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Mathematical manifold theory
In mathematics, Hodge theory, named after W. V. D. Hodge, is a method for studying the cohomology groups of a smooth manifold M using partial differential
Hodge_theory
Unsolved problem in geometry
In mathematics, the Hodge conjecture is a major unsolved problem in algebraic geometry and complex geometry that relates the algebraic topology of a non-singular
Hodge_conjecture
Mathematical theory
In mathematics, p-adic Hodge theory is a theory that provides a way to classify and study p-adic Galois representations of characteristic 0 local fields
P-adic_Hodge_theory
British mathematician
now called Hodge theory and pertaining more generally to Kähler manifolds—has been a major influence on subsequent work in geometry. Hodge was born in
W._V._D._Hodge
Elliptic curves
In mathematics, Hodge–Arakelov theory of elliptic curves is an analogue of classical and p-adic Hodge theory for elliptic curves carried out in the framework
Hodge–Arakelov_theory
Algebraic structure
mathematics, a Hodge structure, named after W. V. D. Hodge, is an algebraic structure at the level of linear algebra, similar to the one that Hodge theory gives
Hodge_structure
Belgian mathematician
functional equations of L-functions. Deligne also focused on topics in Hodge theory. He introduced the concept of weights and tested them on objects in complex
Pierre_Deligne
Manifold with Riemannian, complex and symplectic structure
metrics. Every smooth complex projective variety is a Kähler manifold. Hodge theory is a central part of algebraic geometry, proved using Kähler metrics
Kähler_manifold
sequence degenerates, thereby leading to the Hodge decomposition of the de Rham cohomology. It is a tool in the theory of complex manifolds for expressing the
Hodge–de Rham spectral sequence
Hodge–de_Rham_spectral_sequence
Natural number
1974, pp. 389. Peano 1889, p. 1. Peano 1908, p. 27. Halmos 1974, p. 32. Hodges 2009, p. 14. Hext 1990. Graham, Knuth & Patashnik 1994, p. 381. Blokhintsev
1
Differential operator in mathematics
The Laplacian is the simplest elliptic operator and is at the core of Hodge theory as well as the results of de Rham cohomology. It is also essentially
Laplace_operator
Lie algebras also arise in the theory of mixed Hodge structures. Goodwillie, Thomas G. (1986), "Relative algebraic K-theory and cyclic homology", Annals
Malcev_Lie_algebra
Correspondsnce between Higgs bundles and fundamental group representations
In algebraic geometry and differential geometry, the nonabelian Hodge correspondence or Corlette–Simpson correspondence (named after Kevin Corlette and
Nonabelian Hodge correspondence
Nonabelian_Hodge_correspondence
Brinon, Olivier; Conrad, Brian (2009), CMI Summer School notes on p-adic Hodge theory (PDF), retrieved 2010-02-05 Fontaine, Jean-Marc (1982), "Sur Certains
Fontaine's_period_rings
and the related Hodge-theoretic concepts. This idea is one of the foundations of logarithmic geometry and has applications in the theory of moduli spaces
Log_structure
Meromorphic differential form
In algebraic geometry and the theory of complex manifolds, a logarithmic differential form is a differential form with poles of a certain kind. The concept
Logarithmic_form
group G is said to be of Hodge–Tate type if it is generated by the eigenvectors of integral powers of χ. p-adic Hodge theory Mumford–Tate group Faltings
Hodge–Tate_module
American mathematician
spaces of vector bundles, higher non-abelian de Rham cohomology (Hodge theory), the theory of higher categories and computer verification of mathematical
Carlos_Simpson
French mathematician (born 1962)
of Hodge structures and mirror symmetry, and has written several books on Hodge theory. In 2002, Voisin proved that the generalization of the Hodge conjecture
Claire_Voisin
Chinese mathematician specializing in number theory
specialising in number theory and arithmetic geometry. He is best known for his fundamental contributions to p-adic Hodge theory, for which he was awarded
Ruochuan_Liu
Romanian-American mathematician
algebraic geometry. He is known for his work on complex birational geometry, Hodge theory, abelian varieties, and vector bundles. Popa received his bachelor's
Mihnea_Popa
Objects between rings and their fields of fractions
They were introduced by Gerd Faltings (1988) in his study of p-adic Hodge theory. Let V be a local integral domain with the maximal ideal m, and K a fraction
Almost_ring
Whole number
set. Also in set theory, 0 is the lowest ordinal number, corresponding to the empty set viewed as a well-ordered set. In order theory (and especially its
0
American mathematician (born 1938)
