Difference between revisions of "Graduate Algebraic Geometry Seminar Fall 2021"
(→Fall 2021 Topic Wish List: added some of my own) |
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''' | ''' | ||
− | '''When:''' 6:00 PM Thursdays | + | '''When:''' 5:00-6:00 PM Thursdays |
'''Where:''' TBD | '''Where:''' TBD | ||
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=== Fall 2021 Topic Wish List === | === Fall 2021 Topic Wish List === | ||
− | This was assembled using input from an interest form at the beginning of the semester. Choose one and you will have the rare guarantee of having one interested audience member. | + | This was assembled using input from an interest form at the beginning of the semester. Choose one and you will have the rare guarantee of having one interested audience member. Feel free to add your own. |
− | |||
* Stacks for Kindergarteners | * Stacks for Kindergarteners | ||
− | * Beilinson resolution of diagonal | + | * Motives for Kindergarteners |
+ | * Applications of Beilinson resolution of the diagonal, Fourier Mukai transforms in general | ||
* Wth did June Huh do and what is combinatorial hodge theory? | * Wth did June Huh do and what is combinatorial hodge theory? | ||
+ | * Computing things about Toric varieties | ||
+ | * Reductive groups and flag varieties | ||
+ | * Introduction to arithmetic geometry -- what are some big picture ideas of what "goes wrong" when not over an algebraically closed field? | ||
+ | * Geothendieck '66, "On the de Rham Cohomology of Algebraic Varieties" | ||
+ | * Going from line bundles and divisors to vector bundles and chern classes | ||
+ | * A History of the Weil Conjectures | ||
+ | * Mumford & Bayer, "What can be computed in Algebraic Geometry?" | ||
+ | * A pre talk for any other upcoming talk | ||
== Being an audience member == | == Being an audience member == | ||
Line 37: | Line 45: | ||
|- | |- | ||
| bgcolor="#E0E0E0"| September 30 | | bgcolor="#E0E0E0"| September 30 | ||
− | | bgcolor="#C6D46E"| | + | | bgcolor="#C6D46E"| Yifan Wei |
− | | bgcolor="#BCE2FE"|[[#September 30| | + | | bgcolor="#BCE2FE"|[[#September 30| On Chow groups and K groups]] |
|- | |- | ||
| bgcolor="#E0E0E0"| October 7 | | bgcolor="#E0E0E0"| October 7 | ||
− | | bgcolor="#C6D46E"| | + | | bgcolor="#C6D46E"| Owen Goff |
− | | bgcolor="#BCE2FE"|[[#October 7 | | + | | bgcolor="#BCE2FE"|[[#October 7 | Roguish Noncommutativity and the Onsager Algebra]] |
|- | |- | ||
| bgcolor="#E0E0E0"| October 14 | | bgcolor="#E0E0E0"| October 14 | ||
− | | bgcolor="#C6D46E"| | + | | bgcolor="#C6D46E"| Peter YI WEI |
− | | bgcolor="#BCE2FE"|[[#October 14 | | + | | bgcolor="#BCE2FE"|[[#October 14 | Pathologies in Algebraic Geometry]] |
|- | |- | ||
| bgcolor="#E0E0E0"| October 21 | | bgcolor="#E0E0E0"| October 21 | ||
− | | bgcolor="#C6D46E"| | + | | bgcolor="#C6D46E"| Asvin G |
− | | bgcolor="#BCE2FE"|[[#October 21 | | + | | bgcolor="#BCE2FE"|[[#October 21 | Introduction to Arithmetic Schemes]] |
|- | |- | ||
| bgcolor="#E0E0E0"| October 28 | | bgcolor="#E0E0E0"| October 28 | ||
− | | bgcolor="#C6D46E"| | + | | bgcolor="#C6D46E"| Caitlyn Booms |
− | | bgcolor="#BCE2FE"|[[#October 28 | | + | | bgcolor="#BCE2FE"|[[#October 28 | Classifying Varieties of Minimal Degree]] |
|- | |- | ||
| bgcolor="#E0E0E0"| November 4 | | bgcolor="#E0E0E0"| November 4 | ||
| bgcolor="#C6D46E"| John Cobb | | bgcolor="#C6D46E"| John Cobb | ||
− | | bgcolor="#BCE2FE"|[[#November 4 | Koszul Cohomology]] | + | | bgcolor="#BCE2FE"|[[#November 4 | Syzygies and Koszul Cohomology]] |
|- | |- | ||
| bgcolor="#E0E0E0"| November 11 | | bgcolor="#E0E0E0"| November 11 | ||
− | | bgcolor="#C6D46E"| | + | | bgcolor="#C6D46E"| Colin Crowley |
