Poonens result reduces the complexity of the formula needed to select the integers from. After the talk we will also go to lunch with the speaker. This undecidability result generalises to other number theoretic objects. Undecidability in number theory bjorn poonen h10 polynomial equations hilberts 10th problem diophantine sets listable sets dprm theorem consequences of dprm primeproducing polynomials riemann hypothesis related problems h10 over ok. Undecidability in number theory bjorn poonen h10 polynomial equations hilberts 10th problem diophantine sets listable sets dprm theorem consequences of dprm. Maz94,poo08, the books mat93,dlpvg00,shl07, the website vse, and the movie csi08. H10 was proved by relating it to undecidability results in logic and computability theory from the 1930s. A new method for undecidability proofs of first order theories. Assume that the number of problems can be in oneone correspondence with positive integers. First book i was suggested is introduction to automata theory, languages, and computation by hopcroft, motwani and ullman. Free computation theory books download ebooks online. I am part of the simons collaboration on arithmetic geometry, number theory, and computation. We refer the reader to8for an update on the subject, and to 5 for a general survey on undecidability in number theory. This is a free textbook for an undergraduate course on the theory of computation, which have been teaching at carleton university since 2002.
We construct an infinite family of totally real algebraic extensions of. Nov 08, 2017 undecidable problems have arisen in nearly every major branch of mathematics. A well written book just about the proof is godels proof by nagel and newman. The institute is located at 17 gauss way, on the university of california, berkeley campus, close to grizzly peak, on the. Legendary math writing there are a number of urban legends about various written works in math shortest, longest, funny titles, etc. A beautifully written book on the subject is incompleteness by rebecca goldstein. An introduction to logic and computability is an introduction to the fundamental concepts of contemporary logic those of a proof, a computable function, a model and a set. The ringtheories of qp p is a prime number are decidable results of. Hilberts 10th problem for the ring of integers of a number field that is, a finite exten sion of q. Undecidability in number theory department of mathematics. Today, pure and applied number theory is an exciting mix of simultaneously broad and deep theory, which is constantly informed and motivated.
Undecidability everywhere department of mathematics. Bjorn mikhail poonen is a mathematician, fourtime putnam competition winner and currently the claude shannon professor of mathematics at mit. This chapter lays the foundations for our study of the theory of numbers by weaving together the themes of prime numbers, integer factorization, and the distribution of primes. Autumn 2011 volume 6 massachusetts institute of technology. Poonen is a 1985 alumnus of winchester high school in winchester, massachusetts. Bjorn poonen has many interesting papers on this, see this mo question.
Bjorn poonen received the maa chauvenet prize for his article undecidability in number theory, notices of the ams, 55 2008. Poonen s research focuses mainly on number theory and algebraic geometry. His research is primarily in arithmetic geometry, but he has occasionally published in other subjects such as probability and computer science. Poonen strikes a balance between technicality for the experts and accessibility for the rest of us. Undecidability from todays lecture wont be on the final. In the context of computability theory, to show that acfg is decidable it is. Undecidability in group theory, analysis, and topology. Some lesser known open problems conjectures in number theory. He has received the packard, rosenbaum, and sloan fellowships as well as a miller professorship. This lecture will present a sampling of these and will discuss a few problems whose undecidability status is not yet known, including one about chess. His article hilberts tenth problem over rings of numbertheoretic interest is a. The final version of this survey is published as b. The tas and i will hold office hours during reading period, and up to the day of the day of the exam.
Knot theory analysis inequalities complex analysis integration fundamental group fix a manifold m. So thats not in my lifetime, but a lot of current peoples. Mathematisches forschungsinstitut oberwolfach definability and. Basic index number theory 15 introduction theanswertothequestionwhatisthemeanofagiven setofmagnitudescannotingeneralbefound,unlessthere. Analytic number theory lecture notes by andreas strombergsson.
The film is made all the more timely by current activity related to h10, e. This question can address any given equation, but in the true spirit of mathematics, it can address a general situation. In 2015 poonen received a simons fellowship in mathematics, and in 2016 he was awarded a simons investigatorship. Bjorn poonen is professor of mathematics at the uni. The talk is open to everyone, and includes refreshments. In 2011, poonen received the chauvenet prize for his article, undecidability in number theory, notices ams 55 2008. Undecidability in number theory bjorn poonen h10 polynomial equations hilberts 10th problem diophantine sets listable sets dprm theorem consequences of dprm primeproducing polynomials riemann hypothesis. Absolute undecidability in number theory abstract this paper deals with the concept of absolute undecidability in mathematics. Hilbert, in the list of 23 problems he published after a famous lecture in 1900, asked his audience to.
