Undecidability In Group Theory, Topology, And Analysis
Undecidability ingroup theory,topology, andanalysisBjorn PoonenGroup theoryUndecidability in group theory, topology, andanalysisBjorn PoonenF.p. groupsWord problemMarkov propertiesTopologyFundamental groupHomeomorphismproblemManifold?Knot theoryAnalysisInequalitiesComplex analysisIntegrationRademacher Lecture 2November 7, 2017
Group theoryUndecidability ingroup theory,topology, andanalysisBjorn PoonenQuestionCan a computer decide whethertwo given elements of a group are equal?Group theoryF.p. groupsWord problemMarkov propertiesTopologyFundamental groupHomeomorphismproblemManifold?Knot theoryAnalysisInequalitiesComplex analysisIntegration
Group theoryUndecidability ingroup theory,topology, andanalysisBjorn PoonenQuestionCan a computer decide whethertwo given elements of a group are equala given element of a group equals the identity?Group theoryF.p. groupsWord problemMarkov propertiesTopologyFundamental groupHomeomorphismproblemManifold?Knot theoryAnalysisInequalitiesComplex analysisIntegration
Group theoryUndecidability ingroup theory,topology, andanalysisBjorn PoonenQuestionCan a computer decide whethertwo given elements of a group are equala given element of a group equals the identity?To make sense of this question, we must specify1. how the group is described2. how the element is describedThe descriptions should be suitable for input into a Turingmachine.Group theoryF.p. groupsWord problemMarkov propertiesTopologyFundamental groupHomeomorphismproblemManifold?Knot theoryAnalysisInequalitiesComplex analysisIntegration
Group theoryUndecidability ingroup theory,topology, andanalysisBjorn PoonenQuestionCan a computer decide whethertwo given elements of a group are equala given element of a group equals the identity?To make sense of this question, we must specify1. how the group is described: f.p. group2. how the element is described: wordThe descriptions should be suitable for input into a Turingmachine.Group theoryF.p. groupsWord problemMarkov propertiesTopologyFundamental groupHomeomorphismproblemManifold?Knot theoryAnalysisInequalitiesComplex analysisIntegration
Example: The symmetric group S3Undecidability ingroup theory,topology, andanalysisBjorn PoonenGroup theoryF.p. groupsWord problemMarkov propertiesTopologyFundamental groupHomeomorphismproblemManifold?Knot theoryAnalysisIn cycle notation, r (123) and t (12). These satisfyr 3 1,t 2 1,trt 1 r 1It turns out that r and t generate S3 , and every relationinvolving them is a consequence of the relations above:S3 hr , t r 3 1, t 2 1, trt 1 r 1 i.InequalitiesComplex analysisIntegration
Finitely presented groupsUndecidability ingroup theory,topology, andanalysisBjorn PoonenDefinitionGroup theoryAn f.p. group is a group specified by finitely manygenerators and finitely many relations.F.p. groupsWord problemMarkov propertiesTopologyFundamental groupHomeomorphismproblemManifold?Knot theoryExampleZ Z ha, b ab baiExampleThe free group on 2 (noncommuting) generators isF2 : ha, b iAnalysisInequalitiesComplex analysisIntegration
Representing elements of an f.p. group: wordsUndecidability ingroup theory,topology, andanalysisBjorn PoonenGroup theoryF.p. groupsWord problemMarkov propertiesTopologyS3 hr , t r 3 1, t 2 1, trt 1 r 1 i.DefinitionA word is a sequence of the generator symbols and theirinverses, such astr 1 ttrt 1 rrr .Since r and t generate S3 , every element of S3 is representedby a word, but not necessarily in a unique way.ExampleThe words tr and r 1 t both represent (23).Fundamental groupHomeomorphismproblemManifold?Knot theoryAnalysisInequalitiesComplex analysisIntegration