− | | bgcolor="#BCE2FE"| [[#November 11 | | + | | bgcolor="#BCE2FE"| [[#November 11 | Introduction to Geometric Invariant Theory]] |
|- | |- | ||
− | | bgcolor="#E0E0E0"| November | + | | bgcolor="#E0E0E0"| November 23 |
− | | bgcolor="#C6D46E"| | + | | bgcolor="#C6D46E"| Connor Simpson |
− | | bgcolor="#BCE2FE"|[[#November 18 | | + | | bgcolor="#BCE2FE"|[[#November 18 | Combinatorial Hodge Theory]] |
|- | |- | ||
| bgcolor="#E0E0E0"| December 2 | | bgcolor="#E0E0E0"| December 2 | ||
− | | bgcolor="#C6D46E"| | + | | bgcolor="#C6D46E"| Alex Mine |
− | | bgcolor="#BCE2FE"|[[#December 2 | | + | | bgcolor="#BCE2FE"|[[#December 2 | Fourier-Mukai Transforms]] |
|- | |- | ||
| bgcolor="#E0E0E0"| December 9 | | bgcolor="#E0E0E0"| December 9 | ||
− | | bgcolor="#C6D46E"| | + | | bgcolor="#C6D46E"| Yu Luo |
− | | bgcolor="#BCE2FE"|[[#December 9 | | + | | bgcolor="#BCE2FE"|[[#December 9 | Stacks for Kindergarteners]] |
|} | |} | ||
</center> | </center> | ||
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{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20" | {| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20" | ||
|- | |- | ||
− | | bgcolor="#A6B658" align="center" style="font-size:125%" | ''' | + | | bgcolor="#A6B658" align="center" style="font-size:125%" | '''Yifan Wei''' |
|- | |- | ||
− | | bgcolor="#BCD2EE" align="center" | Title: | + | | bgcolor="#BCD2EE" align="center" | Title: On Chow groups and K groups |
|- | |- | ||
| bgcolor="#BCD2EE" | | | bgcolor="#BCD2EE" | | ||
Abstract: | Abstract: | ||
− | + | ||
+ | We define Chow groups and K groups for non-singular varieties, illustrate some basic properties, and explain how intersection theory is done using K groups (on a smooth surface). Then we proceed to compute the K group of a non-singular curve. On higher dimensions there might be some issues, if time permits we will show how these issues can be mitigated, and why Grothendieck-Riemann-Roch is one of the greatest theorems in algebraic geometry (in my humble opinion). | ||
+ | |||
|} | |} | ||
</center> | </center> | ||
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{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20" | {| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20" | ||
|- | |- | ||
− | | bgcolor="#A6B658" align="center" style="font-size:125%" | ''' | + | | bgcolor="#A6B658" align="center" style="font-size:125%" | '''Owen Goff''' |
|- | |- | ||
− | | bgcolor="#BCD2EE" align="center" | Title: | + | | bgcolor="#BCD2EE" align="center" | Title: Roguish Noncommutativity and the Onsager Algebra |
|- | |- | ||
| bgcolor="#BCD2EE" | | | bgcolor="#BCD2EE" | | ||
Abstract: | Abstract: | ||
− | + | While throughout algebraic geometry and many other fields we like commutative rings, we often wonder what happens if our ring is not commutative. Say, for instance, you have A^2, but instead of xy=yx you have a relation xy = qyx for some constant q. In this talk I will discuss the consequences of this relation and how it relates to an object of combinatorial nature called the q-Onsager algebra. | |
|} | |} | ||
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{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20" | {| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20" | ||
|- | |- | ||
− | | bgcolor="#A6B658" align="center" style="font-size:125%" | ''' | + | | bgcolor="#A6B658" align="center" style="font-size:125%" | '''Peter YI WEI''' |
|- | |- | ||
− | | bgcolor="#BCD2EE" align="center" | Title: | + | | bgcolor="#BCD2EE" align="center" | Title: Pathologies in Algebraic Geometry |
|- | |- | ||
| bgcolor="#BCD2EE" | | | bgcolor="#BCD2EE" | | ||
− | Abstract: | + | Abstract: This talk serves as a brief discussion on pathologies in algebraic geometry, inspired by a short thread of Daniel Litt’s twitter. No hard preliminaries! :) |
