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