• Speaker: Ioannis Zachos, Michigan State
• Title: Gröbner basis and the Ideal Membership problem
• Date: 09/09/2019
• Time: 4:30 PM - 5:30 PM
• Place: C304 Wells Hall

We know from the Hilbert Basis Theorem that any ideal in a polynomial ring over a field is finitely generated. However, there remains question as to the best generators to choose to describe the ideal. Are there generators for a polynomial ideal $I$ that make it easy to see if a given polynomial $f$ belongs to $I$? For instance, does $2x^2z^2+2xyz^2+2xz^3+z^3-1$ belong to $I=(x+y+z, xy+xz+yz, xyz−1)$? Deciding if a polynomial is in an ideal is called the Ideal Membership Problem. In polynomial rings of one variable, we use long division of polynomials to solve this problem. There is a corresponding algorithm for $K[x_1,\ldots, x_n]$, but because there are multiple variables and multiple divisors, the remainder of the division is not unique. Hence a remainder of $0$ is a sufficient condition, but not a necessary condition, to determine ideal membership. However, if we choose the correct divisors, then the remainder is unique regardless of the order of the divisors. These divisors are called a Gröbner basis. In our talk we will define the Gröbner basis and see how it solves the Ideal Membership Problem.

• Speaker: Nick Rekuski, Michigan State
• Title: Splitting Criteria for Vector Bundles on $\mathbb{P}^n$
• Date: 09/23/2019
• Time: 4:30 PM - 5:30 PM
• Place: C304 Wells Hall

Grothendieck's Theorem says that any vector bundle on $\mathbb{P}^1$ can be decomposed as a finite sum of line bundles. In this talk, we will discuss a generalization of this theorem: Horrocks Splitting Criterion. This criterion completely describes when a vector bundle on $\mathbb{P}^n$ splits as a sum of line bundles. We will then discuss an open conjecture of Hartshorne. If time permits, we will also consider the similar question of classifying when a vector bundle on $\mathbb{P}^n$ decompose as line bundles and twists of the tangent bundle.

• Speaker: Zheng Xiao, MSU
• Title: Recent results of GCD problems on almost $S$-units and recurrences
• Date: 10/07/2019
• Time: 4:30 PM - 5:30 PM
• Place: C517 Wells Hall

The GCD problem is one of the major problems in Diophantine Geometry. Corvaja, Zannier and Bugeaud first gave a fundamental result on GCD of integers powers and then generalized to rational numbers and algebraic numbers by many mathematicians. In this talk I will introduce recent GCD results on $S$-units due to Levin and generalize to almost $S$-units. I will give the definition of almost units and present the main theorem of GCD on multivariable polynomials, which is lead to a result about recurrence sequences. If time allows, I will also introduce Silverman’s generalized GCD along the blow up of a closed subscheme and apply to abelian surface case and its connection to Vojta’s conjecture.

• Speaker: Yu Shen, Michigan State
• Title: Serre Duality I
• Date: 10/21/2019
• Time: 4:30 PM - 5:30 PM
• Place: C304 Wells Hall

Serre duality was first proved by Serre in 1950s. It is a very useful tool in algebraic and complex geometry. In this lecture, I will use Čech cohomology to prove Serre duality of projective varieties. If time permits, I would like to talk about some applications of it.

• Speaker: Zheng Xiao, Michigan State
• Title: Arithmetic intersection theory and Arakelov's Hodge Index Theorem
• Date: 11/04/2019
• Time: 4:30 PM - 5:30 PM
• Place: C304 Wells Hall

The famous Mordell-Weil conjecture was first proved by Faltings in a classical way, then Vojta gave an alternative proof using arithmetic Arakelov geometry, which is one big motivation for developing Arakelov theory into a mature tool. In this talk I will introduce Neron functions and divisors, which is an arithmetic approach to define divisors rather than classical algebraic geometry. We shall also cover arithmetic chow groups and the arithmetic intersection number. In the end I will present Neron symbols and use it to give a sketch proof of Arakelov’s Hodge Index Theorem.

• Speaker: Joshua Ruiter, Michigan State
• Title: Split and non-split tori
• Date: 11/18/2019
• Time: 4:30 PM - 5:30 PM
• Place: C517 Wells Hall

Tori are an important structural aspect of algebraic groups, and "split" vs "non-split" tori are especially important. Unfortunately, "non-split" phenomena only occur over non-algebraically closed fields, so not all the traditional tools of classical algebraic geometry apply. Using the generalized field norm map from my talk last week, I'll describe a concrete example of a non-split torus. Then we'll try to use that example to try and understand the importance of non-split tori.