Our first distinguished lecture, by bjorn poonen, will be undecidability in number theory. Two survey articles on undecidability in mathematics by bjorn poonen undecidability in number theory awarded the 2011 chauvenet prize undecidable problems. In these lecture notes we give sketches of classical undecidability results in number theory, like godels first incompleteness theorem that the first order theory of the integers in the language of rings is undecidable, julia robinsons extensions of this result to arbitrary number fields and rings of integers in them, as well as to the ring of totally real integers, and. Undecidability in number theory instituto superior tecnico. In 2014, he received the mit school of science prize in undergraduate teaching.
Undecidability in group theory, topology, and analysis. Julia robinson numbers international journal of number. Presburgers work was published two years before the dramatic. Undecidability in number theory, bjorn poonen, notices ams 55, 2008 an introduction to irrationality and transcendence methods, course and project outline, draft lecture notes for lectures 1, 2, 3, 5, arizona winter school 2008, michel waldschmidt. This problem is not only undecidable, as the halting problem, but highly undecidable. Buy mathematical undecidability, quantum nonlocality and the question of the existence of god on free shipping on qualified orders. Bjorn poonen, march 10, 2010 bjorn poonen mit will present the 22nd isidore and hilda dressler lecture title of talk. It presents basically the same result from his 1936 paperin my opinion, one of the most interesting results in mathematics. Hilberts tenth problem asked for an algorithm that, given a multivariable polynomial equation with integer coefficients, would decide whether there exists a solution in integers. Nov 07, 2017 even before the resolution of hilberts tenth problem, some problems in group theory were proved undecidable. Poonen, hilberts tenth problem over rings of numbertheoretic interest, ob. While written from the point of view of algebraic number theory, the book includes chapters on mazurs conjectures on topology of rational points and poonen s elliptic curve method for constructing a diophatine model of rational integers over a very large subring of the field of rational numbers. A history of interactions between logic and number theory lecture 1 i concentrate on logic and number theory, and the evolution of this interaction.
The full first order theory of r and of c is decidable and hence, in particular. Turings method of proving that this class of problems is undecidable is particularly suggestive. Tarski also proved undecidability of various other firstorder theories, like. What is an example of a firstorder theory that is not. Shlapentokh, diophantine definability and decidability in large subrings of totally real number fields and their totally complex extensions of degree 2, j. Alexandra shlapentokh hilberts tenth problem to find an algorithm to determine whether a polynomial equation in several variables with integer coefficients has integer solutions was shown to be unsolvable in the late. Using elliptic curves of rank one towards the undecidability of hilberts tenth problem over rings of algebraic integers. Simons collaboration on arithmetic geometry, number theory, and computation annual meeting 2020, simons foundation, january 910, 2020 coorganized with jennifer balakrishnan, noam elkies, brendan hassett, andrew sutherland, and john voight. Primes in arithmetic progressions, infinite products, partial summation and dirichlet series, dirichlet characters, l1, x and class numbers, the distribution of the primes, the prime number theorem, the functional equation, the prime number theorem for arithmetic progressions, siegels. Julia robinson numbers international journal of number theory. Thats probably the highest profile recent problem to be shown impossible, but theres a whole bunch of undecidability results from the 20th century following the spirit of godel, turing, and cohen my favourite. Students, postdocs, and other researchers supervised or sponsored. Cisc462, fall 2018, decidability and undecidability 5 note. The relevance of this methodology will be demonstrated using examples from probability theory, number theory, geometry, information theory, and quantum mechanics.
That would be consistent with the statement poonen is asking if the integers has diophantine dimension 2 after all, one would use two quantifiers and a 3variable polynomial fx,yz in the diophantine definition of n, if the answer to poonen s question is yes. Cancelled undecidability in number theory department. Introduction to my research curriculum vitae and publication list mathscinet search for poonen this link works only if your institution subscribes my research is supported by the national science foundation and the simons foundation. A standard example is the theory of real closed fields with the exponential function added in as another primitive operation. This lecture will discuss how undecidability in group theory led to undecidability in topology, and how undecidability in number theory led to undecidability in analysis. They showed that the theory of diophantine equations is rich enough to simulate any computer. His research is primarily in number theory and algebraic geometry, but he has occasionally published in other subjects such as probability and computer science. Geometric quantization, representation theory, and lattice point counting. Undecidability in number theory mathematical association of.
The number of intersection points made by the diagonals of a regular polygon bjorn poonen and michael rubinstein abstract. Jan 09, 2019 our first distinguished lecture, by bjorn poonen, will be undecidability in number theory. Bjorn mikhail poonen is a mathematician, fourtime putnam competition winner and currently the claude shannon professor of mathematics at the massachusetts institute of technology. By the dprm theorem implies a diophantine set that is not computable. The last page of the notes points you to certain sections of the book for additional reading. Peter shor was elected fellow of the american academy of arts and sciences. We give a formula for the number of interior intersection points made by the diagonals of a regular ngon. Undecidability of the halting problem there exists a listable set that is not computable. Adapted waveform analysis, a musical notation for functions. The theory of real closed fields is, in itself, decidable. Undecidability in number theory mathematical association. The universal halting problem, also known in recursion theory as totality, is the problem of determining, whether a given computer program will halt for every input the name totality comes from the equivalent question of whether the computed function is total. In 2009, bjorn poonen p09a managed to find an 89definition with 2 universal and.
