The Life and Work of Caucher Birkar, the Migrant Mathematician

The life and work of Caucher Birkar, the migrant mathematician

Diletta Martinelli

Universiteit van Amsterdam

Nationale Wiskunde Dagen, February 1, 2020 Maths makes the news, 1 August 2018

Internation Congress of Mathematichians, Rio de Janerio. Maths makes the news, 2 August 2018 Oﬃcial announcement He is Kurdish!

1979: Iranian Revolution.

1980-1988: Iraq-Iran war.

Short Bio of Caucher Birkar

1978: Born in Marivan, Iran. 1979: Iranian Revolution.

1980-1988: Iraq-Iran war.

Short Bio of Caucher Birkar

1978: Born in Marivan, Iran.

He is Kurdish! 1980-1988: Iraq-Iran war.

Short Bio of Caucher Birkar

1978: Born in Marivan, Iran.

He is Kurdish!

1979: Iranian Revolution. Short Bio of Caucher Birkar

1978: Born in Marivan, Iran.

He is Kurdish!

1979: Iranian Revolution.

1980-1988: Iraq-Iran war. Short bio of Cauher Birkar

”My brother Haidar started teaching me more advanced mathematics, which was not part of middle school education. He taught me to learn maths for the fun of it rather than passing exams. He is probably the person that had the most inﬂuence on my education.”

Birkar’s family. 2000: He travels to England for a mathematics competition and asks political asylum. The British government places him in Nottingham where he enrolls at the University.

He changes his name to Caucher Birkar, the migrant mathematician in Kurdish.

2002: He meets Vyacheslav Shokurov and starts working in birational geometry.

2010: He proves a major result in collaboration with Cascini, Hacon and McKernan [BCHM], he becomes professor at Cambridge and a maths superstar.

Short bio of Caucher Birkar

He attends the University in Tehran; He changes his name to Caucher Birkar, the migrant mathematician in Kurdish.

2002: He meets Vyacheslav Shokurov and starts working in birational geometry.

2010: He proves a major result in collaboration with Cascini, Hacon and McKernan [BCHM], he becomes professor at Cambridge and a maths superstar.

Short bio of Caucher Birkar

He attends the University in Tehran;

2000: He travels to England for a mathematics competition and asks political asylum. The British government places him in Nottingham where he enrolls at the University. 2002: He meets Vyacheslav Shokurov and starts working in birational geometry.

2010: He proves a major result in collaboration with Cascini, Hacon and McKernan [BCHM], he becomes professor at Cambridge and a maths superstar.

Short bio of Caucher Birkar

He attends the University in Tehran;

2000: He travels to England for a mathematics competition and asks political asylum. The British government places him in Nottingham where he enrolls at the University.

He changes his name to Caucher Birkar, the migrant mathematician in Kurdish. 2010: He proves a major result in collaboration with Cascini, Hacon and McKernan [BCHM], he becomes professor at Cambridge and a maths superstar.

Short bio of Caucher Birkar

He attends the University in Tehran;

2000: He travels to England for a mathematics competition and asks political asylum. The British government places him in Nottingham where he enrolls at the University.

He changes his name to Caucher Birkar, the migrant mathematician in Kurdish.

2002: He meets Vyacheslav Shokurov and starts working in birational geometry. Short bio of Caucher Birkar

He attends the University in Tehran;

2000: He travels to England for a mathematics competition and asks political asylum. The British government places him in Nottingham where he enrolls at the University.

He changes his name to Caucher Birkar, the migrant mathematician in Kurdish.

2002: He meets Vyacheslav Shokurov and starts working in birational geometry.

2010: He proves a major result in collaboration with Cascini, Hacon and McKernan [BCHM], he becomes professor at Cambridge and a maths superstar. Algebraic ⇒ He uses algebraic tools.

Birkar is an Algebraic Geometer

Geometer ⇒ He thinks about geometric objects. Birkar is an Algebraic Geometer

Geometer ⇒ He thinks about geometric objects.

Algebraic ⇒ He uses algebraic tools. The Circle

x2 + y2 = 1 The Circle

x2 + y2 = 1 Polynomials are everywhere

Simple functions. They model many naturally-occurring processes across science. They approximate smooth functions.

Sydney Harbour Bridge We call the space of solutions of polynomial equations algebraic varieties. The zoo of polynomials

X12 = 243S12 − 22Q12 = 0 where 2 2 2 2 6 Q12 = (x + y + z + w ) p − − − S12 = 33 (5)(s2,3 +s3,4 +s4,2)+ + + + 19(s2,3 + s3,4 + +s4,2) + 10s2,3,4 − 14s1,0 + 2s1,1 − 6s1,2 2 − 352s5,1 + 336l5 l1 + 48l2l3l4

4 4 4 4 l1 = x + y + z + w 2 2 2 2 l2 = x y + z w 2 2 2 2 l3 = x z + y w 2 2 2 2 l4 = x w + y z

l5 = xyzw ... The dodecic surface. It was discovered by Alessandra Sarti in 2008. Classiﬁcation

Caucher Birkar: “We want to bring order to an inﬁnitude of equations.” Comparing varieties I

Cylinder vs Hyperboloid Isomorphism

We say that two varieties are isomorphic if there are morphisms f : X → Y and h : Y → X that are inverse of each other.

Example: The cylinder and the hyperboloid are isomorphic. The hyperboloid and the cone are NOT isomorphic.

Comparing varieties II

Hyperboloid vs Cone Comparing varieties II

Hyperboloid vs Cone

The hyperboloid and the cone are NOT isomorphic. Singularities are evil!