a major developer in particular of the theory of variation of Hodge structure in Hodge theory and moduli theory, which forms part of transcendental algebraic
Phillip_Griffiths
Mathematical decompositions of vector fields
from, both the Helmholtz decomposition of certain vector fields, and Hodge theory from algebraic geometry. It has applications to stochastic thermodynamics
Helmholtz–Hodge_decomposition
of the Kähler metric. The Kähler identities combine with results of Hodge theory to produce a number of relations on de Rham and Dolbeault cohomology
Kähler_identities
Kind of homology class in differential geometry
{\displaystyle 2p} , and in the direct sum decomposition of H shown to exist in Hodge theory, x is purely of type ( p , p ) {\displaystyle (p,p)} . Secondly, x is
Hodge_cycle
Mathematical theory by Shinichi Mochizuki
anabelian geometry, and the development of p-adic Teichmüller theory, Hodge–Arakelov theory and Frobenioid categories. It was developed with explicit references
Inter-universal Teichmüller theory
Inter-universal_Teichmüller_theory
Exterior algebraic map taking tensors from p forms to n-p forms
In mathematics, the Hodge star operator or Hodge star is a linear map defined on the exterior algebra of a finite-dimensional oriented vector space endowed
Hodge_star_operator
Japanese mathematician
Mochizuki has also worked in Hodge–Arakelov theory and p-adic Teichmüller theory. Mochizuki developed inter-universal Teichmüller theory, which has attracted
Shinichi_Mochizuki
Australian mathematician
European Congress of Mathematicians in Berlin 2016 (Shadows of Hodge theory in representation theory). In 2016 he was awarded the EMS Prize, for 2017 he was
Geordie_Williamson
Mathematical concept
In mathematics, mixed Hodge modules are the culmination of Hodge theory, mixed Hodge structures, intersection cohomology, and the decomposition theorem
Mixed_Hodge_module
Mathematics award
"For pioneering work in geometric representation theory, including the development of Hodge theory for Soergel bimodules and the proof of the Kazhdan-Lusztig
Breakthrough Prize in Mathematics
Breakthrough_Prize_in_Mathematics
Algebraic structure
It is a generalization of a Hodge structure, which is used to study smooth projective varieties. In mixed Hodge theory, where the decomposition of a
Mixed_Hodge_structure
Japanese mathematician (1915–1997)
During the war years he worked in isolation, but was able to master Hodge theory as it then stood. He obtained his PhD from the University of Tokyo in
Kunihiko_Kodaira
theory Hodge theory Homology theory Homotopy theory Ideal theory Index theory Information theory Intersection theory Invariant theory Iwasawa theory K-theory
List_of_mathematical_theories
Mathematical theory
m}\right)\right].} Hodge–Arakelov theory Hodge theory P-adic Hodge theory Adelic group Leong, Y. K. (July–December 2018). "Shou-Wu Zhang: Number Theory and Arithmetic
Arakelov_theory
Polish mathematician
Her research concerns arithmetic geometry, and in particular p-adic Hodge theory, Galois representations, and p-adic cohomology. Nizioł earned an M.S
Wiesława_Nizioł
Study of complex manifolds and several complex variables
such as Hodge theory of Kähler manifolds inspire understanding of Hodge structures for varieties and schemes as well as p-adic Hodge theory, deformation
Complex_geometry
Japanese mathematician (born 1947)
algebraic analysis, microlocal analysis, D-module theory, Hodge theory, sheaf theory and representation theory. He was awarded the Abel Prize in 2025, and is
Masaki_Kashiwara
French mathematician (1944–2019)
2019) was a French mathematician. He was one of the founders of p-adic Hodge theory. He was a professor at Paris-Sud 11 University from 1988 to his death
Jean-Marc_Fontaine
Differential form on a manifold which is permitted to have complex coefficients
Kähler geometry, and Hodge theory. Over non-complex manifolds, they also play a role in the study of almost complex structures, the theory of spinors, and
Complex_differential_form
foundations of algebraic geometry to include, for example, techniques from Hodge theory. (NB: While analytic geometry as use of Cartesian coordinates is also
List of publications in mathematics
List_of_publications_in_mathematics
Theorem in algebraic geometry
duality theorem is an application of Hodge theory for Dolbeault cohomology, and may be seen as a result in the theory of elliptic operators. These two different
Serre_duality
Number system extending the rational numbers
representation. Hodge–Tate representations are related to certain decompositions of p-adic cohomology theories analogous to the Hodge decomposition, hence
P-adic_number
Theory in number theory
of partial cases of the Grothendieck conjecture without using p-adic Hodge theory. Combinatorial anabelian geometry helps to study various aspects of the
