Caucher Birkar

Caucher Birkar
Born	Fereydoun Derakhshani July 1978
Birthplace	Marivan County, Iran
Nationality	Iranian, British
Occupation	Mathematician, professor
Known for	Proof of boundedness of Fano varieties, contributions to the minimal model program
Education	PhD, University of Nottingham (2004)
Children	1
Awards	Leverhulme Prize (2010) Moore Prize (2016) Fields Medal (2018)

Caucher Birkar (Template:Lang-ku, romanized: Koçer Bîrkar, lit. "migrant mathematician"; born Fereydoun Derakhshani, July 1978) is an Iranian Kurdish and British mathematician who has made fundamental contributions to birational geometry, particularly to the minimal model program and the study of Fano varieties. Born on a subsistence farm near the Kurdish city of Marivan in western Iran, Birkar grew up amid the turmoil of the Iran–Iraq War before eventually seeking asylum in the United Kingdom, where he rebuilt his life and pursued advanced mathematics. He studied at the University of Tehran before completing his doctorate at the University of Nottingham in 2004. Birkar held a position at the University of Cambridge for many years and is currently a professor at Tsinghua University in Beijing, as well as an Honorary Professor at the University of Nottingham. In 2018, he was awarded the Fields Medal, one of the highest honors in mathematics, "for his proof of boundedness of Fano varieties and contributions to the minimal model program." His life story — from a war-torn Kurdish village to the pinnacle of mathematical achievement — has made him a figure of inspiration, and his chosen name, Caucher Birkar, which roughly translates to "the migrating mathematician," reflects both his personal journey and his enduring connection to his Kurdish heritage.

Early Life

Caucher Birkar was born Fereydoun Derakhshani in July 1978 in Marivan County, a predominantly Kurdish region in the western province of Kurdistan in Iran. He grew up on a subsistence farm near the city of Marivan, close to the border with Iraq. His childhood was shaped by the Iran–Iraq War (1980–1988), a conflict that brought significant violence and disruption to the Kurdish border regions. Birkar has spoken publicly about the impact of growing up in a war zone, recalling the hardships faced by his family and community during this period.

Despite the difficult circumstances of his upbringing, Birkar developed an early interest in mathematics. His older brother played an important role in his mathematical education, reportedly teaching him calculus during his formative years. Birkar has described the formative influence of his early exposure to mathematical concepts in an environment where educational resources were limited. The Kurdish region of Iran, while rich in cultural traditions, faced economic challenges and the aftereffects of conflict, which made academic pursuits particularly difficult for young people of his generation.

Birkar later changed his name from Fereydoun Derakhshani to Caucher Birkar, a Kurdish name that translates approximately to "the migrating mathematician." The name reflected both his experience as a migrant and refugee and his deep connection to his Kurdish identity. Birkar has maintained strong ties to his Kurdish heritage throughout his career and has spoken about the importance of Kurdish identity, while also distancing it from nationalism and politics.

Education

Birkar attended the University of Tehran, where he earned a Bachelor of Science degree in mathematics. While studying at Tehran University, Birkar recalled staring at the portraits of Fields Medal winners that hung on the walls of the university, wondering if he would ever reach such heights in mathematics. In 2000, Birkar participated in the International Mathematics Competition for university students.

After completing his undergraduate studies, Birkar left Iran and sought asylum in the United Kingdom. He arrived in the UK speaking little English, which presented an additional barrier to his academic pursuits. He enrolled at the University of Nottingham, where he pursued doctoral studies in mathematics. Under supervision, he completed his PhD thesis titled "Topics in Modern Algebraic Geometry" in 2004. His doctoral work laid the groundwork for his future contributions to birational geometry and the minimal model program. Following his PhD, Birkar was awarded the Cecil King Travel Scholarship by the London Mathematical Society, which supported his further research activities.

Career

Early Academic Career and the Minimal Model Program

After completing his doctorate at the University of Nottingham in 2004, Birkar pursued research in algebraic geometry, with a particular focus on birational geometry and the minimal model program (MMP). The minimal model program, sometimes called the Mori program, is a major research area in algebraic geometry that seeks to classify algebraic varieties — fundamental objects in geometry defined by systems of polynomial equations — by finding their simplest representative forms, or "minimal models."

