Differentials and K3 surfaces Ignacio Barros Humboldt-Universit at - - PowerPoint PPT Presentation

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Differentials and K3 surfaces Ignacio Barros Humboldt-Universit at - - PowerPoint PPT Presentation

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Riemann Surfaces Differentials Geometry of H Differentials and K3 surfaces Ignacio Barros Humboldt-Universit at zu Berlin February 12, 2019 Riemann Surfaces Differentials Geometry of H Overview History Differentials My

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Riemann Surfaces Differentials Geometry of H

Differentials and K3 surfaces

Ignacio Barros

Humboldt-Universit¨ at zu Berlin

February 12, 2019

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Riemann Surfaces Differentials Geometry of H

Overview

  • History
  • Differentials
  • My contributions
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Riemann Surfaces Differentials Geometry of H

Riemann surfaces

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Riemann Surfaces Differentials Geometry of H

Riemann surfaces

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Riemann Surfaces Differentials Geometry of H

Riemann surfaces

A Riemann surface is a one dimensional complex geometric

  • bject.
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Riemann Surfaces Differentials Geometry of H

Riemann surfaces

They are classified by the genus.

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Riemann Surfaces Differentials Geometry of H

Riemann surfaces

They are classified by the genus. genus = 0

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Riemann Surfaces Differentials Geometry of H

Riemann surfaces

They are classified by the genus. genus = 0 genus = 1

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Riemann Surfaces Differentials Geometry of H

Riemann surfaces

They are classified by the genus. genus = 0 genus = 1 genus = 2

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Riemann Surfaces Differentials Geometry of H

Riemann surfaces

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Riemann Surfaces Differentials Geometry of H

Riemann surfaces

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Riemann Surfaces Differentials Geometry of H

Riemann surfaces

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Riemann Surfaces Differentials Geometry of H

Moduli of Riemann surfaces

In 1857 Riemann conceives a geometric space that parameterizes all Riemann surfaces of a given genus.

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Riemann Surfaces Differentials Geometry of H

Moduli of Riemann surfaces

In 1857 Riemann conceives a geometric space that parameterizes all Riemann surfaces of a given genus.

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Riemann Surfaces Differentials Geometry of H

Moduli of Riemann surfaces

In 1857 Riemann conceives a geometric space that parameterizes all Riemann surfaces of a given genus.

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Riemann Surfaces Differentials Geometry of H

Moduli of Riemann surfaces

In 1857 Riemann conceives Mg and computes the dimension, dimCMg = 3g − 3.

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Riemann Surfaces Differentials Geometry of H

Moduli of Riemann surfaces

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Riemann Surfaces Differentials Geometry of H

Moduli of Riemann surfaces

dimMg,n = 3g − 3 + n

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Riemann Surfaces Differentials Geometry of H

Moduli of Riemann surfaces

  • The space Mg,n is a fundamental object in modern

mathematics.

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Riemann Surfaces Differentials Geometry of H

Moduli of Riemann surfaces

  • The space Mg,n is a fundamental object in modern

mathematics.

  • There are still many questions about Mg,n that remain

unanswered.

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Riemann Surfaces Differentials Geometry of H

Moduli of Riemann surfaces

  • The space Mg,n is a fundamental object in modern

mathematics.

  • There are still many questions about Mg,n that remain

unanswered.

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Riemann Surfaces Differentials Geometry of H

Moduli of Riemann surfaces

  • The space Mg,n is a fundamental object in modern

mathematics.

  • There are still many questions about Mg,n that remain

unanswered.

  • It plays also an important role in modern physics.
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Riemann Surfaces Differentials Geometry of H

Moduli of Riemann surfaces

  • The space Mg,n is a fundamental object in modern

mathematics.

  • There are still many questions about Mg,n that remain

unanswered.

  • It plays also an important role in modern physics.
  • Time-unfolding of a fundamental string (string theory)
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Riemann Surfaces Differentials Geometry of H

Moduli of Riemann surfaces

  • The space Mg,n is a fundamental object in modern

mathematics.

  • There are still many questions about Mg,n that remain

unanswered.

  • It plays also an important role in modern physics.
  • Time-unfolding of a fundamental string (string theory)
  • Quantization
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Riemann Surfaces Differentials Geometry of H

Moduli of Riemann surfaces

  • The space Mg,n is a fundamental object in modern

mathematics.

  • There are still many questions about Mg,n that remain

unanswered.

  • It plays also an important role in modern physics.
  • Time-unfolding of a fundamental string (string theory)
  • Quantization
  • Theta functions and the heat equation
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Riemann Surfaces Differentials Geometry of H

Moduli of Riemann surfaces

  • The space Mg,n is a fundamental object in modern

mathematics.

