Quadratic Forms, Milnor K-Theory, and Arithmetic - 2021

Explore the intersection of quadratic forms, Milnor K-theory, and arithmetic through this comprehensive undergraduate summer school program from the Park City Mathematics Institute. Develop a thorough understanding of quadratic form theory and its connections to arithmetic and algebraic geometry, beginning with foundational concepts before specializing in applications to rational numbers. Master key mathematical results including the Hasse-Minkowski theorem, the quadratic reciprocity law and its formulation via Milnor K-theory, and the Siegel mass formula. Engage with experimental mathematics components featuring open-ended problems and computational work alongside daily lectures delivered by Dustin Clausen from the University of Copenhagen and Akhil Mathew from the University of Chicago. Study the algebra and arithmetic of quadratic forms through seven detailed sessions, then delve into Hilbert reciprocity theory across eight comprehensive lectures. Build upon prerequisites in abstract algebra while utilizing concepts from general analysis and topology, particularly in discussions of p-adic numbers and metric space completions.

Syllabus

The algebra and the arithmetic of quadratic forms - Akhil Mathew
The algebra and the arithmetic of quadratic forms II - Akhil Mathew
The algebra and the arithmetic of quadratic forms III - Akhil Mathew
The algebra and the arithmetic of quadratic forms IV - Akhil Mathew
The algebra and the arithmetic of quadratic forms V - Akhil Mathew
The algebra and the arithmetic of quadratic forms VI - Akhil Mathew
The algebra and the arithmetic of quadratic forms VII - Akhil Mathew
Hilbert Reciprocity - Dustin Clausen
Hilbert Reciprocity II - Dustin Clausen
Hilbert Reciprocity III - Dustin Clausen
Hilbert Reciprocity IV - Dustin Clausen
Hilbert Reciprocity V - Dustin Clausen
Hilbert Reciprocity VI - Dustin Clausen
Hilbert Reciprocity VII - Dustin Clausen
Hilbert Reciprocity VIII - Dustin Clausen