Birkar established himself as an important contributor to this field during the late 2000s. One of his most significant early achievements was his collaboration with Paolo Cascini, Christopher Hacon, and James McKernan on the landmark paper "Existence of minimal models for varieties of log general type." This paper, which addressed a fundamental question in the minimal model program, demonstrated the existence of minimal models for a broad class of algebraic varieties known as varieties of log general type. The result represented a major advance in the classification of higher-dimensional algebraic varieties and was considered one of the most important developments in algebraic geometry of the era.

Birkar held a research position at the University of Cambridge, where he was based for a significant portion of his career. At Cambridge, he continued to develop and refine his work on the minimal model program, producing a series of papers that extended the reach of the program and resolved longstanding conjectures in the field.

Contributions to Birational Geometry

Birkar's research has centered on some of the most technically challenging problems in modern birational geometry. Birational geometry studies the relationships between algebraic varieties that share the same "birational type" — meaning they are equivalent in a generalized sense, even if they differ in specific geometric details. The goal is to find the simplest representative within each birational equivalence class, which is the essence of the minimal model program.

Among Birkar's key contributions is his work on the existence of minimal models and on the termination of certain procedures (called "flips") within the minimal model program. His research has addressed questions about how algebraic varieties behave under various geometric operations and has provided foundational results that other mathematicians have built upon.

Birkar published several significant papers through the 2010s that advanced the field. His work appeared in leading mathematical journals, and his preprints on arXiv attracted wide attention from the mathematical community.

Boundedness of Fano Varieties

The work for which Birkar is most celebrated is his proof of the boundedness of Fano varieties, a result that answered a major open conjecture in algebraic geometry known as the Borisov–Alexeev–Borisov conjecture (BAB conjecture). Fano varieties are a fundamental class of algebraic varieties that appear throughout geometry and mathematical physics. They are characterized by having positive curvature in a specific algebraic sense and form one of the three "building blocks" in the classification of algebraic varieties according to the minimal model program — alongside varieties of general type and Calabi–Yau varieties.

The BAB conjecture, which had been open for decades, predicted that Fano varieties of a given dimension with mild singularities form a bounded family — meaning that despite their apparent diversity, they can be parametrized by a finite-dimensional space. Birkar's proof of this conjecture, accomplished in a series of papers in 2016, was a tour de force of modern algebraic geometry. The proof required developing new techniques for analyzing the singularities and geometric properties of these varieties and represented a fundamental advance in mathematicians' understanding of the structure of algebraic varieties.

The proof of the BAB conjecture had wide-ranging implications across algebraic geometry. It confirmed that the class of Fano varieties, despite its apparent complexity, has a deep underlying structure and can be understood in a systematic way. This result was a key factor in Birkar's selection for the Fields Medal in 2018.

The Fields Medal

On August 1, 2018, at the International Congress of Mathematicians (ICM) held in Rio de Janeiro, Brazil, Birkar was awarded the Fields Medal, often described as the most prestigious prize in mathematics. He was one of four mathematicians to receive the award that year. The citation from the International Mathematical Union recognized Birkar "for his proof of boundedness of Fano varieties and contributions to the minimal model program."

The award carried particular significance given Birkar's life story. He was the second mathematician of Iranian origin to receive the Fields Medal, after Maryam Mirzakhani in 2014, and the first Kurdish mathematician to be so honored. Upon receiving the prize, Birkar spoke about the meaning of the award in the context of his background, expressing hope that his achievement would be an inspiration to others from disadvantaged and conflict-affected regions.

In an event that attracted worldwide media attention, Birkar's Fields Medal — the physical gold medal itself — was stolen approximately thirty minutes after it was presented to him at the ICM ceremony in Rio de Janeiro. Birkar had placed the medal in a briefcase, which was taken from a table at the convention center. The theft was reported to Brazilian police and became a major news story. The International Mathematical Union subsequently issued a replacement medal to Birkar.

