From a Refugee to a Fields Medalist: The Remarkable Journey of Caucher Birkar in Mathematics

Birkar is a Kurd born during the Iran-Iraq War in Mariwan County, Kurdistan Province, Iran, in 1978. He had five siblings, and during his early years of school, his brothers taught him a lot of maths. Birkar studied mathematics at the University of Tehran after finishing high school, where he earned a bachelor's degree. He won third place in the 2000 International Mathematics Competition for University Students before moving to the UK as a refugee and requesting political asylum while still a student at the university. Birkar studied for a PhD at the University of Nottingham from 2001 to 2004. As the most promising PhD candidate, he received the Cecil King Travel Scholarship from the London Mathematical Society in 2003. His name was changed to Caucher Birkar when he immigrated to the UK; this name means "migrant mathematician" in Kurdish.

Birkar became interested in mathematics at school. "I had a sensation. Nothing major, just a passing thought that I'm brilliant at mathematics, he remarked. His brother served as his primary mentor in his formative years and taught him mathematics' foundational ideas. He was left to learn mathematics on his own in high school, borrowing books from the neighbourhood library. "After finishing all these books, I got the impression that reading isn't enough. I also wanted to make my own things and come up with something fresh," he added. The UK government sent Birkar to Nottingham, where he enrolled in the University of Nottingham. There, he met mathematician Vyacheslav Shokurov, who ignited his interest in birational geometry, a sparsely studied branch of algebraic geometry. Birkar received the Cecil King Travel Scholarship while pursuing his PhD from the London Mathematical Society for being the most promising candidate. More recently, Birkar has contributed significantly to the modernization of ideas in the area through his work at Cambridge, particularly through his proof of the boundedness of Fano varieties, which was published in 2016 and for which he was awarded the Fields Medal in 2018. His groundbreaking work on minimal model theory was also mentioned in the citation for the award. When Birkar's Fields Medal was regrettably stolen after the ceremony in Rio, news of the theft spread throughout the world. Thankfully, a few days later the committee gave him a replacement medal. At the time, Birkar stated, "This has received significant media attention, making me more renowned than I would have been... And far more individuals than last week are aware of what a Fields Medal is. "I just couldn't imagine that this would come true," he said after receiving the medal. "To go from the point that I didn't imagine meeting these people to the point where someday I hold a medal myself — I just couldn't imagine that this would come true." Birkar has won various honours throughout his career, including the Philip Leverhulme Prize (2010), the Moore Research Article Prize from the American Mathematical Society (2016), and the Whitehead Prize from the London Mathematical Society (2018). Two images of the mathematician Alexander Grothendieck are on display at Birkar's office at the University of Cambridge. Like Birkar, Grothendieck was a Fields medalist and a refugee who left Nazi Germany. He is also usually acknowledged as the mathematician who had the greatest single impact throughout the second half of the 20th century. In one of the images, Grothendieck is seated alongside a diverse collection of activists who were active in the early 1970s French environmental movement. Birkar, a Kurd who was raised in England and is now wed to a Thai woman, admires Grothendieck's mathematical vision as well as the ease with which he interacted with various social groups.

“All these cultures make things more interesting to me. All these cultures give you a sense of pleasure,” he said. His four-year-old son Zanko is a living example of this diversity because he is fluent in English, his mother's native Thai language, and his father's native Kurdish.

A mingling of civilizations can also be seen in algebraic geometry. Algebra, the study of equations, is on one side, and geometry, the study of shapes, is on the other. The two present various angles on the same issues. Consider the mathematical formula y = 2x - 3. If you plot the solutions, you get a straight line, which is a geometric shape. The two viewpoints work well together. You may use mathematics to determine the answer or you could graph the two equations to see where they overlap if you wanted to find a solution that was shared by two equations, such as y = 2x - 3 and y = 3x + 5.

János Kollár, a mathematician at Princeton University, stated that "sometimes an algebraic question can be solved by geometric methods and sometimes a geometric question can be solved by algebraic methods." You can travel between these two sides and benefit them both. The simplest algebraic equations are linear ones. There are a lot more varieties. They may contain additional variables, and those additional variables may be increased to various levels. You might also consider the set of solutions that a collection of equations shares. This collection is referred to as an "algebraic variety." An infinite number of algebraic varieties exist; each has a unique geometric representation.

“The most important thing is the shape, the form, the structure of the set of solutions,” Birkar said. “The set of solutions is what we call roughly an algebraic variety.”

