Variations of Hodge Structure Considered as an Exterior Differential System: Old and New Results

Landau-Ginzburg models -- old and new

In the last three years a new concept -- the concept of wall crossing has emerged. The current situation with wall crossing phenomena, after papers of Seiberg-Witten, Gaiotto-Moore-Neitzke, Vafa-Cecoti and seminal works by Donaldson-Thomas, Joyce-Song, Maulik-Nekrasov-Okounkov-Pandharipande, Douglas, Bridgeland, and Kontsevich-Soibelman, is very similar to the situation with Higgs Bundles after the works of Higgs and Hitchin -- it is clear that a general "Hodge type" of theory exists and needs to be developed. Nonabelian Hodge theory did lead to strong mathematical applications -- uniformizati…

Characteristic classes of mixed Hodge modules

This paper gives an introduction and overview about recent developments on the interaction of the theories of characteristic classes and mixed Hodge theory for singular spaces in the complex algebraic context. It uses M. Saito's deep theory of mixed Hodge modules as a "black box", thinking about them as "constructible or perverse sheaves of Hodge structures", having the same functorial calculus of Grothendieck functors. For the "constant Hodge sheaf", one gets the "motivic characteristic classes" of Brasselet-Schuermann-Yokura, whereas the classes of the "intersection homology Hodge sheaf" wer…

$L^p$ 有界子解析流形的 Hodge 理论

$L^p$ Hodge theory for bounded subanalytic manifolds

本文研究有界子解析流形上满足 Dirichlet 或 Neumann 条件的 $L^p$ 微分形式的上同调，证明了 $L^p$ Hodge 分解、de Rham 定理及 Lefschetz 对偶，并通过稠密性结果给出边界条件的显式刻画。

Machine learning for complete intersection Calabi-Yau manifolds: a methodological study

We revisit the question of predicting both Hodge numbers $h^{1,1}$ and $h^{2,1}$ of complete intersection Calabi-Yau (CICY) 3-folds using machine learning (ML), considering both the old and new datasets built respectively by Candelas-Dale-Lutken-Schimmrigk / Green-Hübsch-Lutken and by Anderson-Gao-Gray-Lee. In real world applications, implementing a ML system rarely reduces to feed the brute data to the algorithm. Instead, the typical workflow starts with an exploratory data analysis (EDA) which aims at understanding better the input data and finding an optimal representation. It is followed b…