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DTSTART;TZID=America/Los_Angeles:20230926T121500
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DTSTAMP:20230925T071646
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SUMMARY:Chromatic numbers of abelian Cayley graphs (Michael Krebs\, Cal State LA)
DESCRIPTION:A classic problem in graph theory is to find the chromatic number of a given graph: that is\, to find the smallest number of colors needed to assign every vertex a color such that whenever two vertices are adjacent\, they receive different colors. This problem has been studied for many families of graphs\, including cube-like graphs\, unit-distance graphs\, circulant graphs\, integer distance graphs\, Paley graphs\, unit-quadrance graphs\, etc. All of those examples just listed can be regarded as “abelian Cayley graphs\,” that is\, Cayley graphs whose underlying group is abelian. Our goal is to create a unified\, systematic approach for dealing with problems of this sort\, rather than attacking each individually with ad hoc methods. Building upon the work of Heuberger\, we associate an integer matrix to each abelian Cayley graph. In certain cases\, such as when the matrix is small enough\, we can more or less read the chromatic number directly from the entries of the matrix. In this way we immediately recover both Payan’s theorem (that cubelike graphs cannot have chromatic number 4) as well as Zhu’s theorem (which determines the chromatic number of six-valent integer distance graphs). The proofs utilize only elementary group theory\, elementary graph theory\, elementary number theory\, and elementary linear algebra. This is joint work with J. Cervantes.
URL:https://colleges.claremont.edu/ccms/event/antc-talk-michael-krebs-cal-state-la/
LOCATION:Roberts North 102\, CMC
CATEGORIES:Algebra / Number Theory / Combinatorics Seminar
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DTSTART;TZID=America/Los_Angeles:20230926T150000
DTEND;TZID=America/Los_Angeles:20230926T160000
DTSTAMP:20230925T071646
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LAST-MODIFIED:20230922T154321Z
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SUMMARY:Claremont Topology Seminar: Reginald Anderson (CMC)
DESCRIPTION:Title: Cellular resolutions of the diagonal and exceptional collections for toric Deligne-Mumford stacks (Continued) \nAbstract: Beilinson gave a resolution of the diagonal for complex projective space which yields a strong\, full exceptional collection of line bundles. Bayer-Popescu-Sturmfels generalized Beilinson’s result to a cellular resolution of the diagonal for what they called “unimodular” toric varieties (a more restrictive condition than being smooth)\, which can also be extended to smooth toric varieties and global quotient toric DM stacks of a smooth toric variety by a finite abelian group\, if we allow our resolution to have cokernel which is supported only along the vanishing of the irrelevant ideal. Here we show implications for exceptional collections of line bundles and a positive example for the modified King’s conjecture by giving a strong\, full exceptional collection of line bundles on a smooth\, non-unimodular nef-Fano complete toric surface.
URL:https://colleges.claremont.edu/ccms/event/claremont-topology-seminar-reginald-anderson-cmc-2/
LOCATION:Fletcher 110\, Pitzer College\, 1050 N Mills Ave\, Claremont\, CA\, 91711\, United States
CATEGORIES:Topology Seminar
ORGANIZER;CN="Bahar%20Acu":MAILTO:Bahar_Acu@pitzer.edu
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BEGIN:VEVENT
DTSTART;TZID=America/Los_Angeles:20230927T161500
DTEND;TZID=America/Los_Angeles:20230927T173000
DTSTAMP:20230925T071646
CREATED:20230912T031043Z
LAST-MODIFIED:20230913T074346Z
UID:3193-1695831300-1695835800@colleges.claremont.edu
SUMMARY:Fourier-Mukai Transforms and Resolutions of the Diagonal (Professor Reginald Anderson\, Claremont McKenna College)
DESCRIPTION:Title: Fourier-Mukai Transforms and Resolutions of the Diagonal \nSpeaker: Reginald Anderson\, Department of Mathematical Sciences\, Claremont McKenna College \nAbstract: Fourier-Mukai transforms provide functors between derived categories of coherent sheaves on smooth projective varieties over the complex numbers. Using Beilinson’s resolution of the diagonal for complex projective space\, we encounter a description of any object in the derived category of projective space in terms of fundamental building blocks\, called an exceptional collection. An emphasis will be given on examples of Fourier-Mukai transforms of smooth projective varieties. If time permits\, we will also discuss a combinatorial\, cellular resolution of the diagonal due to Bayer-Popescu-Sturmfels for a proper subclass of smooth toric varieties. \n\n\n\n\n\nReginald Anderson received his PhD in mathematics from Kansas State University in May and studies derived categories of toric DM stacks. His research areas include algebraic geometry\, homological algebra\, and category theory.
URL:https://colleges.claremont.edu/ccms/event/fourier-mukai-transforms-and-resolutions-of-the-diagonal-professor-reginald-anderson-claremont-mckenna-college/
LOCATION:Argue Auditorium\, Pomona College\, 610 N. College Ave.\, Claremont\, CA\, 91711\, United States
CATEGORIES:Colloquium
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