̽��ֱ�� of Cambridge - geometry

Cambridge mathematician awarded 2018 Fields Medal

sc604 — Wed, 01 Aug 2018 14:25:33 +0000

Professor Birkar, who originally came to the UK as a Kurdish refugee, was given the award today at the International Congress of Mathematicians in Rio de Janeiro, Brazil.

̽��ֱ��Fields medals, often called the Nobel Prize of mathematics, are awarded every four years. Medallists must be under the age of 40 by the start of the year they receive the award, with up to four mathematicians honoured at a time. Awarded for the first time in 1936, the medal is recognition for works of excellence and an incentive for new outstanding achievements.

Caucher Birkar from simonsfoundation.org on Vimeo.

Birkar, a member of Cambridge’s Department of Pure Mathematics and Mathematical Statistics, won the award for his work on categorising different kinds of polynomial equations. He proved that the infinite variety of such equations can be split into a finite number of classifications, a major breakthrough in the field of birational geometry. Born in a Kurdish village in pre-revolutionary Iran, Birkar sought and obtained political asylum in the UK while finishing his undergraduate degree in Iran.

“War-ridden Kurdistan was an unlikely place for a kid to develop an interest in mathematics,” Birkar told the ICM today. “I'm hoping that this news will put a smile on the faces of those 40 million people.”

Birkar, who just this year received recognition for his work as one of the London Mathematical Society Prize winners, was born in 1978 in Marivan, a Kurdish province in Iran bordering Iraq with about 200,000 inhabitants. His curiosity was awakened by algebraic geometry, the same interest that, in that same region, centuries earlier, had attracted the attention of Omar Khayyam (1048-1131) and Sharaf al-Din al-Tusi (1135-1213).

After graduating in Mathematics from Tehran ̽��ֱ��, Birkar went to live in the UK, where he became a British citizen. In 2004, he completed his PhD at the ̽��ֱ�� of Nottingham with the thesis “Topics in modern algebraic geometry”. Throughout his career, birational geometry has stood out as his main area of interest. He has devoted himself to the fundamental aspects of key problems in modern mathematics – such as minimal models, Fano varieties, and singularities. His theories have solved long-standing conjectures.

In 2010, the year in which he was awarded by the Foundation Sciences Mathématiques de Paris, Birkar wrote, alongside Paolo Cascini (Imperial College London), Christopher Hacon ( ̽��ֱ�� of Utah) and James McKernan ( ̽��ֱ�� of California, San Diego), an article called “Existence of minimal models for varieties of general log type” that revolutionised the field. ̽��ֱ��article earned the quartet the AMS Moore Prize in 2016.

Founded by the Canadian mathematician John Charles Fields to celebrate outstanding achievements, the Fields Medal has already been awarded to 56 scholars of the most diverse nationalities, among them, Brazilian Fields laureate Artur Avila, an extraordinary researcher from IMPA, awarded in 2014 in South Korea. Due to its importance and prestige, the medal is often likened to a Nobel Prize of Mathematics.

“This is absolutely phenomenal, both for Caucher and for mathematics at Cambridge,” said Professor Gabriel Paternain, Head of the Department of Pure Mathematics and Mathematical Statistics. “Caucher was already an exceptional young researcher when he came to Cambridge, and he's now one of the most remarkable people in this field. At Cambridge, we want to give all of our young researchers the opportunity to really explore their field early in their career: it can lead to some truly amazing things.”

̽��ֱ��winners of the Fields medal are selected by a group of specialists nominated by the Executive Committee of the International Mathematical Union (IMU), which organize the ICMs. Every four years, between two and four researchers under the age of 40 are chosen. Since 2006, a cash prize of 15 thousand Canadian dollars accompanies the medal.

In an interview with Quanta Magazine, Birkar spoke of the math club at Tehran ̽��ֱ��, where pictures of Fields medallists lined the walls. “I looked at them and said to myself, ‘Will I ever meet one of these people?’ At that time in Iran, I couldn’t even know that I’d be able to go to the West.

“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.”

Professor Birkar is Cambridge’s 11^th Fields medallist.

