Pierre René, Viscount Deligne (; born 3 October 1944) is a Belgian mathematician. He is best known for work on the
Weil conjectures
In mathematics, the Weil conjectures were highly influential proposals by . They led to a successful multi-decade program to prove them, in which many leading researchers developed the framework of modern algebraic geometry and number theory.
Th ...
, leading to a complete proof in 1973. He is the winner of the 2013
Abel Prize
The Abel Prize ( ; no, Abelprisen ) is awarded annually by the King of Norway to one or more outstanding mathematicians. It is named after the Norwegian mathematician Niels Henrik Abel (1802–1829) and directly modeled after the Nobel Prizes. ...
, 2008
Wolf Prize
The Wolf Prize is an international award granted in Israel, that has been presented most years since 1978 to living scientists and artists for ''"achievements in the interest of mankind and friendly relations among people ... irrespective of natio ...
, 1988
Crafoord Prize
The Crafoord Prize is an annual science prize established in 1980 by Holger Crafoord, a Swedish industrialist, and his wife Anna-Greta Crafoord. The Prize is awarded in partnership between the Royal Swedish Academy of Sciences and the Crafoord Foun ...
, and 1978
Fields Medal
The Fields Medal is a prize awarded to two, three, or four mathematicians under 40 years of age at the International Congress of the International Mathematical Union (IMU), a meeting that takes place every four years. The name of the award ho ...
.
Early life and education
Deligne was born in
Etterbeek
Etterbeek (French: ; Dutch: ) is one of the 19 municipalities of the Brussels-Capital Region, Belgium. Located in the eastern part of the region, it is bordered by the municipalities of Auderghem, the City of Brussels, Ixelles, Schaerbeek, Wolu ...
, attended school at
Athénée Adolphe Max
Athénée Adolphe Max is a secondary school of the City of Brussels which is part of the official education network;. It is located to the east of the center of Brussels, near the Squares district .
Historical
A first building was designed in 1 ...
and studied at the
Université libre de Bruxelles (ULB), writing a dissertation titled ''Théorème de Lefschetz et critères de dégénérescence de suites spectrales'' (Theorem of Lefschetz and criteria of degeneration of spectral sequences). He completed his doctorate at the
University of Paris-Sud
Paris-Sud University (French: ''Université Paris-Sud''), also known as University of Paris — XI (or as Université d'Orsay before 1971), was a French research university distributed among several campuses in the southern suburbs of Paris, in ...
in
Orsay
Orsay () is a commune in the Essonne department in Île-de-France in northern France. It is located in the southwestern suburbs of Paris, France, from the centre of Paris.
A fortified location of the Chevreuse valley since the 8th centur ...
1972 under the supervision of
Alexander Grothendieck, with a thesis titled ''Théorie de Hodge''.
Career
Starting in 1972, Deligne worked with Grothendieck at the
Institut des Hautes Études Scientifiques
The Institut des hautes études scientifiques (IHÉS; English: Institute of Advanced Scientific Studies) is a French research institute supporting advanced research in mathematics and theoretical physics. It is located in Bures-sur-Yvette, jus ...
(IHÉS) near Paris, initially on the generalization within
scheme theory
In mathematics, a scheme is a mathematical structure that enlarges the notion of algebraic variety in several ways, such as taking account of multiplicities (the equations ''x'' = 0 and ''x''2 = 0 define the same algebraic variety but different sc ...
of
Zariski's main theorem
In algebraic geometry, Zariski's main theorem, proved by , is a statement about the structure of birational morphisms stating roughly that there is only one branch at any normal point of a variety. It is the special case of Zariski's connectedness ...
. In 1968, he also worked with
Jean-Pierre Serre
Jean-Pierre Serre (; born 15 September 1926) is a French mathematician who has made contributions to algebraic topology, algebraic geometry, and algebraic number theory. He was awarded the Fields Medal in 1954, the Wolf Prize in 2000 and the ina ...
