sabharwal, Ashutosh (September 2024, IEEE)

   In this paper, we propose a method to repurpose the multi-user MIMO downlink transmission for joint wireless communication and imaging. The key idea is to introduce the concept of virtual users in the communication coverage area and use the existing MUMIMO beamforming methods to jointly beamform towards real and virtual users. The virtual users are placed to complement the locations of actual users, with the objective to illuminate the scene as uniformly as possible. We study a single-parameter tradeoff, introduced by a power split parameter between real and virtual users. We demonstrate via simulated examples that the virtual user concept is effective in providing a scalable imaging and communications performance tradeoff for cases where the real users are clustered in small geographical areas. 

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