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DTSTART;TZID=America/Los_Angeles:20240213T121500
DTEND;TZID=America/Los_Angeles:20240213T131000
DTSTAMP:20260413T065611
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SUMMARY:Quiver categorification of quandle invariants (Sam Nelson\, CMC)
DESCRIPTION:Quiver structures are naturally associated to subsets of the endomorphism sets of quandles and other knot-coloring structures\, providing a natural form of categorification of homset invariants and their enhancements. In this talk we will survey recent work in this area.
URL:https://colleges.claremont.edu/ccms/event/antc-seminar-sam-nelson-cmc-3/
LOCATION:Estella 2099
CATEGORIES:Algebra / Number Theory / Combinatorics Seminar
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BEGIN:VEVENT
DTSTART;TZID=America/Los_Angeles:20240220T121500
DTEND;TZID=America/Los_Angeles:20240220T131000
DTSTAMP:20260413T065611
CREATED:20231127T045722Z
LAST-MODIFIED:20240219T164238Z
UID:3328-1708431300-1708434600@colleges.claremont.edu
SUMMARY:Point-counting and topology of algebraic varieties (Siddarth Kannan\, UCLA)
DESCRIPTION:A projective algebraic variety X is the zero locus of a collection of homogeneous polynomials\, in projective space. When the polynomials have integer coefficients\, we can think of the k-valued points X(k) of the variety\, for any field k. Now suppose we have two different fields k and k’. How does the behavior of X(k) inform the behavior of X(k’)? It turns out that this is a rich line of inquiry. I will present a particularly pleasing example which relates the topology of the complex-valued points of X with the number of points it has over finite fields.
URL:https://colleges.claremont.edu/ccms/event/antc-seminar-siddarth-kannan-ucla/
LOCATION:Estella 2099
CATEGORIES:Algebra / Number Theory / Combinatorics Seminar
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DTSTART;TZID=America/Los_Angeles:20240227T121500
DTEND;TZID=America/Los_Angeles:20240227T131000
DTSTAMP:20260413T065611
CREATED:20240126T230120Z
LAST-MODIFIED:20240221T014138Z
UID:3354-1709036100-1709039400@colleges.claremont.edu
SUMMARY:The restricted variable Kakeya problem (Pete Clark\, University of Georgia)
DESCRIPTION:For a finite field F_q\, a subset of F_q^N is a Kakeya set if it contains a line in every direction (i.e.\, a coset of every one-dimensional linear subspace).  The finite field Kakeya problem is to determine the minimal size K(N\,q) of a Kakeya set in F_q^N.  This problem was posed by Wolff in 1999 as an analogue to the Kakeya problem in Euclidean N-space\, which was (and still is) one of the major open problems in harmonic analysis.  It caused quite a stir in 2008 when Zeev Dvir showed that for each fixed N\, as q -> oo\, K(N\,q) is bounded below by a constant times q^N: the Euclidean analogue of this result is not only proved but known to be false.\n\nBut what about the constant?  In 2009 Dvir-Kopparty-Saraf-Sudan gave a lower bound on K(N\,q) that was within a factor of 2 of an upper bound due to Dvir-Thas.  (I will briefly mention recent work of Bukh-Chao giving a decisive further improvement\, but that is not the focus of the talk.) The key to this improved lower bound is a multiplicity enhancement of a 1922 result of Ore. In this talk I want to give my own exposition of this work together with a mild generalization: if X is a subset of F_q^N \ {0}\, then an X-Kakeya set is a subset that contains a translate of the line generated by x for all x in X.  Putting K_X(N\,q) to be the minimal size of an X-Kakeya set in F_q^N\, I will give a lower bound on K_X(N\,q) that recovers the DKSS bound when X = F_q^N \ {0}.  This is similar in spirit to  “statistical Kakeya” results of Dvir and DKSS but not overlapping much; in fact\, I will give a statistical generalization of my result as well.
URL:https://colleges.claremont.edu/ccms/event/antc-seminar-pete-clark-university-of-georgia/
LOCATION:Estella 2099
CATEGORIES:Algebra / Number Theory / Combinatorics Seminar
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