Tomer Schlank (U Chicago): Random Links and Arithmetic Statistics
- Event Type
- Seminar/Symposium
- Sponsor
- UIUC math department
- Location
- Altgeld 241
- Date
- Sep 10, 2024 11:00 am
- Views
- 100
- Originating Calendar
- Topology Seminar
Étalé homotopy is a theory that enables the study of algebraic and arithmetic concepts through geometric perspectives. Barry Mazur observed that, from this viewpoint, square-free integers are analogous to links in three-dimensional space. We will explore this analogy and propose a way to give it a statistical flavor. Informally, we assert that a random square-free integer of size approximately X resembles the closure of a random braid on roughly logX strands. We will make this statement more precise by introducing a number-theoretical analog for a family of numerical link invariants (called Kei's) and use this analogy to propose a conjecture regarding their asymptotic behavior. We will also present a few cases where this conjecture has been proven.