| Week | Date | Topic | Relevant Katok Reading | Background/Alt/Further Reading | Notes/Slides/Recordings | | --- | --- | --- | --- | --- | --- | | 0 | 1/23 | Organizational Meet + Greet | | | | | 1 | 1/28 | Introducing P-adics Part I | 1.1, 1.2, 1.3 | Equivalence of Absolute Values [K. Conrad] Equivalence of Norms [K. Conrad] Metric Spaces [K. Conrad] The P-adic Expanson of Rational Numbers [K. Conrad] | https://www.youtube.com/watch?v=dmlAERBPkqI&list=PL6nyQ4Ix4QG2jt4bWaAxOSj1cQiYMh6qJ&index=1 | | 2 | 2/4 | Introducing P-Adics Part II | 1.4 | Modular Arithmetic [K. Conrad] | | | 3 | 2/11 | The Arithmetic and Algebra of Q_p and Z_p | 1.5, 1.6, 1.8 | Standard Definitions for Rings [K. Conrad] Modular Arithmetic [K. Conrad] The Local-Global Principle [K. Conrad] | | | 4 | 2/18 | Theorems of Hensel and Ostrowski | 1.7, 1.9, 1.10 (optionally) | Modular Arithmetic [K. Conrad] Hensel’s Lemma [K. Conrad]
Ostrowski’s Theorem for Number Fields [K. Conrad] Compare to Ostrowski’s Theorem for Q [K. Conrad[ | | | 5 | 2/25 | Basic Topology towards the Baire Category Theorem for P-adics | 2.1, 4.3 | Metric Spaces [K. Conrad] | | | | | MIDTERMS | | | | | | | SPRING BREAK | | | | | 6 | 3/18 | Introducing P-adic Power Series | 3.1, 3.2 | Infinite Series in P-adic Fields [K. Conrad] | | | 7 | 3/25 | P-Adic Geo Part I | N/A | | | | 8 | 4/1 | P-Adic Geo Part II | N/A | | | | 9 | 4/8 | Algebraic Implications of Valuations | N/A | Local Fields [C. Hsu] | | | 10 | 4/15 | Continuity and Locally Constant Functions | 4.1, 4.2, 4.4 | | | | 11 | 4/22 | Mahler Expansions | N/A | Mahler Expansions [K. Conrad] | | | 12 | 4/29 | TBD | TBD | TBD | | | | | FINALS | | Some Further Directions with P-adics:

History of Class Field Theory [K. Conrad] Representations of p-adic Groups [J. Fintzen] Adic Spaces [S. Morel] Several Approaches to Non-archimedian Geometry [B. Conrad] | |