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1. The Hodge Laplacian on axisymmetric domains and its discretization

   https://doi.org/10.1093/imanum/draa048  

   Oh, Minah (July 2020, IMA Journal of Numerical Analysis)

   null (Ed.)

   Abstract We study the mixed formulation of the abstract Hodge Laplacian on axisymmetric domains with general data through Fourier finite element methods (Fourier-FEMs) in weighted function spaces. Closed Hilbert complexes and commuting projectors are used as in the work of Arnold, Falk & Winther, (2010, Finite element exterior calculus: from Hodge theory to numerical stability. Bull. Amer. Math. Soc. (N.S.), 47, 281–354), by using the new family of finite element spaces for general axisymmetric problems introduced in Oh, (2015, de Rham complexes arising from Fourier-FEMs in axisymmetric domains. Comput. Math. Appl., 70, 2063–2073). In order to get stability results and error estimates for the discrete mixed formulation, we construct commuting projectors that can be applied to functions with low regularity. 

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2. Perverse Leray filtration and specialisation with applications to the Hitchin morphism

   https://doi.org/10.1017/S0305004121000293  

   DE CATALDO, MARK ANDREA (March 2022, Mathematical Proceedings of the Cambridge Philosophical Society)

   Abstract We initiate and develop a framework to handle the specialisation morphism as a filtered morphism for the perverse, and for the perverse Leray filtration, on the cohomology with constructible coefficients of varieties and morphisms parameterised by a curve. As an application, we use this framework to carry out a detailed study of filtered specialisation for the Hitchin morphisms associated with the compactification of Dolbeault moduli spaces in [de 2018]. 

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3. Balanced Hodge Laplacians optimize consensus dynamics over simplicial complexes

   https://doi.org/10.1063/5.0080370  

   Ziegler, Cameron; Skardal, Per Sebastian; Dutta, Haimonti; Taylor, Dane (February 2022, Chaos: An Interdisciplinary Journal of Nonlinear Science)

   Despite the vast literature on network dynamics, we still lack basic insights into dynamics on higher-order structures (e.g., edges, triangles, and more generally, [Formula: see text]-dimensional “simplices”) and how they are influenced through higher-order interactions. A prime example lies in neuroscience where groups of neurons (not individual ones) may provide building blocks for neurocomputation. Here, we study consensus dynamics on edges in simplicial complexes using a type of Laplacian matrix called a Hodge Laplacian, which we generalize to allow higher- and lower-order interactions to have different strengths. Using techniques from algebraic topology, we study how collective dynamics converge to a low-dimensional subspace that corresponds to the homology space of the simplicial complex. We use the Hodge decomposition to show that higher- and lower-order interactions can be optimally balanced to maximally accelerate convergence and that this optimum coincides with a balancing of dynamics on the curl and gradient subspaces. We additionally explore the effects of network topology, finding that consensus over edges is accelerated when two-simplices are well dispersed, as opposed to clustered together. 

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4. Hodge Representations

   https://doi.org/10.1017/exp.2020.55  

   Han, Xiayimei; Robles, Colleen (January 2020, Experimental Results)

   Clingher, Adrian (Ed.)

   Abstract Green–Griffiths–Kerr introduced Hodge representations to classify the Hodge groups of polarized Hodge structures, and the corresponding Mumford–Tate subdomains. We summarize how, given a fixed period domain $$ \mathcal{D} $$ , to enumerate the Hodge representations and corresponding Mumford–Tate subdomains $$ D \subset \mathcal{D} $$ . The procedure is illustrated in two examples: (i) weight two Hodge structures with $$ {p}_g={h}^{2,0}=2 $$ ; and (ii) weight three CY-type Hodge structures. 

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5. General multiple Dirichlet series from perverse sheaves

   https://doi.org/10.1016/j.jnt.2024.03.020  

   Sawin, Will (September 2024, Journal of Number Theory)

   We give an axiomatic characterization of multiple Dirichlet series over the function field Fq(T), generalizing a set of axioms given by Diaconu and Pasol. The key axiom, relating the coefficients at prime powers to sums of the coefficients, formalizes an observation of Chinta. The existence of multiple Dirichlet series satisfying these axioms is proved by exhibiting the coefficients as trace functions of explicit perverse sheaves and using properties of perverse sheaves. The multiple Dirichlet series defined this way include, as special cases, many that have appeared previously in the literature. 

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