May 27, 2016 @BKC, Forest House F105, 16:30-18:00
1. Time: 16:30-17:10
Keita Owari (Ritsumeikan University)
Title: On Convex Functions on Orlicz Spaces with $\Delta_2$-Conjugates
Abstract: We show that in an Orlicz space \(L^\Phi\) with the conjugate Young function \(\Psi\) being \(\Delta_2\) (so \(L^\Phi\) is the dual of \(L^\Psi\)), a proper convex function has a \(\langle L^\Phi,L^\Psi\rangle\)-dual representation iff it is order lower semicontinuous; more precisely, a convex set \(C\subset L^\Phi\) is \(\sigma(L^\Phi,L^\Psi)\)-closed iff for each order interval \([-\eta,\eta]=\{\xi:-\eta\leq \xi\leq \eta\}\) (\(0\leq \eta\in L^\Phi\)), the intersection \(C\cap [-\eta,\eta]\) is closed in \(L^0\). The result is based on the following technical lemma: for any norm bounded sequence \((\xi_n)_n\) in \(L^\Phi\) which converges in probability to \(0\), there exist forward convex combinations \(\zeta_n\in \mathrm{conv}(\xi_n,\xi_{n+1},…)\) as well as an element \(\eta\in L^\Phi_+\) such that \(\zeta_n\rightarrow 0\) and \(|\zeta_n|\leq \eta\). We show also that a finite-valued convex function on \(L^\Phi\) is \(\tau(L^\Phi,L^\Psi)\)-continuous iff it is sequentially \(\tau(L^\Phi,L^\Psi)\)-continuous on order intervals, and the condition is equivalent to the order continuity of the function.
This is a Joint work with Freddy Delbaen (ETH Zürich and Univ. Zürich).
2. Time: 17:20-18:00
Daisuke Tarama (Ritsumeikan University)
Title: Elliptic brations associated to free rigid bodies
Abstract: The free rigid body dynamics stands for a dynamical system describing the rotational motion of a rigid body under no external force, which is a typical solvable example in analytical mechanics. Mathematically, the dynamics is formulated as a Hamiltonian system on the cotangent bundle of the three-dimensional rotation group. One can also regard it as the geodesic flow on the rotation group with respect to a left-invariant metric. Because of its symmetry, the dynamics can essentially be described by Euler equation on the threedimensional Euclidean space, which is completely integrable. The integral curve of this equation is in fact a smooth elliptic curve. It is also known that one can associate a Lax equation whose spectral curve is again a smooth elliptic curve.
Varying the parameters of the system, one obtains two elliptic fibrations over threedimensional projective space. In this talk, the singular fibres and the monodromy of these fibrations, as well as their mutual relation, are analysed and the link to Birkhoff normal forms, introduced from the viewpoint of Hamiltonian systems, will be also mentioned.
English