"Arithmetic statistics" concerns counting arithmetic objects: number fields, class groups, rational points on algebraic varieties, and so on. I will give an overview of some interesting results in this subject -- starting with Gauss and continuing through ongoing work of many (including your colleagues Arul Shankar and Jacob Tsimerman!) in the present day.
The subject is interesting both from an algebraic point of view (how does one relate these objects to something which is more easily counted?) and an analytic point of view (how does one get the sharpest counting results, and then what applications do they have?), and in this talk I will focus mostly on the latter. I will outline a program with Takashi Taniguchi to sharpen our tools, and I will describe the kinds of applications that these tools are good for.