Anabelian_geometry
Branch of algebraic geometry
provides topological invariants associated to algebraic varieties. p-adic Hodge theory gives tools to examine when cohomological properties of varieties over
Arithmetic_geometry
American mathematician (born 1983)
mathematics. Huh was awarded the 2022 Fields Medal for "bringing the ideas of Hodge theory to combinatorics, the proof of the Dowling–Wilson conjecture for geometric
June_Huh
Field of higher mathematics
topology. The use of linear elliptic PDEs dates at least as far back as Hodge theory. More recently, it refers largely to the use of nonlinear partial differential
Geometric_analysis
Mathematics award
four." June Huh Princeton University, US "For bringing the ideas of Hodge theory to combinatorics, the proof of the Dowling–Wilson conjecture for geometric
Fields_Medal
mathematics, the intermediate Jacobian of a compact Kähler manifold or Hodge structure is a complex torus that is a common generalization of the Jacobian
Intermediate_Jacobian
German algebraic number theorist
research interests include L-functions, modular forms, p-adic Hodge theory, and Iwasawa theory, and her work has led to new insights towards the Birch and
Sarah_Zerbes
Russian mathematician
diagrams". Barannikov is known for his work on mirror symmetry, Morse theory, and Hodge theory. In mirror symmetry, he is a co-author of construction of Frobenius
Serguei_Barannikov
Supposition or system of ideas intended to explain something
theory — Galois theory — Game theory — Gauge theory — Graph theory — Group theory — Hodge theory — Homology theory — Homotopy theory — Ideal theory —
Theory
Cohomology with real coefficients computed using differential forms
appropriate definition of harmonic forms and of the Hodge theorem. For further details see Hodge theory. If M {\displaystyle M} is a compact Riemannian manifold
De_Rham_cohomology
} This is shown using the Kodaira–Spencer map. In Hodge theory, there are objects called real Hodge structures which are the data of a real vector space
Jacobian_ideal
Topics referred to by the same term
Hodge theorem may refer to: Hodge theory Hodge index theorem This disambiguation page lists mathematics articles associated with the same title. If an
Hodge_theorem
American actor (born 1985)
Edwin Martel Basil Hodge (born January 26, 1985) is an American actor. He is recognized for portraying Dante Bishop in The Purge film series, and is the
Edwin_Hodge
Mathematical term
&p=q\\0&{\text{otherwise}}\end{cases}}} We apply the following well-known fact from Hodge theory: H d R k ( P C n , C ) = ⨁ p + q = k H ∂ ¯ p , q ( P C n ) {\displaystyle
Dolbeault_cohomology
American mathematician
Brosnan is an American mathematician, known for his work on motives, Hodge theory, and algebraic groups. He received his Ph.D. from the University of Chicago
Patrick_Brosnan
Italian mathematician (1921–1987)
with the introduction of the "theta-like class of functions." In p-adic Hodge theory, so-called period rings are denoted by the letter B, a nod to Barsotti's
Iacopo_Barsotti
American mathematician
research in commutative algebra, algebraic geometry, Hodge theory, differential geometry, and the theory of several complex variables. He is known for Green's
Mark_Lee_Green
Moldavian mathematician
In particular, he studies dynamics on Teichmüller spaces and studies Hodge theory in complex geometry. "Students honored at Opening Exercises". Princeton
Simion_Filip
structures. Hodge theory a method for studying the cohomology groups of a smooth manifold M using partial differential equations. Hodge–Arakelov theory Holomorphic
Glossary of areas of mathematics
Glossary_of_areas_of_mathematics
Chern class c1 gives a map from holomorphic line bundles to H2(X, Z). By Hodge theory, the de Rham cohomology group H2(X, C) decomposes as a direct sum H0
Lefschetz theorem on (1,1)-classes
Lefschetz_theorem_on_(1,1)-classes
(1978). "Théorie de Hodge à coefficients dégénérescents". Compt. Rend. Acad. Sci. 286: 1137–1140. Zucker, Steven (1979). "Hodge theory with degenerating
L²_cohomology
Hodge, a Scottish mathematician. Hodge algebra Hodge–Arakelov theory Hodge bundle Hodge conjecture Hodge cycle Hodge–de Rham spectral sequence Hodge diamond
List of things named after W. V. D. Hodge
List_of_things_named_after_W._V._D._Hodge
Theorem in complex geometry
¯ {\displaystyle \partial {\bar {\partial }}} -lemma is a result of Hodge theory and the Kähler identities on a compact Kähler manifold. Sometimes it
Ddbar_lemma
Mathematics concept
In algebraic geometry, the Mumford–Tate group (or Hodge group) MT(F) constructed from a Hodge structure F is a certain algebraic group G. When F is given
