This seminar is designed to create an environment for graduate students working on algebraic geometry and neighboring fields to give talks about their research, advanced topics they are studying on. During Autumn 2020 Semester, we meet 11:30am-12:30pm on Friday and due to Covid-19, the meeting will take place on Zoom. If you are interested please send me an e-mail (email@example.com).
Previously Yilong Zhang and Aniket Shah were the organizers. You can find the talks given during the previous semesters on the following link:
Title: Vanishing Cycles and Geometry of Cubic Threefolds
Abstract: For a smooth complex variety X embedded in projective space P^n, a general hyperplane intersects X transversely and the hyperplane section is smooth. As the hyperplane moves towards a special position, the hyperplane section can acquire with a node. Along with this process, there is a topological sphere specializing into the node, whose cohomology class is called a vanishing cycle. It was shown by the speaker recently that for X being certain threefold, the monodromy of a vanishing cycle can be used to generate the 3rd cohomology of X over rational numbers, which generalizes a result by Schnell in 2011. In this talk, we will restrict to the case for X being a cubic threefold, and we will see how the vanishing cycles on cubic surfaces describe the geometry of X.
Abstract: This is a talk on higher structures and stacky formulations of classical field theories. In this talk, we always consider higher spaces in algebraic geometry in a functorial perspective. In the first part of the talk, we shall provide a suitable language in order to define the notion of a pre-stack in a functorial manner and try to stress the role of higher spaces in moduli theory. In that respect, the basics of higher category theory will also be introduced. In order to encode the so-called local-to-global properties, on the other hand, we shall employ the homotopy theory of stacks. In the second part of the talk, we shall study the main ingredients of this framework. Having established enough background material, we will present two important examples from gauge theories and Einstein gravity together with some related results from our work.
Speaker: Ian Cavey
Title: Newton-Okounkov Bodies, with Examples
Abstract: Newton-Okounkov bodies are convex bodies that one can associate to a line bundle on a variety. These convex bodies are well-behaved as one varies the line bundle, and the collection of all these convex bodies on a particular space encodes geometric information about the space which can be quite subtle. In this talk I will try to answer the question "what is a Newton-Okounkov body?", state some of the main properties, and show by example how to extract nontrivial information out of them.
Speaker: Aziz Burak Gülen
Title: Linear Systems of Divisors on Metric Graphs
Abstract: Abstract tropical curves are basically metric graphs. Given a divisor D on a metric graph, the linear system |D| has the structure of a cell complex. In this talk, we will introduce anchor divisors and anchor cells, which help us find the f-vector and the cells of |D|. We will see some examples for genus 3 metric graphs, i.e. abstract tropical quartics.
Speaker: Deniz Genlik
Title: What is a Stack?
Abstract: Stacks are inevitable part of algebraic geometry; they appear mostly in moduli theory as natural objects to consider. In this talk, starting from the very basics we will give the definition of a stack. If time permits, we will define algebraic stacks and stack quotients. To be able to follow the talk fully, an exposure to the definition of a vector bundle and basic category theory is enough. Knowledge of algebraic geometry will not be assumed in most of the talk.
Speaker: Jonghoo Lee
Title: Galois Categories and Étale Fundamental Groups
Abstract: Galois category provides a uniform way of describing the Galois correspondences such as Galois theory for fields or the covering space theory for topological spaces. In this talk, we will give the definition of Galois category and the main theorem. After that, we will see how the fundamental group of a scheme is defined using this idea.