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1. One reason to believe the Hodge conjecture is that it suggests a close relation between Hodge theory and algebraic cycles, and this hope has led to a long series of discoveries about algebraic cycles.
2. Hodge theory has become an important tool in algebraic geometry, particularly through its connection to the study of algebraic cycles.
3. While Hodge theory is intrinsically dependent upon the real and complex numbers, it can be applied to questions in number theory.
4. In arithmetic situations, the tools of p-adic Hodge theory have given alternative proofs of, or analogous results to, classical Hodge theory.
5. This book provides a comprehensive and up-to-date introduction to Hodge theory—one of the central and most vibrant areas of contemporary mathematics—from leading specialists on the subject.
6. The topics range from the basic topology of algebraic varieties to the study of variations of mixed Hodge structure and the Hodge theory of maps.
7. This exposition of Hodge theory is a slightly retooled version of the authors Harvard minor thesis, advised by Professor Joe Harris.
8. Introduction Our objective in this exposition is to state and prove the main theorems of Hodge theory.
9. We then conclude the exposition by showing that Hodge theory can be used to give elegant proofs of the Date: March 8, 2018.
10. Next, we dene an operator that is, as its name will suggest, central to Hodge theory.
11. The recent work of Adiprasito–Huh–Katz establishes the Rota–Welsh Conjecture by developing a combinatorial Hodge theory for matroids in general.
12. Hodge–Tate representations are related to certain decompositions of p-adic cohomology theories analogous to the Hodge decomposition, hence the name p-adic Hodge theory.
13. This third symposium in the series focused on recent advances in all aspects of p-adic Hodge theory.
14. Another recurring theme at the symposium was the connection between p-adic Hodge theory and the local Langlands correspondence, modularity and number-theoretic applications.
15. The goal of the course is to give an introduction to the basic results in Hodge theory.
16. Hodge theory of compact, oriented, Riemannian manifolds 6.
17. Consider R C t (cid:55) g(tu + (1 t)w) MATH 731: HODGE THEORY 5 where w Cn is such that g(w) (cid:54)= 0.
18. We introduce real and complex Hodge theory to study topological invariants using harmonic analysis.
19. To do so, we review Riemannian and complex geometry, intro- duce de Rham cohomology, and give the basic theorems of real and complex Hodge theory.
20. Different faces of the Hodge theory Let M be a smooth compact manifold.
21. Hodge theory claims H k Each de Rham cohomology class contains a unique harmonic representative.
22. 1 2 LECTURE 26: THE HODGE THEORY The fact that these two statements are equivalent follows from Lemma 1.1.
23. 4 LECTURE 26: THE HODGE THEORY Now let , (M ) be two dierential forms on M .
24. The Banff International Research Station will host the "Equivariant Stable Homotopy Theory and p-adic Hodge Theory" workshop in Banff from March 1 to March 06, 2020.
25. The goal of p-adic Hodge theory is to classify and study p-adic representations of G p (i.e. l = p above) and to this end p-adic Hodge theory is astonishingly successful.
26. In this course, we will develop p-adic Hodge theory from the beginning and will provide complete proofs of many of the key results and theorems.
27. Working seminar on Hodge theory Spring 2020 Hodge theory is a beautiful and powerful subject, and the algebraic geometry prerequisites are minimal.
28. We recall that the almost purity theorem is one key technical ingredient in Faltings approach to p-adic Hodge theory, and we follow many of his ideas in this paper.
29. This is of course in analogy with classical Hodge theory, which is formulated most naturally in terms of complex-analytic spaces.
30. p-adic Hodge theory for rigid-analytic varieties 3 Let us rst explain our proof of this theorem.
31. We note that the idea of extracting many p-power roots is common to all known proofs of comparison theorems in p-adic Hodge theory.