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VERSION:2.0
PRODID:-//University of Liverpool Computer Science Seminar System//v2//EN
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DTSTAMP:20260408T211811Z
UID:Seminar-ACTO-753@lxserverA.csc.liv.ac.uk.csc.liv.ac.uk
ORGANIZER:CN=Nikhil Mande:MAILTO:Nikhil.Mande@liverpool
DTSTART:20200701T140000
DTEND:20200701T150000
SUMMARY:Algorithms, Complexity Theory and Optimisation Series
DESCRIPTION:: Implementing geometric complexity theory: On the separation of orbit closures via symmetries\n\nUnderstanding the difference between group orbits and their closures is\na key difficulty in geometric complexity theory (GCT): While the GCT\nprogram is set up to separate certain orbit closures, many beautiful\nmathematical properties are only known for the group orbits, in\nparticular close relations with symmetry groups and invariant spaces,\nwhile the orbit closures seem much more difficult to understand.\nHowever, in order to prove lower bounds in algebraic complexity theory,\nconsidering group orbits is not enough.\nIn this paper we tighten the relationship between the orbit of the power\nsum polynomial and its closure, so that we can separate this orbit\nclosure from the orbit closure of the product of variables by just\nconsidering the symmetry groups of both polynomials and their\nrepresentation theoretic decomposition coefficients. In a natural way\nour construction yields a multiplicity obstruction that is neither an\noccurrence obstruction, nor a so-called vanishing ideal occurrence\nobstruction. All multiplicity obstructions so far have been of one of\nthese two types.\nOur paper is the first implementation of the ambitious approach that was\noriginally suggested in the first papers on geometric complexity theory\nby Mulmuley and Sohoni (SIAM J Comput 2001, 2008): Before our paper, all\nexistence proofs of obstructions only took into account the symmetry\ngroup of one of the two polynomials (or tensors) that were to be\nseparated. In our paper the multiplicity obstruction is obtained by\ncomparing the representation theoretic decomposition coefficients of\nboth symmetry groups.\nOur proof uses a semi-explicit description of the coordinate ring of the\norbit closure of the power sum polynomial in terms of Young tableaux,\nwhich enables its comparison to the coordinate ring of the orbit.\nThis is joint work with Umangathan Kandasamy\n\nhttps://www.csc.liv.ac.uk/research/seminars/abstract.php?id=753
LOCATION:Microsoft Teams
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