Venue：https://nthu-meeting.webex.com/join/hyliao
Tea time：因遠距暫停茶會
Abstract:
The study of Cremona transformations has a history of more than one hundred years, and it is still very active nowadays. The simplest yet interesting example of such a transformation is the map f(x : y : z) = [yz : xz : xy] on the projective plane, which is defined everywhere but the three points [1 : 0 : 0], [0 : 1 : 0], and [0 : 0 : 1]. In general, Cremona transformations are, roughly speaking, invertible self-maps of a projective space defined by polynomials, and they are allowed to be undefined at loci of higher codimensions.
Over a finite field, there are only finitely many points in a projective space, and we call a Cremona transformation "bijective" if it induces a permutation on these points. Here comes the question: on a projective space over a finite field, could we realize all possible permutations on its points via bijective Cremona transformations? In this talk, I will give a full answer to this question in the case of the projective plane. This is a joint work with Shamil Asgarli, Masahiro Nakahara, and Susanna Zimmermann. The talk is intended for a general audience in mathematics.
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