This talk aims to share a “mystery for the 21st century”. I will begin with a broad overview of the Kahler packages (Poincare duality, Hard Lefschetz, and Hodge—Riemann relations) from Poincare, Lefschetz, and Hodge’s work. Then I will briefly review several shadows of Hodge theory: Weil conjectures and Grothendieck’s standard conjectures, Goresky—MacPherson’s intersection homology and Beilinson—Bernstein—Deligne(—Gabber)’s perverse sheaf, McMullen’s g-conjecture and Stanley’s toric h-vector, Kazhdan—Lusztig positivity conjectures and Elias—Williamson’s work, Heron—Rota—Welsh conjecture and Adiprasito—Huh—Katz’s work, Top—Heavy conjecture and Braden—Huh—Matherne—Proudfoot—Wang’s work. Finally, I will make a guess that two shadows of Hodge theory should help us to solve two (series of) long-standing open conjectures: Stanley—Stembridge conjecture / Haiman’s conjecture / Shareshian—Wachs conjecture, and Lusztig—Dyer conjecture / Brenti’s conjecture. This talk will be a non-technical colloquium talk: nothing will be assumed other than elementary linear algebra.