Peter Scholze

Peter Scholze
Scholze in Berkeley, 2014
Peter Scholze
Born	11 12, 1987
Birthplace	Dresden, East Germany
Nationality	German
Occupation	Mathematician
Title	Professor; Co-director
Employer	University of Bonn, Max Planck Institute for Mathematics
Known for	Perfectoid spaces, contributions to arithmetic geometry, Langlands program
Education	Ph.D. in Mathematics (2012)
Children	1
Awards	Fields Medal (2018), Breakthrough Prize in Mathematics (2016), Ostrowski Prize (2015)

Peter Scholze (Template:IPA-de; born 11 December 1987) is a German mathematician whose work in arithmetic geometry has reshaped the landscape of modern mathematics. Born in Dresden, then part of East Germany, Scholze rose to international prominence at a remarkably young age, completing his doctoral thesis on perfectoid spaces at the University of Bonn in 2012 at the age of 24 and becoming one of the youngest full professors in Germany the same year. His theory of perfectoid spaces provided a powerful new framework for translating problems between different areas of mathematics, particularly between number theory and algebraic geometry, unlocking solutions to long-standing open problems. In 2018, at the age of 30, Scholze was awarded the Fields Medal—often described as the highest professional honor in mathematics—becoming only the second German mathematician to receive the prize in over 80 years. Since 2018, he has served as a co-director at the Max Planck Institute for Mathematics in Bonn, where he continues to advance research at the frontiers of arithmetic geometry and the Langlands program.

Early Life

Peter Scholze was born on 11 December 1987 in Dresden, which at the time was part of East Germany. He grew up in Berlin, where his mathematical talent became apparent at an early age. German media outlets noted his exceptional abilities in mathematics during his school years, and he distinguished himself early in mathematical competitions.

According to profiles published in major German newspapers, Scholze's intellectual development was characterized by an unusually rapid progression through mathematical concepts. The Berliner Zeitung described him as a "Mathematikgenie aus Berlin" (mathematics genius from Berlin), highlighting the speed at which he absorbed and mastered advanced mathematical material. The Bild newspaper also profiled Scholze as a prodigious mathematical talent, drawing attention to his youth and the pace of his academic achievements.

Scholze participated in the International Mathematical Olympiad (IMO) as a teenager, earning three gold medals and one silver medal in successive years of competition. His performance at these international competitions placed him among the top young mathematical talents in the world and foreshadowed the groundbreaking contributions he would make in professional mathematics. According to a profile in Quanta Magazine, Scholze taught himself several advanced areas of mathematics during his teenage years, moving far beyond the standard school curriculum.

Education

Scholze enrolled at the University of Bonn to study mathematics, an institution long regarded as one of the leading centers for mathematical research in Germany. He progressed through his undergraduate and graduate studies at an exceptional pace. His doctoral research was conducted under the supervision of Michael Rapoport, a distinguished figure in arithmetic geometry and algebraic number theory.

Scholze completed his doctoral thesis, titled "Perfectoid Spaces," in 2012 at the age of 24. The thesis introduced the concept of perfectoid spaces, a new class of mathematical objects that provided a unifying framework for problems in p-adic Hodge theory, algebraic geometry, and number theory. The work was immediately recognized as a major breakthrough by the mathematical community. Upon completion of his doctorate, Scholze was appointed as a full professor (W3) at the University of Bonn, making him one of the youngest professors in Germany at the time.

Career

Perfectoid Spaces

The central contribution of Scholze's early career was the development of the theory of perfectoid spaces, introduced in his 2012 doctoral thesis. Perfectoid spaces are a type of mathematical structure in p-adic geometry that allow mathematicians to move between different mathematical settings—specifically between objects in characteristic zero (such as the p-adic numbers) and objects in positive characteristic (such as fields related to finite fields). This "tilting" operation, as it came to be known, enabled the transfer of techniques and results between contexts that had previously seemed quite separate.

The theory built upon earlier work in p-adic Hodge theory by Jean-Marc Fontaine, Gerd Faltings, and others, but represented a significant conceptual leap. As described in Scholze's own expository writing, perfectoid spaces provided a way to systematically exploit a deep connection between mixed characteristic and equal characteristic algebra, yielding new proofs of existing theorems and solutions to problems that had resisted previous approaches.

