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Professor's work figures in 2018 Fields medal

Alan Flurry

Finding a proof for a mathematical conjecture – a strong guess illuminating a way forward – has helped shape the course of mathematical history. The mathematics world recently celebrated the achievements of a proof that proved a conjecture partially formulated by UGA faculty member Valery Alexeev.

Professor Caucher Birkar of the University of Cambridge was awarded the 2018 Fields medal, widely considered to be an equivalent of a Nobel prize for mathematicians, at the ICM-2018, International Congress of Mathematicians held in August 2018 in Rio de Janeiro, Brazil. While there are prizes larger in monetary terms, the Fields medal remains the oldest and one of the most prestigious. It is awarded by the International Mathematical Union whose membership includes math societies of over 80 countries, including the American Mathematical Society. The Congress meets once every 4 years, when the main achievements are presented and the future agenda is set. 

Birkar grew up in the Kurdish region of Iran during the Iran-Iraq war, and as a refugee fled with his family to the UK. The short citation for Birkar's medal is: "For the proof of the boundedness of Fano varieties and for contributions to the minimal model program." The first half refers to his solution of the B-A-B conjecture, attributed to Alexeev and twin brothers, Lev and Alexander Borisov.

Valery Alexeev is the David C. Barrow Professor of Mathematics at UGA. He was a speaker at the ICM-2006 in Madrid.

The main object of algebraic geometry, Birkar's and Alexeev's field, is algebraic varieties, solutions of polynomial equations in many variables. 

“These varieties fall into three main categories depending on how they are "curved": Fano (positively), general type (negatively), and Calabi-Yau (zero curvature),” Alexeev said. “On the most basic level, one would like to know if they are "bounded", i.e. if under suitable conditions there are only finitely many types of them.”

For varieties of general type this was established by Alexeev in dimension two nearly twenty years ago, just after he first arrived at UGA, and by Hacon-McKernan-Xu for higher dimensions in 2014. Hacon and McKernan received the 2018 Breakthrough Prize. For Fano varieties the result was established by Alexeev in dimension two and by Borisovs for special, "toric" varieties, in the 1990s. After decades of hard work by many contributors, finally in 2016 Birkar proved the BAB conjecture in all dimensions. For Calabi-Yau varieties -- famously appearing in many modern theories of the Universe -- the result is still open. 

Congratulations to Alexeev for these extraordinary contributions at the very pinnacle of his field.

Image: The obverse of a Fields Medal, via wikimedia commons, made by Stefan Zachow for the International Mathematical Union (IMU), showing a bas relief of Archimedes (in Greek text). The Latin phrase states: Transire suum pectus mundoque potiri. To overcome one's human limitations and become master of the universe.

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