Algebra, Geometry, and Number Theory - Student Seminar

Abstracts

Pat Lank - $F$-thick varieties: A survey - Within this talk, I introduce a notion of Frobenius generation in the bounded category of coherent sheaves for a variety. This will cover known examples, and some new.

Baiely Heath - A Walk On the Weyl Side: An Introduction to Root Systems and Weyl Groups - We will begin this talk by defining root systems, which are finite subsets of Euclidean space satisfying nice properties, such as invariance under the reflections they determine. These sets and their automorphisms give rise to very special and ubiquitous groups called Weyl groups, as well as interesting invariant lattices under the action of the Weyl group. Finally, we will discuss how Weyl groups may be used to construct maximal finite subgroups of $\operatorname{GL}_n\left(\mathbb{Z}\right)$, an important application of these ideas.

Anirban Bhaduri - Representation of Quiver algebras - We discuss the Quiver representation, Path algebra, the indecomposable modules over path algebra, the simple and the projective modules. We specially talk about Finite representation type with an Example of $kA_n$.

Pat Lank - Dimension and generation for derived categories - Within this talk, I will discuss a method for constructing objects within the derived category of complexes of finitely generated modules from a single object, called a classical generator, and three operations (i.e. cones, shifts, and direct summands). This gives one a meaningful understanding to the structure of the bounded derived category by isolating an object and studying it thru these operations. From this, we can determine whether or not there is a uniform bound on the number of steps required to build any object from a given classical generator. In the case that there is a finite number, we call such objects strong generators. This introduces the question, what is the smallest number of steps required to build an object? We answer these questions and explore further notions of dimension which have been introduced for derived categories.

Shreya Sharma - Intersection numbers - Given two polynomials $f$ and $g$ in variables $x$ and $y$, it is possible to find the number of common solutions by solving $f=g=0$. In some more generality, one can ask if there are any common solutions to homogeneous polynomials $f_1, \ldots f_m$ in $n$ variables. And if yes, how many? Intersection theory deals with the generalization of these questions in higher dimensions and other spaces besides $\mathbb{P}^n$. In this talk, I will motivate and describe the ‘intersection number’ for two curves(effective divisors) on a surface and compute it in some cases. All the necessary notations and definitions will be given as required and are taken from Hartshorne’s Algebraic Geometry. The talk is roughly based on Chapter 5.1 from the book.