; their work led to important results on the l-adic representations attached to
modular form
In mathematics, a modular form is a (complex) analytic function on the upper half-plane satisfying a certain kind of functional equation with respect to the Group action (mathematics), group action of the modular group, and also satisfying a grow ...
s, and the conjectural
functional equation
In mathematics, a functional equation
is, in the broadest meaning, an equation in which one or several functions appear as unknowns. So, differential equations and integral equations are functional equations. However, a more restricted meaning ...
s of
L-function
In mathematics, an ''L''-function is a meromorphic function on the complex plane, associated to one out of several categories of mathematical objects. An ''L''-series is a Dirichlet series, usually convergent on a half-plane, that may give ris ...
s. Deligne's also focused on topics in
Hodge 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 equations. The key observation is that, given a Riemannian metric on ''M'', every cohom ...
. He introduced the concept of weights and tested them on objects in
complex geometry
In mathematics, complex geometry is the study of geometric structures and constructions arising out of, or described by, the complex numbers. In particular, complex geometry is concerned with the study of spaces such as complex manifolds and c ...
. He also collaborated with
David Mumford
David Bryant Mumford (born 11 June 1937) is an American mathematician known for his work in algebraic geometry and then for research into vision and pattern theory. He won the Fields Medal and was a MacArthur Fellow. In 2010 he was awarded t ...
on a new description of the
moduli space
In mathematics, in particular algebraic geometry, a moduli space is a geometric space (usually a scheme or an algebraic stack) whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects. Such spac ...
s for curves. Their work came to be seen as an introduction to one form of the theory of
algebraic stack
In mathematics, an algebraic stack is a vast generalization of algebraic spaces, or schemes, which are foundational for studying moduli theory. Many moduli spaces are constructed using techniques specific to algebraic stacks, such as Artin's repr ...
s, and recently has been applied to questions arising from
string theory
In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings. String theory describes how these strings propagate through space and interac ...
. But Deligne's most famous contribution was his proof of the third and last of the
Weil conjectures
In mathematics, the Weil conjectures were highly influential proposals by . They led to a successful multi-decade program to prove them, in which many leading researchers developed the framework of modern algebraic geometry and number theory.
Th ...
. This proof completed a programme initiated and largely developed by
Alexander Grothendieck lasting for more than a decade. As a corollary he proved the celebrated
Ramanujan–Petersson conjecture In mathematics, the Ramanujan conjecture, due to , states that Ramanujan's tau function given by the Fourier coefficients of the cusp form of weight
:\Delta(z)= \sum_\tau(n)q^n=q\prod_\left (1-q^n \right)^ = q-24q^2+252q^3- 1472q^4 + 4830q^5-\ ...
for
modular form
In mathematics, a modular form is a (complex) analytic function on the upper half-plane satisfying a certain kind of functional equation with respect to the Group action (mathematics), group action of the modular group, and also satisfying a grow ...
s of weight greater than one; weight one was proved in his work with Serre. Deligne's 1974 paper contains the first proof of the
Weil conjectures
In mathematics, the Weil conjectures were highly influential proposals by . They led to a successful multi-decade program to prove them, in which many leading researchers developed the framework of modern algebraic geometry and number theory.
Th ...
. Deligne's contribution was to supply the estimate of the
eigenvalues
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
of the
Frobenius endomorphism
In commutative algebra and field theory, the Frobenius endomorphism (after Ferdinand Georg Frobenius) is a special endomorphism of commutative rings with prime characteristic , an important class which includes finite fields. The endomorphism ma ...
, considered the geometric analogue of the
Riemann hypothesis
In mathematics, the Riemann hypothesis is the conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part . Many consider it to be the most important unsolved problem in ...
. It also led to the proof of
Lefschetz hyperplane theorem In mathematics, specifically in algebraic geometry and algebraic topology, the Lefschetz hyperplane theorem is a precise statement of certain relations between the shape of an algebraic variety and the shape of its subvarieties. More precisely, the ...
and the old and new estimates of the classical exponential sums, among other applications. Deligne's 1980 paper contains a much more general version of the Riemann hypothesis.
From 1970 until 1984, Deligne was a permanent member of the IHÉS staff. During this time he did much important work outside of his work on algebraic geometry. In joint work with
George Lusztig
George Lusztig (born ''Gheorghe Lusztig''; May 20, 1946) is an American-Romanian mathematician and Abdun Nur Professor at the Massachusetts Institute of Technology (MIT). He was a Norbert Wiener Professor in the Department of Mathematics from 1 ...
