This spring, no longer lengthy after Caucher Birkar realized that he can be receiving the Fields Medal, the best honor in arithmetic, he shared a reminiscence from his undergraduate years. Even via that point, he had come a great distance. Born and raised in a rural subsistence farming village within the Kurdish area of western Iran, Birkar had made his solution to the University of Tehran, one of the pre-eminent universities within the nation. There, on the math membership, he recalled finding out the photographs of Fields medalists lining the partitions. “I checked out them and mentioned to myself, ‘Will I ever meet one of those folks?’ At that point in Iran, I couldn’t even know that I’d be capable of cross to the West.”

There was once a lot about his long term that Birkar couldn’t have predicted at the moment: his flight from Iran, his request for political asylum, the frenzy to re-ignite a just about deserted box of arithmetic. And this month, at a rite in Rio de Janeiro, the honour of being decided on as one of four winners of the Fields Medal, a prize conferred each and every four years via the International Mathematical Union at the maximum achieved mathematicians on the earth beneath the age of 40 originally of the yr by which the prize is awarded. “To cross from the purpose that I didn’t consider assembly those folks to the purpose the place sooner or later I cling a medal myself–I simply couldn’t consider that this could come true,” mentioned Birkar, who became 40 in July. (He additionally more than likely couldn’t have imagined that lower than an hour after the award rite his 14-karat gold medal would cross lacking when his briefcase was once stolen; inside of days, Birkar was once promised an similar alternative.)

As a mathematician, Birkar has helped deliver order to the countless selection of polynomial equations–the ones equations that consist of other variables raised to quite a lot of powers. No two equations are precisely alike, however Birkar has helped disclose that many can also be well categorised into a small quantity of households. In two papers printed in 2016, he confirmed that a vast quantity of other polynomials can also be outlined via a finite quantity of traits–a consequence which demonstrated that this bewildering array of reputedly unrelated algebraic equations stocks one thing in commonplace.

Birkar’s contemporary paintings on Fano types is one of his maximum influential mathematical effects, nevertheless it sprang from an impulse that has ruled him since his self-guided initiation into arithmetic virtually three a long time in the past: an impulse to create one thing new.

## The First Revolution

When Birkar was once very younger, he appreciated to be close to his mom, Sakina, whilst she baked bread. She recalls a scene from when Birkar was once about three. “He was once sitting subsequent to me, seeking to get very on the subject of me, seeking to let his ft contact my ft,” she mentioned.

Birkar’s mom instructed me this tale over Skype. He and I have been sitting in the lounge of his small area in Cambourne, a group 10 miles out of doors of Cambridge, England, that was once constructed from scratch two a long time in the past. His mom was once within the circle of relatives’s area in Marivan, a Kurdish village in a mountainous area alongside the Iran-Iraq border the place Birkar’s circle of relatives has lived for generations. She wore a white headband and was once sitting on a sofa between Birkar’s oldest brother and his father, Majid.

Caucher Birkar was once born in 1978, the 3rd of six kids. His circle of relatives lived on a subsistence farm rising rice, wheat and greens. They additionally stored cows and the occasional horse. The circle of relatives was once self-sufficient in a manner that helped refuge Birkar from the worst of the tumult that engulfed the area all the way through his early life–the Islamic Revolution of 1979 and the brutal eight-year struggle with Iraq that quickly adopted.

“Just after I used to be born there was once a revolution and struggle, however the truth that we produced the whole thing we would have liked intended shall we live on,” he mentioned.

Birkar’s father had attended faculty for a few years. His mom had no formal schooling in any respect. But Birkar and his siblings went to the village faculty. Around 5th grade, Birkar began to note arithmetic. “I felt one thing. Nothing severe, it was once simply a little feeling that I’m excellent in arithmetic,” he mentioned.