The introduction of perfectoid spaces had immediate applications in several areas. Scholze used them to prove new cases of the weight-monodromy conjecture, a problem in arithmetic geometry that had been open for decades. The theory also provided new insights into the structure of Shimura varieties, objects of central importance in the Langlands program. The mathematical community recognized the theory as opening a broad new avenue of research, and perfectoid spaces quickly became one of the most active areas of study in modern arithmetic geometry.

Professorship at Bonn

Following the completion of his doctorate, Scholze was appointed to a Hausdorff Chair at the University of Bonn in 2012, a position he has held since. At 24 years of age, he became one of the youngest full professors in the history of the German university system, a fact that attracted considerable media attention in Germany.

At Bonn, Scholze continued to develop and extend the theory of perfectoid spaces and its applications. He also became a prominent figure in the global mathematical community, delivering invited lectures and talks at major conferences and institutions worldwide. In 2015, the University of Bonn issued a press release highlighting Scholze's continued achievements and his contributions to the university's reputation as a leading center for mathematical research.

Cours Peccot at the Collège de France

In recognition of his contributions, Scholze was invited to deliver the Cours Peccot at the Collège de France in Paris, a prestigious lecture series historically given by outstanding young mathematicians. The Cours Peccot has been awarded since 1899, and its recipients constitute a distinguished roster of mathematicians who made significant contributions early in their careers.

Co-directorship at the Max Planck Institute

In 2018, Scholze was appointed as a co-director of the Max Planck Institute for Mathematics (MPIM) in Bonn, one of the foremost research institutes for mathematics in the world. The appointment, which Scholze took up alongside his professorship at the University of Bonn, made him one of the youngest directors in the history of the Max Planck Society. In this role, Scholze has helped shape the research direction of the institute and has attracted international scholars to Bonn.

The Langlands Program and Fargues–Scholze

A major focus of Scholze's research has been the Langlands program, a vast web of conjectures and results that seeks to connect number theory, algebraic geometry, and representation theory. The Langlands program, initiated by Robert Langlands in the late 1960s, is considered one of the deepest and most ambitious research programs in modern mathematics.

In collaboration with the French mathematician Laurent Fargues, Scholze developed a new geometric framework for understanding aspects of the Langlands program in the context of p-adic mathematics. Their work introduced a new geometric object—the Fargues–Fontaine curve—and used it to establish a "geometrization" of the local Langlands correspondence, providing a new and more powerful way of connecting number theory and geometry. A detailed account of this collaboration was published by Quanta Magazine in 2021, which described the Fargues–Scholze work as opening a "wormhole" between numbers and geometry.

The Fargues–Scholze collaboration represented a significant advance in the understanding of the local Langlands program and was seen as one of the most important developments in arithmetic geometry in recent years.

Formal Verification and the Lean Project

In late 2020, Scholze posed a challenge to the formal verification community: he asked whether a key result from his work on condensed mathematics—a theorem he called the "liquid tensor experiment"—could be formally verified using the Lean proof assistant. The theorem in question was a technically demanding result that Scholze himself had described as being at the boundary of what he was confident could be verified.

In June 2021, a team of mathematicians and computer scientists, led by Johan Commelin, announced that they had successfully formalized and verified the proof using Lean. The verification was described as a landmark event in the intersection of mathematics and computer science, demonstrating that computer-based proof assistants could handle results at the cutting edge of research mathematics. Quanta Magazine covered the achievement in detail, noting that it represented a significant step forward for formal verification in mathematics.

The liquid tensor experiment attracted attention not only for its mathematical content but also for what it suggested about the future of mathematical proof. By subjecting a difficult and novel result to formal computer verification, Scholze helped catalyze broader interest in the use of proof assistants among research mathematicians.