, Deligne applied
étale cohomology
In mathematics, the étale cohomology groups of an algebraic variety or scheme are algebraic analogues of the usual cohomology groups with finite coefficients of a topological space, introduced by Grothendieck in order to prove the Weil conjecture ...
to construct representations of
finite groups of Lie type
In mathematics, specifically in group theory, the phrase ''group of Lie type'' usually refers to finite groups that are closely related to the group of rational points of a reductive linear algebraic group with values in a finite field. The phra ...
; with
Michael Rapoport
Michael Rapoport (born 2 October 1948) is an Austrian mathematician.
Career
Rapoport received his PhD from Paris-Sud 11 University in 1976, under the supervision of Pierre Deligne. He held a chair for arithmetic algebraic geometry at the Univ ...
, Deligne worked on the moduli spaces from the 'fine' arithmetic point of view, with application to
modular form
In mathematics, a modular form is a (complex) analytic function on the upper half-plane satisfying a certain kind of functional equation with respect to the Group action (mathematics), group action of the modular group, and also satisfying a grow ...
s. He received a
Fields Medal
The Fields Medal is a prize awarded to two, three, or four mathematicians under 40 years of age at the International Congress of the International Mathematical Union (IMU), a meeting that takes place every four years. The name of the award ho ...
in 1978. In 1984, Deligne moved to the
Institute for Advanced Study
The Institute for Advanced Study (IAS), located in Princeton, New Jersey, in the United States, is an independent center for theoretical research and intellectual inquiry. It has served as the academic home of internationally preeminent scholar ...
in Princeton.
Hodge cycles
In terms of the completion of some of the underlying Grothendieck program of research, he defined
absolute Hodge cycles, as a surrogate for the missing and still largely conjectural theory of
motives. This idea allows one to get around the lack of knowledge of the
Hodge conjecture
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 complex algebraic variety to its subvarieties.
In simple terms, the Hodge conjectu ...
, for some applications. The theory of
mixed Hodge structures, a powerful tool in algebraic geometry that generalizes classical Hodge theory, was created by applying weight filtration, Hironaka's
resolution of singularities
In algebraic geometry, the problem of resolution of singularities asks whether every algebraic variety ''V'' has a resolution, a non-singular variety ''W'' with a proper birational map ''W''→''V''. For varieties over fields of characterist ...
and other methods, which he then used it to prove the Weil conjectures. He reworked the
Tannakian category
In mathematics, a Tannakian category is a particular kind of monoidal category ''C'', equipped with some extra structure relative to a given field ''K''. The role of such categories ''C'' is to approximate, in some sense, the category of linear re ...
theory in his 1990 paper for the "Grothendieck Festschrift", employing
Beck's theorem – the Tannakian category concept being the categorical expression of the linearity of the theory of motives as the ultimate
Weil cohomology. All this is part of the ''yoga of weights'', uniting
Hodge 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 equations. The key observation is that, given a Riemannian metric on ''M'', every cohom ...
and the l-adic
Galois representations. The
Shimura variety In number theory, a Shimura variety is a higher-dimensional analogue of a modular curve that arises as a quotient variety of a Hermitian symmetric space by a congruence subgroup of a reductive algebraic group defined over Q. Shimura varieties a ...
theory is related, by the idea that such varieties should parametrize not just good (arithmetically interesting) families of Hodge structures, but actual motives. This theory is not yet a finished product, and more recent trends have used
K-theory
In mathematics, K-theory is, roughly speaking, the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology, it is a cohomology theory known as topological K-theory. In algebra and algebraic geometry, ...
approaches.
Perverse sheaves
With
Alexander Beilinson
Alexander A. Beilinson (born 1957) is the David and Mary Winton Green University professor at the University of Chicago and works on mathematics. His research has spanned representation theory, algebraic geometry and mathematical physics. In 1 ...
,
Joseph Bernstein
Joseph Bernstein (sometimes spelled I. N. Bernshtein; he, יוס(י)ף נאומוביץ ברנשטיין; russian: Иосиф Наумович Бернштейн; born 18 April 1945) is a Soviet-born Israeli mathematician working at Tel Aviv Univ ...