Variety in algebra is a wild horde. Mathematicians want to put some kind of order on them. This tendency is similar to the urge to categorise biological life in that it makes the living world seem more manageable to our thoughts and more significant in its structure when we consider phyla and families rather than each individual organism separately. Algebraic varieties can be transformed using birational geometry in order to be categorised. It's similar to surgery in that you start with an algebraic variety that has a unique shape and then remove some of its lumps and creases to get a more general shape. There are rigorous restrictions on what you can chop, ensuring that you don't completely eliminate the variety you started with. Following surgery, many formerly distinct varieties would resemble one another; this group is referred to as the "birational equivalence class." Christopher Hacon, a prominent figure in birational geometry and a mathematician at the University of Utah, will speak on Birkar's work at the Fields Medal ceremony in Rio. "We're focusing on the big picture and not worrying about varieties disagreeing on some small subset of points," he said. Fano varieties, Calabi-Yau variations, and varieties of general type are the three major birational equivalence classes. Like the name "insect" is relative to the distinct species that belong under that heading, the three classes are generic forms. Each class has a distinct uniform curvature, which can be uniformly flat, uniformly positive, or uniformly negative. Through the process of birational transformation, mathematicians hope to be able to demonstrate how every algebraic variety may be reduced to one of these three categories of generic shapes. Kollár stated, "We are aiming to locate objects that have the same type of curvature everywhere. "We don't want anything that occasionally resembles a saddle, occasionally like a sphere, and occasionally has flat areas. That is too intricate.

Birkar has completed most of his significant mathematical work alone in his dining room. He usually works at the table, standing up occasionally to pace and contemplate, prepare tea, or play music, frequently classical or modern Kurdish tunes. "A lot of it is going through my head. Sometimes I might only write one page all day," he remarked. He enjoys taking a bike ride along the trails that wind through his neighbourhood after spending a few hours pondering. It could be difficult to discern from a distance how humming with intellectual activity Birkar's days are. He visited his wife's family in Thailand a few years ago. Her grandfather asked her one afternoon," What kind of profession does your spouse have? He was lounging around in the lawn, gazing at a mango tree. However, Birkar is renowned among mathematicians for the ferocity with which he approaches issues. He is quite successful at mastering the technical details of the current challenge. Algebraic variety research by Birkar is a component of an ongoing project called the minimum model programme. The objective is to demonstrate that all algebraic varieties can be binary transformed into one of the three fundamental types. The minimal model programme dates back more than a century to the classification of two-dimensional algebraic varieties (varieties with three variables) by a team of Italian mathematicians. Shigefumi Mori demonstrated that all three-dimensional algebraic varieties—varieties with four variables—reduce to one of these three types more recently, in the 1980s. The Fields Medal for this work was awarded in 1990 to Mori, a mathematician at Kyoto University and the current president of the International Mathematical Union. However, after Mori's finding, there was a lull in the birational geometry community. The extension of this to higher dimensions presented enormous hurdles, according to Birkar. One of the few who remained in the field throughout the 1990s was Shokurov. He was largely responsible for the revival of birational geometry in the early 2000s. Since then, the classification of algebraic varieties in every dimension has advanced significantly, and Birkar is one of the few mathematicians at the forefront of this work. He co-authored a paper in 2006 with Hacon, James McKernan of the University of California, San Diego, and Paolo Cascini of the Imperial College London that examined the classification scheme for general type variety types. In part because of their work, Hacon and McKernan later received the $3 million Breakthrough Prize. In 2016, Birkar contributed the most to mathematics as an individual. He released two papers that year that resolved one of the most crucial questions surrounding the characteristics of particular Fano kinds. The studies demonstrated how the birational transformation of Fano variants results in a tidy family that can be identified by just a few traits. Consider the flat plane to begin understanding what this entails. Imagine all the lines that cross a plane point at this moment. There are an endless number of such lines. Next, make a circle with that point in the centre. The circle is crossed by each line at two different locations. This means that any line can be defined, or "parameterized," by any one of these points. The fact that every line can be parameterized by points on the same tidy geometric object—the circle—indicates that your family of lines is organised. You couldn't make any type of neat remark about all the lines if they were instead scattered over the plane.

Birkar developed a similar type of parameterization in the 2016 publications for particular Fano varieties (those of fixed dimension with "mild singularities"). He demonstrated how, after birational transformation, Fano varieties resemble lines that pass through a point in that they are in a sufficiently organised relationship to one another to be parameterized by the same tidy geometric object. Birkar demonstrated that any dimension's Fano varieties may be described by a finite set of parameters. Infinitely many algebraic varieties share a fundamental trait when it is claimed that they may all be reduced to a small number of features. Create an organism category, but if you need an unlimited number of traits to describe every organism in that category, your category is worthless. However, you've made some strides if you can characterise a seemingly disparate collection of things using a limited number of traits.

"When you can parameterize things with finitely many parameters, this means the family shares many properties," explained Birkar. Due to the fact that you can sort of talk about all of them at once, finiteness is significant.

Although Birkar has demonstrated that there are a finite set of properties that characterise Fano varieties, there is still a great deal he would like to learn about this enormous family of algebraic equations. He will start working on investigating more particular aspects of their geometry. He will do it knowing how unusual a person he is to have attained his newfound reputation among mathematics. At the same time, he continues to be inspired by the same drive that led him to mathematics when he first encountered the subject in Mariwan decades ago.

“You get ideas from someone, somewhere, and you create something new,” he said. “You contribute something new and create more beautiful things.”