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DTSTART;TZID=America/Los_Angeles:20181009T121500
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DTSTAMP:20260404T025409
CREATED:20180912T160739Z
LAST-MODIFIED:20181001T220127Z
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SUMMARY:State Polytopes of Combinatorial Neural Codes (Rob Davis\, HMC)
DESCRIPTION:Combinatorial neural codes are 0/1 vectors that are used to model the co-firing patterns of a set of place cells in the brain. One wide-open problem in this area is to determine when a given code can be algorithmically drawn in the plane as a Venn diagram-like figure. A sufficient condition to do so is for the code to have a property called k-inductively pierced. Gross\, Obatake\, and Youngs recently used toric algebra to show that a code on three neurons is 1-inductively pierced if and only if the toric ideal is trivial or generated by quadratics. No result is known for additional neurons in the same generality. \nIn this talk\, we study two infinite classes of combinatorial neural codes in detail. For each code\, we explicitly compute its universal Gröbner basis. This is done for the first class by recognizing that the codewords form a Lawrence-type matrix. With the second class\, this is done by showing that the matrix is totally unimodular. These computations allow one to compute the state polytopes of the corresponding toric ideals\, from which all distinct initial ideals may be computed efficiently. Moreover\, we show that the state polytopes are combinatorially equivalent to well-known polytopes: the permutohedron and the stellohedron.
URL:https://colleges.claremont.edu/ccms/event/antc-talk-by-rob-davis-hmc/
LOCATION:Millikan 2099\, Pomona College\, 610 N. College Ave.\, Claremont\, CA\, 91711\, United States
CATEGORIES:Algebra / Number Theory / Combinatorics Seminar
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DTSTART;TZID=America/Los_Angeles:20181010T041500
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CREATED:20180928T170449Z
LAST-MODIFIED:20181005T213928Z
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SUMMARY:Applications of Cayley Digraphs to Waring's Problem and Sum-Product Formulas (Yesim Demiroglu\, Harvey Mudd)
DESCRIPTION:Abstract: In this talk\, we first present some elementary new proofs (using Cayley digraphs and spectral graph theory) for Waring’s problem over finite fields\, and explain how in the process of re-proving these results\, we obtain an original result that provides an analogue of Sarkozy’s theorem in the finite field setting (showing that any subset E of a finite field Fq for which |E| >  (qk)/sqrt{q – 1}must contain at least two distinct elements whose difference is a kth power). Once we have our results for finite fields\, we apply some classical mathematics to extend our Waring’s problem results to the context of general (not  necessarily commutative) finite rings. In the second half of our talk\, we present our sum-product results related to matrix rings over finite fields\, which can again be proven using Cayley digraphs and spectral graph theory in an efficient way.
URL:https://colleges.claremont.edu/ccms/event/yesim-demiroglu-harvey-mudd/
LOCATION:Argue Auditorium\, Pomona College\, 610 N. College Ave.\, Claremont\, CA\, 91711\, United States
CATEGORIES:Colloquium
ORGANIZER;CN="Ali Nadim":MAILTO:ali.nadim@cgu.edu
GEO:34.0999157;-117.7142668
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