Invited addresses mathematical association of america. Undecidability in number theory tuesday, april 20, 2010 2. I need help with finding literature about turing machine and undecidability. Undecidability in number theory bjorn poonen mit novos talentos em matem. By definition, this means that we have a polynomial pt, x. Undecidability in group theory, topology, and analysis bjorn poonen group theory f. I also found some notes from different courses about this topic online, but not books that have something specific about this. The number of intersection points made by the diagonals of a. Some plane truths page 2 discrete geometry and other typos. This undecidability result generalises to other number theoretic objects, like all number. Undecidability in number theory jochen koenigsmann oxford.
A diophantine equation in two variables is decidable. Its hard to pick a unique favorite, but i rather like gentzens 1938 new version of the consistency proof for elementary number theory. Undecidability in number theory bjorn poonen rademacher lecture 1 november 6, 2017. Also, please feel free to share it with friends who might find the list helpful. Even for arithmetic, sometimes regarded as one of the simpler branches of mathematics, the set of true sentences is not a decidable set. The answer is a polynomial on each residue class modulo 2520. Moderate level of formality, also covers some other things, but all godel. He is a fellow of the american academy of arts and sciences and of the american mathematical society. This gives new examples of algebraic extensions of. Algebraic number theory, the book includes chapters on mazurs. Of course, number theory does not end with the study of cubic equations in three variables. Which books will give me an understanding of godels.
A set x of algebraic numbers is said to have the northcott property if. List is ongoing, please like if you found it useful or let me know if you found other free books. A basic, old result is the decidability of presburger arithmetic, i. The unsolvability of the halting problem provides a listable set for. Regular schedule see syllabus, or slides of lecture 1. Finite automata and regular languages, contextfree languages, turing machines and the churchturing thesis, decidable and undecidable languages and complexity theory. Jackson, library of mathematics, routlege and kegan paul, 1975 michael jacobson jr. Undecidability in number theory bjorn poonen h10 polynomial equations hilbert s 10th problem diophantine sets listable sets dprm theorem consequences of dprm primeproducing polynomials riemann hypothesis related problems h10 over q firstorder sentences subrings of q status of knowledge examples of polynomial equations do there exist integers x,y,z such that. On basis of this result, it is argued that it is not possible to decide d.
The number of problems is then, the number of ways of assigning yes or no to each integer. Hilberts tenth problem asked for an algorithm that, given a multivariable polynomial equation with integer coefficients, would. Theory of npcompleteness wont be on the final but it is on hw 8. Let f and k be number fields, with f contained in k. One such result concerns the halting problem, which asks for an algorithm that takes as input a. Symbolic computation 1992 14, 43 7457 a new method for undecidability proofs of first order theories ralf treinen dfki, stuhlsatzenhausweg s, w6600 saarbrucken germany received 21 march 1991 we claim that the reduction of posts correspondence problem to the decision problem of a theory provides a useful tool for proving undecidability of first order theories given by some interpretation. Undecidability in number theory bjorn poonen university of california at berkeley on sabbatical at harvard and mit in fall 2007 september 20, 2007. Limits of computation course recap princeton university. Hilberts tenth problem over rings of numbertheoretic interest. He has edited two books, and his research articles have been cited by approximately 1,000 distinct authors. These undecidability results were proved using diagonalization arguments reminiscent of g. The main technical result concerns a particular sentence d expressed in the language of peano arithmetic.
Of course, if instead of calling it arithmetic, we call it number theory, then its undecidability comes as less of a surprise. Rademacher lecturers 1978 department of mathematics. In computability theory and computational complexity theory, an undecidable problem is a decision problem for which it is proved to be impossible to construct an algorithm that always leads to a correct yesorno answer. Poo02 bjorn poonen, using elliptic curves of rank one towards the undecidability of hilberts tenth problem over rings of algebraic integers, algorithmic number theory sydney, 2002, springer lecture notes in computer science 2369, berlin, 2002, pp. Undecidability in number theory andrew gilroy june 23, 2008 in the study of number theory the question often arises. Number theory, logic, algebraic geometry, computability, model theory, arith metic of fields, valuation. Poonens diophantine definition of quaternionic semilocal rings. The origin of probability and entropy, neurocomputing 67 c 2005 245274. Once the concept of mechanical procedure was crystallized, it was relatively easy to find absolutely unsolvable problemse. Poonen s research focuses on number theory and algebraic geometry, but occasionally he has worked also 39 in combinatorics, probability, and computer science. Prior to coming to mit, poonen received the guggenheim, packard, rosenbaum, simons, and sloan fellowships, as well as a miller professorship. Undecidability in number theory, notices of the american mathematical. Mathematical undecidability, quantum nonlocality and the.
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