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1. On extremizing sequences for adjoint Fourier restriction to the sphere

   https://doi.org/10.1016/j.aim.2024.109854  

   Flock, Taryn C; Stovall, Betsy (September 2024, Advances in Mathematics)

   In this article, we develop a linear profile decomposition for the $$L^p \to L^q$$ adjoint Fourier restriction operator associated to the sphere, valid for exponent pairs $p\max\{p,\tfrac{d+2}d p'\}$$, or if $$q=\tfrac{d+2}d p'$$ and the operator norm exceeds a certain constant times the operator norm of the parabolic extension operator. 

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2. On the Korányi spherical maximal function on Heisenberg groups

   https://doi.org/10.1007/s00208-022-02533-2  

   Srivastava, Rajula (December 2022, Mathematische Annalen)

   Abstract We prove$$L^p\rightarrow L^q$$ $L^p\to L^q$estimates for the local maximal operator associated with dilates of the Kóranyi sphere in Heisenberg groups. These estimates are sharp up to endpoints and imply new bounds on sparse domination for the corresponding global maximal operator. We also prove sharp$$L^p\rightarrow L^q$$ $L^p\to L^q$estimates for spherical means over the Korányi sphere, which can be used to improve the sparse domination bounds in (Ganguly and Thangavelu in J Funct Anal 280(3):108832, 2021) for the associated lacunary maximal operator. 

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3. Iwahori‐metaplectic duality

   https://doi.org/10.1112/jlms.12896  

   Brubaker, Ben; Buciumas, Valentin; Bump, Daniel; Gustafsson, Henrik_P A (June 2024, Journal of the London Mathematical Society)

   Abstract We construct a family of solvable lattice models whose partition functions include ‐adic Whittaker functions for general linear groups from two very different sources: from Iwahori‐fixed vectors and from metaplectic covers. Interpolating between them by Drinfeld twisting, we uncover unexpected relationships between Iwahori and metaplectic Whittaker functions. This leads to new Demazure operator recurrence relations for spherical metaplectic Whittaker functions. In prior work of the authors it was shown that the row transfer matrices of certain lattice models for spherical metaplectic Whittaker functions could be represented as ‘half‐vertex operators’ operating on the ‐Fock space of Kashiwara, Miwa and Stern. In this paper the same is shown for all the members of this more general family of lattice models including the one representing Iwahori Whittaker functions. 

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4. More about the lattice Hamiltonian for Adjoint QCD2

   https://doi.org/10.1007/JHEP06(2025)260  

   Dempsey, Ross; Pufu, Silviu S; Søgaard, Benjamin T; Klebanov, Igor R (June 2025, Journal of High Energy Physics)

   A<sc>bstract</sc> In our earlier work [1], we introduced a lattice Hamiltonian for Adjoint QCD₂using staggered Majorana fermions. We found the gauge invariant space of states explicitly for the gauge group SU(2) and used them for numerical calculations of observables, such as the spectrum and the expectation value of the fermion bilinear. In this paper, we carry out a more in-depth study of our lattice model, extending it to any compact and simply-connected gauge groupG. We show how to find the gauge invariant space of states and use it to study various observables. We also use the lattice model to calculate the mixed ’t Hooft anomalies of Adjoint QCD₂for arbitraryG. We show that the matrix elements of the lattice Hamiltonian can be expressed in terms of the Wigner 6j-symbols ofG. ForG= SU(3), we perform exact diagonalization for lattices of up to six sites and study the low-lying spectrum, the fermion bilinear condensate, and the string tension. We also show how to write the lattice strong coupling expansion for ground state energies and operator expectation values in terms of the Wigner 6j-symbols. For SU(3) we carry this out explicitly and find good agreement with the exact diagonalizations, and for SU(4) we give expansions that can be compared with future numerical studies. 

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5. System norm regularization methods for Koopman operator approximation

   https://doi.org/10.1098/rspa.2022.0162  

   Dahdah, Steven; Forbes, James R. (September 2022, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences)

   Approximating the Koopman operator from data is numerically challenging when many lifting functions are considered. Even low-dimensional systems can yield unstable or ill-conditioned results in a high-dimensional lifted space. In this paper, Extended Dynamic Mode Decomposition (DMD) and DMD with control, two methods for approximating the Koopman operator, are reformulated as convex optimization problems with linear matrix inequality constraints. Asymptotic stability constraints and system norm regularizers are then incorporated as methods to improve the numerical conditioning of the Koopman operator. Specifically, the H ∞   norm is used to penalize the input–output gain of the Koopman system. Weighting functions are then applied to penalize the system gain at specific frequencies. These constraints and regularizers introduce bilinear matrix inequality constraints to the regression problem, which are handled by solving a sequence of convex optimization problems. Experimental results using data from an aircraft fatigue structural test rig and a soft robot arm highlight the advantages of the proposed regression methods. 

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