Can you compute a conformal map? Ilia Binder University of Toronto - PowerPoint PPT Presentation

Can you compute a conformal map? Ilia Binder University of Toronto Based on joint work with M. Braverman (Princeton), C. Rojas (Universidad Andres Bello), and M. Yampolsky (University of Toronto) Modern Aspects of Complex Analysis and Its Applications August 20, 2019

How to compute a conformal map? 2

Computability: a crash course Inside the domain: computability and complexity Boundary behaviour: Caratheodory extension Boundary behaviour: harmonic measure

Computability of natural numbers and functions on natural numbers A function f : N → N is called computable if there exists an algorithm (a Turing machine) which, upon input n , outputs f ( n ) . 1

Computability of natural numbers and functions on natural numbers A function f : N → N is called computable if there exists an algorithm (a Turing machine) which, upon input n , outputs f ( n ) . A set E ⊆ N is called computable if its characteristic function is computable. 1

Computability of natural numbers and functions on natural numbers A function f : N → N is called computable if there exists an algorithm (a Turing machine) which, upon input n , outputs f ( n ) . A set E ⊆ N is called computable if its characteristic function is computable. An explicit example of non-computable set: Halting set H : i ∈ H iff the i -th Turing machine halts. 1

Computability of natural numbers and functions on natural numbers A function f : N → N is called computable if there exists an algorithm (a Turing machine) which, upon input n , outputs f ( n ) . A set E ⊆ N is called computable if its characteristic function is computable. An explicit example of non-computable set: Halting set H : i ∈ H iff the i -th Turing machine halts. A set E ⊆ N is called lower-computable if there exists an algorithm that enumerates E , i.e. on an input n it halts if n ∈ E , and never halts otherwise. 1

Computability of natural numbers and functions on natural numbers A function f : N → N is called computable if there exists an algorithm (a Turing machine) which, upon input n , outputs f ( n ) . A set E ⊆ N is called computable if its characteristic function is computable. An explicit example of non-computable set: Halting set H : i ∈ H iff the i -th Turing machine halts. A set E ⊆ N is called lower-computable if there exists an algorithm that enumerates E , i.e. on an input n it halts if n ∈ E , and never halts otherwise. The algorithm can verify the inclusion n ∈ E , but not the inclusion n ∈ E c . 1

Computability of natural numbers and functions on natural numbers A function f : N → N is called computable if there exists an algorithm (a Turing machine) which, upon input n , outputs f ( n ) . A set E ⊆ N is called computable if its characteristic function is computable. An explicit example of non-computable set: Halting set H : i ∈ H iff the i -th Turing machine halts. A set E ⊆ N is called lower-computable if there exists an algorithm that enumerates E , i.e. on an input n it halts if n ∈ E , and never halts otherwise. The algorithm can verify the inclusion n ∈ E , but not the inclusion n ∈ E c . Halting set is a classical example of a lower-computable non computable set. 1

Computability of natural numbers and functions on natural numbers A function f : N → N is called computable if there exists an algorithm (a Turing machine) which, upon input n , outputs f ( n ) . A set E ⊆ N is called computable if its characteristic function is computable. An explicit example of non-computable set: Halting set H : i ∈ H iff the i -th Turing machine halts. A set E ⊆ N is called lower-computable if there exists an algorithm that enumerates E , i.e. on an input n it halts if n ∈ E , and never halts otherwise. The algorithm can verify the inclusion n ∈ E , but not the inclusion n ∈ E c . Halting set is a classical example of a lower-computable non computable set. A complement of lower-computable set is called upper-computable . 1

Computability of natural numbers and functions on natural numbers A function f : N → N is called computable if there exists an algorithm (a Turing machine) which, upon input n , outputs f ( n ) . A set E ⊆ N is called computable if its characteristic function is computable. An explicit example of non-computable set: Halting set H : i ∈ H iff the i -th Turing machine halts. A set E ⊆ N is called lower-computable if there exists an algorithm that enumerates E , i.e. on an input n it halts if n ∈ E , and never halts otherwise. The algorithm can verify the inclusion n ∈ E , but not the inclusion n ∈ E c . Halting set is a classical example of a lower-computable non computable set. A complement of lower-computable set is called upper-computable . A set is computable iff it is simultaneously upper- and lower-computable. 1

Computability of reals and functions x ∈ R is called • computable if there is a computable function f : N → Q such that | f ( n ) − x | < 2 − n ; • lower-computable if there is a computable function f : N → Q such that f ( n ) ↑ x ; • upper-computable if there is a computable function f : N → Q such that f ( n ) ↓ x. 2

