NT Learning Seminar

During Spring 2020, I organized the number theory learning seminar focused on O-minimality and Diophantine Geometry. We looked at how the Pila-Zannier strategy has been used to prove (and sometimes reprove) important conjectures on the transcendence properties of interesting functions (e.g. the complex exponential, the Weierstrass p function, the j function).

Overview of talks

• 1/22: Introduction and overview of topics to be covered (see sections 1 and 2 here).

• 1/29: Manin-Mumford for the exponential function.

• 2/5: Introduction to o-minimality (see section 3 here).

• 2/12: Pila-Wilkie point counting theorem (see Pila and Wilkie's original paper, Pila's strengthening to algebraic points, or section 4 here for an overview).

• 2/19: Proof of the Ax-Schanuel theorem for the exponential function using o-minimality (following Tsimerman, see here).

• 2/26: Manin-Mumford conjecture for powers of an elliptic curve (following Habegger*).

• 3/4: Rational Points On Grassmanians and Unlikely Intersections in Tori (following Capuano, Masser, Pila, Zannier, see here).

• 3/11: No seminar.

• 3/18: Galois orbits of torsion points and heights (following Habegger*). Slides by Roy Zhao.

• 3/25: Spring break.

• 4/1: Ax-Schanuel for the j function (following Pila-Tsimerman, see here). Slides by Roy Zhao.

• 4/8: André-Oort conjecture for powers of the modular curve (following Pila, see here). Slides.

• 4/15: Introduction to the Zilber-Pink conjecture. Slides by Vahagn Aslanyan.

• 4/22: Ax-Schanuel theorem for pure Shimura varieties (following Mok-Pila-Tsimerman, see here).

*This refers to P. Habegger's chapter "The Manin-Mumford Conjecture, an elliptic Curve, its Torsion Points & their Galois Orbits" from O-Minimality and Diophantine Geometry, London Mathematical Society Lecture Note Series 421, edited by G. O. Jones and A. J. Wilkie, Cambridge University Press 2015.