Move to Tsinghua University

Following his time at Cambridge, Birkar took a professorship at Tsinghua University in Beijing, China, where he continues his research in birational geometry and algebraic geometry. He also holds the title of Honorary Professor at the University of Nottingham, his doctoral alma mater. In his office, Birkar keeps two photographs of Alexander Grothendieck, whom he considers his favorite mathematician. Grothendieck, like Birkar, was a refugee who went on to win the Fields Medal, and Birkar has spoken about the parallels between their life experiences.

Personal Life

Birkar arrived in the United Kingdom as an asylum seeker after completing his undergraduate studies at the University of Tehran in Iran. He was granted refugee status in the UK and subsequently obtained British citizenship. He has one child.

Birkar has spoken publicly about the challenges he faced as a refugee adjusting to life in a new country, including learning English and navigating the British academic system. He arrived in the UK with limited English proficiency, which made his early doctoral studies at the University of Nottingham particularly challenging.

Throughout his career, Birkar has maintained a strong connection to his Kurdish identity and heritage. He has spoken about the importance of Kurdish identity and culture while emphasizing that his sense of Kurdish identity is distinct from nationalism or political movements. This stance is reflected in his choice of name — Caucher Birkar, meaning "the migrating mathematician" in Kurdish — which he adopted to honor both his mathematical vocation and his experience as a migrant.

In 2019, Birkar was voted the world's "top thinker" in a poll conducted by Prospect Magazine, in which tens of thousands of readers selected from a list of fifty nominees. In an interview with the magazine, Birkar discussed his views on mathematics, migration, and the role his brother played in his early mathematical education.

Recognition

Birkar has received several major awards and honors in recognition of his contributions to mathematics.

In 2010, Birkar was awarded the Philip Leverhulme Prize in mathematics and statistics by the Leverhulme Trust. The prize recognized his contributions to algebraic geometry, particularly his work on the minimal model program.

In 2016, Birkar shared the AMS Moore Prize (the E.H. Moore Research Article Prize) awarded by the American Mathematical Society. The prize was given for the article "Existence of minimal models for varieties of log general type," co-authored with Paolo Cascini, Christopher Hacon, and James McKernan. This paper was recognized as a major breakthrough in birational geometry.

Birkar's most prominent recognition came in 2018 with the Fields Medal, awarded at the International Congress of Mathematicians in Rio de Janeiro. The Fields Medal is awarded every four years to mathematicians under the age of 40 and is often described as the equivalent of the Nobel Prize in mathematics. Birkar's citation specifically highlighted his proof of the boundedness of Fano varieties and his broader contributions to the minimal model program.

In addition to these major prizes, Birkar received the Cecil King Travel Scholarship from the London Mathematical Society early in his career, supporting his postdoctoral research activities.

Legacy

Birkar's mathematical contributions have had a significant impact on the field of algebraic geometry, particularly in advancing the classification of algebraic varieties through the minimal model program. His proof of the BAB conjecture on the boundedness of Fano varieties resolved a problem that had been open for decades and provided new tools and perspectives that continue to influence research in birational geometry.

Beyond his technical achievements, Birkar's life story has carried resonance in the broader public sphere. As a Kurdish refugee who overcame the hardships of war, displacement, and language barriers to reach the summit of mathematical achievement, his story has been cited as an example of the potential that can be unlocked when refugees and displaced persons are given opportunities to pursue their talents. His Fields Medal in 2018 generated widespread media coverage not only because of the mathematical significance of his work but also because of the compelling arc of his personal journey from a subsistence farm in war-torn Kurdistan to the world stage of mathematics.

Birkar is the second person of Iranian origin to win the Fields Medal, after Maryam Mirzakhani, who received the award in 2014. Both laureates studied at the University of Tehran before moving abroad, and their achievements have been a source of pride within the Iranian and Kurdish mathematical communities.

Birkar has expressed the view that mathematics is a universal language that transcends borders and that his work belongs to a tradition of mathematical inquiry that connects people across cultures and nationalities. His admiration for Alexander Grothendieck — another refugee and Fields medalist whose life followed a trajectory from displacement to mathematical greatness — underscores Birkar's awareness of the broader human dimensions of mathematical achievement.

His selection as the world's "top thinker" by Prospect Magazine readers in 2019 reflected the degree to which his story and achievements had captured public imagination beyond the mathematical community.