, and
Ofer Gabber
Ofer Gabber (עופר גאבר; born May 16, 1958) is a mathematician working in algebraic geometry.
Life
In 1978 Gabber received a Ph.D. from Harvard University for the thesis ''Some theorems on Azumaya algebras,'' written under the supervis ...
, Deligne made definitive contributions to the theory of
perverse sheaves The mathematical term perverse sheaves refers to a certain abelian category associated to a topological space ''X'', which may be a real or complex manifold, or a more general topologically stratified space, usually singular. This concept was intro ...
. This theory plays an important role in the recent proof of the
fundamental lemma by
Ngô Bảo Châu
Ngô Bảo Châu (, born June 28, 1972) is a Vietnamese-French mathematician at the University of Chicago, best known for proving the fundamental lemma for automorphic forms (proposed by Robert Langlands and Diana Shelstad). He is the first Vie ...
. It was also used by Deligne himself to greatly clarify the nature of the
Riemann–Hilbert correspondence In mathematics, the term Riemann–Hilbert correspondence refers to the correspondence between regular singular flat connections on algebraic vector bundles and representations of the fundamental group, and more generally to one of several generaliz ...
, which extends
Hilbert's twenty-first problem
The twenty-first problem of the 23 Hilbert problems, from the celebrated list put forth in 1900 by David Hilbert, concerns the existence of a certain class of linear differential equations with specified singular points and monodromic group.
S ...
to higher dimensions. Prior to Deligne's paper,
Zoghman Mebkhout
Zoghman Mebkhout (born 1949 ) (مبخوت زغمان) is a French-Algerian mathematician. He is known for his work in algebraic analysis, geometry and representation theory, more precisely on the theory of ''D''-modules.
Career
Mebkhout is c ...
's 1980 thesis and the work of
Masaki Kashiwara
is a Japanese mathematician. He was a student of Mikio Sato at the University of Tokyo. Kashiwara made leading contributions towards algebraic analysis, microlocal analysis, D-module, ''D''-module theory, Hodge theory, sheaf theory and represent ...
through
D-modules
In mathematics, a ''D''-module is a module over a ring ''D'' of differential operators. The major interest of such ''D''-modules is as an approach to the theory of linear partial differential equations. Since around 1970, ''D''-module theory has ...
theory (but published in the 80s) on the problem have appeared.
Other works
In 1974 at the IHÉS, Deligne's joint paper with
Phillip Griffiths
Phillip Augustus Griffiths IV (born October 18, 1938) is an American mathematician, known for his work in the field of geometry, and in particular for the complex manifold approach to algebraic geometry. He was a major developer in particul ...
,
John Morgan and
Dennis Sullivan
Dennis Parnell Sullivan (born February 12, 1941) is an American mathematician known for his work in algebraic topology, geometric topology, and dynamical systems. He holds the Albert Einstein Chair at the City University of New York Graduate Ce ...
on the real
homotopy theory
In mathematics, homotopy theory is a systematic study of situations in which maps can come with homotopies between them. It originated as a topic in algebraic topology but nowadays is studied as an independent discipline. Besides algebraic topolog ...
of compact
Kähler manifolds was a major piece of work in complex differential geometry which settled several important questions of both classical and modern significance. The input from Weil conjectures, Hodge theory, variations of Hodge structures, and many geometric and topological tools were critical to its investigations. His work in complex
singularity theory
In mathematics, singularity theory studies spaces that are almost manifolds, but not quite. A string can serve as an example of a one-dimensional manifold, if one neglects its thickness. A singularity can be made by balling it up, dropping it ...
generalized
Milnor map In mathematics, Milnor maps are named in honor of John Milnor, who introduced them to topology and algebraic geometry in his book ''Singular Points of Complex Hypersurfaces'' (Princeton University Press, 1968) and earlier lectures. The most studied ...
s into an algebraic setting and extended the
Picard-Lefschetz formula beyond their general format, generating a new method of research in this subject. His paper with
Ken Ribet
Kenneth Alan Ribet (; born June 28, 1948) is an American mathematician working in algebraic number theory and algebraic geometry. He is known for the Herbrand–Ribet theorem and Ribet's theorem, which were key ingredients in the proof of Fermat's ...