Mumford–Tate_group
Australian academic, author, theorist and critic
social and political criticism. Hodge's increasingly interdisciplinary approach has grown to include history, chaos theory, critical management studies,
Bob_Hodge_(linguist)
the period mapping relates families of Kähler manifolds to families of Hodge structures. Let f : X → B be a holomorphic submersive morphism. For a point
Period_mapping
British mathematician and author (born 1949)
1975 for research on twistor theory supervised by Roger Penrose. Since the early 1970s, Hodges has worked on twistor theory, which is the approach to the
Andrew_Hodges
French mathematician
Shimura varieties, p-divisible groups and their moduli spaces, and p-adic Hodge theory. One of his most significant contributions has been to link the local
Laurent_Fargues
topologies. It has subsequently been used by Beilinson to study p-adic Hodge theory, in Bhatt and Scholze's work on projectivity of the affine Grassmannian
H_topology
German-American mathematician
1943) is a German-born American mathematician who works in Hodge theory, representation theory, and automorphic forms. Schmid was raised in Bonn where his
Wilfried_Schmid
Structure in algebraic geometry
a unifying framework for open problems such as the Hodge conjecture and Tate conjecture. The theory of motives was originally conjectured as an attempt
Motive_(algebraic_geometry)
Mathematical group of loops in a Lie group
interaction with Hodge theory. In work of Jeremy Daniel, a loop Hodge structure is defined as an infinite-dimensional analogue of a Hodge structure incorporating
Loop_group
Toponogov theorem Sphere theorem Hodge theory Uniformization theorem Yamabe problem Killing vector field Myers-Steenrod theorem Hodge star operator Weitzenböck
List of differential geometry topics
List_of_differential_geometry_topics
Algebraic structure used in topology
over a field of characteristic 0 {\displaystyle 0} . Tools from Hodge theory, called Hodge structures, help give computations of cohomology of these types
Cohomology
Analogs of homology groups for algebraic varieties
Intersection Theory, section 3.2 and Example 8.3.3. Voisin, Hodge Theory and Complex Algebraic Geometry, v. 2, Conjecture 11.21. Voisin, Hodge Theory and Complex
Chow_group
Concept in algebraic geometry
first Betti number) when k {\displaystyle k} is the complex numbers. Hodge theory has inspired a large body of work on the topological properties of complex
Coherent_sheaf_cohomology
Concept in mathematics
Zoghman Mebkhout (1980, 1984) independently. In the setting of nonabelian Hodge theory, the Riemann-Hilbert correspondence provides a complex analytic isomorphism
Riemann–Hilbert correspondence
Riemann–Hilbert_correspondence
American mathematician
whose focus has been on geometric analysis, Yang–Mills theory, Hodge theory, and string theory. One of Stern's foremost accomplishments is his proof (joint
Mark_Stern
American mathematician
Retrieved 2017-07-01. "Hodge theory of matroids" (PDF), Notices of the AMS, retrieved 2017-07-03 Baker, Matt. "Hodge Theory and Combinatorics" (PDF)
Eric_Katz
combinatorics, and considered the main contributor to the combinatorial Hodge theory Vonnie McLoyd (MA 1973, Ph.D. (1975), developmental psychologist Natalia
List of University of Michigan alumni
List_of_University_of_Michigan_alumni
Theorem in algebraic geometry
p+q<n-1} and is injective if p + q = n − 1 {\displaystyle p+q=n-1} . By Hodge theory, these cohomology groups are equal to the sheaf cohomology groups H q
Lefschetz_hyperplane_theorem
Mathematics concept
following Hodge diamonds. Mirror symmetry conjecture - more mathematically based article Topological quantum field theory Category theory Floer homology
Homological_mirror_symmetry
Used to compare mixed characteristic situations with purely finite characteristic ones
It's a generalization of Faltings's almost purity theorem in p-adic Hodge theory. The name is alluding to almost mathematics, which is used in a proof
Perfectoid_space
complex and the Dolbeault complex which are essential for performing Hodge theory. They also arise in connection with the Atiyah-Singer index theorem and
Elliptic_complex
the Hodge bundle, named after W. V. D. Hodge, appears in the study of families of curves, where it provides an invariant in the moduli theory of algebraic
Hodge_bundle
Hungarian mathematician
manifolds, Yang–Mills instantons, non-Abelian Hodge theory, Geometric Langlands program, and representation theory of quivers and Kac–Moody algebras. Hausel
Tamás_Hausel
Algebraic variety in a projective space
algebraic methods for complex projective varieties lead to areas such as Hodge theory. Let k be an algebraically closed field. The basis of the definition
Projective_variety