Early on, the principle steward of Birkar’s mathematical hobby was once Haidar, his oldest brother, who offered him to the elemental ideas of calculus. “He was once very inquisitive about mathematical and bodily questions,” Haidar mentioned over Skype. “I take into account he’d pick out up my textbooks and take a look at to resolve the issues.” Birkar recalls his brother educating him one thing else as neatly: that wisdom can also be beautiful.

“He was once more than likely the one particular person I knew who adopted his pursuits, no longer handiest to do excellent at school. It was once a lot more than that, he in reality was once all in favour of positive issues,” Birkar mentioned. “He would do issues no one else would do in that atmosphere.”

By highschool, Birkar had outrun his older brother’s mathematical wisdom and was once left to be informed the topic on his personal. He took out books from the native library with titles like *Men of Mathematics* and *What Is Mathematics*. His circle of relatives recalls him studying overdue into the night time, regularly whilst taking note of track–a dependancy he maintains nowadays.

Even in his first brushes with skilled arithmetic, Birkar sought after to do extra than simply appreciate people’s discoveries. “I learn these kinds of books and I had the sensation that simply studying issues isn’t sufficient. I additionally sought after to create my very own stuff, to create one thing new,” he mentioned.

As a highschool pupil, he began writing his personal mathematical proofs. In faculty, he started to publish them to mathematical journals. Later on, after he’d had some formal coaching, he discovered that the proofs he’d discovered have been found out way back. “Maybe I didn’t turn out anything else vital, however simply the enjoy, that sort of perspective, proved to be helpful within the later levels of my schooling,” he mentioned.

Birkar examined into the University of Tehran. In his ultimate yr as an undergraduate he traveled to England. While there, he sought political asylum based on the “political issues in Iran for folks like me,” he mentioned, relating to his identification as a Kurd, a minority ethnic staff that has regularly suffered state-sponsored repression. The British executive positioned him in Nottingham, a town in central England. During the yr it took for the federal government to procedure his case, Birkar met college individuals on the University of Nottingham, the place he enrolled after his asylum request was once granted.

There was once no one on the University of Nottingham who specialised in algebraic geometry, the sphere Birkar was hoping to go into. But Birkar’s authentic adviser there, a quantity theorist named Ivan Fesenko, inspired him to wait math occasions out of doors the college. At a convention in 2002 in Cambridge, Birkar met Vyacheslav Shokurov, a mathematician at Johns Hopkins University. Early on of their courting, Shokurov took Birkar snowboarding a few hours from Baltimore. Shokurov was once inspired via how briefly Birkar picked it up, and likewise via his bearing.

“He is a shy and respectable particular person, perhaps, as a result of he grew up in a village and in a conventional circle of relatives,” Shokurov mentioned via e mail.

At the time he met Birkar, Shokurov have been running for years in a just about deserted subfield of algebraic geometry referred to as birational geometry. Birational geometry had observed some primary advances greater than a decade previous, however the box had fizzled out for lack of new concepts. “The entire factor virtually just about stopped. Most folks in reality gave up,” Birkar mentioned.

Shokurov was once one of the few mathematicians who didn’t. In Birkar he discovered a gifted younger mathematician who may lend a hand reinvigorate the sphere.

## Three Types of the Infinite

In his place of work on the University of Cambridge, Birkar presentations two images of the mathematician Alexander Grothendieck. Grothendieck was once a refugee–he fled Nazi Germany–and a Fields medalist, identical to Birkar. He could also be extensively considered the one maximum influential mathematician of the second one part of the 20th century. In one of the footage, Grothendieck sits with a various staff of activists concerned within the French environmental motion of the early 1970s. Birkar, who grew up Kurdish in Iran and now lives in England the place he’s married to a Thai girl, admires each facets of Grothendieck: his mathematical imaginative and prescient and the best way he moved simply amongst other kinds of folks.

“All those cultures make issues extra attention-grabbing to me. All those cultures provide you with a sense of excitement,” he mentioned. His four-year-old son, Zanko, is a residing mirrored image of this range: He speaks his mom’s local Thai, his father’s local Kurdish, and his house nation’s English.

