Pierre Deligne

2.1. Birth and Early Background

Pierre René Deligne was born on October 3, 1944, in Etterbeek, a municipality in Brussels, Belgium Etterbeek_Etterbeek^Dutch. From an early age, Deligne displayed exceptional mathematical aptitude. By the age of 14, he was already proficient in reading and understanding the rigorous mathematical texts of Nicolas Bourbaki's Éléments de mathématique. His talent was so advanced that by the time he entered university, he had reportedly already mastered the entire undergraduate mathematics curriculum.

2.2. Education

Deligne pursued his higher education at the Université libre de Bruxelles (ULB), where he completed his master's degree around 1966 and his doctorate around 1968. His dissertation at ULB was titled Théorème de Lefschetz et critères de dégénérescence de suites spectrales_{Theorem of Lefschetz and criteria of degeneration of spectral sequences}^French. He continued his doctoral studies at the University of Paris-Sud in Orsay, France, where he earned his second doctorate in 1972. His thesis, titled Théorie de Hodge_{Hodge Theory}^French, was supervised by the eminent mathematician Alexander Grothendieck.

3.1. Early Career at IHÉS

In 1965, Deligne began his association with the Institut des Hautes Études Scientifiques (IHÉS), located near Paris, initially working as a visiting professor and later becoming a permanent member of the staff from 1970 until 1984. During this period, he collaborated closely with his doctoral supervisor, Alexander Grothendieck, on the generalization of Zariski's main theorem within scheme theory.

In 1968, Deligne also commenced a productive collaboration with Jean-Pierre Serre, which yielded important results concerning the l-adic representations associated with modular forms and the conjectural functional equations of L-functions. He also delved into topics within Hodge theory, introducing the concept of weights and testing them on objects in complex geometry. His work with David Mumford focused on a novel description of the moduli spaces for curves, which became a foundational introduction to a form of the theory of algebraic stacks. This work has subsequently found applications in questions arising from string theory in theoretical physics.

3.2. Work at the Institute for Advanced Study

In 1984, Deligne moved to the Institute for Advanced Study (IAS) in Princeton, New Jersey, where he continued his extensive research. Even while at IHÉS, Deligne had already gained significant recognition for his work on the Weil conjectures, which he completed in 1973, and for introducing the concept of mixed Hodge structures, an extension of classical Hodge theory. His move to IAS marked a new phase of his career, during which he continued to develop new mathematical theories and explore deep connections between different areas of mathematics.

3.3. Collaborations and Key Partnerships

Throughout his career, Deligne engaged in numerous significant collaborations that enriched his mathematical output:

• Alexander Grothendieck:** Their early work at IHÉS focused on generalizing Zariski's main theorem within scheme theory.
• Jean-Pierre Serre:** Their joint efforts led to crucial results on l-adic representations linked to modular forms and the conjectural functional equations of L-functions.
• David Mumford:** Their collaboration resulted in a new description of the moduli spaces for curves, which introduced the theory of Deligne-Mumford stacks, finding applications in algebraic geometry and string theory.
• George Lusztig:** Together, they applied étale cohomology to construct representations of finite groups of Lie type, a significant contribution to representation theory.
• Michael Rapoport:** Deligne worked with Rapoport on the moduli spaces from an arithmetic perspective, with important applications to modular forms.
• Phillip Griffiths, John Morgan, and Dennis Sullivan:** Their joint 1974 paper on the real homotopy theory of compact Kähler manifolds was a major achievement in complex differential geometry, resolving several important questions.
• Alexander Beilinson, Joseph Bernstein, and Ofer Gabber:** Deligne made definitive contributions to the theory of perverse sheaves in collaboration with these mathematicians.
• Ken Ribet:** Their work on abelian L-functions and their extensions to Hilbert modular surfaces and p-adic L-functions formed an important part of Deligne's contributions to arithmetic geometry.
• George Mostow:** Deligne co-authored a book on monodromy with Mostow, and they collaborated on examples of non-arithmetic lattices and the monodromy of hypergeometric differential equations in complex hyperbolic spaces.
• David Kazhdan:** Their joint work includes the Deligne-Kazhdan trace formula.

4.1. Proof of the Weil Conjectures

Deligne's most celebrated achievement is his complete proof of the Weil conjectures in 1973, published in 1974 and a more general version in 1980. These conjectures, formulated by André Weil, described properties of the number of points on algebraic varieties over finite fields, analogous to the Riemann hypothesis for the Riemann zeta function. The proof of the Weil conjectures was the culmination of a vast program initiated and largely developed by Alexander Grothendieck over more than a decade. Deligne's critical contribution was to provide the estimate of the eigenvalues of the Frobenius endomorphism, which is considered the geometric analogue of the Riemann hypothesis. As a direct consequence, he proved the renowned Ramanujan-Petersson conjecture for modular forms of weight greater than one; the case for weight one had been established earlier in his work with Jean-Pierre Serre. The proof also led to a demonstration of the Lefschetz hyperplane theorem and provided new estimates for classical exponential sums, among other significant applications.

