| Talk_id | Date | Speaker | Title |
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19613
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Monday 9/9
4:30 PM
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Ioannis Zachos, Michigan State
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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 $2x^2z^2+2xyz^2+2xz^3+z^3-1$ 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.
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19610
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Monday 9/16
4:30 PM
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Chuangtian Guan, MSU
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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.
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19631
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Monday 9/23
4:30 PM
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Nick Rekuski, Michigan State
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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.
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19612
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Monday 9/30
4:30 PM
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Joshua Ruiter, Michigan State
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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 $\mathbb{R}^n$.
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19614
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Monday 10/14
4:30 PM
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Dan Normand, Harvard University
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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 homology---and 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
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