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This paper consists of four parts. In the first part, we explain what eigenvalues we are interested in and show the difficulties of the study on the first (non-trivial) eigenvalue through examples. In the second part, we present some (dual) variational formulas and explicit bounds for the first eigenvalue of Laplacian on Riemannian manifolds or Jacobi matrices (Markov chains). Here, a probabilistic approach--the coupling methods is adopted. In the third part, we introduce recent lower bounds of several basic inequalities; these are based on a generalization of Cheeger's approach which comes from Riemannian geometry. In the last part, a diagram of nine different types of ergodicity and a table of explicit criteria for them are presented. These criteria are motivated by the weighted Hardy inequality which comes from Harmonic analysis.

This paper surveys the main results obtained during the period 1992-1999 on three aspects mentioned at the title. The first result is a new and general variational formula for the lower bound of spectral gap (i.e., the first non-trivial eigenvalue) of elliptic operators in Euclidean space, Laplacian on Riemannian manifolds or Markov chains (\S 1). Here, a probabilistic method-coupling method is adopted. The new formula is a dual of the classical variational formula. The last formula is actually equivalent to Poincar\'e inequality. To which, there are closely related logarithmic Sobolev inequality, Nash inequality, Liggett inequality and so on. These inequalities are treated in a unified way by using Cheeger's method which comes from Riemannian geometry. This consists of \S 2. The results on these two aspects are mainly completed by the author joint with F. Y. Wang. Furthermore, a diagram of the inequalities and the traditional three types of ergodicity is presented (\S 3). The diagram extends the ergodic theory of Markov processes. The details of the methods used in the paper will be explained in a subsequent paper under the same title.