Algebraic geometry is a mixing of cultures, too. On one aspect there’s algebra, the find out about of equations, and at the different aspect there’s geometry, the find out about of shapes. The two be offering alternative ways of taking a look on the identical issues. Take the algebraic equation *y* = 2*x* – 3. If you plot its answers you get a geometric object–a immediately line. The two views are complementary. If you sought after to spot a resolution commonplace to two equations, like *y* = 2*x* – 3 and *y* = 3*x* + 5, you will discover the solution thru algebra, or it’s good to graph each equations and notice the place they intersect.

“Sometimes a geometric query can also be solved via algebraic strategies and once in a while an algebraic query can also be solved via geometric strategies,” mentioned János Kollár, a mathematician at Princeton University. “You can cross between those two aspects and enrich each of them.”

Linear equations are the most simple algebraic equations. There are many different sorts. They will have extra variables, and the ones variables can also be raised to other levels. You too can take into consideration the set of answers commonplace to a staff of equations. This set is referred to as an “algebraic selection.” An countless quantity of algebraic types exist; every has a distinctive geometric illustration.

“The maximum essential factor is the form, the shape, the construction of the set of answers,” Birkar mentioned. “The set of answers is what we name more or less an algebraic selection.”

Algebraic types are an unruly mob. Mathematicians wish to impose some order on them. This impulse isn’t so other from the urge to categorise organic existence–the residing international feels extra tractable to our minds, and extra significant in its form, if we predict in phrases of phyla and households moderately than reckon with every organism by itself.

Birational geometry is a manner of reworking algebraic types in order that they are able to be categorized. It’s a kind of surgical procedure: You get started with an algebraic selection, which has its personal idiosyncratic shape, then bring to a halt some of its bumps and easy out some of its creases, till you find yourself with a extra generic form. There are strict limits on what you’re allowed to chop, which make sure that you don’t utterly efface the variability you began with. Post-surgery, many in the past distinct types will now glance the similar; those are mentioned to belong to the similar “birational equivalence elegance.”

“We’re specializing in the large image and no longer being concerned about types disagreeing on some small subset of issues,” mentioned Christopher Hacon, a mathematician on the University of Utah and a main determine in birational geometry who will probably be giving a speech on Birkar’s paintings on the Fields Medal rite in Rio.

There are three huge birational equivalence categories: Fano types, Calabi-Yau types, and types of normal kind. The three categories are generic shapes, in the similar manner that the time period “insect” is generic relative to the precise organisms that fall beneath that heading. Each elegance has a other kind of uniform curvature (both uniformly sure, uniformly flat or uniformly detrimental, respectively). Mathematicians hope with the intention to display that each and every algebraic selection reduces to one of those three sorts of generic shapes during the procedure of birational transformation.

“We are hoping to search out gadgets that experience the similar sort of curvature far and wide,” Kollár mentioned. “We don’t need one thing that once in a while seems like a saddle, once in a while like a sphere, and once in a while has flat portions. That’s too sophisticated.”

## Points of Order

Birkar has completed his maximum essential arithmetic in his eating room, running on my own. On a conventional day, he’s going to paintings on the desk, getting as much as tempo and suppose, make tea, or placed on track, regularly classical or recent Kurdish songs. “So much of it is going on in my thoughts. Sometimes the entire day perhaps I write simply one web page,” he mentioned. After a few hours of considering he loves to transparent his thoughts with a motorcycle trip alongside the paths that weave thru his community.

From a distance, it could be exhausting to discover the roiling highbrow process that characterizes Birkar’s days. A couple of years in the past he was once in Thailand, visiting his spouse’s circle of relatives. One afternoon her grandfather mentioned to her, “What sort of process does your husband have? He was once within the lawn watching a mango tree doing not anything.”

But amongst mathematicians, Birkar is understood for the depth with which he assaults issues. “My influence is he has a tendency to be a bit solitary, targeted and made up our minds at the issues he’s running on,” Hacon mentioned. “He’s in reality into mastering the technical main points of the issue to hand, and with nice good fortune.”