The word problemGiven an f.p. group G , we haveWord problem for GFind an algorithm withinput: a word w in the generators of Goutput: YES or NO, according to whether w 1 in G .Undecidability ingroup theory,topology, andanalysisBjorn PoonenGroup theoryF.p. groupsWord problemMarkov propertiesTopologyFundamental groupHomeomorphismproblemManifold?Knot theoryAnalysisHarder problem:Uniform word problemFind an algorithm withinput: an f.p. group G , and a word w in thegenerators of Goutput: YES or NO, according to whether w 1 in G .InequalitiesComplex analysisIntegration
Undecidability ingroup theory,topology, andanalysisWord problem for FnTheoremBjorn PoonenThe word problem for the free group Fn is decidable.Group theoryAlgorithm to decide whether a given word w represents 1:1. Repeatedly cancel adjacent inverses until there isnothing left to cancel.2. Check if the end result is the empty word.TopologyExampleAnalysisIn the free group F2 ha, bi, given the wordaba 1 bb 1 abb,cancellation leads toabbb,which is not the empty word,so aba 1 bb 1 abb does not represent the identity.F.p. groupsWord problemMarkov propertiesFundamental groupHomeomorphismproblemManifold?Knot theoryInequalitiesComplex analysisIntegration
Undecidability of the word problemUndecidability ingroup theory,topology, andanalysisBjorn PoonenTheorem (P. S. Novikov and Boone,independently in the 1950s)There exists an f.p. group G such that the word problem forG is undecidable.The strategy of the proof, as for Hilbert’s tenth problem, isto build a group G such that solving the word problem for Gis at least as hard as solving the halting problem.CorollaryThe uniform word problem is undecidable.Group theoryF.p. groupsWord problemMarkov propertiesTopologyFundamental groupHomeomorphismproblemManifold?Knot theoryAnalysisInequalitiesComplex analysisIntegration
Markov propertiesUndecidability ingroup theory,topology, andanalysisBjorn PoonenDefinitionA property of f.p. groups is called a Markov property if1. there exists an f.p. group G1 with the property, and2. there exists an f.p. group G2 that cannot be embeddedin any f.p. group with the property.Group theoryF.p. groupsWord problemMarkov propertiesTopologyFundamental groupHomeomorphismproblemManifold?Knot theoryAnalysisExampleThe property of being finite is a Markov property, because1. There exists a finite group!2. Z cannot be embedded in any finite group.Other Markov properties: trivial, abelian, free, . . . .InequalitiesComplex analysisIntegration
Theorem (Adian & Rabin 1955–1958)For each Markov property P, the problem of decidingwhether an arbitrary f.p. group has P is undecidable.Sketch of proof.Embed the uniform word problem in this P problem:Given an f.p. group G and a word w in its generators,build another f.p. group K such thatK has P w 1 in G .ExampleThere is no algorithm to decide whether an f.p. group istrivial.Undecidability ingroup theory,topology, andanalysisBjorn PoonenGroup theoryF.p. groupsWord problemMarkov propertiesTopologyFundamental groupHomeomorphismproblemManifold?Knot theoryAnalysisInequalitiesComplex analysisIntegration
Fundamental groupFix a manifold M.Undecidability ingroup theory,topology, andanalysisBjorn PoonenGroup theoryF.p. groupsWord problemMarkov propertiesTopologyFundamental groupHomeomorphismproblemManifold?Knot theoryAnalysisInequalitiesComplex analysisIntegration
Fundamental groupFix a manifold M and a point p.Undecidability ingroup theory,topology, andanalysisBjorn PoonenGroup theoryF.p. groupsWord problemMarkov propertiesTopologyFundamental groupHomeomorphismproblemManifold?Knot theoryAnalysisInequalitiesComplex analysisIntegration
