With support from this award, the Southwest Center for Arithmetic Geometry will continue its series of annual "Winter Schools" in 2019 taking place March 2-6, 2019 at the University of Arizona in Tucson, AZ. Since its founding in 1997, the primary activity of the Southwest Center is the Arizona Winter School (AWS), an annual meeting which has become a prominent national event and provides high-level training and research experience for graduate students in arithmetic geometry and related areas. The AWS is organized around a different central topic each year and features a set of courses and accompanying research projects carefully designed and delivered by leading and emerging experts. The result is a unique fusion of traditional mathematics conference and intensive research workshop: the speakers organize courses of four or five lectures, provide lecture notes in advance, and propose research projects for graduate students to work on during the meeting. Nightly working sessions on these projects and on separate problem sets are run by the speakers and postdoctoral fellows. On the last day, students present their findings to the participants of the meeting. The result is a particularly intense and focused five days of mathematical activity for everyone involved. At the AWS, connections among peers are formed, and mentoring relationships between students and senior researchers are developed. Subsequent collaborations between participants at all levels are the norm. Students make concrete strides toward becoming research mathematicians, post-doctoral assistants gain valuable mentoring experience in their academic careers, and faculty develop new interests and see new connections that lead to important published results. The Southwest Center website shares reusable content from the Winter Schools, including lecture notes, project descriptions, and audio and video of lectures (both live-streamed and permanently archived). Through these thorough records, the dialogues begun at the AWS are extended to the greater community, and the efforts of the AWS participants are made freely and indefinitely available to all. More information about the upcoming and past Arizona Winter School programs can be found at the Southwest Center's website: http://swc.math.arizona.edu/ The 2019 AWS will be held on the topic of Topology and Arithmetic, exploring the boundary of higher homotopy theory and arithmetic geometry. The development and maturation of higher category/homotopy theory is one of the most significant achievements in topology in the last decade, and has recently found striking applications to number theory and arithmetic geometry. The earliest applications were to topological quantum field theories, cobordisms, and elliptic cohomology; even here, the interaction between generalized cohomology theories and formal groups yielded results of interest to arithmetic geometry (especially, the elucidation of the geometry of Lubin--Tate and Gross--Hopkins moduli spaces of formal groups). More recent progress abounds. Gaitsgory--Lurie study Tamagawa numbers (a fundamental invariant of algebraic groups over global fields) via topological techniques. Bhatt--Morrow--Scholze establish the existence of a weight filtration on topological Hochschild homology whose graded pieces are related both to classical cohomology theories (de Rham, crystalline) and to integral p-adic Hodge theory. Work of Wickelgren and others develop arithmetic aspects of A1-homotopy theory, extending some enumerative problems from the complex numbers to other fields. The 2019 Winter School will focus on exposition of applications and connections to arithmetic geometry, rather than the development of a general theory. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.