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1. Regular Supercuspidal Representations

   Tasho Kaletha (July 2019, Journal of the American Mathematical Society)

   We show that, in good residual characteristic, most supercuspidal representations of a tamely ramified reductive p-adic group G arise from pairs (S,\theta), where S is a tame elliptic maximal torus of G, and \theta is a character of S satisfying a simple root-theoretic property. We then give a new expression for the roots of unity that appear in the Adler-DeBacker-Spice character formula for these supercuspidal representations and use it to show that this formula bears a striking resemblance to the character formula for discrete series representations of real reductive groups. Led by this, we explicitly construct the local Langlands correspondence for these supercuspidal representations and prove stability and endoscopic transfer in the case of toral representations. In large residual characteristic this gives a construction of the local Langlands correspondence for almost all supercuspidal representations of reductive p-adic groups. 

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2. Quasi-Locality Bounds for Quantum Lattice Systems. Part II. Perturbations of Frustration-Free Spin Models with Gapped Ground States

   https://doi.org/10.1007/s00023-021-01086-5  

   Nachtergaele, Bruno; Sims, Robert; Young, Amanda (August 2021, Annales Henri Poincaré)

   Abstract We study the stability with respect to a broad class of perturbations of gapped ground-state phases of quantum spin systems defined by frustration-free Hamiltonians. The core result of this work is a proof using the Bravyi–Hastings–Michalakis (BHM) strategy that under a condition of local topological quantum order (LTQO), the bulk gap is stable under perturbations that decay at long distances faster than a stretched exponential. Compared to previous work, we expand the class of frustration-free quantum spin models that can be handled to include models with more general boundary conditions, and models with discrete symmetry breaking. Detailed estimates allow us to formulate sufficient conditions for the validity of positive lower bounds for the gap that are uniform in the system size and that are explicit to some degree. We provide a survey of the BHM strategy following the approach of Michalakis and Zwolak, with alterations introduced to accommodate more general than just periodic boundary conditions and more general lattices. We express the fundamental condition known as LTQO by means of an indistinguishability radius, which we introduce. Using the uniform finite-volume results, we then proceed to study the thermodynamic limit. We first study the case of a unique limiting ground state and then also consider models with spontaneous breaking of a discrete symmetry. In the latter case, LTQO cannot hold for all local observables. However, for perturbations that preserve the symmetry, we show stability of the gap and the structure of the broken symmetry phases. We prove that the GNS Hamiltonian associated with each pure state has a non-zero spectral gap above the ground state. 

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3. On asymptotic stability of connective groups

   https://doi.org/doi:10.2969/aspm/08010000  

   Dadarlat, Marius (April 2019, Operator algebras and mathematical physics)

   In their study of almost group representations, Manuilov and Mishchenko introduced and investigated the notion of asymptotic stability of a finitely presented discrete group. In this paper we establish connections between connectivity of amenable groups and asymptotic stability and exhibit new classes of asymptotically stable groups. In particular, we show that if G is an amenable and connective discrete group whose classifying space BG is homotopic to a finite simplicial complex, then G is asymptotically stable. 

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4. Self-organization of modular activity in immature cortical networks

   https://doi.org/10.1038/s41467-024-48341-x  

   Mulholland, Haleigh_N; Kaschube, Matthias; Smith, Gordon_B (May 2024, Nature Communications)

   Abstract During development, cortical activity is organized into distributed modular patterns that are a precursor of the mature columnar functional architecture. Theoretically, such structured neural activity can emerge dynamically from local synaptic interactions through a recurrent network with effective local excitation with lateral inhibition (LE/LI) connectivity. Utilizing simultaneous widefield calcium imaging and optogenetics in juvenile ferret cortex prior to eye opening, we directly test several critical predictions of an LE/LI mechanism. We show that cortical networks transform uniform stimulations into diverse modular patterns exhibiting a characteristic spatial wavelength. Moreover, patterned optogenetic stimulation matching this wavelength selectively biases evoked activity patterns, while stimulation with varying wavelengths transforms activity towards this characteristic wavelength, revealing a dynamic compromise between input drive and the network’s intrinsic tendency to organize activity. Furthermore, the structure of early spontaneous cortical activity – which is reflected in the developing representations of visual orientation – strongly overlaps that of uniform opto-evoked activity, suggesting a common underlying mechanism as a basis for the formation of orderly columnar maps underlying sensory representations in the brain. 

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5. Analysis and application of a local discontinuous Galerkin method for the electromagnetic concentrator model

   https://doi.org/10.1002/num.23069  

   Huang, Yunqing; Li, Jichun; Liu, Xin (September 2023, Numerical Methods for Partial Differential Equations)

   Abstract In this paper, we develop a local discontinuous Galerkin (LDG) method to simulate the wave propagation in an electromagnetic concentrator. The concentrator model consists of a coupled system of four partial differential equations and one ordinary differential equation. Discrete stability and error estimate are proved for both semi‐discrete and full‐discrete LDG schemes. Numerical results are presented to justify the theoretical analysis and demonstrate the interesting wave concentration property by the electromagnetic concentrator. 

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