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Mathematics Calendar

November 09, 2012
Friday, November 09
Algebra Seminar
Time: 14:30
Speaker: Jaydeep Chipalkatti (University of Manitoba)
Title: "On Hilbert covariants"
Room: MC 108

Abstract: Consider a binary form

F=a0xd1+a1xd11x2++adxd2,(aiC)

of order d in the variables {x1,x2}. Its Hessian is defined to be

He(F)=2Fx212Fx22(2Fx1x2)2.

It is classical that F is the perfect d-th power of a linear form, if and only if He(F) vanishes identically. Moreover, He(F) is a covariant of F, in the sense that its construction commutes with a linear change of variables in {x1,x2}. Now assume that d=rm, and suppose we ask for a covariant whose vanishing is equivalent to F being the perfect power of an order r form. In 1885, Hilbert constructed such a covariant, to be denoted by Hr,d(F). In geometric terms, the variety of perfect powers of order r forms defines a subvariety XrPd, and the coefficients of Hr,d give defining equations for this variety.

In this talk, I will outline a wholly different construction of this covariant, which leads to a generalisation called the G¨ottingen covariants. Moreover, we have the theorem that the ideal generated by the coefficients of the Hilbert covariant generates Xr as a scheme, and not merely as a variety. This is joint work with Abdelmalek Abdesselam from the University of Virginia.

Noncommutative Geometry
Time: 14:30
Speaker: Jason Haradyn (Western)
Title: "NCG Learning Seminar: Isospectral and Nonisometric Domains in the Euclidean Plane"
Room: MC 107

Abstract: In 1964, Milnor discovered flat tori in dimension 16 that are isospectral but not isometric. As amazing a result as this is, it still took about thirty years to construct isospectral plane domains that are not isometric. In this talk, I will review Sunada's method, as extended by Berard, to give an example of a pair of simply-connected nonisometric domains in the Euclidean plane that are both Dirichlet isospectral and Neumann isospectral.