Undecidability everywhere Wang tiles Integer matrices 3 x + 1 - PowerPoint PPT Presentation

Undecidability everywhere Bjorn Poonen Two kinds of undecidability Integration Undecidability everywhere Wang tiles Integer matrices 3 x + 1 problem Bjorn Poonen Algebraic geometry Varieties Isomorphism problem Automorphisms Commutative algebra Rademacher Lecture 3 F.g. algebras F.g. fields November 8, 2017 Noncommutative algebra Games Abstract games Chess

Undecidability Undecidability of a single question? everywhere So far, we’ve been considering families of questions with Bjorn Poonen YES/NO answers, and we wanted to know if there is a Two kinds of undecidability computer program that gets the right answer on all of them. Integration Question Wang tiles Can a single question be undecidable? Integer matrices 3 x + 1 problem Algebraic geometry Example Varieties Isomorphism problem Could the Riemann hypothesis be undecidable? Automorphisms Commutative algebra F.g. algebras F.g. fields Noncommutative algebra Games Abstract games Chess

Undecidability Undecidability of a single question? everywhere So far, we’ve been considering families of questions with Bjorn Poonen YES/NO answers, and we wanted to know if there is a Two kinds of undecidability computer program that gets the right answer on all of them. Integration Question Wang tiles Can a single question be undecidable? Integer matrices 3 x + 1 problem Algebraic geometry Example Varieties Isomorphism problem Could the Riemann hypothesis be undecidable? Automorphisms Commutative Answer: Not in the sense we’ve been considering, because algebra F.g. algebras there is a computer program that correctly answers the F.g. fields Noncommutative question algebra Games Is the Riemann hypothesis true? Abstract games Chess

Undecidability Undecidability of a single question? everywhere So far, we’ve been considering families of questions with Bjorn Poonen YES/NO answers, and we wanted to know if there is a Two kinds of undecidability computer program that gets the right answer on all of them. Integration Question Wang tiles Can a single question be undecidable? Integer matrices 3 x + 1 problem Algebraic geometry Example Varieties Isomorphism problem Could the Riemann hypothesis be undecidable? Automorphisms Commutative Answer: Not in the sense we’ve been considering, because algebra F.g. algebras there is a computer program that correctly answers the F.g. fields Noncommutative question algebra Games Is the Riemann hypothesis true? Abstract games Chess Program 1: PRINT “YES” Program 2: PRINT “NO”

Undecidability Independence everywhere Bjorn Poonen But it could be that Two kinds of undecidability neither the Riemann hypothesis nor its negation is provable Integration (within the ZFC axiom system, say). Wang tiles Integer matrices In that case, one would say 3 x + 1 problem Algebraic geometry “The Riemann hypothesis is independent of ZFC.” Varieties Isomorphism problem Automorphisms Commutative Example algebra F.g. algebras The continuum hypothesis, that there is no set S such that F.g. fields Noncommutative algebra # N < # S < # R , Games Abstract games is independent of ZFC (G¨ odel 1940, Cohen 1963). Chess (The fine print: we’re assuming that ZFC is consistent.)

Undecidability Undecidability vs. independence everywhere If a family of problems is undecidable, at least one instance Bjorn Poonen is independent of ZFC. For example, Two kinds of undecidability Theorem Integration There exists a polynomial p such that the statement Wang tiles Integer matrices ∃ x 1 , . . . , x n ∈ Z such that p ( x 1 , . . . , x n ) = 0 3 x + 1 problem Algebraic geometry is independent of ZFC, neither provable nor disprovable. Varieties Isomorphism problem (The fine print: we’re assuming that ZFC is consistent and that Automorphisms ZFC theorems about integers are true.) Commutative algebra F.g. algebras Proof. F.g. fields Noncommutative Suppose that each such statement were either provable or algebra disprovable. Then Hilbert’s tenth problem is solvable: search Games Abstract games for a proof by day, and for a disproof by night; stop when Chess one or the other is found! There is a different proof that is constructive —one can write down a specific polynomial with this property! (Post 1944)

Undecidability Integration everywhere Bjorn Poonen Question Two kinds of undecidability Can a computer, given an explicit function f ( x ) , Integration � 1. decide whether there is a formula for f ( x ) dx, Wang tiles 2. and if so, find it? Integer matrices 3 x + 1 problem Algebraic geometry Varieties Isomorphism problem Automorphisms Commutative algebra F.g. algebras F.g. fields Noncommutative algebra Games Abstract games Chess