Fundamental groupFix a manifold M and a point p.Consider paths in M that start and end at p.Undecidability ingroup theory,topology, andanalysisBjorn PoonenGroup theoryF.p. groupsWord problemMarkov propertiesTopologyFundamental groupHomeomorphismproblemManifold?Knot theoryAnalysisInequalitiesComplex analysisIntegration
Fundamental groupFix a manifold M and a point p.Consider paths in M that start and end at p.Paths are homotopic if one can be deformed to the other.Undecidability ingroup theory,topology, andanalysisBjorn PoonenGroup theoryF.p. groupsWord problemMarkov propertiesTopologyFundamental groupHomeomorphismproblemManifold?Knot theoryAnalysisInequalitiesComplex analysisIntegration
Fundamental groupFix a manifold M and a point p.Consider paths in M that start and end at p.Paths are homotopic if one can be deformed to the other.Fundamental group π1 (M) : {paths}/homotopy.Group law: concatenation of paths.Undecidability ingroup theory,topology, andanalysisBjorn PoonenGroup theoryF.p. groupsWord problemMarkov propertiesTopologyFundamental groupHomeomorphismproblemManifold?Knot theoryAnalysisInequalitiesComplex analysisIntegration
Examples of fundamental groupsUndecidability ingroup theory,topology, andanalysisBjorn PoonenGroup theoryF.p. groupsWord problemMarkov propertiesTopologyFundamental groupHomeomorphismproblemManifold?Knot theoryAnalysisπ1 (torus) Z Zπ1 (sphere) {1}This gives one way to prove that the torus and the sphereare not homeomorphic, i.e., that they do not have the sameshape even after stretching.InequalitiesComplex analysisIntegration
The homeomorphism problemUndecidability ingroup theory,topology, andanalysisBjorn PoonenGroup theoryQuestionGiven two manifolds, can one decide whether they arehomeomorphic?F.p. groupsWord problemMarkov propertiesTopologyFundamental groupHomeomorphismproblemManifold?Knot theoryAnalysisTo make sense of this question, we must specify how amanifold is described.This will be done using the notion of simplicial complex.InequalitiesComplex analysisIntegration
Simplicial complexesUndecidability ingroup theory,topology, andanalysisBjorn PoonenDefinitionGroup theoryRoughly speaking, a finite simplicial complex is a finite unionof simplices (points, segments, triangles, tetrahedra, . . . )together with data on how they are glued. The description ispurely combinatorial.TopologyExampleAnalysisThe icosahedron is a finite simplicial complexhomeomorphic to the 2-sphere S 2 .From now on, manifold means “compact manifoldrepresented by a particular finite simplicial complex”,so that it can be the input to a Turing machine.F.p. groupsWord problemMarkov propertiesFundamental groupHomeomorphismproblemManifold?Knot theoryInequalitiesComplex analysisIntegration
Undecidability of the homeomorphism problemTheorem (Markov 1958)The problem of deciding whether two manifolds arehomeomorphic is undecidable.Sketch of proof.Let n 5. Given an f.p. group G and a word w in itsgenerators, one can construct a n-manifold ΣG ,w such that1. If w 1 in G , then ΣG ,w S n .2. If w 6 1, then π1 (ΣG ,w ) is nontrivial (so ΣG ,w 6 S n ).Thus, if the homeomorphism problem were decidable, thenthe uniform word problem would be too. But it isn’t.In fact, the homeomorphism problem is known to bedecidable in dimensions 3, andundecidable in dimensions 4.Undecidability ingroup theory,topology, andanalysisBjorn PoonenGroup theoryF.p. groupsWord problemMarkov propertiesTopologyFundamental groupHomeomorphismproblemManifold?Knot theoryAnalysisInequalitiesComplex analysisIntegration