on abelian L-functions and their extensions to
Hilbert modular surface
In mathematics, a Hilbert modular surface or Hilbert–Blumenthal surface is an algebraic surface obtained by taking a quotient of a product of two copies of the upper half-plane by a Hilbert modular group. More generally, a Hilbert modular varie ...
s and p-adic L-functions form an important part of his work in
arithmetic geometry
In mathematics, arithmetic geometry is roughly the application of techniques from algebraic geometry to problems in number theory. Arithmetic geometry is centered around Diophantine geometry, the study of rational points of algebraic variety, alg ...
. Other important research achievements of Deligne include the notion of cohomological descent, motivic L-functions, mixed sheaves, nearby
vanishing cycle In mathematics, vanishing cycles are studied in singularity theory and other parts of algebraic geometry. They are those homology cycles of a smooth fiber in a family which vanish in the singular fiber.
For example, in a map from a connected compl ...
s, central extensions of
reductive group
In mathematics, a reductive group is a type of linear algebraic group over a field. One definition is that a connected linear algebraic group ''G'' over a perfect field is reductive if it has a representation with finite kernel which is a direct ...
s, geometry and topology of
braid group
A braid (also referred to as a plait) is a complex structure or pattern formed by interlacing two or more strands of flexible material such as textile yarns, wire, or hair.
The simplest and most common version is a flat, solid, three-strande ...
s, the work in collaboration with
George Mostow
George Daniel Mostow (July 4, 1923 – April 4, 2017) was an American mathematician, renowned for his contributions to Lie theory. He was the Henry Ford II (emeritus) Professor of Mathematics at Yale University, a member of the National Academy o ...
on the examples of non-arithmetic lattices and monodromy of
hypergeometric differential equations in two- and three-dimensional complex
hyperbolic space
In mathematics, hyperbolic space of dimension n is the unique simply connected, n-dimensional Riemannian manifold of constant sectional curvature equal to -1. It is homogeneous, and satisfies the stronger property of being a symmetric space. Th ...
s, etc.
Awards
He was awarded the
Fields Medal
The Fields Medal is a prize awarded to two, three, or four mathematicians under 40 years of age at the International Congress of the International Mathematical Union (IMU), a meeting that takes place every four years. The name of the award ho ...
in 1978, the
Crafoord Prize
The Crafoord Prize is an annual science prize established in 1980 by Holger Crafoord, a Swedish industrialist, and his wife Anna-Greta Crafoord. The Prize is awarded in partnership between the Royal Swedish Academy of Sciences and the Crafoord Foun ...
in 1988, the
Balzan Prize
The International Balzan Prize Foundation awards four annual monetary prizes to people or organizations who have made outstanding achievements in the fields of humanities, natural sciences, culture, as well as for endeavours for peace and the br ...
in 2004, the
Wolf Prize
The Wolf Prize is an international award granted in Israel, that has been presented most years since 1978 to living scientists and artists for ''"achievements in the interest of mankind and friendly relations among people ... irrespective of natio ...
in 2008, and the
Abel Prize
The Abel Prize ( ; no, Abelprisen ) is awarded annually by the King of Norway to one or more outstanding mathematicians. It is named after the Norwegian mathematician Niels Henrik Abel (1802–1829) and directly modeled after the Nobel Prizes. ...
in 2013, "for seminal contributions to algebraic geometry and for their transformative impact
on number theory, representation theory, and related fields". He was elected a foreign member of the Academie des Sciences de Paris in 1978.
In 2006 he was ennobled by the Belgian king as
viscount
A viscount ( , for male) or viscountess (, for female) is a title used in certain European countries for a noble of varying status.
In many countries a viscount, and its historical equivalents, was a non-hereditary, administrative or judicial ...
.
