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Richard Hain

Professor of Mathematics
Mathematics
Box 90320, Durham, NC 27708-0320
107 Physics Bldg, Durham, NC 27708

Selected Publications


Remembering Steve Zucker

Journal Article Notices of the American Mathematical Society · August 2, 2021 Cite

Hodge theory of the Turaev cobracket and the Kashiwara-Vergne problem

Journal Article Journal of the European Mathematical Society · January 1, 2021 In this paper we show that, after completing in the I -adic topology, the Turaev cobracket on the vector space freely generated by the closed geodesics on a smooth, complex algebraic curve X with a quasi-algebraic framing is a morphism of mixed Hodge struc ... Full text Open Access Cite

Johnson homomorphisms

Journal Article EMS Surveys in Mathematical Sciences · January 1, 2021 Torelli groups are subgroups of mapping class groups that consist of those diffeomorphism classes that act trivially on the homology of the associated closed surface. The Johnson homomorphism, defined by Dennis Johnson, and its generalization, defined by S ... Full text Cite

Hodge theory of the Goldman bracket

Journal Article Geometry & Topology · November 10, 2020 Full text Open Access Cite

Universal Mixed Elliptic Motives

Journal Article Journal of the Institute of Mathematics of Jussieu · May 1, 2020 Featured Publication In this paper we construct a-linear tannakian category of universal mixed elliptic motives over the moduli space of elliptic curves. It contains , the category of mixed Tate motives unramified over the integers. Each object of is an object of endowed with ... Full text Cite

Notes on the Universal Elliptic KZB Equation

Journal Article Pure and Applied Mathematics Quarterly · January 30, 2020 The universal elliptic KZB equation is the integrable connection on the pro-vector bundle over M_{1,2} whose fiber over the point corresponding to the elliptic curve E and a non-zero point x of E is the unipotent completion of \pi_1(E-{0},x). This was writ ... Open Access Link to item Cite

Notes on the universal elliptic KZB connection

Journal Article Pure and Applied Mathematics Quarterly · January 1, 2020 In this paper, we give an exposition of the elliptic KZB connection over the universal elliptic curve and use it to compute the limit mixed Hodge structure on the unipotent fundamental group of the first order Tate curve. We also give an explicit algebraic ... Full text Cite

Algebraic de Rham theory for weakly holomorphic modular forms of level one

Journal Article Algebra and Number Theory · January 1, 2018 We establish an Eichler–Shimura isomorphism for weakly modular forms of level one. We do this by relating weakly modular forms with rational Fourier coefficients to the algebraic de Rham cohomology of the modular curve with twisted coefficients. This leads ... Full text Cite

Deligne-Beilinson Cohomology of Affine Groups

Chapter · 2017 The goal of this paper is to develop the theory of Deligne-Beilinson cohomology of affine groups with a mixed Hodge structure. The motivation comes from Hodge theory and the study of motives, where such groups appear. Several of Francis Brown's period comp ... Link to item Cite

On the fundamental groups of normal varieties

Journal Article Communications in Contemporary Mathematics · August 1, 2016 We show that the fundamental groups of normal complex algebraic varieties share many properties of the fundamental groups of smooth varieties. The jump loci of rank one local systems on a normal variety are related to the jump loci of a resolution and of a ... Full text Cite

The Hodge-de Rham theory of modular groups

Chapter · February 4, 2016 Featured Publication Full text Cite

Genus 3 mapping class groups are not Kähler

Journal Article Journal of Topology · March 2015 Full text Cite

The abelianization of the Johnson kernel

Journal Article Journal of the European Mathematical Society · January 1, 2014 We prove that the first complex homology of the Johnson subgroup of the Torelli group Tg is a non-trivial, unipotent Tg-module for all g ≥ 4 and give an explicit presentation of it as a Sym H 1(Tg,C)-module when g ≥ 6. We do this by proving that, for a fin ... Full text Open Access Cite

