The analysis is extended to nonorientable surfaces and to surfaces with boundaries. A remarkable duality for the average of characteristic polynomials is essential for obtaining such topological invariants. The main interest is to use this freedom to compute various topological invariants for surfaces such as the intersection numbers for curves drawn on a surface of given genus with marked points, Euler characteristics, and the Gromov–Witten invariants. We show that results from the theory of random matrices are potentially of great interest to understand the statistical structure of the empirical. The freedom given by the external source allows for various tunings to different classes of universality. In such models the correlation functions are known exactly for an arbitrary source and for any size of the matrices. This dissertation uses random matrix theory (RMT). We consider Gaussian random matrix models in the presence of a deterministic matrix source. How to deal with these new features often plays a key role in modern statistical and machine learning research. Introduction: A Computational Trick Can Also Be a Theoretical Trick 1 2. This is a first book to show that the theory of the Gaussian random matrix is essential to understand the universal correlations with random fluctuations and to demonstrate that it is useful to evaluate topological universal quantities. random matrix techniques as the stochastic operator approach, the method of ghosts and shadows, and the method of \Riccatti Di usion/Sturm Sequences,' giving new insights into the deeper mathematics underneath random matrix theory.
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