  • There are still many questions about Mg,n that remain

unanswered.

  • It plays also an important role in modern physics.
  • Time-unfolding of a fundamental string (string theory)
  • Quantization
  • Theta functions and the heat equation
  • Witten Conjecture
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Riemann Surfaces Differentials Geometry of H

Moduli of Riemann surfaces

  • The space Mg,n is a fundamental object in modern

mathematics.

  • There are still many questions about Mg,n that remain

unanswered.

  • It plays also an important role in modern physics.
  • Time-unfolding of a fundamental string (string theory)
  • Quantization
  • Theta functions and the heat equation
  • Witten Conjecture / Kontsevich theorem
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Riemann Surfaces Differentials Geometry of H

Moduli of Riemann surfaces

KdV equation,

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Riemann Surfaces Differentials Geometry of H

Moduli of Riemann surfaces

KdV equation, motion of water.

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Riemann Surfaces Differentials Geometry of H

Moduli of Riemann surfaces

KdV equation, motion of water.

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Riemann Surfaces Differentials Geometry of H

Moduli of Riemann surfaces

KdV equation, motion of water. Geometry of Mg

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Riemann Surfaces Differentials Geometry of H

Moduli of Riemann surfaces

Mg

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Riemann Surfaces Differentials Geometry of H

Moduli of Riemann surfaces

Mg

Algebraic Geometry

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Riemann Surfaces Differentials Geometry of H

Moduli of Riemann surfaces

Mg

Complex Analysis Algebraic Geometry Dynamics

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Riemann Surfaces Differentials Geometry of H

Moduli of Riemann surfaces

Mg

Complex Analysis Algebraic Geometry Dynamics Physics

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Riemann Surfaces Differentials Geometry of H

Moduli of Riemann surfaces

Algebraic Geometry Dynamics

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Riemann Surfaces Differentials Geometry of H

Holomorphic differentials

In mathematics it is very common to study enriched objects.

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Riemann Surfaces Differentials Geometry of H

Holomorphic differentials

In mathematics it is very common to study enriched objects.

  • In my thesis I study the global geometry of the space of

Riemann surfaces together a differential form. (C, ω)

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Riemann Surfaces Differentials Geometry of H

Holomorphic differentials

In mathematics it is very common to study enriched objects.

  • In my thesis I study the global geometry of the space of

Riemann surfaces together a differential form. (C, ω) ∈ Hg,n(µ)

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Riemann Surfaces Differentials Geometry of H

Holomorphic differentials

In mathematics it is very common to study enriched objects.

  • In my thesis I study the global geometry of the space of

Riemann surfaces together a differential form. (C, ω) ∈ Hg,n(µ)

  • The differential form ω realizes C as a flat polygon with
  • pposite edges identified.
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Riemann Surfaces Differentials Geometry of H

Examples of flat surfaces in genus g = 2

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Riemann Surfaces Differentials Geometry of H

Holomorphic differentials

The group GL(2, R) acts on Hg,n(µ).

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Riemann Surfaces Differentials Geometry of H

Holomorphic differentials

The group GL(2, R) acts on Hg,n(µ).

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Riemann Surfaces Differentials Geometry of H

Holomorphic differentials

The group GL(2, R) acts on Hg,n(µ). This is called Teichm¨ uller Dynamics

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Riemann Surfaces Differentials Geometry of H

Holomorphic differentials

The group GL(2, R) acts on Hg,n(µ). This is called Teichm¨ uller Dynamics

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Riemann Surfaces Differentials Geometry of H

Holomorphic differentials

Figure: Maryam Mirzakhani (1977 - 2017)

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Riemann Surfaces Differentials Geometry of H

Geometry of Hg,n(µ)

  • My thesis is about the global geometry of Hg,n(µ).
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Riemann Surfaces Differentials Geometry of H

Geometry of Hg,n(µ)

  • My thesis is about the global geometry of Hg,n(µ).
  • In Algebraic Geometry we have ways of measuring complexity.
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Riemann Surfaces Differentials Geometry of H

Geometry of Hg,n(µ)

  • My thesis is about the global geometry of Hg,n(µ).
  • In Algebraic Geometry we have ways of measuring complexity.

Uniruled General type

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Riemann Surfaces Differentials Geometry of H

Geometry of Hg,n(µ)

  • My thesis is about the global geometry of Hg,n(µ).
  • In Algebraic Geometry we have ways of measuring complexity.