The previous proof showed that for n 5, the manifold S n isunrecognizable: the problem of deciding whether a givenn-manifold is homeomorphic to S n is undecidable.Theorem (S. P. Novikov 1974)Each n-manifold M with n 5 is unrecognizable.QuestionIs S 4 recognizable? (The answer is not known.)To explain the idea of the proof of the theorem, we need thenotion of connected sum.Undecidability ingroup theory,topology, andanalysisBjorn PoonenGroup theoryF.p. groupsWord problemMarkov propertiesTopologyFundamental groupHomeomorphismproblemManifold?Knot theoryAnalysisInequalitiesComplex analysisIntegration
Connected sumThe connected sum of n-manifolds M and N is then-manifold obtained by cutting a small disk out of each andconnecting them with a tube.Undecidability ingroup theory,topology, andanalysisBjorn PoonenGroup theoryF.p. groupsWord problemMarkov propertiesTopologyFundamental groupHomeomorphismproblemManifold?Knot theoryAnalysisInequalitiesComplex analysisIntegration
Undecidability ingroup theory,topology, andanalysisBjorn PoonenGroup theoryF.p. groupsWord problemMarkov propertiesTopologyFundamental groupHomeomorphismproblemManifold?Knot theoryAnalysisInequalitiesComplex analysisIntegration
Am I a manifold?TheoremIt is impossible to decide whether a finite simplicial complexis homeomorphic to a manifold.Proof.SΣG ,w : suspension over our possibly fake sphere ΣG ,w .If w 1 in G , then ΣG ,w S n , so SΣG ,w S n 1 .If w 6 1, then SΣG ,w is not locally Euclidean.Undecidability ingroup theory,topology, andanalysisBjorn PoonenGroup theoryF.p. groupsWord problemMarkov propertiesTopologyFundamental groupHomeomorphismproblemManifold?Knot theoryAnalysisInequalitiesComplex analysisIntegration
Knot theoryDefinitionA knot is an embedding of the circle S 1 in R3 .Undecidability ingroup theory,topology, andanalysisBjorn PoonenGroup theoryF.p. groupsWord problemMarkov propertiesTopologyFundamental groupHomeomorphismproblemManifold?Knot theoryAnalysisInequalitiesComplex analysisIntegrationDefinitionTwo knots are equivalent if one can be deformed into theother within R3 , without crossing itself.
From now on, knot means “a knot obtained by connecting afinite sequence of points in Q3 ”, so that it admits a finitedescription.Undecidability ingroup theory,topology, andanalysisBjorn PoonenGroup theoryF.p. groupsWord problemMarkov propertiesTopologyFundamental groupHomeomorphismproblemManifold?Knot theoryAnalysisInequalitiesComplex analysisIntegrationTheorem (Haken 1961 and Hemion 1979)There is an algorithm that takes as input two knots in R3and decides whether they are equivalent.
Higher-dimensional knotsUndecidability ingroup theory,topology, andanalysisBjorn PoonenThough the knot equivalence problem is decidable, ahigher-dimensional analogue is not:Group theoryTheorem (Nabutovsky & Weinberger 1996)TopologyIf n 3, the problem of deciding whether two embeddings ofS n in Rn 2 are equivalent is undecidable.F.p. groupsWord problemMarkov propertiesFundamental groupHomeomorphismproblemManifold?Knot theoryAnalysisInequalitiesComplex analysisIntegrationQuestionWhat about n 2? Not known.
Polynomial inequalitiesQuestionWhich of the following inequalities are true for all real valuesof the variables?22a b 2abUndecidability ingroup theory,topology, andanalysisBjorn PoonenGroup theoryF.p. groupsWord problemMarkov propertiesTopologyx 4 4x 5 0Fundamental groupHomeomorphismproblemManifold?Knot theoryAnalysisInequalitiesComplex analysisIntegration
Undecidability ingroup theory,topology, andanalysisPolynomial inequalitiesQuestionWhich of the following inequalities are true for all real valuesof the variables?22a b 2abTRUEBjorn PoonenGroup theoryF.p. groupsWord problemMarkov propertiesTopologyx 4 4x 5 0Fundamental groupHomeomorphismproblemManifold?Knot theoryAnalysisInequalitiesComplex analysisIntegration