Singular vs Smooth

Going from the hyperboloid to the cone we created a singularity, i.e. a point where all the partial derivatives vanish. Singular vs Smooth

Going from the hyperboloid to the cone we created a singularity, i.e. a point where all the partial derivatives vanish.

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We want to allow small modiﬁcations. ned syusol hc oref h map the yourself, check should ( you as Indeed, ,y z y, x, A R E N ) E 7! . ⇡ S onto M ( I z z z yz, xz, complex T H h oei uhawy hti is it that way, a such in cone the ⇡ o h oe hr sasimple a is there cone: the for z bv rdcstefunction the produces above 1 a 1 )doesthetrick! z ubr,ee fsm of some if even numbers, 2 a 2 ...z x n a 2 n always + nlclanalytic local in y 2 efound, be z 2 n3- on From the cylinder to the cone: the black loop collapses into the singularity. From the cone to the cylinder: we are resolving the singularity adding the black loop that is parametrizing all the colliding directions. This is what we call a blow-up.

Birational Equivalence

Deﬁnition X and Y are birationally equivalent if there are closed subsets Z ⊂ X and W ⊂ Y such that X \Z is isomorphic to Y \W .

Example The hyperboloid and the cone are birationally equivalent. From the cone to the cylinder: we are resolving the singularity adding the black loop that is parametrizing all the colliding directions. This is what we call a blow-up.

Birational Equivalence

Deﬁnition X and Y are birationally equivalent if there are closed subsets Z ⊂ X and W ⊂ Y such that X \Z is isomorphic to Y \W .

Example The hyperboloid and the cone are birationally equivalent.

From the cylinder to the cone: the black loop collapses into the singularity. Birational Equivalence

Deﬁnition X and Y are birationally equivalent if there are closed subsets Z ⊂ X and W ⊂ Y such that X \Z is isomorphic to Y \W .

Example The hyperboloid and the cone are birationally equivalent.

From the cylinder to the cone: the black loop collapses into the singularity. From the cone to the cylinder: we are resolving the singularity adding the black loop that is parametrizing all the colliding directions. This is what we call a blow-up. The birational classiﬁcation

Hironaka’s resolution of singularities, 1964 Over C, every algebraic variety is birationally equivalent to a smooth variety obtained via a series of blow-ups.

Our goal We want to classify smooth projective varieties deﬁned over the complex numbers up to birational equivalence. What are we hoping to obtain?

For smooth projective curves we have a satisfactory classiﬁcation.

deg(f) g(X) Variety Universal cover Curvature 1 2 1,2 0 P S > 0 2 3 1 Elliptic curves R = 0 2 > 3 ≥ 2 Hyperbolic curves |x| ≤ 1 ⊂ R < 0 We are looking for a representative that is minimal.

Higher dimension?

As soon as the dimension of the variety is greater than one, there are too many smooth varieties in the birational equivalence class. We need a more subtle representative, one that is simpler than all the others. Higher dimension?

We are looking for a representative that is minimal. Birational surfaces I

All the time that we blow up a point on a smooth surface, we produce a special curve E. We call E an exceptional curve. Minimal surfaces

Deﬁnition A smooth surface is minimal if it does not contain any exceptional curves.

Deﬁnition Let X be a smooth surface. A minimal surface Y birational to X is called a minimal model for X .

How do we ﬁnd the minimal model inside the birational equivalence class? Minimal Model Program in dimension 2

Start

S: smooth surface

Is there an : exceptional no S = End curve E on Smin S?

yes

Castelnuovo Theo- rem φ: S → T the S := T contraction of E. T smooth surface Birational surfaces II Minimal Model Program in dimension 2

Start

S: smooth surface

Is there an : exceptional no S = End curve E on Smin S?

yes

Castelnuovo Theo- rem φ: S → T the S := T contraction of E. T smooth surface Greatest achievement of the Italian School of algebraic geometry.

Classiﬁcation of surfaces

In a ﬁnite number of steps we can reach the minimal surface in the birational equivalence class.

Minimal surfaces are classiﬁed. Classiﬁcation of surfaces

In a ﬁnite number of steps we can reach the minimal surface in the birational equivalence class.

Minimal surfaces are classiﬁed. Greatest achievement of the Italian School of algebraic geometry. MMP: state of the art

The program has been completely solved in dimension 3 [Mori ’80s, . . . ], 4 [Shokurov ’90s, . . . ], 5 [Birkar2009], and for an important class of varieties called of general type [Birkar, Cascini, Hacon, McKernan, 2010].

McKernan and Hacon, Birkar, Fields medal in Breakthrough prize in Mori, Fields medal in 2018. 2017. 1990. The building blocks

Fano varieties Canonically polarized Calabi-Yau varieties

Courtesy of The FanoSearch Project. Hurwitz surface. Kummer surface. Trichotomy in terms of curvature The periodic table of shapes

Watercolour by London artist Gemma Anderson. Oﬃcial announcement Boundedness of Fano varieties

In 2016, Birkar proved the BAB (Borisov-Alexeev-Borisov) conjecture that states that Fano varietes with a special class of singularities form a bounded family. What does being a bounded family mean?

Bounded means you can describe all the (inﬁnitely many) shapes in the family with a ﬁnite number of parameters. Geometric idea

Lines passing through the origin Geometric idea

Lines passing through the origin are parametrized by a circle. In terms of algebra

We can describe the elements of the family with a ﬁnite set of variables and equations of a bounded degree.

Different elements in the family corresponds to different choices of coefficients:

7 5y + 3x = 0 2y + 9 x = 0. Thank you!