American foundation
Probabilistic and Extremal Combinatorics The P=W Conjecture in Nonabelian Hodge Theory Daniel Graham from the University of Surrey won the gold medal for Mathematical
Clay_Mathematics_Institute
American mathematician (1949–2019)
de l'Académie des Sciences. 286: 1137–1140. Zucker, Steven (1979). "Hodge theory with degenerating coefficients: L2-cohomology in the Poincaré metric"
Steven_Zucker
Set of conjectures in algebraic geometry
characteristic zero the Hodge standard conjecture holds, being a consequence of Hodge theory. In positive characteristic the Hodge standard conjecture is
Standard conjectures on algebraic cycles
Standard_conjectures_on_algebraic_cycles
International Congress of Mathematicians award
transcendental methods in complex geometry, particularly his seminal work in Hodge theory and periods of algebraic varieties." 2018 Masaki Kashiwara "For his outstanding
Chern_Medal
Seven mathematical problems with a US$1 million prize for each solution
unsolved mathematical problems, the Birch and Swinnerton-Dyer conjecture, Hodge conjecture, Navier–Stokes existence and smoothness, P versus NP problem
Millennium_Prize_Problems
deprecated parameter |citeseerx= (help) Voisin, Claire (2003). "Nori's Work". Hodge Theory and Complex Algebraic Geometry II. pp. 215–242. doi:10.1017/CBO9780511615177
Griffiths_group
French mathematician (1975–2024)
International congress of mathematicians in Rio with a talk on "Hodge theory and cycle theory of locally symmetric spaces" Bergeron was a Takagi lecturer
Nicolas_Bergeron
American mathematician (1925–2019)
analogue of Hodge theory, now called Hodge–Tate theory, which has blossomed into another central technique of modern algebraic number theory. Other innovations
John_Tate_(mathematician)
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HODGE THEORY
HODGE THEORY
Surname or Lastname
English
English : patronymic from Hodgen.
Boy/Male
British, Christian, English, French
Famed Spear; Renowned Spearman; Diminutive of Rodger
Surname or Lastname
North German and Dutch
North German and Dutch : variant of Otto.English : variant of Hood 1.
Surname or Lastname
English (northern)
English (northern) : patronymic from Hodge.
Surname or Lastname
English (northern Ireland)
English (northern Ireland) : from a pet form of Hodge.
Surname or Lastname
English
English : probably a variant of Budge.
Surname or Lastname
English
English : local name for someone who lived in a small cottage or temporary dwelling, Middle English logge (Old French loge, of Germanic origin). The term was used in particular of a cabin erected by masons working on the site of a particular construction project, such as a church or cathedral, and so it was probably in many cases equivalent to an occupational name for a mason. Reaney suggests that one early form, atte Logge, might sometimes have denoted the warden of a masons’ lodge.Henry Cabot Lodge (1850–1924), the influential U.S. senator from MA, was born in Boston, the only son of John Ellerton Lodge, a prosperous merchant and owner of swift clipper ships engaged in commerce with China, one of several Lodges who emigrated from England in the 18th and 19th centuries.
Female
Yiddish
(×”Ö¸×דֶע) Yiddish form for Hebrew Hadaccah, HODE means "myrtle tree."
Surname or Lastname
English
English : variant of Hedge.
Surname or Lastname
English
English : topographic name for someone who lived by a hedge, Middle English hegg(e).
Surname or Lastname
English (mainly Lancashire and Staffordshire)
English (mainly Lancashire and Staffordshire) : patronymic from Hodge.
Surname or Lastname
English (northern England)
English (northern England) : from the Middle English personal name Dogge, a pet form of Roger.English (northern England) : possibly a nickname from Middle English dogge ‘dog’ (Old English docga, dogga).
Boy/Male
English
Famed spear.
Male
English
Old pet form of English Rodger, DODGE means "famous spear."
Surname or Lastname
English and Scottish
English and Scottish : variant spelling of Hogg.
Surname or Lastname
English
English : from a pet form of Hodge.
Surname or Lastname
English
English : patronymic from Hodge.
Male
English
Middle English pet form of Anglo-Saxon Hroðgar, HODGE means "famous spear."Â
Male
English
Short form of English Rodger, RODGE means "famous spear."
Surname or Lastname
English
English : from the medieval personal name Hodge, a short form of Roger. (For the change of initial, compare Hick.)English : nickname from Middle English hodge ‘hog’, which occurs as a dialect variant of hogge, for example in Cheshire place names.
HODGE THEORY
HODGE THEORY
HODGE THEORY
HODGE THEORY
HODGE THEORY
HODGE THEORY
HODGE THEORY
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