4.2. Hodge Theory and Motives

Deligne made foundational contributions to Hodge theory, a powerful tool in algebraic geometry that generalizes classical Hodge theory. He introduced the theory of mixed Hodge structures, which provided a framework for understanding the cohomology of complex algebraic varieties, even singular ones. This theory, built upon concepts like weight filtration and resolution of singularities, was instrumental in his proof of the Weil conjectures.

In the context of completing Grothendieck's research program, Deligne defined absolute Hodge cycles. This concept served as a surrogate for the still largely conjectural theory of motives, allowing mathematicians to circumvent the lack of a complete understanding of the Hodge conjecture for certain applications. He also reworked the theory of Tannakian categories in his 1990 paper for the "Grothendieck Festschrift," employing Beck's theorem. The Tannakian category concept provides a categorical expression of the linearity of the theory of motives, viewed as the ultimate Weil cohomology. This entire framework is part of the "yoga of weights," which unifies Hodge theory and l-adic Galois representations. The theory of Shimura varieties is related to this, based on the idea that such varieties should parametrize not just arithmetically interesting families of Hodge structures, but actual motives.

4.3. Moduli Spaces and Algebraic Stacks

In collaboration with David Mumford, Deligne contributed significantly to the theory of moduli spaces, particularly their compactification. Their work led to the development of Deligne-Mumford stacks, which provide a powerful framework for studying families of algebraic curves and other geometric objects, even those with mild singularities. This theory has found wide-ranging applications in algebraic geometry, including in the study of Gromov-Witten theory, and has also proven relevant in theoretical physics, particularly in string theory.

4.4. Representation Theory and Algebraic Groups

Deligne made significant contributions to the representation theory of algebraic groups. In joint work with George Lusztig, he pioneered the application of étale cohomology to construct representations of finite groups of Lie type. This groundbreaking approach provided new insights into the structure and classification of these groups, and the resulting theory is known as Deligne-Lusztig theory.

4.5. Perverse Sheaves and Related Theories

In collaboration with Alexander Beilinson, Joseph Bernstein, and Ofer Gabber, Deligne made definitive contributions to the theory of perverse sheaves. This theory is a powerful tool in algebraic geometry and topology, providing a framework for studying constructible sheaves on singular spaces. Perverse sheaves have played an important role in various areas, including the recent proof of the fundamental lemma by Ngô Bảo Châu in the context of the Langlands program. Deligne himself used this theory to greatly clarify the nature of the Riemann-Hilbert correspondence, which extends Hilbert's twenty-first problem to higher dimensions. His work built upon earlier contributions by Zoghman Mebkhout and Masaki Kashiwara using D-modules theory.

4.6. Other Significant Research

Beyond his most famous contributions, Deligne's research encompasses a wide array of other important mathematical achievements:

• Kähler Manifolds:** In 1974, his joint paper with Phillip Griffiths, John Morgan, and Dennis Sullivan on the real homotopy theory of compact Kähler manifolds was a major work in complex differential geometry, resolving several significant questions. This work drew heavily on insights from the Weil conjectures, Hodge theory, and variations of Hodge structures.
• Singularity Theory:** His work in complex singularity theory generalized Milnor maps into an algebraic setting and extended the Picard-Lefschetz formula beyond their general format, generating new research methods in this subject.
• L-functions and Arithmetic Geometry:** His paper with Ken Ribet on abelian L-functions and their extensions to Hilbert modular surfaces and p-adic L-functions forms an important part of his work in arithmetic geometry.
• Cohomological Descent and Motivic L-functions:** Other notable achievements include the notion of cohomological descent, motivic L-functions, mixed sheaves, nearby vanishing cycles, and central extensions of reductive groups.
• Geometry and Topology:** He contributed to the geometry and topology of braid groups and provided the modern axiomatic definition of Shimura varieties.
• Non-arithmetic Lattices and Monodromy:** In collaboration with George Mostow, he worked on examples of non-arithmetic lattices and the monodromy of hypergeometric differential equations in two- and three-dimensional complex hyperbolic spaces.
• Deligne-Kazhdan Trace Formula:** This formula is another significant contribution stemming from his collaborations.
• Deligne-Mostow Classification:** He contributed to the classification of configuration spaces of projective lines.
• Deformation Quantization:** He introduced the relative cohomology between two deformation quantizations.
• Multiple Zeta Values:** He explored the relationship between multiple zeta values and motives.