Undecidability ingroup theory,topology, andanalysisPolynomial inequalitiesQuestionWhich of the following inequalities are true for all real valuesof the variables?22a b 2abTRUEBjorn PoonenGroup theoryF.p. groupsWord problemMarkov propertiesTopologyx 4 4x 5 0TRUEFundamental groupHomeomorphismproblemManifold?Knot theoryAnalysisInequalitiesComplex analysisIntegration
Undecidability ingroup theory,topology, andanalysisPolynomial inequalitiesQuestionWhich of the following inequalities are true for all real valuesof the variables?22a b 2abTRUEBjorn PoonenGroup theoryF.p. groupsWord problemMarkov propertiesTopologyx 4 4x 5 0TRUE536x 287196896 210y 287196896 777x 3 y 16 z 4732987 1111x 54987896 2823y 927396 27x 94572 y 9927 z 999 936718x 726896 887236y 726896 9x 24572 y 7827 z 13 89790876x 26896 30y 26896 987x 245 y 6 z 6876 9823709709790790x 28 1987y 28 1467890461986x 2 y 6 z 4 80398600x 2 z 12 27980186xy 3789720156y 2 9328769x 1956820y 275893249827098790768645846898z 389?Fundamental groupHomeomorphismproblemManifold?Knot theoryAnalysisInequalitiesComplex analysisIntegration
Undecidability ingroup theory,topology, andanalysisPolynomial inequalitiesQuestionWhich of the following inequalities are true for all real valuesof the variables?22a b 2abTRUEBjorn PoonenGroup theoryF.p. groupsWord problemMarkov propertiesTopologyx 4 4x 5 0Fundamental groupHomeomorphismproblemManifold?Knot theoryTRUE536x 287196896 210y 287196896 777x 3 y 16 z 4732987 1111x 54987896 2823y 927396 27x 94572 y 9927 z 999 936718x 726896 887236y 726896 9x 24572 y 7827 z 13 89790876x 26896 30y 26896 987x 245 y 6 z 6876 9823709709790790x 28 1987y 28 1467890461986x 2 y 6 z 4 80398600x 2 z 12 27980186xy 3789720156y 2 9328769x 1956820y 275893249827098790768645846898z 389?FALSEAnalysisInequalitiesComplex analysisIntegration
Polynomial inequalities, continuedUndecidability ingroup theory,topology, andanalysisBjorn PoonenQuestionCan a computer decide, given a polynomial inequalityf (x1 , . . . , xn ) 0with rational coefficients, whether it is true for all realnumbers x1 , . . . , xn ?Group theoryF.p. groupsWord problemMarkov propertiesTopologyFundamental groupHomeomorphismproblemManifold?Knot theoryAnalysisInequalitiesComplex analysisIntegration
Polynomial inequalities, continuedUndecidability ingroup theory,topology, andanalysisBjorn PoonenQuestionCan a computer decide, given a polynomial inequalityf (x1 , . . . , xn ) 0with rational coefficients, whether it is true for all realnumbers x1 , . . . , xn ?Group theoryF.p. groupsWord problemMarkov propertiesTopologyFundamental groupHomeomorphismproblemManifold?Knot theoryAnalysisYES! (Tarski 1951) More generally, it can decide the truth ofany first-order sentence involving polynomial inequalities.How? For example, how could it decide whether a given setdefined by a Boolean combination of inequalities is empty?InequalitiesComplex analysisIntegration
Inequalities: induction on the number of variablesUndecidability ingroup theory,topology, andanalysisBjorn Poonenx2 y2 1Group theoryF.p. groupsWord problemMarkov propertiesTopologyFundamental groupHomeomorphismproblemManifold?Knot theoryx 1 x 1AnalysisInequalitiesComplex analysisIntegrationIn general, the projection (x1 , . . . , xn ) 7 (x1 , . . . , xn 1 )maps a set S defined by an explicit Booleancombination of inequalities to another such set S 0 .S 6 if and only if S 0 6 .Keep projecting until only 1 variable is left; then usecalculus.
