With support from this award, the Southwest Center for Arithmetic Geometry will continue its series of annual "Winter Schools" from 2023 to 2025 at the University of Arizona in Tucson, AZ, the first taking place March 4-8, 2023. 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 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. This cycle the Southwest Center will expand its programming to include Preliminary Arizona Winter School (PAWS). This virtual program aimed at advanced undergraduates and junior graduate students from underrepresented groups will feature two six-week-long courses during the fall semester organized around the same topic as the AWS. Participants will engage in weekly problem sessions run by advanced graduate students and participate in community building and mentorship activities. At both AWS and PAWS, 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 both AWS and PAWS, including lecture notes, project descriptions, and audio and video of lectures. Through these thorough records, the efforts of the Southwest Center participants are made freely 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/ In 2022-2023, PAWS and AWS will be on "Unlikely Intersections." This topic concerns the expected finiteness of certain rare arithmetically or geometrically interesting intersections. For example, given a curve and a countable collection of "special" points in its ambient space, one expects the intersection of the curve with this set of points to be finite unless the curve is also "special." There has been substantial recent progress on the main conjectures and complementary problems of this field, especially on the Andr�-Oort/Lang-Manin-Mumford/Zilber-Pink conjectures and analogues from arithmetic dynamics. This work draws together not just classical tools from algebraic and arithmetic geometry, but also tools from logic, heights, analytic number theory, dynamics, and p-adic geometry. The program will introduce students to the central problems, applications, and unique tools of this flourishing field. 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.