Remarks on non-abelian cohomology of proalgebraic groups

Journal Article Journal of Algebraic Geometry · June 18, 2013 In this paper we develop a theory of non-abelian cohomology for proalgebraic groups which is used in J. Amer. Math. Soc. 24 (2011), 709-769 to study the unipotent section conjecture. The non-abelian cohomology H 1nab(G,P) is a scheme. The argument G is a p ... Full text Cite

Rational points of universal curves

Journal Article Journal of the American Mathematical Society · July 1, 2011 Full text Open Access Cite

Remembering Steve Zucker

Journal Article Notices of the American Mathematical Society · August 2, 2021 Cite

Hodge theory of the Turaev cobracket and the Kashiwara-Vergne problem

Journal Article Journal of the European Mathematical Society · January 1, 2021 In this paper we show that, after completing in the I -adic topology, the Turaev cobracket on the vector space freely generated by the closed geodesics on a smooth, complex algebraic curve X with a quasi-algebraic framing is a morphism of mixed Hodge struc ... Full text Open Access Cite

Johnson homomorphisms

Journal Article EMS Surveys in Mathematical Sciences · January 1, 2021 Torelli groups are subgroups of mapping class groups that consist of those diffeomorphism classes that act trivially on the homology of the associated closed surface. The Johnson homomorphism, defined by Dennis Johnson, and its generalization, defined by S ... Full text Cite

Hodge theory of the Goldman bracket

Journal Article Geometry & Topology · November 10, 2020 Full text Open Access Cite

Universal Mixed Elliptic Motives

Journal Article Journal of the Institute of Mathematics of Jussieu · May 1, 2020 Featured Publication In this paper we construct a-linear tannakian category of universal mixed elliptic motives over the moduli space of elliptic curves. It contains , the category of mixed Tate motives unramified over the integers. Each object of is an object of endowed with ... Full text Cite

Notes on the Universal Elliptic KZB Equation

Journal Article Pure and Applied Mathematics Quarterly · January 30, 2020 The universal elliptic KZB equation is the integrable connection on the pro-vector bundle over M_{1,2} whose fiber over the point corresponding to the elliptic curve E and a non-zero point x of E is the unipotent completion of \pi_1(E-{0},x). This was writ ... Open Access Link to item Cite

Notes on the universal elliptic KZB connection

Journal Article Pure and Applied Mathematics Quarterly · January 1, 2020 In this paper, we give an exposition of the elliptic KZB connection over the universal elliptic curve and use it to compute the limit mixed Hodge structure on the unipotent fundamental group of the first order Tate curve. We also give an explicit algebraic ... Full text Cite

Algebraic de Rham theory for weakly holomorphic modular forms of level one

Journal Article Algebra and Number Theory · January 1, 2018 We establish an Eichler–Shimura isomorphism for weakly modular forms of level one. We do this by relating weakly modular forms with rational Fourier coefficients to the algebraic de Rham cohomology of the modular curve with twisted coefficients. This leads ... Full text Cite

Deligne-Beilinson Cohomology of Affine Groups

Chapter · 2017 The goal of this paper is to develop the theory of Deligne-Beilinson cohomology of affine groups with a mixed Hodge structure. The motivation comes from Hodge theory and the study of motives, where such groups appear. Several of Francis Brown's period comp ... Link to item Cite

On the fundamental groups of normal varieties

Journal Article Communications in Contemporary Mathematics · August 1, 2016 We show that the fundamental groups of normal complex algebraic varieties share many properties of the fundamental groups of smooth varieties. The jump loci of rank one local systems on a normal variety are related to the jump loci of a resolution and of a ... Full text Cite

The Hodge-de Rham theory of modular groups

Chapter · February 4, 2016 Featured Publication Full text Cite

Genus 3 mapping class groups are not Kähler

Journal Article Journal of Topology · March 2015 Full text Cite

The abelianization of the Johnson kernel

Journal Article Journal of the European Mathematical Society · January 1, 2014 We prove that the first complex homology of the Johnson subgroup of the Torelli group Tg is a non-trivial, unipotent Tg-module for all g ≥ 4 and give an explicit presentation of it as a Sym H 1(Tg,C)-module when g ≥ 6. We do this by proving that, for a fin ... Full text Open Access Cite

