Talk_id  Date  Speaker  Title 
19613

Monday 9/9 4:30 PM

Ioannis Zachos, Michigan State

Gröbner basis and the Ideal Membership problem
 Ioannis Zachos, Michigan State
 Gröbner basis and the Ideal Membership problem
 09/09/2019
 4:30 PM  5:30 PM
 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 $2x^2z^2+2xyz^2+2xz^3+z^31$ 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.

19610

Monday 9/16 4:30 PM

Chuangtian Guan, MSU

P.D. rings with a view towards Crystals
 Chuangtian Guan, MSU
 P.D. rings with a view towards Crystals
 09/16/2019
 4:30 PM  5:30 PM
 C304 Wells Hall
In this talk we will define P.D. rings, which are triples consisting a ring, an ideal of the ring and a map on an ideal mimicking $x^n/n!$. We will give some examples of P.D. rings and discuss their properties. Then we will use the P.D. structures to define the crystalline site of schemes and crystals. If time admits we will talk about some examples of crystals and explain why we care about them.

19631

Monday 9/23 4:30 PM

Nick Rekuski, Michigan State

Splitting Criteria for Vector Bundles on $\mathbb{P}^n$
 Nick Rekuski, Michigan State
 Splitting Criteria for Vector Bundles on $\mathbb{P}^n$
 09/23/2019
 4:30 PM  5:30 PM
 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.

19612

Monday 9/30 4:30 PM

Joshua Ruiter, Michigan State

Root systems  a powerful tool for classification
 Joshua Ruiter, Michigan State
 Root systems  a powerful tool for classification
 09/30/2019
 4:30 PM  5:30 PM
 C304 Wells Hall
Root systems arose historically as a tool for classifying semisimple Lie algebras, but they can also be understood without that context. I will describe several concrete examples of root systems, with plenty of pictures. I will describe how to associate a special graph called a Dynkin diagram to a root system, and briefly describe the classification of root systems. If time allows, I will describe some of the applications to classifying semisimple Lie algebras and reductive algebraic groups. All you need to know to understand my talk is how to compute dot products on $\mathbb{R}^n$.

19671

Monday 10/7 4:30 PM

Zheng Xiao, MSU

Recent results of GCD problems on almost $S$units and recurrences
 Zheng Xiao, MSU
 Recent results of GCD problems on almost $S$units and recurrences
 10/07/2019
 4:30 PM  5:30 PM
 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.

19614

Monday 10/14 4:30 PM

Dan Normand, Harvard University

The Isomorphism Theorems in an Abelian Category
 Dan Normand, Harvard University
 The Isomorphism Theorems in an Abelian Category
 10/14/2019
 4:30 PM  5:30 PM
 C304 Wells Hall
It is often said that abelian categories are where homology can "naturally" occur. As the notion of an isomorphism is indispensable to the study of homologyand an innate aspect of a category, one would hope that there are analogues to the usual three isomorphism theorems of algebra in an arbitrary abelian category. In this [talk] we show that there are indeed such analogues, and we spend time developing the machinery to implement them

19629

Monday 10/21 4:30 PM

Yu Shen, Michigan State

Serre Duality I
 Yu Shen, Michigan State
 Serre Duality I
 10/21/2019
 4:30 PM  5:30 PM
 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.

19630

Monday 10/28 4:30 PM

Yu Shen, Michigan State

Serre Duality II
 Yu Shen, Michigan State
 Serre Duality II
 10/28/2019
 4:30 PM  5:30 PM
 C517 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.

19672

Monday 11/4 4:30 PM

Zheng Xiao, Michigan State

Arithmetic intersection theory and Arakelov's Hodge Index Theorem
 Zheng Xiao, Michigan State
 Arithmetic intersection theory and Arakelov's Hodge Index Theorem
 11/04/2019
 4:30 PM  5:30 PM
 C304 Wells Hall
The famous MordellWeil 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.

20699

Monday 11/11 4:30 PM

Joshua Ruiter, Michigan State

Field norm for algebraic groups, with a view towards nonsplit tori
 Joshua Ruiter, Michigan State
 Field norm for algebraic groups, with a view towards nonsplit tori
 11/11/2019
 4:30 PM  5:30 PM
 C304 Wells Hall
Field norm maps are useful in many areas of algebra, such as Galois theory. Using the language of (affine) algebraic groups, I will place the field norm in a larger context, as a particular instance of a certain natural transformation. This will set us up for my talk the following week, on special subgroups of algebraic groups called tori, and what it means for such tori to be split or nonsplit. In particular, the generalized norm will provide a (somewhat) concrete example of a nonsplit torus.

20700

Monday 11/18 4:30 PM

Joshua Ruiter, Michigan State

Split and nonsplit tori
 Joshua Ruiter, Michigan State
 Split and nonsplit tori
 11/18/2019
 4:30 PM  5:30 PM
 C517 Wells Hall
Tori are an important structural aspect of algebraic groups, and "split" vs "nonsplit" tori are especially important. Unfortunately, "nonsplit" phenomena only occur over nonalgebraically 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 nonsplit torus. Then we'll try to use that example to try and understand the importance of nonsplit tori.
