Topics in random graphs, combinatorial optimization, and statistical inference

The manuscript is made of three chapters presenting three differenttopics on which I worked with Ph.D. students. Each chapter can be read independently of the others andshould be relatively self-contained. Chapter 1 is a gentle introduction to the theory of random graphswith an emphasis on contagions on such networks. In Chapter 2, I explain the main ideas of the objectivemethod developed by Aldous and Steele applied to the spectral measure of random graphs and themonomer-dimer problem. This topic is dear to me and I hope that this chapter will convince the readerthat it is an exciting field of research. Chapter 3 deals with problems in high-dimensional statistics whichnow occupy a large proportion of my time. Unlike Chapters 1 and 2 which could be easily extended inlecture notes, I felt that the material in Chapter 3 was not ready for such a treatment. This field ofresearch is currently very active and I decided to present two of my recent contributions