Uniruled General type Hg,n(µ)

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Riemann Surfaces Differentials Geometry of H

Geometry of Hg,n(µ)

  • My thesis is about the global geometry of Hg,n(µ).
  • In Algebraic Geometry we have ways of measuring complexity.
  • We can understand Hg,n(µ) when g is small!!

Uniruled General type Hg,n(µ)

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Riemann Surfaces Differentials Geometry of H

Geometry of Hg,n(µ)

Theorem (–, 2018)

For all partitions µ of 2g − 2, the moduli spaces Hg,n(µ) are uniruled when g ≤ 11.

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Riemann Surfaces Differentials Geometry of H

Geometry of Hg,n(µ)

For a given g, there are many Hg,n(µ)’s.

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Riemann Surfaces Differentials Geometry of H

Geometry of Hg,n(µ)

For a given g, there are many Hg,n(µ)’s.

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Riemann Surfaces Differentials Geometry of H

Geometry of Hg,n(µ)

For a given g, there are many Hg,n(µ)’s. g Possible µ′s

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Riemann Surfaces Differentials Geometry of H

Geometry of Hg,n(µ)

For a given g, there are many Hg,n(µ)’s. g Possible µ′s 8 → 135

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Riemann Surfaces Differentials Geometry of H

Geometry of Hg,n(µ)

For a given g, there are many Hg,n(µ)’s. g Possible µ′s 8 → 135 9 → 231

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Riemann Surfaces Differentials Geometry of H

Geometry of Hg,n(µ)

For a given g, there are many Hg,n(µ)’s. g Possible µ′s 8 → 135 9 → 231 10 → 385

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Riemann Surfaces Differentials Geometry of H

Geometry of Hg,n(µ)

For a given g, there are many Hg,n(µ)’s. g Possible µ′s 8 → 135 9 → 231 10 → 385 11 → 627

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Riemann Surfaces Differentials Geometry of H

Geometry of Hg,n(µ)

For a given g, there are many Hg,n(µ)’s. They have very different geometries. g Possible µ′s 8 → 135 9 → 231 10 → 385 11 → 627

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Riemann Surfaces Differentials Geometry of H

Geometry of Hg,n(µ)

Theorem (–, 2018)

For all partitions µ of 2g − 2, the moduli spaces Hg,n(µ) are uniruled when g ≤ 11.

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Riemann Surfaces Differentials Geometry of H

Geometry of Hg,n(µ)

Theorem (–, 2018)

For all partitions µ of 2g − 2, the moduli spaces Hg,n(µ) are uniruled when g ≤ 11.

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Riemann Surfaces Differentials Geometry of H

Geometry of Hg,n(µ)

Theorem (–, 2018)

For all partitions µ of 2g − 2, the moduli spaces Hg,n(µ) are uniruled when g ≤ 11.

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Riemann Surfaces Differentials Geometry of H

Geometry of Hg,n(µ)

Theorem (–, 2018)

For all partitions µ of 2g − 2, the moduli spaces Hg,n(µ) are uniruled when g ≤ 11.

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Riemann Surfaces Differentials Geometry of H

Geometry of Hg,n(µ)

Theorem (–, 2018)

For all partitions µ of 2g − 2, the moduli spaces Hg,n(µ) are uniruled when g ≤ 11.

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Riemann Surfaces Differentials Geometry of H

Geometry of Hg,n(µ)

  • Uniruled means that the space can be covered by Riemann

spheres.

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Riemann Surfaces Differentials Geometry of H

Geometry of Hg,n(µ)

  • Uniruled means that the space can be covered by Riemann

spheres.

  • We can parameterize the space with free parameters.
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Riemann Surfaces Differentials Geometry of H

Geometry of Hg,n(µ)

  • Uniruled means that the space can be covered by Riemann

spheres.

  • We can parameterize the space with free parameters.
  • It has positive scalar curvature.
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Riemann Surfaces Differentials Geometry of H

Geometry of Hg,n(µ)

  • Uniruled means that the space can be covered by Riemann

spheres.

  • We can parameterize the space with free parameters.
  • It has positive scalar curvature.
  • Uniruledness is a very spacial property that most varieties

don’t have.

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Riemann Surfaces Differentials Geometry of H

Geometry of Hg,n(µ)

  • Uniruled means that the space can be covered by Riemann

spheres.

  • We can parameterize the space with free parameters.
  • It has positive scalar curvature.
  • Uniruledness is a very spacial property that most varieties

don’t have.

  • It was very surprising to find that it does not depend on µ.
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Riemann Surfaces Differentials Geometry of H

“Empires die, but Euclid’s theorems keep their youth forever”

Vito Volterra