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

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 contributions

Riemann Surfaces Differentials Geometry of H Riemann surfaces

Riemann Surfaces Differentials Geometry of H Riemann surfaces

Riemann Surfaces Differentials Geometry of H Riemann surfaces A Riemann surface is a one dimensional complex geometric object.

Riemann Surfaces Differentials Geometry of H Riemann surfaces They are classified by the genus .

Riemann Surfaces Differentials Geometry of H Riemann surfaces They are classified by the genus . genus = 0

Riemann Surfaces Differentials Geometry of H Riemann surfaces They are classified by the genus . genus = 0 genus = 1

Riemann Surfaces Differentials Geometry of H Riemann surfaces They are classified by the genus . genus = 0 genus = 1 genus = 2

Riemann Surfaces Differentials Geometry of H Riemann surfaces

Riemann Surfaces Differentials Geometry of H Riemann surfaces

Riemann Surfaces Differentials Geometry of H Riemann surfaces

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.

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.

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.

Riemann Surfaces Differentials Geometry of H Moduli of Riemann surfaces In 1857 Riemann conceives M g and computes the dimension, dim C M g = 3 g − 3.

Riemann Surfaces Differentials Geometry of H Moduli of Riemann surfaces

Riemann Surfaces Differentials Geometry of H Moduli of Riemann surfaces dim M g , n = 3 g − 3 + n

Riemann Surfaces Differentials Geometry of H Moduli of Riemann surfaces • The space M g , n is a fundamental object in modern mathematics.

Riemann Surfaces Differentials Geometry of H Moduli of Riemann surfaces • The space M g , n is a fundamental object in modern mathematics. • There are still many questions about M g , n that remain unanswered.

Riemann Surfaces Differentials Geometry of H Moduli of Riemann surfaces • The space M g , n is a fundamental object in modern mathematics. • There are still many questions about M g , n that remain unanswered.

Riemann Surfaces Differentials Geometry of H Moduli of Riemann surfaces • The space M g , n is a fundamental object in modern mathematics. • There are still many questions about M g , n that remain unanswered. • It plays also an important role in modern physics.

Riemann Surfaces Differentials Geometry of H Moduli of Riemann surfaces • The space M g , n is a fundamental object in modern mathematics. • There are still many questions about M g , n that remain unanswered. • It plays also an important role in modern physics. • Time-unfolding of a fundamental string (string theory)

Riemann Surfaces Differentials Geometry of H Moduli of Riemann surfaces • The space M g , n is a fundamental object in modern mathematics. • There are still many questions about M g , n that remain unanswered. • It plays also an important role in modern physics. • Time-unfolding of a fundamental string (string theory) • Quantization

Riemann Surfaces Differentials Geometry of H Moduli of Riemann surfaces • The space M g , n is a fundamental object in modern mathematics. • There are still many questions about M g , 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

Riemann Surfaces Differentials Geometry of H Moduli of Riemann surfaces • The space M g , n is a fundamental object in modern mathematics. • There are still many questions about M g , 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

Riemann Surfaces Differentials Geometry of H Moduli of Riemann surfaces • The space M g , n is a fundamental object in modern mathematics. • There are still many questions about M g , 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

Riemann Surfaces Differentials Geometry of H Moduli of Riemann surfaces KdV equation,

Riemann Surfaces Differentials Geometry of H Moduli of Riemann surfaces KdV equation, motion of water.

Riemann Surfaces Differentials Geometry of H Moduli of Riemann surfaces KdV equation, motion of water.

Riemann Surfaces Differentials Geometry of H Moduli of Riemann surfaces KdV equation, Geometry of M g motion of water.

Riemann Surfaces Differentials Geometry of H Moduli of Riemann surfaces M g

Riemann Surfaces Differentials Geometry of H Moduli of Riemann surfaces M g Algebraic Geometry

Riemann Surfaces Differentials Geometry of H Moduli of Riemann surfaces M g Complex Analysis Algebraic Geometry Dynamics

Riemann Surfaces Differentials Geometry of H Moduli of Riemann surfaces M g Complex Analysis Algebraic Geometry Dynamics Physics

Riemann Surfaces Differentials Geometry of H Moduli of Riemann surfaces Dynamics Algebraic Geometry

Riemann Surfaces Differentials Geometry of H Holomorphic differentials In mathematics it is very common to study enriched objects.

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 )

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 ) ∈ H g , n ( µ )

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 ) ∈ H g , n ( µ ) • The differential form ω realizes C as a flat polygon with opposite edges identified.

Riemann Surfaces Differentials Geometry of H Examples of flat surfaces in genus g = 2

Riemann Surfaces Differentials Geometry of H Holomorphic differentials The group GL (2 , R ) acts on H g , n ( µ ).

Riemann Surfaces Differentials Geometry of H Holomorphic differentials The group GL (2 , R ) acts on H g , n ( µ ).

Riemann Surfaces Differentials Geometry of H Holomorphic differentials The group GL (2 , R ) acts on H g , n ( µ ). This is called Teichm¨ uller Dynamics

Riemann Surfaces Differentials Geometry of H Holomorphic differentials The group GL (2 , R ) acts on H g , n ( µ ). This is called Teichm¨ uller Dynamics

Riemann Surfaces Differentials Geometry of H Holomorphic differentials Figure: Maryam Mirzakhani (1977 - 2017)

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

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

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

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

Riemann Surfaces Differentials Geometry of H Geometry of H g , n ( µ ) • My thesis is about the global geometry of H g , n ( µ ). • In Algebraic Geometry we have ways of measuring complexity. • We can understand H g , n ( µ ) when g is small!! Uniruled General type H g , n ( µ )

Riemann Surfaces Differentials Geometry of H Geometry of H g , n ( µ ) Theorem (–, 2018) For all partitions µ of 2 g − 2, the moduli spaces H g , n ( µ ) are uniruled when g ≤ 11.

Riemann Surfaces Differentials Geometry of H Geometry of H g , n ( µ ) For a given g , there are many H g , n ( µ )’s.

Riemann Surfaces Differentials Geometry of H Geometry of H g , n ( µ ) For a given g , there are many H g , n ( µ )’s.