− | |||
− | |||
|} | |} | ||
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{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20" | {| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20" | ||
|- | |- | ||
− | | bgcolor="#A6B658" align="center" style="font-size:125%" | ''' | + | | bgcolor="#A6B658" align="center" style="font-size:125%" | '''Asvin G''' |
|- | |- | ||
− | | bgcolor="#BCD2EE" align="center" | Title: | + | | bgcolor="#BCD2EE" align="center" | Title: Introduction to Arithmetic Schemes |
|- | |- | ||
| bgcolor="#BCD2EE" | | | bgcolor="#BCD2EE" | | ||
− | Abstract: | + | Abstract: Many of us are comfortable working with varieties over the complex numbers (or other fields) but part of the magic is that it's almost as easy to consider varieties over more exotic rings like the integers or the p-adics. |
+ | |||
+ | I'll explain how to think about such varieties and then use them to prove the birational invariance of Hodge numbers for Calabi-Yau's over the complex numbers using results from finite fields and p-adic analysis! | ||
TBD | TBD | ||
Line 144: | Line 154: | ||
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20" | {| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20" | ||
|- | |- | ||
− | | bgcolor="#A6B658" align="center" style="font-size:125%" | ''' | + | | bgcolor="#A6B658" align="center" style="font-size:125%" | '''Caitlyn Booms''' |
|- | |- | ||
− | | bgcolor="#BCD2EE" align="center" | Title: | + | | bgcolor="#BCD2EE" align="center" | Title: Classifying Varieties of Minimal Degree |
|- | |- | ||
| bgcolor="#BCD2EE" | | | bgcolor="#BCD2EE" | | ||
Abstract: | Abstract: | ||
− | + | The degree of a variety embedded in projective space is a well-defined invariant, and there is a sense in which some varieties have minimal degree. Long ago, Del Pezzo and Bertini classified geometrically all possible projective varieties of minimal degree. More recently, Eisenbud and Goto gave an algebraic notion that classifies such varieties. In this talk, we will introduce the necessary background and explore these two theorems and the ways they are connected. | |
|} | |} | ||
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| bgcolor="#A6B658" align="center" style="font-size:125%" | '''John Cobb''' | | bgcolor="#A6B658" align="center" style="font-size:125%" | '''John Cobb''' | ||
|- | |- | ||
− | | bgcolor="#BCD2EE" align="center" | Title: Koszul Cohomology | + | | bgcolor="#BCD2EE" align="center" | Title: Syzygies and Koszul Cohomology |
|- | |- | ||
| bgcolor="#BCD2EE" | | | bgcolor="#BCD2EE" | | ||
Abstract: | Abstract: | ||
− | + | Early on in the history of algebraic geometry it was recognized that many properties/invariants of projective varieties could be deduced by looking at their hyperplane sections. Starting in the 1950s, this classical picture was gradually refined into general theory by people like Serre and Kodaira — many hard-earned numbers could now be obtained by more brainless methods. I hope to motivate a few ideas introduced in the 1980’s as a continuation of this story beginning from Serre’s vanishing theorem. | |
− | |||
|} | |} | ||
</center> | </center> | ||
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{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20" | {| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20" | ||
|- | |- | ||
− | | bgcolor="#A6B658" align="center" style="font-size:125%" | '' | + | | bgcolor="#A6B658" align="center" style="font-size:125%" | '''Colin Crowley''' |
|- | |- | ||
− | | bgcolor="#BCD2EE" align="center" | Title: | + | | bgcolor="#BCD2EE" align="center" | Title: Introduction to Geometric Invariant Theory |
