In the first half of this paper, all the limits of irreducible characters of $G_{n} = {\mathfrak S}_{n}(T)$ as $n \to \infty$ are calculated. The set of all \emph{continuous} limit functions on $G = {\mathfrak S}_{\infty}(T)$ is exactly equal to the set of all characters of $G$ determined in [HH6]. We give a necessary and sufficient condition for a series of irreducible characters of $G_{n}$ to have a continuous limit and also such a condition to have a discontinuous limit. In the second half, we study the limits of characters of certain induced representations of $G_{n}$ which are usually reducible. The limits turn out to be characters of $G$, and we analyse which of irreducible components are responsible to these limits.

We compute the equation of the 7-secant variety to the Veronese variety $({\bf P}^{4}, \mathcal{O}(3))$, its degree is 15. This is the last missing invariant in the Alexander-Hirschowitz classification. It gives the condition to express a homogeneous cubic polynomial in 5 variables as the sum of 7 cubes (Waring problem). The interesting side in the construction is that it comes from the determinant of a matrix of order 45 with linear entries, which is a cube. The same technique allows to express the classical Aronhold invariant of plane cubics as a pfaffian.

The generalized test ideals introduced in [HY] are related to multiplier ideals via reduction to characteristic $p$. In addition, they satisfy many of the subtle properties of the multiplier ideals, which in characteristic zero follow via vanishing theorems. In this note we give several examples to emphasize the different behavior of test ideals and multiplier ideals. Our main result is that every ideal in an $F$-finite regular local ring can be written as a generalized test ideal. We also prove the semicontinuity of $F$-pure thresholds (though the analogue of the Generic Restriction Theorem for multiplier ideals does not hold).

The classification of isoparametric hypersurfaces with four principal curvatures in spheres in [2] hinges on a crucial characterization, in terms of four sets of equations of the 2nd fundamental form tensors of a focal submanifold, of an isoparametric hypersurface of the type constructed by Ferus, Karcher and Münzner. The proof of the characterization in [2] is an extremely long calculation by exterior derivatives with remarkable cancellations, which is motivated by the idea that an isoparametric hypersurface is defined by an over-determined system of partial differential equations. Therefore, exterior differentiating sufficiently many times should gather us enough information for the conclusion. In spite of its elementary nature, the magnitude of the calculation and the surprisingly pleasant cancellations make it desirable to understand the underlying geometric principles.

In this paper, we give a conceptual, and considerably shorter, proof of the characterization based on Ozeki and Takeuchi's expansion formula for the Cartan-Münzner polynomial. Along the way the geometric meaning of these four sets of equations also becomes clear.

In this paper, we prove some Bernstein type results for $n$-dimensional minimal Lagrangian graphs in quaternion Euclidean space $H^{n} \cong R^{4n}$. In particular, we also get a new Bernstein Theorem for special Lagrangian graphs in $C^{n}$.