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Starting with a commutative ring $R$ and an ideal $I$ , it is possible to define a family of rings $R(I)_{a,b}$ , with $a,b \in R$ , as quotients of the Rees algebra $\oplus_{n \geq0} I^{n}t^{n}$ ; among the rings appearing in this family we find Nagata’s idealization and amalgamated duplication. Many properties of these rings depend only on $R$ and $I$ and not on $a$ , $b$ ; in this paper we show that the Gorenstein and the almost Gorenstein properties are independent of $a$ , $b$ . More precisely, we characterize when the rings in the family are Gorenstein, complete intersection, or almost Gorenstein and we find a formula for the type.

Due to the length this work is published in two parts. The second part will appear in Vol 23: 1 of this journal.

For a random quantum state on $\cH=\C^d \otimes \C^d$ obtained by partial tracing a random pure state on $\cH \otimes \C^s$, we consider the question whether it is typically separable or typically entangled. For this problem, we show the existence of a sharp threshold $s_0=s_0(d)$ of order roughly $d^3$. More precisely, for any $\e>0$ and for $d$ large enough, such a random state is entangled with very large probability when $s \leq (1-\e)s_0$, and separable with very large probability when $s \geq (1+\e) s_0$. One consequence of this result is as follows: for a system of $N$ identical particles in a random pure state, there is a threshold $k_0 = k_0(N)\sim N/5$ such that two subsystems of $k$ particles each typically share entanglement if $k>k_0$, and typically do not share entanglement if $k<k_0$. Our methods work also for multipartite systems and for ``unbalanced'' systems such as $\C^{d_1} \otimes \C^{d_2} $, ${d_1} \neq {d_2} $. The arguments rely on random matrices, classical convexity, high-dimensional probability and geometry of Banach spaces; some of the auxiliary results may be of reference value.

It is a great honor for me to be invited by Tsinghua University to talk today. Tsinghua has, in its history, made many contributions to mathematics. The two most famous mathematicians in modern Chinese history have both been closely related to Tsinghua University. One is Professor Hua Lo-Keng, the other is Prof essor Chern Shiing-Sheng. Especially since Professor Chern is my teacher, I feel proud to be able to say some words in the opening ceremony for the Center for Advanced Study. I chose the title of my talk also for this reason. Due to my critical comments, I have decided to spend most of my time in giving a general talk in English. The remainder of the talk will be in Chinese.