RTG: Algebraic Geometry and Representation Theory @ Northeastern
Samuel Grushevsky (Stony Brook University), Mirzakhani's recursion for Weil-Petersson volumes of moduli spaces.
- Time and rooms:
Tuesday, April 17, 10:30am-12:00pm, Room: Lake Hall -- LA 509 (LA is building 34 on this map).
Wednesday, April 18, 10:30am-12:00pm, LA 509.
Thursday, April 19, 10:30am-12:00pm, Richards Hall -- RI 458 (RI is building 42 on this map).
The Weil-Petersson metric is a natural Kaehler metric on the moduli space of Riemann surfaces. We will start by outlining the proof of Wolpert's formula that expresses the metric in terms of the "length and twist" parameters for cutting up a Riemann surface into a collection of spheres with three holes (i.e. in terms of Fenchel-Nielsen coordinates). We will then state Mirzakhani's recursive formula for the Weil-Petersson volumes of the moduli spaces of bordered Riemann surfaces, and explain some ingredients of the proof, and also how this recursion implies the Witten's conjecture that the intersection numbers of tautological classes on moduli satisfies the KdV integrable hierarchy.
Alex Küronya (Goethe-Universität Frankfurt), Geometric aspects of Newton-Okounkov bodies.
- Time and rooms:
Monday, March 12, 2:30-3:30pm. Room: Lake Hall -- LA 509 (LA is building 34 on this map).
Tuesday, March 13, 10:30-11:30am, LA 509.
Wednesday, March 14, 2:30-3:30pm, LA 509.
Thursday, March 15, 10:30-11:30am, LA 509.
Recent years have witnessed a new way to introduce convex geometric methods to areas of mathematics around algebraic geometry: based on earlier works of Newton and Okounkov, Kaveh-Khovanskii and Lazarsfeld-Mustata defined convex bodies (so-called Newton-Okounkov bodies), which capture the vanishing behaviour of sections of line bundles.
As a first approximation, the theory of Newton-Okounkov bodies is an attempt to create a correspondence between line bundles and convex bodies known from toric geometry, except that in the absence of a large torus action, one has to make do with an infinite collection of bodies for every line bundle.
This point of view has been fairly successful in that Newton-Okounkov bodies has been shown to encode positivity of line bundles, and they also serve as targets for completely integrable systems analogous to moment maps. The theory has exciting connections with symplectic geometry, representation theory, and combinatorics for instance, nevertheless, in these lectures we will focus on its applications to projective geometry.
After reviewing fundamental notions of positivity for line bundles and introducing Newton-Okounkov bodies along with their basic theory, we will discuss the case of surfaces, where there is a particularly satisfying theory, and the connection to (local) positivity of line bundles. If time permits, we will look at how to define interesting functions on Newton-Okounkov bodies which yield an interesting connection to Diophantine approximation.
François Charles (Université Paris-Sud), Algebraic cycles and Arakelov geometry.
- Time and rooms:
Tuesday, October 10, 9-10:30am, Behrakis 204
Wednesday, October 11, 5-6:30pm, Cargill 097
Friday, October 13, 10-11:30am, Ryder 155
Arakelov geometry gives a way to work geometrically with schemes defined over the integers. We will discuss some applications of Arakelov geometry to some problems in algebraic cycles and periods, trying to emphasize how geometric ideas can be translated in the setting of arithmetic geometry. The plan is:
Lecture 1: general setting of Arakelov geometry, relationship to geometry of numbers.
Lecture 2: application of arithmetic intersection theory to isogenies of elliptic curves.
Lecture 3: application to transcendance problems, theta-invariants.