Remarks on non-abelian cohomology of proalgebraic groups

Journal Article Journal of Algebraic Geometry · June 18, 2013 In this paper we develop a theory of non-abelian cohomology for proalgebraic groups which is used in J. Amer. Math. Soc. 24 (2011), 709-769 to study the unipotent section conjecture. The non-abelian cohomology H 1nab(G,P) is a scheme. The argument G is a p ... Full text Cite

Rational points of universal curves

Journal Article Journal of the American Mathematical Society · July 1, 2011 Full text Open Access Cite

Relative pro-ℓ completions of mapping class groups

Journal Article Journal of Algebra · June 2009 Full text Cite

Galois actions on fundamental groups of curves and the cycle

Journal Article Journal of the Institute of Mathematics of Jussieu · January 1, 2005 Suppose that [formula omitted] is a subfield of [formula omitted] for which the [formula omitted] -adic cyclotomic character has infinite image. Suppose that [formula omitted] is a curve of genus [formula omitted] defined over [formula omitted], and that [ ... Full text Cite

The Hyodo-Kato theorem for rational homotopy types

Journal Article Mathematical Research Letters · January 1, 2005 The Hyodo-Kato theorem relates the De Rham cohomology of a variety over a local field with semi-stable reduction to the log crystalline cohomology of the special fiber. In this paper we prove an analogue for rational homotopy types. In particular, this giv ... Full text Open Access Cite

A De Rham–Witt approach to crystalline rational homotopy theory

Journal Article Compositio Mathematica · September 2004 Full text Open Access Cite

On the arakelov geometry of moduli spaces of curves

Journal Article Journal of Differential Geometry · January 1, 2004 In this paper we compute the asymptotics of the natural metric on the line bundle over the moduli spaceMg associated to the algebraic cycle C − C− in the jacobian Jac C of a smooth projective curve C of genus g ≤ 3. The asymptotics are related to the struc ... Full text Cite

Weighted completion of galois groups and galois actions on the fundamental group of ℙ1 -{0, 1, ∞}

Journal Article Compositio Mathematica · November 1, 2003 Fix a prime number l. We prove a conjecture stated by Ihara, which he attributes to Deligne, about the action of the absolute Galois group on the pro-l completion of the fundamental group of the thrice punctured projective line. Similar techniques are also ... Full text Cite

The rational cohomology ring of the moduli space of abelian 3-folds

Journal Article Mathematical Research Letters · 2002 Featured Publication Cite

Geometric proofs of some results of Morita

Journal Article Journal of Algebraic Geometry · 2001 Cite

The Hodge De Rham theory of relative Malcev completion

Journal Article Annales Scientifiques de l'Ecole Normale Superieure · 1998 Featured Publication Suppose that X is a smooth manifold and ρ : π1 (X,N) → S is a representation of the fundamental group of X into a real reductive group with Zariski dense image. To such data one can associate the Malcev completion G of π1(X,x) relative to ρ. In this paper ... Cite

Infinitesimal presentations of the Torelli groups

Journal Article Journal of the American Mathematical Society · 1997 Featured Publication Cite

Chow varieties of abelian varieties

Journal Article Sociedad Matemática Mexicana. Boletí n. Tercera Serie · 1996 Cite