Engagement with the ABC Conjecture

Scholze has also been a prominent figure in the mathematical community's response to Shinichi Mochizuki's claimed proof of the abc conjecture, a major open problem in number theory. In 2018, Scholze and Jakob Stix of Goethe University Frankfurt traveled to Kyoto to discuss the proof with Mochizuki and subsequently published a report identifying what they described as a serious gap in the argument. According to Quanta Magazine, Scholze and Stix pointed to a specific step in Mochizuki's inter-universal Teichmüller theory (IUT) that they argued was flawed, stating that the relevant inequality in the proof appeared to reduce to a trivial statement when examined closely.

Mochizuki rejected the criticism, and the disagreement has persisted. As of 2025, the controversy remains unresolved, with some mathematicians exploring whether formal computer verification might provide a path to settling the dispute. The episode highlighted both the rigor of Scholze's mathematical standards and his willingness to engage publicly with contentious questions in the field.

Personal Life

Scholze has one child. He has maintained a relatively private personal life, with public information about him largely confined to his professional achievements. He resides in Bonn, Germany, where both the University of Bonn and the Max Planck Institute for Mathematics are located.

In interviews and profiles, Scholze has been described as modest and unassuming despite his achievements. Deutsche Welle noted upon his Fields Medal award in 2018 that he was "a 30-year-old German" who had become one of the most celebrated mathematicians in the world.

Recognition

Scholze has received numerous awards and honors for his mathematical contributions:

• SASTRA Ramanujan Prize (2013): Awarded by the SASTRA University in Kumbakonam, India, for outstanding contributions by a mathematician not exceeding the age of 32 in an area influenced by Srinivasa Ramanujan.

• Clay Research Award (2014): Given by the Clay Mathematics Institute in recognition of Scholze's contributions to arithmetic algebraic geometry, particularly his development of the theory of perfectoid spaces.

• Ostrowski Prize (2015): One of the most significant prizes in mathematics, awarded for outstanding mathematical achievement. The award committee cited Scholze's development of the theory of perfectoid spaces and its applications.

• Breakthrough Prize in Mathematics (2016): Awarded by the Breakthrough Prize foundation. The citation recognized Scholze "for pioneering in arithmetic algebraic geometry, in particular his invention of perfectoid spaces and their application to the weight-monodromy conjecture and the discovery of new instances of the local Langlands correspondence."

• Fields Medal (2018): Awarded at the International Congress of Mathematicians in Rio de Janeiro, Brazil. The Fields Medal is awarded every four years to mathematicians under the age of 40 and is considered the highest honor in the discipline. Scholze received the medal for "transforming arithmetic algebraic geometry over p-adic fields through his introduction of perfectoid spaces, with application to Galois representations, and for the development of new cohomology theories." He was the second German to receive the Fields Medal, the first being Gerd Faltings in 1986.

As noted by Quanta Magazine in 2016, Scholze was considered a strong favorite for the Fields Medal well before the official announcement, reflecting the mathematical community's assessment of the depth and influence of his work.

Legacy

Peter Scholze's contributions have had a transformative impact on several areas of mathematics. The theory of perfectoid spaces, introduced in his doctoral thesis, has become a foundational tool in arithmetic geometry and p-adic Hodge theory, and has been adopted and extended by mathematicians worldwide. The framework has opened new avenues of research and enabled progress on problems that had been considered intractable.

Scholze's work on the geometrization of the local Langlands program, particularly his collaboration with Laurent Fargues, has advanced one of the central research programs in contemporary mathematics. By providing a geometric interpretation of deep number-theoretic phenomena, the Fargues–Scholze framework has offered new perspectives and tools for understanding the connections between arithmetic and geometry.

Beyond his technical contributions, Scholze has influenced the mathematical community's engagement with formal verification and computer-assisted proof. His challenge to the Lean proof assistant community to verify a result in condensed mathematics—and the successful completion of that challenge—has been cited as a pivotal moment in the acceptance of formal methods among research mathematicians.

Scholze's critical engagement with the claimed proof of the abc conjecture has also had a lasting impact on mathematical discourse, underscoring the importance of rigor and verifiability in the evaluation of complex proofs.

At the Max Planck Institute for Mathematics and the University of Bonn, Scholze continues to shape the direction of research in arithmetic geometry and related fields. His career, marked by the rapid development of powerful new mathematical theories and a commitment to the highest standards of proof, has established him as a central figure in 21st-century mathematics.