| 摘要: | 我們可以把線性變換大概分類為在有限維空間上（即歐式空間）與無窮維空間上（巴拿赫空間/希爾伯特空間）。在有限維空間上，我們稱之為矩陣理論。在無窮維空間上，我們稱之為算子理論。當然，這兩個分支的研究方法大相徑庭。若我們讓矩陣的元素取成隨機變量，則我們得到所謂的隨機矩陣。這已經發展成為數學中一門深刻的學問。大家輩出。而無窮維空間上對應的隨機算子理論則甚少被理解。當然有些特殊情況，如隨機薛定諤算子，因為其物理背景，被認真的研究。隨機Toeplitz算子也有一些（不系統）的研究。目前，就一般算子理論而言，其隨機版本基本上是空白的。這需要對抽象算子理論與機率論都有較深刻的理解，從而找到合適的問題來切入。在這項計劃裡，我們試圖從最重要的非自伴算子的隨機版本開始，建立一套新的，系統的理論。然後第二個問題我們將考慮點過程與解析函數的隨機零點。 ;We divide linear analysis roughly into two categories: on finite dimensional spaces (Euclidean spaces) and infinite dimensional spaces (Banach spaces/Hilbert spaces, e.g.). On a finite dimensional space, we have the familiar "Matrix Theory". On an infinite dimensional space, we usually call it “Operator Theory”. Of course, these two areas are largely different in methodology. If we ask the entries in a matrix to be random variables, then we have the so-called random matrices. This has developed into a deep area in mathematics. On infinite dimensional spaces, however, the random theory is poorly understood so far. Of course, for some special cases, such as random Schrodinger operators, due to their physical background, are carefully analyzed. Random Toeplitz operators are also scarely studied, but far from being systematic. Currently, as far as the general operator theory is concerned, the random theory is essentially untapped. This demands a good understanding of both operator theory and probability theory, and, moreover, one needs to find the right questions to ask. In this proposal, we attempt to start with the most important non-self-adjoint operator：the unilateral shift, and develop a random theory for it. Then the second problem we consider will be the connection between point processes and random zero sets of analytic functions. |
