Hurwitz spaces, Hurwitz numbers, and the double ramification cycleLecture 1: From Hurwitz to ELSV. In this lecture we cover the basic notions in Hurwitz theory from a geometric and representation theoretic point of view, and we introduce and discuss the ELSV formula, which creates a connection between Hurwitz theory and tautological intersection theory in the moduli space of curves Lecture 2: Double Hurwitz Numbers. In this lecture we study the piecewise polynomiality and wall crossings behavior of double Hurwitz numbers, following work of Goulden-Jackson-Vakil, Shadrin-Shapiro-Vainshtein and Cavalieri-Johnson-Markwig. We introduce and discuss an ELSV type conjecture for double Hurwitz numbers. Lecture 3: Geometry behind Double Hurwitz Numbers. Here we explore the geometry underlying the piecewise polyomiality of double Hurwitz numbers. We develop an intersection theoretic formula for DhN's that explains piecewise polynomiality as a consequence of the polynomiality of the double ramification cycle. We then take a peek into the connection between classical and tropical Hurwitz theory. Lecture 4: The Double Ramification Cycle. We introduce and take a survey of the existing work on the Double Ramification Cycle, starting from the partial solution to Eliashberg's question given by Hain, the extension of Grushevski-Zakharov, and the work of Zvonkine and coauthors on "using" the DRC as a distinguished tautological class. We conclude by presenting an appealing conjecture by Pixton giving a closed form for the DRC. |

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