We prove new results concerning the topology and Hodge theory of singular varieties. A common theme is that concrete conditions on the complexity of the singularities, from a number of different perspectives, are closely related to the symmetries of...
We prove Hodge-theoretic formulas for the $\mathbb{Q}$-factoriality defect of a normal projective variety, and for the local analytic $\mathbb{Q}$-factoriality defect of an analytic germ of a normal variety. These formulas lead to consequences rangin...
We study the mixed formulation of the abstract Hodge Laplacian on axisymmetric domains with general data through Fourer-finite-element-methods in weighted functions spaces. Closed Hilbert complexes and commuting projectors are used through a family o...
There is an isomorphism between the moduli spaces of $σ$-stable holomorphic triples and some of the critical submanifolds of the moduli space of $k$-Higgs bundles of rank three, whose elements $(E,\varphi^k)$ correspond to variations of Hodge struct...
We study the asymptotic behaviour of polarization form in the variation of mixed Hodge structure associated to isolated hypersurface singularities. The contribution characterizes a modification of Grothendieck residue as the polarization on the exten...
In one of the fundamental results of Arakelov's arithmetic intersection theory, Faltings and Hriljac (independently) proved the Hodge Index Theorem for arithmetic surfaces by relating the intersection pairing to the negative of the Néron-Tate height...
We define defect for hypersurfaces with A-D-E singularities in complex projective normal Cohen-Macaulay fourfolds having some vanishing properties of Bott-type and prove formulae for Hodge numbers of big resolutions of such hypersurfaces. We comput...
We construct a geometric cycle model for a Hodge filtered extension of complex cobordism for every smooth manifold with a filtration on its de Rham complex with complex coefficients. Using a refinement of the Pontryagin-Thom construction, we construc...
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The hidden symmetry and an infinite set non-local conserved currents of the Green-Schwarz superstring on $AdS_5\otimes S^5$ have been pointed out by Bena et al. In this paper, we shown that the Hodge dual between the Maurer-Cartan equation and the...
Language Models on the AI Executive Order — Deploying text embeddings and Llama2 to answer questions about the Executive Order on Artificial Intelligence issued by President Biden (⭐ 10)
We study the universal PGL_n$character variety over M_g whose fiber over a point [C] is the space of PGL_n-local systems on the curve C. We use nonabelian Hodge theory and properties of Saito's mixed Hodge modules to show that the Leray-Serre spectra...
We prove by induction on dimension the Hodge conjecture for smooth complex projective varieties. Let $X$ be a smooth complex projective variety. Then $X$ is birational to a possibly singular projective hypersurface, hence to a smooth projective varie...
Jupyter Notebook that introduces BIDEN: Binary Inference Dictionaries for Electoral NLP. It demonstrates a compression-based binary classification technique that is fast at both training and inference on common CPU hardware in Python (⭐ 10)
We compute the Hochschild and negative cyclic homology of the nodal curves, and we show that the (noncommutative) Hodge to de Rham spectral sequence degenerates at the second page. We also classify all the Hochschild classes that can be lifted to neg...
We study the Variations of mixed Hodge structures (VMHS) associated to a pencil ${\cal X}$ (parametrised by an open set $B \subset {\Bbb P}^1$) of equisingular hypersurfaces of degree $d$ in ${\Bbb P}^{4}$ with exactly $m$ ordinary double points as s...
Firstly, we provide a different proof of an important lemma in Buzzard and Calegari's work on slopes of overconvergent 2-adic modular forms via nonarchimedean linear Hodge-Newton decomposition. The lemma shows that two equivalent matrices with coeffi...
We collect evidence in support of a conjecture of Griffiths, Green and Kerr on the arithmetic of extension classes of limiting mixed Hodge structures arising from semistable degenerations over a number field. After briefly summarizing how a result of...
The Hodge conjecture is a major open problem in complex algebraic geometry. In this survey, we discuss the main cases where the conjecture is known, and also explain an approach by Griffiths-Green to solve the problem....
A short proof of convergence for the discretization of the Hodge-Dirac operator in the framework of discrete exterior calculus (DEC) is provided using the techniques established in [Johnny Guzmán and Pratyush Potu, A Framework for Analysis of DEC Ap...