Events

Paul Cusson, University of Waterloo

The Kodaira embedding theorem and background material

The Kodaira embedding theorem is a crucial result in complex geometry that forms a nice bridge between differential and algebraic geometry, giving a necessary and sufficient condition for a compact complex manifold to be a smooth projective variety, that is, a complex submanifold of a complex projective space. The material and proof will follow the exposition in Griffiths & Harris's classic textbook.

MC 5479

Nikolay Bogachev, University of Toronto

Commensurability classes and quasi-arithmeticity of hyperbolic reflection groups

In the first part of the talk I will give an intro to the theory of hyperbolic reflection groups initiated by Vinberg in 1967. Namely, we will discuss the old remarkable and fundamental theorems and open problems from that time. The second part will be devoted to recent results regarding commensurability classes of finite-covolume reflection groups in the hyperbolic space H^n. We will also discuss the notion of quasi-arithmeticity (introduced by Vinberg in 1967) of hyperbolic lattices, which has recently become a subject of active research. The talk is partially based on a joint paper with S. Douba and J. Raimbault.

MC 5417

Matthew Harrison-Trainor, University of Illinois at Chicago

Back-and-forth games to characterize countable structures

Given two countable structures A and B of the same type, such as graphs, linear orders, or groups, two players Spoiler and Copier can play a back-and-forth games as follows. Spoiler begins by playing a tuple from A, to which Copier responds by playing a tuple of the same size from B. Spoiler then plays a tuple from B (adding it to the tuple from B already played by Copier), and Copier responds by playing a tuple from B (adding it to the tuple already played by Spoiler). They continue in this way, alternating between the two structures. Copier loses if at any point the tuples from A and B look different, e.g., if A and B are linear orders then the two tuples must be ordered in the same way. If Copier can keep copying forever, they win. A and B are isomorphic if and only if Copier has a winning strategy for this game.   Even if Copier does not have a winning strategy, they may be able to avoid losing for some (ordinal) amount of time. This gives a measure of similarity between A and B. A classical theorem of Scott says that for every structure A, there is an α such that if B is any countable structure, A is isomorphic to B if and only if Copier can avoid losing for α steps of the back-and-forth game, that is, when A is involved we only need to play the back-and-forth game for α many steps rather than the full infinite game. This gives a measure of complexity for A, called the Scott rank. I will introduce these ideas and talk about some recent results.

MC 5501

Candace Bethea, Duke University

The local equivariant degree and equivariant rational curve counting

I will talk about joint work with Kirsten Wickelgren on defining a global and local degree in stable equivariant homotopy theory. We construct the degree of a proper G-map between smooth G-manifolds and show a local to global property holds. This allows one to use the degree to compute topological invariants, such as the equivariant Euler characteristic and Euler number. I will discuss the construction of the equivariant degree and local degree, and I will give an application to counting orbits of rational plane cubics through 8 general points invariant under a finite group action on CP^2. This gives the first equivariantly enriched rational curve count, valued in the representation ring and Burnside ring. I will also show this equivariant enrichment recovers a Welchinger invariant in the case when Z/2 acts on CP^2 by conjugation.

MC 5417