RTG: Algebraic Geometry and Representation Theory @ Northeastern

Advanced Mini-Courses:

Fall 2018:

Owen Gwilliam (University of Massachusetts Amherst), Chern-Simons theory & quantum groups approached through factorization algebras (Click on the link for a more detailed syllabus of the series).

  • Time and rooms:

Lecture 1: Tuesday, October 23, 10:30 AM – 12:00 PM.

Lecture 2: Wednesday, October 24, 4:00 PM – 5:30 PM.

Lecture 3: Thursday, October 25, 1:00 PM – 2:30 PM.

  • Abstract:

Over the last few decades, various mathematical communities have studied the three-dimensional topological field theory called Chern-Simons theory. Recently, ideas and methods from higher algebra have been applied to it. These lectures will give an introduction to such work, using it as an excuse to discuss some sophisticated ideas, like En algebras, factorization homology, or filtered Koszul duality. (We will not assume comfort with higher algebra on the part of the audience.) The focus will be on joint work with John Francis and Kevin Costello, which directly connects the Feynman diagrammatic approach (think: Vassiliev invariants of knots) to the quantum group approach (think: HOMFLYPT polynomials). We will also relate our approach to the recent work of Ben-Zvi, Brochier, and Jordan on “integrating quantum groups” and of Calaque, Pantev, Toen, Vaquie, and Vezzosi on categorical deformation quantization.

Spring 2018:

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).

  • Abstract:

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.

  • Abstract:

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.

Fall 2017:

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

  • Abstract:

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.