Real Grassmann polylogarithms and Chern classes

Journal Article Mathematische Annalen · 1996 Full text Cite

The existence of higher logarithms

Journal Article Compositio Mathematica · 1996 Cite

Nil-manifolds as links of isolated singularities

Journal Article Compositio Mathematica · 1992 Cite

Biextensions and heights associated to curves of odd genus

Journal Article Duke Mathematical Journal · 1990 Full text Cite

The Life and Work of Kuo Tsai Chen

Journal Article Illinois Journal of Mathematics · 1990 Cite

Higher Logarithms

Journal Article Illinois Journal of Mathematics · 1990 Cite

Mixed Hodge Structures on the Homotopy of Links

Journal Article Mathematische Annalen · 1988 Full text Cite

Truncations of Mixed Hodge Complexes

Conference Proceedings of the U.S. Spain Workshop · 1987 Full text Cite

Iterated Integrals and Mixed Hodge Structures on Homotopy Groups

Conference Proceedings of the U.S. Spain Workshop · 1987 Full text Cite

Higher Albanese Manifolds

Conference Proceedings of the U.S. Spain Workshop · 1987 Full text Cite

A Guide to Unipotent Variations of Mixed Hodge Structure

Conference Proceedings of the U.S. Spain Workshop · 1987 Full text Cite

The de Rham homotopy theory of complex algebraic varieties. II

Journal Article $K$-Theory. An Interdisciplinary Journal for the Development, Application, and Influence of $K$-Theory in the Mathematical Sciences · 1987 Full text Cite

The de rham homotopy theory of complex algebraic varieties I

Journal Article K-Theory · 1987 In this paper we use Chen's iterated integrals to put a mixed Hodge structure on the homotopy Lie algebra of an arbitrary complex algebraic variety, generalizing work of Deligne and Morgan. Similar techniques are used to put a mixed Hodge structure on othe ... Full text Cite

Unipotent variations of mixed Hodge structure

Journal Article Inventiones Mathematicae · 1987 Featured Publication Full text Cite

On a Generalization of Hilbert’s 21st Problem

Journal Article Annales Scientifiques de l’École Normale Supérieure. Quatrième Série · 1986 Cite

On the Indecomposable Elements of the Bar Construction

Journal Article Proceedings of the American Mathematical Society · 1986 Full text Cite

Mixed Hodge structures on homotopy groups

Journal Article American Mathematical Society. Bulletin. New Series · 1986 Full text Cite

On the indecomposable elements of the bar construction

Journal Article Proceedings of the American Mathematical Society · January 1, 1986 An explicit formula for a canonical splitting s: Qℬ(ℰ.)⟶ℬ(ℰ.) of the projection ℬ(ℰ.)⟶Qℬ(ℰ.) of the bar construction on a commutative d.g. algebraℰ.onto its indecomposables is given. We prove that s induces a d.g. algebra isomorphism Λ(Qℬ(ℰ.))⟶ℬ(ℰ.) and th ... Full text Cite

CORRECTION

Journal Article TOPOLOGY · 1986 Cite

Iterated integrals, intersection theory and link groups

Journal Article Topology. An International Journal of Mathematics · 1985 Full text Cite

ITERATED INTEGRALS AND HOMOTOPY PERIODS

Journal Article MEMOIRS OF THE AMERICAN MATHEMATICAL SOCIETY · January 1, 1984 Link to item Cite

Primitive Elements in Rings of Holomorphic Functions

Journal Article Journal für die Reine und Angewandte Mathematik. [Crelle’s Journal] · 1984 Full text Cite

Twisting Cochains and Duality Between Minimal Algebras and Minimal Lie Algebras

Journal Article Transactions of the American Mathematical Society · May 1983 Full text Cite

Twisting Cochains and Duality Between Minimal Algebras and Minimal Lie Algebras

Journal Article Transactions of the American Mathematical Society · 1983 Full text Cite

A Characterization of Smooth Functions Defined on a Banach Space

Journal Article Proceedings of the American Mathematical Society · October 1979 Full text Cite

A characterization of smooth functions defined on a Banach space

Journal Article Proceedings of the American Mathematical Society · 1979 A sufficient condition for a function defined on a Banach space to be Full text Cite

On Circulant Weighing Matrices

Journal Article Ars Combinatoria · 1976 Cite