UC San Diego

In this dissertation we will first give an exposition for some topics on Perelman's entropy and some results related to the analysis of the entropy, and then present the content of two papers among the author's publication list.

Chapter 1 is an introduction to Perelman's entropy and the author's main results. The statement of these main results can be found in section 1.3.

Chapter 2 and chapter 3 are expository materials on Perelman's entropy and its related analytic tools; these results are included because of their importance to our main theorems.

In chapter 4, we prove some estimates for the Nash entropy on ancient solutions and thereby prove a gap theorem for the asymptotic entropy.

In chapter 5, we prove an assertion made by Perelman, saying that for an ancient solution to the Ricci flow with bounded and nonnegative curvature operator, bounded entropy is equivalent to $\kappa$-noncollapsing on all scales. This proof is based on accurate gaussian upper and lower estimates for the conjugate heat kernel.