Undecidability Integration everywhere Bjorn Poonen Question Two kinds of undecidability Can a computer, given an explicit function f ( x ) , Integration � 1. decide whether there is a formula for f ( x ) dx, Wang tiles 2. and if so, find it? Integer matrices 3 x + 1 problem Algebraic geometry Theorem (Risch) Varieties Isomorphism problem Automorphisms YES. Commutative algebra F.g. algebras F.g. fields Noncommutative algebra Games Abstract games Chess

Undecidability Integration everywhere Bjorn Poonen Question Two kinds of undecidability Can a computer, given an explicit function f ( x ) , Integration � 1. decide whether there is a formula for f ( x ) dx, Wang tiles 2. and if so, find it? Integer matrices 3 x + 1 problem Algebraic geometry Theorem (Risch) Varieties Isomorphism problem Automorphisms YES. Commutative algebra Theorem (Richardson) F.g. algebras F.g. fields Noncommutative NO. algebra Games Abstract games Chess

Undecidability Integration everywhere Bjorn Poonen Question Two kinds of undecidability Can a computer, given an explicit function f ( x ) , Integration � 1. decide whether there is a formula for f ( x ) dx, Wang tiles 2. and if so, find it? Integer matrices 3 x + 1 problem Algebraic geometry Theorem (Risch) Varieties Isomorphism problem Automorphisms YES. Commutative algebra Theorem (Richardson) F.g. algebras F.g. fields Noncommutative NO. algebra Another answer: MAYBE; it’s not known yet. Games Abstract games Chess

Undecidability Integration everywhere Bjorn Poonen Question Two kinds of undecidability Can a computer, given an explicit function f ( x ) , Integration � 1. decide whether there is a formula for f ( x ) dx, Wang tiles 2. and if so, find it? Integer matrices 3 x + 1 problem Algebraic geometry Theorem (Risch) Varieties Isomorphism problem Automorphisms YES. Commutative algebra Theorem (Richardson) F.g. algebras F.g. fields Noncommutative NO. algebra Another answer: MAYBE; it’s not known yet. Games Abstract games Chess All of these answers are correct!

Undecidability Elementary functions everywhere Bjorn Poonen Warmup question � e x 2 dx exist? Two kinds of Does undecidability Integration Wang tiles Yes, but Liouville proved in 1835 that it cannot be Integer matrices represented by an elementary formula. 3 x + 1 problem What does elementary mean? Algebraic geometry Varieties Isomorphism problem Automorphisms Example Commutative algebra √ � x 3 + log x 2 + 2 e x F.g. algebras 3 F.g. fields x + √ e x + log x is elementary. Noncommutative algebra Games In general: any function that can be built up from constants Abstract games Chess and x by arithmetic operations, adjoining roots of polynomials whose coefficients are previously constructed functions, and adjoining e f or log f for previously constructed functions f .

Undecidability YES: Risch’s algorithm for integration everywhere Bjorn Poonen Question Two kinds of undecidability Can a computer decide, given an elementary function f , Integration whether it has an elementary antiderivative? Wang tiles Integer matrices MAYBE: This runs into sticky questions about constants : 3 x + 1 problem � � e e 3 / 2 + e 5 / 3 − 13396 � e x 2 dx elementary? e.g., is Algebraic geometry 143 Varieties Isomorphism problem Automorphisms Theorem (Risch) Commutative algebra Let K be a field of functions built up from constants whose F.g. algebras F.g. fields algebraic relations are known by adjoining x, by making Noncommutative finite extensions, and by adjoining functions e f and log f algebra Games such that the field of constants does not grow. Abstract games Then a computer can decide, given f ∈ K, Chess � whether f is elementary (and can compute it if so).

Undecidability NO: Undecidability of integration everywhere Bjorn Poonen Theorem (Richardson) Two kinds of undecidability If one enlarges the class of elementary functions by including Integration | | among the building blocks, then there is no algorithm for Wang tiles deciding whether an elementary function has an elementary Integer matrices antiderivative. 3 x + 1 problem Algebraic geometry Sketch of proof. Varieties Isomorphism problem Automorphisms Using undecidability of trigonometric inequalities, and using Commutative | | , build a function g ( x ) that is either 0 everywhere, or that algebra F.g. algebras is 1 on some interval, but such that we can’t tell which. F.g. fields Then it is impossible to decide whether Noncommutative algebra g ( x ) e x 2 dx Games � Abstract games Chess is elementary.