In 2009, Deligne was elected a foreign member of the
Royal Swedish Academy of Sciences
The Royal Swedish Academy of Sciences ( sv, Kungliga Vetenskapsakademien) is one of the Swedish Royal Academies, royal academies of Sweden. Founded on 2 June 1739, it is an independent, non-governmental scientific organization that takes special ...
and a residential member of the
American Philosophical Society
The American Philosophical Society (APS), founded in 1743 in Philadelphia, is a scholarly organization that promotes knowledge in the sciences and humanities through research, professional meetings, publications, library resources, and communit ...
. He is a member of the
Norwegian Academy of Science and Letters
The Norwegian Academy of Science and Letters ( no, Det Norske Videnskaps-Akademi, DNVA) is a learned society based in Oslo, Norway. Its purpose is to support the advancement of science and scholarship in Norway.
History
The Royal Frederick Univer ...
.
Selected publications
*
*
*
*
*
* ''Quantum fields and strings: a course for mathematicians''. Vols. 1, 2. Material from the Special Year on Quantum Field Theory held at the Institute for Advanced Study, Princeton, NJ, 1996–1997. Edited by Pierre Deligne,
Pavel Etingof,
Daniel S. Freed,
Lisa C. Jeffrey,
David Kazhdan
David Kazhdan ( he, דוד קשדן), born Dmitry Aleksandrovich Kazhdan (russian: Дми́трий Александро́вич Кажда́н), is a Soviet and Israeli mathematician known for work in representation theory. Kazhdan is a 1990 Ma ...
,
John W. Morgan,
David R. Morrison and
Edward Witten
Edward Witten (born August 26, 1951) is an American mathematical and theoretical physicist. He is a Professor Emeritus in the School of Natural Sciences at the Institute for Advanced Study in Princeton. Witten is a researcher in string theory, q ...
. American Mathematical Society, Providence, RI; Institute for Advanced Study (IAS), Princeton, NJ, 1999. Vol. 1: xxii+723 pp.; Vol. 2: pp. i–xxiv and 727–1501. .
Hand-written letters
Deligne wrote multiple hand-written letters to other mathematicians in the 1970s. These include
*
*
*
*
Concepts named after Deligne
The following mathematical concepts are named after Deligne:
*
Deligne–Lusztig theory
In mathematics, Deligne–Lusztig theory is a way of constructing linear representations of finite groups of Lie type using ℓ-adic cohomology with compact support, introduced by .
used these representations to find all representations of all ...
*
Deligne–Mumford moduli space of curves
In algebraic geometry, a moduli space of (algebraic) curves is a geometric space (typically a scheme or an algebraic stack) whose points represent isomorphism classes of algebraic curves. It is thus a special case of a moduli space. Depending ...
*
Deligne–Mumford stack
In algebraic geometry, a Deligne–Mumford stack is a stack ''F'' such that
Pierre Deligne and David Mumford introduced this notion in 1969 when they proved that moduli spaces of stable curves of fixed arithmetic genus are proper smooth Deligne ...
s
*
Fourier–Deligne transform In algebraic geometry, the Fourier–Deligne transform, or ℓ-adic Fourier transform, or geometric Fourier transform, is an operation on objects of the derived category of ''ℓ''-adic sheaves over the affine line. It was introduced by Pierre Deli ...
*
Deligne cohomology In mathematics, Deligne cohomology is the hypercohomology of the Deligne complex of a complex manifold. It was introduced by Pierre Deligne in unpublished work in about 1972 as a cohomology theory for algebraic varieties that includes both ordin ...
* Deligne
motive
Motive(s) or The Motive(s) may refer to:
* Motive (law)
Film and television
* ''Motives'' (film), a 2004 thriller
* ''The Motive'' (film), 2017
* ''Motive'' (TV series), a 2013 Canadian TV series
* ''The Motive'' (TV series), a 2020 Israeli T ...
* Deligne tensor product of abelian categories (denoted
)
*
Deligne's theorem
*
Langlands–Deligne local constant
In mathematics, the Langlands–Deligne local constant, also known as the local epsilon factor or local Artin root number (up to an elementary real function of ''s''), is an elementary function associated with a representation of the Weil group of ...
*
Weil-Deligne group
Additionally, many different conjectures in mathematics have been called the Deligne conjecture:
* ''The Deligne conjecture in
deformation theory
In mathematics, deformation theory is the study of infinitesimal conditions associated with varying a solution ''P'' of a problem to slightly different solutions ''P''ε, where ε is a small number, or a vector of small quantities. The infinitesim ...
