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Mathematics Department

Combinatorics, Algebra, & Topology Seminar

Spring 2019

All talks are from 1200-1300 in the designated room unless otherwise specified.

  • Jan
    14
  • Classifying moduli spaces of line arrangements to find Zariski pairs
    Moshe Cohen
    Vassar College
    Time: 12:00 PM

    View Abstract

    Zariski gave a pair of degree-six polynomials with the same set of singularities but whose complements are not homeomorphic. This motivates the search for similar "Zariski pairs" of line arrangements: two collections of lines with the same combinatorial intersection data but whose (complex projective) complements are not homeomorphic. Rybnikov produced one with thirteen lines in 1998, and only a small number have been found since. Zariski gave a pair of degree-six polynomials with the same set of singularities but whose complements are not homeomorphic. This motivates the search for similar "Zariski pairs" of line arrangements: two collections of lines with the same combinatorial intersection data but whose (complex projective) complements are not homeomorphic. Rybnikov produced one with thirteen lines in 1998, and only a small number have been found since. The literature states that no such pair exists on nine lines or fewer. Together with Amram, Sun, Teicher, Ye, and Zarkh, we investigate arrangements of ten lines. Together with undergraduate students Liu and then Buhmann, May, and Shu, we begin to investigate arrangements of eleven lines. By performing both combinatorial classifications of line arrangements and also algebraic classifications of moduli spaces (or realization spaces) of arrangements from the combinatorial results, we produce new candidates for Zariski pairs.
  • Man-Wai Cheung
    Harvard University
    Time: 12:00 PM
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