Abstracts

Given a polynomial f(z), how can we describe a root of f in terms of its coefficients? Trying to answer this question leads to an invariant known as resolvent degree. In this talk, we will address the original question, discuss what we know about resolvent degree, and use resolvent degree to connect questions about solving polynomials to other geometric and representation theoretic problems.

Broadly speaking, Castelnuovo–Mumford regularity should be thought of as a measure of how complicated a (homogenous) ideal in a polynomial ring can be. It turns out that understanding this complexity highlights the deep connections between the algebra of an ideal and the geometry of the corresponding algebraic variety.

Classical examples of valuation fields are the field of $p$-adic numbers $\Q_p$ for a prime $p$ and the fraction field $k((X))$ of the power series ring $k[[X]]$ for a perfect field $k$ (such as $\F_p$ or $\C$). The valuation, obtained by looking at the exact power of $p$ (respectively $X$) dividing an element of the field, is discrete. Moreover, these fields are complete under the non-Archimedean metric induced by this valuation. Classical invariants of ramification theory capture important information about extensions of these fields.

This nice structure is lost when we allow non-discrete valuations and relax the conditions on the field $k$. For example, many interesting objects in the classical case are generated by a single element. However, in the general case, even finite generation is not guaranteed. In particular, we may encounter the mysterious phenomenon of the {\em defect} (or ramification deficiency). The occurrence of a non-trivial defect is one of the main obstacles to long-standing problems, such as obtaining resolution of singularities in positive characteristic.

In this talk, we will introduce the notion of the defect and discuss how the study of ramification theory leads to a better understanding of the defect.

Using a single point as a toy example, we will take the scenic route for describing the derived category of coherent sheaves. Along the way, we will see familiar examples of free resolutions coming from commutative algebra and peek at strange resolutions over exterior algebras, until we realize that the real treasures are the exceptional collections that helped us make derived categories a little more explicit.

A Cayley diagram is a visualization of a group presentation. These arise in fields such as geometry group theory and topology, but I will focus on their role in elementary abstract algebra. I hope to convince you that learning or teaching algebra without these tools is akin to learning or teaching calculus without graphs. This talk will be filled with colorful pictures and visualizations, and I promise that you will leave with new ideas that will forever change how you think about algebra.

In the theory of elliptic curves, understanding the behavior of rank is a central problem. In light of the Birch--Swinnerton-Dyer and Tate-Shafarevich Conjectures, there are three avenues for understanding rank of a given elliptic curve E/K: by the structure of the Mordell-Weil group E(K), by the vanishing of the associated L-function L(E/K, s), or by the structure of the associated Selmer group Sel(E/K). We will discuss some of the big ideas for attacking the rank problem over number fields via the Selmer group approach, as well as methods of comparing parallel tools in the L-function approach.

Slides for Sunil's talk are now available here.