'' is about the
operad
In mathematics, an operad is a structure that consists of abstract operations, each one having a fixed finite number of inputs (arguments) and one output, as well as a specification of how to compose these operations. Given an operad O, one define ...
ic structure on
Hochschild cochain complex. Various proofs have been suggested by
Dmitry Tamarkin
Dmitri (russian: Дми́трий); Church Slavic form: Dimitry or Dimitri (); ancient Russian forms: D'mitriy or Dmitr ( or ) is a male given name common in Orthodoxy, Orthodox Christian culture, the Russian version of Greek language, Greek De ...
,
Alexander A. Voronov,
James E. McClure
James is a common English language surname and given name:
*James (name), the typically masculine first name James
* James (surname), various people with the last name James
James or James City may also refer to:
People
* King James (disambiguat ...
and
Jeffrey H. Smith
Jeffrey Henderson Smith is a former professor of mathematics at Purdue University in Lafayette, Indiana. He received his Ph.D. from the Massachusetts Institute of Technology in 1981, under the supervision of Daniel Kan, and was promoted to full pr ...
,
Maxim Kontsevich
Maxim Lvovich Kontsevich (russian: Макси́м Льво́вич Конце́вич, ; born 25 August 1964) is a Russian and French mathematician and mathematical physicist. He is a professor at the Institut des Hautes Études Scientifiques an ...
and
Yan Soibelman
Iakov (Yan) Soibelman (Russian: Яков Семенович Сойбельман) born 15 April 1956 (Kiev, USSR) is a Russians, Russian American mathematician, professor at Kansas State University (Manhattan, USA), member of thKyiv Mathematical ...
, and others, after an initial input of construction of homotopy algebraic structures on the Hochschild complex. It is of importance in relation with
string theory
In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings. String theory describes how these strings propagate through space and interac ...
.
* The ''
Deligne conjecture on special values of L-functions
In mathematics, the study of special values of L-functions is a subfield of number theory devoted to generalising formulae such as the Leibniz formula for pi, namely
:1 \,-\, \frac \,+\, \frac \,-\, \frac \,+\, \frac \,-\, \cdots \;=\; \frac,\!
...
'' is a formulation of the hope for
algebraicity of ''L''(''n'') where ''L'' is an
L-function
In mathematics, an ''L''-function is a meromorphic function on the complex plane, associated to one out of several categories of mathematical objects. An ''L''-series is a Dirichlet series, usually convergent on a half-plane, that may give ris ...
and ''n'' is an integer in some set depending on ''L''.
* There is a ''Deligne conjecture on 1-motives'' arising in the theory of
motives in
algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
.
* There is a ''Gross–Deligne conjecture'' in the theory of
complex multiplication
In mathematics, complex multiplication (CM) is the theory of elliptic curves ''E'' that have an endomorphism ring larger than the integers. Put another way, it contains the theory of elliptic functions with extra symmetries, such as are visible wh ...
.
* There is a ''Deligne conjecture on
monodromy
In mathematics, monodromy is the study of how objects from mathematical analysis, algebraic topology, algebraic geometry and differential geometry behave as they "run round" a singularity. As the name implies, the fundamental meaning of ''mono ...
'', also known as the ''weight monodromy conjecture'', or purity conjecture for the monodromy filtration.
* There is a ''
Deligne conjecture
Pierre René, Viscount Deligne (; born 3 October 1944) is a Belgian mathematician. He is best known for work on the Weil conjectures, leading to a complete proof in 1973. He is the winner of the 2013 Abel Prize, 2008 Wolf Prize, 1988 Crafoord P ...
'' in the
representation theory
Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essen ...
of
exceptional Lie group
In mathematics, a simple Lie group is a connected non-abelian Lie group ''G'' which does not have nontrivial connected normal subgroups. The list of simple Lie groups can be used to read off the list of simple Lie algebras and Riemannian symm ...
s.