Exponential inequalitiesCan a computer decide the truth of inequalities likeUndecidability ingroup theory,topology, andanalysisBjorn Poonenee x y 20 5x 4y ?Group theoryF.p. groupsWord problemMarkov propertiesTopologyFundamental groupHomeomorphismproblemManifold?Knot theoryAnalysisInequalitiesComplex analysisIntegration
Exponential inequalitiesCan a computer decide the truth of inequalities likeUndecidability ingroup theory,topology, andanalysisBjorn Poonenee x y 20 5x 4y ?13396?143This should be easy: compute both sides to high precision,but. . .Warmup: What about e e3/2 e 5/3 Group theoryF.p. groupsWord problemMarkov propertiesTopologyFundamental groupHomeomorphismproblemManifold?Knot theoryAnalysisInequalitiesComplex analysisIntegration
Exponential inequalitiesCan a computer decide the truth of inequalities likeUndecidability ingroup theory,topology, andanalysisBjorn Poonenee x y 20 5x 4y ?13396?143This should be easy: compute both sides to high precision,but. . .Warmup: What about e e3/2 e 5/3 What if they turn out to be exactly equal?Schanuel’s conjecture in transcendental number theorypredicts that “coincidences” like these never occur, but ithas not been proved.Theorem (Macintyre and Wilkie)If Schanuel’s conjecture is true, then exponential inequalitiesin any number of variables are decidable.Group theoryF.p. groupsWord problemMarkov propertiesTopologyFundamental groupHomeomorphismproblemManifold?Knot theoryAnalysisInequalitiesComplex analysisIntegration
Trigonometric inequalitiesQuestionCan a computer decide the truth of inequalities involvingexpressions built up from x and sin x?Undecidability ingroup theory,topology, andanalysisBjorn PoonenGroup theoryF.p. groupsWord problemMarkov propertiesTopologyFundamental groupHomeomorphismproblemManifold?Knot theoryAnalysisInequalitiesComplex analysisIntegration
Undecidability ingroup theory,topology, andanalysisTrigonometric inequalitiesQuestionBjorn PoonenCan a computer decide the truth of inequalities involvingexpressions built up from x and sin x?NO! (Richardson 1968)Group theoryF.p. groupsWord problemMarkov propertiesTopologyIdea: Let p, L Z[x1 , . . . , xn ] be such that L( x )p( x )2 .f ( x ) : 1 4p( x )2 L( x )(sin2 πx1 · · · sin2 πxn ).Fundamental groupHomeomorphismproblemManifold?Knot theoryAnalysisInequalitiesComplex analysisIntegrationIf f ( x ) 0, thensin2 πxi 0, so xi is very close to an integer ai , andp( x ) 1/2, which forces p(a1 , . . . , an ) 0Conclusion:f 0 somewhere p( x ) 0 has an integer solution(undecidable)
Inequalities in one variableUndecidability ingroup theory,topology, andanalysisBjorn PoonenQuestionCan a computer at least decide the truth of trigonometricinequalities in one variable?Group theoryF.p. groupsWord problemMarkov propertiesTopologyFundamental groupHomeomorphismproblemManifold?Knot theoryAnalysisInequalitiesComplex analysisIntegration
Inequalities in one variableUndecidability ingroup theory,topology, andanalysisBjorn PoonenQuestionGroup theoryCan a computer at least decide the truth of trigonometricinequalities in one variable?NO! In fact, the one-variable inequality problem is just ashard as the many-variable inequality problem.The proof uses the parametrized curveG (t) : (t sin t, t sin t 3 ).What does this curve in R2 look like?F.p. groupsWord problemMarkov propertiesTopologyFundamental groupHomeomorphismproblemManifold?Knot theoryAnalysisInequalitiesComplex analysisIntegration
As t ranges over real numbers,G (t) : (t sin t, t sin t 3 )traces outUndecidability ingroup theory,topology, andanalysisBjorn PoonenGroup theoryF.p. groupsWord problemMarkov propertiesTopologyFundamental groupHomeomorphismproblemManifold?Knot theoryAnalysisInequalitiesComplex analysisIntegration
As t ranges over real numbers,G (t) : (t sin t, t sin t 3 )traces outUndecidability ingroup theory,topology, andanalysisBjorn PoonenGroup theoryF.p. groupsWord problemMarkov propertiesTopologyFundamental groupHomeomorphismproblemManifold?Knot theoryAnalysisInequalitiesComplex analysisIntegrationFor a continuous function f (x, y ),f (x, y ) 0 on R2 f (G (t)) 0 for all t
Equality of functionsUndecidability ingroup theory,topology, andanalysisBjorn PoonenBad news for automated homework checkers:TheoremIt is impossible for a computer to decide,given two functions built out of x, sin x, ,whether they are equal.Group theoryF.p. groupsWord problemMarkov propertiesTopologyFundamental groupHomeomorphismproblemManifold?Knot theoryAnalysisInequalitiesComplex analysisIntegrationProof: If you can’t decide whether f (x) 0, then you can’tdecide whether f (x) and f (x) are the same function.