̽��ֱ��other three winners of the 2018 Fields medals are Peter Scholze from the ̽��ֱ�� of Bonn, Akshay Venkatesh from the Institute of Advanced Studies and Alessio Figalli from ETH Zurich.

̽��ֱ�� of Cambridge mathematician Caucher Birkar has been named one of four recipients of the 2018 Fields medals, the most prestigious awards in mathematics.

Kurdistan was an unlikely place for a kid to develop an interest in mathematics - I'm hoping that this news will put a smile on the faces of those 40 million people.

Caucher Birkar

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A new twist on soap films

sc604 — Fri, 23 May 2014 15:13:03 +0000

̽��ֱ��way in which soap films collapse and re-form when twisted or stretched could hold the key to predicting the formation and location of mathematical singularities, which can be seen in the motion of solar flares and other natural phenomena.

Research on the processes by which soap films undergo transitions from one stable state to another has led to conjectures on the nature and location of the singular events that occur during the change of form, connecting two previously separate areas in mathematics.

In mathematics, singularities occur when an equation or surface breaks down and ‘explodes’. In surfaces such as soap films, singularities occur when the surface collides with itself, changing shape in the blink of an eye.

Researchers from the ̽��ֱ�� of Cambridge have shown that identifying a special type of curve on the surface can help predict where these singularities are likely to occur in soap films, which could in turn aid in the understanding of singularities in the natural world. ̽��ֱ��results are published in the journal Proceedings of the National Academy of Sciences (PNAS).

We are all familiar with the simplest soap films, which are formed by dipping a wire loop into a soap solution: the flat surface that spans the wire and the bubbles which are formed when we blow on the film. With suitably shaped wires however, much more complex structures can be formed, such as Möbius strips.

All static soap films are ‘minimal surfaces’, for they have the least area of all possible surfaces that span a given wire frame.

What is less understood are the dynamic processes which occur when a minimal surface like a soap film is made unstable by deforming the supporting wire. ̽��ֱ��film typically moves in a fraction of a second to a new configuration through a singular point, at which the surface collides with itself and changes its connectivity.

These kinds of violent events also occur in the natural world – in fluid turbulence and in the motion of solar flares emanating from the sun – and one of the great challenges has been to predict where they will occur.

In research supported by the EPSRC, a team from the Department of Applied Mathematics and Theoretical Physics attempted to understand how to predict where the singularity will occur when soap films are twisted or stretched to a point of instability. For example, it is well-known that the surface spanning two separate wire loops will collapse to a singularity in between the loops.

In previous work, the group had shown that Möbius strip singularity occurs not between the loops but at the wire frame, where there is a complex rearrangement of the surface. “What was unclear was whether there was an underlying mathematical principle by which this striking difference could be explained,” said Professor Raymond Goldstein, who collaborated with Dr Adriana Pesci, Professor Keith Moffatt, and James McTavish, a maths undergraduate, on the research.

̽��ֱ��team recognised that a geometric concept known as a systole might be the key to understanding where singularities will occur. A systole is the length of the shortest closed curve on surface that cannot be shrunk to a point while remaining on the surface. An example of this is found on a bagel, where the shortest such curve encircles the bagel like a handle. Mathematicians have studied the geometric properties of these curves in recent decades, establishing constraints on the relationship between the length of a systole and the area of the surface on which they lie.

Using new laboratory experiments and computations, the researchers found evidence that the ultimate location of the singularities that occur when soap films collapse can be deduced from the properties of the systole. If the systolic curve loops around the wire frame, then the singularity occurs at the boundary, while if there is no such linking the singularity occurs in the bulk.

“This is an example of experimental mathematics, in the sense that we are using laboratory studies to inform conjectures on mathematical connections,” said Professor Goldstein. “While they are certainly not rigorous, we hope they will stimulate further research into this new, developing area.”

Soap films with complex shapes shed light on the formation of mathematical singularities, which occur in a broad range of fields.

This is an example of experimental mathematics, in the sense that we are using laboratory studies to inform conjectures on mathematical connections

Raymond Goldstein

Singularity in a soap film

Raymond Goldstein

Soap film singularity

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