*There is a conjecture named the Deligne–Grothendieck conjecture for the discrete
Riemann–Roch theorem
The Riemann–Roch theorem is an important theorem in mathematics, specifically in complex analysis and algebraic geometry, for the computation of the dimension of the space of meromorphic functions with prescribed zeros and allowed poles. It rel ...
in characteristic 0.
* There is a conjecture named the Deligne–Milnor conjecture for the differential interpretation of a formula of Milnor for Milnor fibres, as part of the extension of nearby cycles and their Euler numbers.
* The Deligne–Milne conjecture is formulated as part of motives and Tannakian categories.
* There is a ''Deligne–Langlands conjecture'' of historical importance in relation with the development of the
Langlands philosophy
In representation theory and algebraic number theory, the Langlands program is a web of far-reaching and influential conjectures about connections between number theory and geometry. Proposed by , it seeks to relate Galois groups in algebraic num ...
.
* ''Deligne's conjecture on the Lefschetz trace formula'' (now called Fujiwara's theorem for equivariant correspondences).
[Martin Olsson]
"Fujiwara's Theorem for Equivariant Correspondences"
p. 1.
See also
*
Brumer–Stark conjecture The Brumer–Stark conjecture is a conjecture in algebraic number theory giving a rough generalization of both the analytic class number formula for Dedekind zeta functions, and also of Stickelberger's theorem about the factorization of Gauss sums ...
*
E7½
In mathematics, the Lie algebra E7½ is a subalgebra of E8 containing E7 defined by Landsberg and Manivel in order
to fill the "hole" in a dimension formula for the exceptional series E''n'' of simple Lie algebras. This hole was observed by ...
*
Hodge–de Rham spectral sequence
*
Logarithmic form In contexts including complex manifolds and algebraic geometry, a logarithmic differential form is a meromorphic differential form with poles of a certain kind. The concept was introduced by Deligne.
Let ''X'' be a complex manifold, ''D'' ⊂ ''X'' ...
*
Kodaira vanishing theorem
In mathematics, the Kodaira vanishing theorem is a basic result of complex manifold theory and complex algebraic geometry, describing general conditions under which sheaf cohomology groups with indices ''q'' > 0 are automatically zero. The implica ...
*
Moduli of algebraic curves
In algebraic geometry, a moduli space of (algebraic) curves is a geometric space (typically a scheme or an algebraic stack) whose points represent isomorphism classes of algebraic curves. It is thus a special case of a moduli space. Depending on ...
*
Motive (algebraic geometry)
In algebraic geometry, motives (or sometimes motifs, following French usage) is a theory proposed by Alexander Grothendieck in the 1960s to unify the vast array of similarly behaved cohomology theories such as singular cohomology, de Rham cohomo ...
*
Perverse sheaf The mathematical term perverse sheaves refers to a certain abelian category associated to a topological space ''X'', which may be a real or complex manifold, or a more general topologically stratified space, usually singular. This concept was intro ...
*
Riemann–Hilbert correspondence In mathematics, the term Riemann–Hilbert correspondence refers to the correspondence between regular singular flat connections on algebraic vector bundles and representations of the fundamental group, and more generally to one of several generaliz ...
*
Serre's modularity conjecture
In mathematics, Serre's modularity conjecture, introduced by , states that an odd, irreducible, two-dimensional Galois representation over a finite field arises from a modular form. A stronger version of this conjecture specifies the weight and ...
*
Standard conjectures on algebraic cycles
References
External links
*
*
* – Biography and extended video interview.
Pierre Delignes home page at Institute for Advanced Study
* An introduction to his work at the time of his Fields medal award.
{{DEFAULTSORT:Deligne, Pierre
Living people
1944 births
21st-century Belgian mathematicians
Arithmetic geometers
Fields Medalists
Abel Prize laureates
Wolf Prize in Mathematics laureates
Viscounts of Belgium
Scientists from Brussels
Free University of Brussels (1834–1969) alumni
Institute for Advanced Study faculty
Members of the French Academy of Sciences
Members of the Royal Swedish Academy of Sciences
Members of the Norwegian Academy of Science and Letters
Foreign associates of the National Academy of Sciences
Foreign Members of the Russian Academy of Sciences
Paris-Saclay University people
Paris-Saclay University alumni