日時:2024年5月7日(火),15:00-16:30
場所:理学部A201号室
講師:Víctor Pérez-Valdés 氏 (龍谷大学)
題目:Construction and classification of vector-valued differential symmetry breaking operators from $S^3$ to $S^2$
Any equivariant irreducible vector bundle for the conformal group $SO_0(4,1)$ on the $3$-sphere $S^3$ is parametrized by an odd number ($=2N+1$) and a complex number $\lambda$. On the other hand, any equivariant irreducible vector bundle for the conformal group $SO_0(3,1)$ on the $2$-sphere $S^2$ is a line bundle,
and is parametrized by an integer number $m$ and a complex number $\nu$.
In the present talk, we consider the problems of construction and classification of all the differential operators,
that are symmetry breaking operators with respect to the conformal pair $(SO_0(4,1), SO_0(3,1))$ between the spaces of smooth sections of a vector bundle $V_\lambda^{2N+1}$ over the $3$-sphere $S^3$ and a line bundle $L_{m, \nu}$ over the $2$-sphere $S^2$.
In particular, we solve these problems when the rank of the vector bundle is less than or equal to $7$ (i.e., for $N = 1,2,3$), and for $|m| > N$, and propose a strategy to solve them for $N > 3$. The method we use is the F-method of T. Kobayashi, which in our setting allows us to reduce the problem of constructing differential symmetry breaking operators to the problem of solving an overdetermined system of $2(2N+1)$ ordinary differential equations on $2N+1$ unknown polynomials.
日時:2024年5月28日(火),15:00-16:30
場所:理学部A201号室
講師:Thomas Raujouan 氏 (神戸大学)
題目:The Loop Weierstrass Representation
We will talk about two classes of surfaces.
On the one hand, minimal surfaces, such as the catenoid, are critical points of the area functional. They can be constructed with the help of holomorphic functions via the Weierstrass representation (1866). On the other hand, constant mean curvature (CMC) surfaces, such as the cylinder, are critical points of the area functional under volume constraint. They can be constructed with the help of loop groups via the method of Dorfmeister, Pedit and Wu (DPW, 1998).
Recently, several achievements in the theory of CMC surfaces have been made using DPW. Inspired by them, we will re-interpret the Weierstrass representation and introduce a new framework for the construction of minimal surfaces: the Loop-Weierstrass Representation (LWR). We will then show that some methods of the DPW method can be applied to the study of minimal surfaces, shedding a new light on ancient results. This talk is based on a joint work with N. Schmitt and J. Ziefle.