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Fingerprinting Richard Stanley Bridget Eileen Tenner DePaul - PowerPoint PPT Presentation

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What is a fingerprint? What is a mathematical fingerprint? Fingerprint databases for theorems Accumulation of evidence Judgement time Fingerprinting Richard Stanley Bridget Eileen Tenner DePaul University June 23, 2014 Bridget Eileen Tenner

  1. What is a fingerprint? What is a mathematical fingerprint? Fingerprint databases for theorems Accumulation of evidence Judgement time Fingerprinting Richard Stanley Bridget Eileen Tenner DePaul University June 23, 2014 Bridget Eileen Tenner Fingerprinting Richard Stanley

  2. What is a fingerprint? What is a mathematical fingerprint? Fingerprint databases for theorems Accumulation of evidence Judgement time What is a fingerprint? Bridget Eileen Tenner Fingerprinting Richard Stanley

  3. What is a fingerprint? What is a mathematical fingerprint? Fingerprint databases for theorems Accumulation of evidence Judgement time Example Properties of a fingerprint: • small a unique identifier • canonical • language-independent Bridget Eileen Tenner Fingerprinting Richard Stanley

  4. What is a fingerprint? What is a mathematical fingerprint? Fingerprint databases for theorems Accumulation of evidence Judgement time What do we do with fingerprints? 1. Catalogue them in a searchable fingerprint database. 2. Query the database. 3. Solve crimes! Who stole the cookies from the cookie jar? Bridget Eileen Tenner Fingerprinting Richard Stanley

  5. What is a fingerprint? What is a mathematical fingerprint? Fingerprint databases for theorems Accumulation of evidence Judgement time What is a mathematical fingerprint? Bridget Eileen Tenner Fingerprinting Richard Stanley

  6. What is a fingerprint? What is a mathematical fingerprint? Fingerprint databases for theorems Accumulation of evidence Judgement time Example 0 , 1 , 1 , 2 , 3 , 5 , 8 , 13 , . . . Properties of a mathematical fingerprint: a unique identifier • small • canonical The n th term enumerates • subsets of [ n − 2 ] containing no • language-independent consecutive integers e.g., numerical • domino tilings of a 2 × ( n − 1 ) rectangle Bridget Eileen Tenner Fingerprinting Richard Stanley

  7. What is a fingerprint? What is a mathematical fingerprint? Fingerprint databases for theorems Accumulation of evidence Judgement time What do we do with mathematical fingerprints? 1. Catalogue them in a searchable fingerprint database. 2. Query the database. 3. Solve mathematical problems! What is the relationship between subsets of [ n − 2 ] containing no consecutive integers, and domino tilings of a 2 × ( n − 1 ) rectangle? They are in bijection with each other. Bridget Eileen Tenner Fingerprinting Richard Stanley

  8. What is a fingerprint? What is a mathematical fingerprint? Fingerprint databases for theorems Accumulation of evidence Judgement time Fingerprint databases for theorems Bridget Eileen Tenner Fingerprinting Richard Stanley

  9. What is a fingerprint? What is a mathematical fingerprint? Fingerprint databases for theorems Accumulation of evidence Judgement time Why bother? Suppose mathematician M has just proved theorem T . How can M determine if her result is truly new, or if T (or some equivalent reformulation of T ) already exists in the literature? Wouldn’t it be great if M could encode T in some small, canonical, language-independent manner, and then search for that encoding in a database?! Bridget Eileen Tenner Fingerprinting Richard Stanley

  10. What is a fingerprint? What is a mathematical fingerprint? Fingerprint databases for theorems Accumulation of evidence Judgement time How does a database encode theorems? In addition to the fingerprint, each entry includes fields like name, comments, formula, references, . . . . Thus each entry in the database is a theorem, or a collection of theorems, associated to its particular fingerprint. Bridget Eileen Tenner Fingerprinting Richard Stanley

  11. What is a fingerprint? What is a mathematical fingerprint? Fingerprint databases for theorems Accumulation of evidence Judgement time A fingerprint database for theorems should be . . . • language-independent numerical, canonical, no special vocab • full of references literature, cross-reference among databases, code, links • collaborative accessible, open • optimistic false positives are okay, but no false negatives And the fingerprints should be small, canonical, and language-independent. Bridget Eileen Tenner Fingerprinting Richard Stanley

  12. What is a fingerprint? What is a mathematical fingerprint? Fingerprint databases for theorems Accumulation of evidence Judgement time What are some good mathematical fingerprints? • integer sequences OEIS • permutation patterns DPPA • statistics for combinatorial objects FindStat • finite simple groups ATLAS • smooth, projective, irred curves of genus g over field of q elts manYPoints • hypergeometric series in canonical form WZ method and Digital Library of Mathematical Functions • graphs what should be the fingerprint?? Bridget Eileen Tenner Fingerprinting Richard Stanley

  13. What is a fingerprint? What is a mathematical fingerprint? Fingerprint databases for theorems Accumulation of evidence Judgement time Accumulation of evidence Bridget Eileen Tenner Fingerprinting Richard Stanley

  14. What is a fingerprint? What is a mathematical fingerprint? Fingerprint databases for theorems Accumulation of evidence Judgement time Integer sequences – OEIS 420 results found for query ref:stanley in OEIS as of June 16, 2014 A003431: number of connected irreducible posets with n labeled points [RPS] A192721: number of pairs of permutations in S n × S n with k common descents [RPS] In a different direction: 207 combinatorial interpretations of Catalan numbers as of June 16, 2014 Bridget Eileen Tenner Fingerprinting Richard Stanley

  15. What is a fingerprint? What is a mathematical fingerprint? Fingerprint databases for theorems Accumulation of evidence Judgement time Permutation patterns – DPPA Av(2143) = permutations w for which Red( w ) and standard Young tableaux of shape λ ( w ) are equinumerous [RPS] Av(321) = fully commutative permutations [Billey-Jockusch-RPS] Bridget Eileen Tenner Fingerprinting Richard Stanley

  16. What is a fingerprint? What is a mathematical fingerprint? Fingerprint databases for theorems Accumulation of evidence Judgement time Not-yet-existing databases A.k.a. homework for the audience! • Combinatorial Hopf algebras via OEIS/FindStat hybrid? • Chess problems via pieces? positions? restrictions? • Catalan objects via symmetry? • Tiling problems via tiles? symmetry? topology? • Conjectures, open problems, attempts and ideas! Bridget Eileen Tenner Fingerprinting Richard Stanley

  17. What is a fingerprint? What is a mathematical fingerprint? Fingerprint databases for theorems Accumulation of evidence Judgement time Judgement time Bridget Eileen Tenner Fingerprinting Richard Stanley

  18. What is a fingerprint? What is a mathematical fingerprint? Fingerprint databases for theorems Accumulation of evidence Judgement time The verdict mathematics GUILTY AS CHARGED Bridget Eileen Tenner Fingerprinting Richard Stanley

  19. What is a fingerprint? What is a mathematical fingerprint? Fingerprint databases for theorems Accumulation of evidence Judgement time Bridget Eileen Tenner Fingerprinting Richard Stanley

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