Birkar’s paintings on algebraic types is a component of an ongoing effort referred to as the minimum fashion program. The function is to turn out that every one algebraic types can also be lowered to one of the three elementary sorts thru birational transformation. The minimum fashion program is going again greater than 100 years to a staff of Italian mathematicians who first categorized two-dimensional algebraic types (types with three variables). More lately, within the 1980s, Shigefumi Mori proved that every one three-dimensional algebraic types (types with four variables) cut back to one of those three sorts. Mori, a mathematician at Kyoto University and the present president of the International Mathematical Union, gained the Fields Medal in 1990 for this paintings.

But after Mori’s consequence, the sphere of birational geometry went quiet. “There have been large demanding situations to extending this to raised dimensions,” Birkar mentioned.

Shokurov was once one of the few who caught with the sphere during the 1990s. Due largely to his efforts, birational geometry reawakened within the early 2000s. Since then, mathematicians have made nice development in classifying algebraic types in each and every size, and Birkar is amongst a small quantity of mathematicians on the middle of all of it. In 2006, at the side of Hacon, James McKernan of the University of California, San Diego, and Paolo Cascini of Imperial College London, he was once co-author of a paper that explored the classification construction for types of normal kind. Hacon and McKernan later gained the $3 million Breakthrough Prize partly for that paintings.

Birkar made his greatest particular person contribution to arithmetic in 2016. That yr he printed two papers that settled one of an important problems in regards to the nature of explicit sorts of Fano types. The papers proved that during the procedure of birational transformation, Fano types shape a neat circle of relatives that may be outlined via a small quantity of traits.

To perceive what this implies, first image the flat aircraft. Now consider the entire traces that cross thru a level at the aircraft. An countless quantity of such traces exists. Now draw a circle focused on that time. Each of the traces intersects the circle at two distinctive puts. Because of this, every line can also be outlined, or “parameterized,” via those issues (or simply one of those issues). The proven fact that the entire traces can also be parameterized via issues at the identical neat geometric object–the circle–suggests there’s one thing orderly about your circle of relatives of traces. If the traces have been as a substitute strewn randomly concerning the aircraft, you wouldn’t be capable of make any sort of tidy remark about all of them.

In the 2016 papers, Birkar created a an identical sort of parameterization for explicit types of Fano types (the ones of fastened size with “gentle singularities”). He proved that, following birational transformation, Fano types grow to be just like the traces passing thru a level: They’re in an orderly sufficient relation to one any other that they are able to be parameterized via the similar neat geometric object. Birkar proved that every one Fano types in any size can also be outlined via a finite quantity of parameters.

To say that infinitely many algebraic types can also be outlined via a finite set of traits is to mention they proportion one thing elemental. If you create a class for organisms, and you wish to have a vast quantity of traits to account for the entire organisms in that class, your class will probably be meaningless. But if you’ll be able to outline a reputedly discordant set of gadgets via way of a finite set of characteristics, you’ve made some development.

“When you’ll be able to parameterize issues with finitely many parameters, this implies the circle of relatives stocks many homes,” Birkar mentioned. “That’s why finiteness is essential, as a result of in a manner you’ll be able to say one thing about all of them on the identical time.”

Birkar has proved there’s a finite quantity of traits that outline Fano types, however there’s nonetheless a lot more about this huge circle of relatives of algebraic equations he’d like to grasp. When he returns house from Rio this month, he’ll get to paintings exploring extra explicit options of their geometry. He’ll achieve this with a newfound standing amongst mathematicians and an consciousness of simply how not likely a particular person he’s to have completed it. At the similar time, he stays motivated via the similar urge that drew him into arithmetic when he first found out the sphere a long time in the past in Marivan.

“You get concepts from any person, someplace, and also you create one thing new,” he mentioned. “You give a contribution one thing new and create extra gorgeous issues.”