Computability of reals and functions x ∈ R is called • computable if there is a computable function f : N → Q such that | f ( n ) − x | < 2 − n ; • lower-computable if there is a computable function f : N → Q such that f ( n ) ↑ x ; • upper-computable if there is a computable function f : N → Q such that f ( n ) ↓ x. A function φ : N → Q 2 is an oracle for x ∈ C if | φ ( n ) − x | < 2 − n . 2

Computability of reals and functions x ∈ R is called • computable if there is a computable function f : N → Q such that | f ( n ) − x | < 2 − n ; • lower-computable if there is a computable function f : N → Q such that f ( n ) ↑ x ; • upper-computable if there is a computable function f : N → Q such that f ( n ) ↓ x. A function φ : N → Q 2 is an oracle for x ∈ C if | φ ( n ) − x | < 2 − n . A function f : S → C ( S ⊂ C ) is called computable if there exists an algorithm with an oracle for x ∈ S and an input n ∈ N which outputs a rational point s n such that | s n − f ( x ) | < 2 − n . An algorithm may query an oracle by reading the values of the function φ for an arbitrary m ∈ N . 2

Computability of planar sets. Let B n be an enumeration of all planar rational balls. Lower-computable open set. An open set U is called lower-computable if there is a computable function f : N → N such that U = � n ∈ N B f ( n ) . 3

Computability of planar sets. Let B n be an enumeration of all planar rational balls. Lower-computable open set. An open set U is called lower-computable if there is a computable function f : N → N such that U = � n ∈ N B f ( n ) . Lower-computable closed set. Closed set K is called lower computable if the set S ⊂ N with i ∈ S iff B i ∩ K � = ∅ is lower-computable. 3

Computability of planar sets. Let B n be an enumeration of all planar rational balls. Lower-computable open set. An open set U is called lower-computable if there is a computable function f : N → N such that U = � n ∈ N B f ( n ) . Lower-computable closed set. Closed set K is called lower computable if the set S ⊂ N with i ∈ S iff B i ∩ K � = ∅ is lower-computable. A compact is computable iff it and its complement are lower-computable. 3

Computability of planar sets. Let B n be an enumeration of all planar rational balls. Lower-computable open set. An open set U is called lower-computable if there is a computable function f : N → N such that U = � n ∈ N B f ( n ) . Lower-computable closed set. Closed set K is called lower computable if the set S ⊂ N with i ∈ S iff B i ∩ K � = ∅ is lower-computable. A compact is computable iff it and its complement are lower-computable. Equivalently, K ⊂ R d is computable if there exists an algorithm A with a single input n ∈ N which outputs a finite set C n of points with rational coordinates (or a rational polygon) such that Hdist( C n , K ) < 2 − n . 3

Computability of planar sets. Let B n be an enumeration of all planar rational balls. Lower-computable open set. An open set U is called lower-computable if there is a computable function f : N → N such that U = � n ∈ N B f ( n ) . Lower-computable closed set. Closed set K is called lower computable if the set S ⊂ N with i ∈ S iff B i ∩ K � = ∅ is lower-computable. A compact is computable iff it and its complement are lower-computable. Equivalently, K ⊂ R d is computable if there exists an algorithm A with a single input n ∈ N which outputs a finite set C n of points with rational coordinates (or a rational polygon) such that Hdist( C n , K ) < 2 − n . � � Hdist( X, Y ) = max sup x ∈ X dist( x, Y ) , sup y ∈ Y dist( y, X ) . 3

Computing the Riemann map Let Ω � C be a simply connected planar domain with w 0 ∈ Ω . g = g Ω ,w 0 is the unique conformal map of Ω onto the unit disk D with g ( w 0 ) = 0 , g ′ ( w 0 ) > 0 . f := g − 1 . 4

Computing the Riemann map Let Ω � C be a simply connected planar domain with w 0 ∈ Ω . g = g Ω ,w 0 is the unique conformal map of Ω onto the unit disk D with g ( w 0 ) = 0 , g ′ ( w 0 ) > 0 . f := g − 1 . Constructive Riemann Mapping Theorem.(Hertling, 1997) The following are equivalent: (i) Ω is a lower-computable open set, ∂ Ω is a lower-computable closed set, and w 0 ∈ Ω is a computable point; (ii) The maps g and f are both computable conformal bijections. 4