Undecidability ingroup theory,topology, andanalysisComplex analysisBjorn PoonenExampleGroup theoryDoesF.p. groupsWord problemMarkov propertiese z w 3 5z 4ew24 w 3z 74w z 9 z 5 2.have a solution in complex numbers z and w ?QuestionCan a computer decide whether a system of equationsinvolving the complex exponential function has a complexsolution?TopologyFundamental groupHomeomorphismproblemManifold?Knot theoryAnalysisInequalitiesComplex analysisIntegration
Complex analysisQuestionCan a computer decide whether a system of equationsinvolving the complex exponential function has a complexsolution?NO! (Adler 1969)Proof: The 3 steps below characterize Z in C by equations:1. 2πiZnis the set of solutions to e zo 1a: a, b 2πiZ and b 6 02. Q b3. Z is the set of q Q such that 2q Q; thusZ : {q Q : z C such that e z 2 and e qz Q}.ThusHilbert’s tenth problem the complex analysis problem.Hilbert’s tenth problem is undecidable,so the complex analysis problem is undecidable.Undecidability ingroup theory,topology, andanalysisBjorn PoonenGroup theoryF.p. groupsWord problemMarkov propertiesTopologyFundamental groupHomeomorphismproblemManifold?Knot theoryAnalysisInequalitiesComplex analysisIntegration
IntegrationQuestionCan a computer, given an explicit function f (x),R1. decide whether there is a formula for f (x) dx,2. and if so, find it?Undecidability ingroup theory,topology, andanalysisBjorn PoonenGroup theoryF.p. groupsWord problemMarkov propertiesTopologyFundamental groupHomeomorphismproblemManifold?Knot theoryAnalysisInequalitiesComplex analysisIntegration
IntegrationQuestionCan a computer, given an explicit function f (x),R1. decide whether there is a formula for f (x) dx,2. and if so, find it?Theorem (Risch)YES.Undecidability ingroup theory,topology, andanalysisBjorn PoonenGroup theoryF.p. groupsWord problemMarkov propertiesTopologyFundamental groupHomeomorphismproblemManifold?Knot theoryAnalysisInequalitiesComplex analysisIntegration
IntegrationQuestionCan a computer, given an explicit function f (x),R1. decide whether there is a formula for f (x) dx,2. and if so, find it?Theorem (Risch)YES.Theorem (Richardson)NO.Undecidability ingroup theory,topology, andanalysisBjorn PoonenGroup theoryF.p. groupsWord problemMarkov propertiesTopologyFundamental groupHomeomorphismproblemManifold?Knot theoryAnalysisInequalitiesComplex analysisIntegration
IntegrationQuestionCan a computer, given an explicit function f (x),R1. decide whether there is a formula for f (x) dx,2. and if so, find it?Theorem (Risch)YES.Theorem (Richardson)NO.Another answer: MAYBE; it’s not known yet.Undecidability ingroup theory,topology, andanalysisBjorn PoonenGroup theoryF.p. groupsWord problemMarkov propertiesTopologyFundamental groupHomeomorphismproblemManifold?Knot theoryAnalysisInequalitiesComplex analysisIntegration
Undecidability ingroup theory,topology, andanalysisIntegrationQuestionBjorn PoonenCan a computer, given an explicit function f (x),R1. decide whether there is a formula for f (x) dx,2. and if so, find it?Theorem (Risch)Group theoryF.p. groupsWord problemMarkov propertiesTopologyFundamental groupHomeomorphismproblemManifold?Knot theoryAnalysisYES.InequalitiesComplex analysisIntegrationTheorem (Richardson)NO.Another answer: MAYBE; it’s not known yet.All of these answers are correct!
analysis Bjorn Poonen Group theory F.p. groups Word problem Markov properties Topology Fundamental group Homeomorphism problem Manifold? Knot theory Analysis Inequalities Complex analysis Integration Group theory Question Can a computer decide whether two given elements of a group are equal?
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