We will be holding JGMRT this semester on Fridays from 14:00-15:00.
Title: A gentle introduction to stacks
Abstract: This talk will be a very gentle introduction to some of the ideas of stacks. I will not claim to be comprehensive, nor super precise, but my goal is to introduce some of the ideas behind stacks and why people thought of these things. Stacks can be thought of as spaces where instead of having an underlying set with a topology, we replace the set with a site \(\mathcal{C}\) which is a category equipped with what's called a Grothendieck topology. In Part I of the talk, I will talk about some of the motivation behind stacks coming from moduli theory. In Part II I will set up some technology regarding Grothendieck topologies, and in Part III, I will define what a stack is and show off some examples.
Title: Gentle introduction to stacks part 2
Abstract: We will continue from our talk on 2/16. This time I will actually define what a stack is along with discussing some example as well as some further topics.
Title: Six Functor Formalisms
Abstract: Many cohomology theories (e.g. singular cohomology of topological spaces, de Rham cohomology of smooth manifolds, Betti/l-adic/etale cohomology of schemes, etc.) share similar structural enhancements generalizing the usual functoriality, cup product, Kunneth formula, and Poincare duality of singular cohomology. In many cases, these structures can be seen as consequences of the existence of six 'distinguished' functors on categories giving rise to these theories, together with various relationships among these functors and properties they satisfy. We'll spend a bit of time discussing six functor formalisms in general, then specialize our attention to the case of singular cohomology of sufficiently nice topological spaces. Here the relevant category is sheaves of abelian groups on a space; we'll construct each of the six functors in this case in detail and sketch the passage from the properties of these functors to familiar structures in singular cohomology.
Title: Mori's Bend and Break
Abstract: The purpose of this talk is to give a glimpse at a very interesting piece of mathematics. Mori's Bend and Break technique incorporate some of the most interesting features of algebraic geometry, namely deformation theory techniques and passing to characteristic \(p\). The main goal of this exposition is to use the bend and break technique to prove that on every point on a (smooth) Fano variety there is a rational curve passing through it.