Genus six curves, K3 surfaces, and stable pairs
In this talk, we will explore some of the ideas surrounding the relationship between genus six curves, K3 surfaces, and stable pairs. We will begin with an informal conversation about moduli spaces in general and then later specialize the discussion to the above particular examples. Along the way, we will talk about basic techniques from birational geometry.
Categorification with examples from Topology
Examples of categorifiction appear in multiple branches of mathematics. In this talk we will see how categorification can provide a framework to attack new problems via two examples from Topology. The first example is the categorification of Betti numbers by singular homology which provides a framework to answer questions about maps between topological spaces. The second example is the categorification of the Jones polynomial by Khovanov homology which provides a framework to answer questions about link cobordisms.
“Algebraic” Dynamics: Where geometry meets number theory
Invariants are helpful across many different branches of mathematics for helping distinguish different objects. In dynamics, one such example is (topological) entropy. For continuous maps of a compact interval, theorems of Thurston reduce the study of this invariant to the study of a special set of algebraic integers, called “weak Perron numbers.” Moreover, we can further restrict our attention to “simple” maps: piecewise linear functions f with |f’|=\lambda wherever f’ exists, \lambda being a weak Perron number. Investigating these maps quickly takes us into the realm of Galois theory, melding algebraic and geometric notions together to give some pretty fantastic results. In this talk I will discuss some basic results in this field, as well as some of the constructions that I am currently studying. Time permitting, we’ll have a look at pretty pictures and ponder some open questions.
Effective divisors in moduli spaces
In this talk, I will discuss effective divisors in the moduli space of curves and in the projectivized Hodge bundle over the moduli space of curves. In particular, I will discuss one method to compute the class of these divisors in terms of the basis of the Picard group. We will focus on the example of the Weierstrass divisor in \bar{M}_{g,1} and an analogous divisor in the projectivized Hodge bundle over \bar{M}_g. Unlike this abstract, the talk will not require any background in algebraic geometry.
Spectra
There is a notion of homotopy in the world of CW complexes and a notion of chain homotopy in the world of chain complexes. As it turns out, spectra, with the fancier name of stable homotopy category to go by, is a larger world in where CW complexes and chain complexes both live and meet. Starting from the historical motivation of stable homotopy groups, I will talk about a few examples and properties of spectra, while giving some explanation as to why this seemingly topological gadget is useful for commutative algebraists. If time permits (though unlikely), I will end with a very brief excursion of Topological Hochschild Homology (THH), for which a rigorous setup such as spectra is needed, and how THH provides glimpses of the crystalline cohomology and other exotic cohomologies (work due to Bhatt, Scholze and Morrow).
Graphs, Lattices, and Graph Lattices
Graphs and lattices are combinatorial tools that show up in many branches of math. In this talk, I'll present a way to associate a lattice to a graph and a graph to a lattice in such a way that properties of one correspond to properties of the other (in a meaningful way). (For anyone coming to my talk in the Topology Reading Group on April 4, this material is supplementary but not requisite.)
Mackey Theory and its applications
Mackey Theory is concerned with intertwining operators between a pair of induced representations. In this talk, we will first introduce the statement of Mackey Theory for finite groups. We will then see how this can be applied to study the representations of the group GL(2) over finite fields.
Mori's Bend and Break
In this talk I will prove Mori’s famous Bend and Break theorem(s). It’s a wonderfully simple geometric argument and shows a very interesting relationship between algebraic geometry in characteristic p and characteristic 0.