|- | |- | ||
| bgcolor="#BCD2EE" | | | bgcolor="#BCD2EE" | | ||
− | Abstract: | + | Abstract: |
+ | |||
+ | Given a group action on a variety, is there a quotient variety? How do you construct it? Geometric invariant theory gives partial answers to these questions for projective varieties and a particular class of groups (reductive groups). I’ll give an overview of how GIT quotients work, which will be in the language of Hartshorne chapter one and does not require any knowledge of schemes. (Although I may need to talk a little about ample line bundles. I haven't decided yet.) | ||
+ | With the remaining time I'll sketch how these ideas are used in constructing (coarse) moduli spaces of semistable vector bundles, and mention which areas of math use these ideas today. | ||
|} | |} | ||
</center> | </center> | ||
− | === November | + | === November 23 === |
<center> | <center> | ||
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20" | {| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20" | ||
|- | |- | ||
− | | bgcolor="#A6B658" align="center" style="font-size:125%" | ''' | + | | bgcolor="#A6B658" align="center" style="font-size:125%" | '''Connor Simpson''' |
|- | |- | ||
− | | bgcolor="#BCD2EE" align="center" | Title: | + | | bgcolor="#BCD2EE" align="center" | Title: Combinatorial Hodge Theory |
|- | |- | ||
| bgcolor="#BCD2EE" | | | bgcolor="#BCD2EE" | | ||
− | Abstract: | + | Abstract: We'll talk about log-concave sequences of natural numbers and the |
+ | evolution of methods for proving log-concavity over the past decade. | ||
+ | In particular, we'll talk about "what June Huh did", then talk about | ||
+ | alternative roads to similar results, including recent work by Chan & Pak. | ||
|} | |} | ||
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{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20" | {| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20" | ||
|- | |- | ||
− | | bgcolor="#A6B658" align="center" style="font-size:125%" | ''' | + | | bgcolor="#A6B658" align="center" style="font-size:125%" | '''Alex Mine''' |
|- | |- | ||
− | | bgcolor="#BCD2EE" align="center" | Title: | + | | bgcolor="#BCD2EE" align="center" | Title: Fourier-Mukai Transforms |
|- | |- | ||
| bgcolor="#BCD2EE" | | | bgcolor="#BCD2EE" | | ||
Abstract: | Abstract: | ||
− | + | I'll say a few things about derived category of sheaves and talk about Fourier-Mukai transforms, which are certain functors between the derived categories of sheaves on two schemes. In particular, I will try to elucidate what is so "Fourier" about them. | |
|} | |} | ||
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{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20" | {| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20" | ||
|- | |- | ||
− | | bgcolor="#A6B658" align="center" style="font-size:125%" | ''' | + | | bgcolor="#A6B658" align="center" style="font-size:125%" | '''Yu Luo''' |
|- | |- | ||
− | | bgcolor="#BCD2EE" align="center" | Title: | + | | bgcolor="#BCD2EE" align="center" | Title: Stacks for Kindergarteners |
|- | |- | ||
| bgcolor="#BCD2EE" | | | bgcolor="#BCD2EE" | | ||
− | Abstract: | + | Abstract: |
+ | |||
+ | After struggling for a while, a kindergartner manages to build a LEGO Death Star. One day, while our kindergartner is at school, their father manages to break it. He hurriedly buys a new set, builds it, and secretly replaces the broken Death Star. Even though our kindergartner does not know it, we know that two Death Stars are not the same. That is, even though the two Death Stars are isomorphic, they are not canonically isomorphic. Motivated by this, we define the stacks in algebraic geometry to study the moduli problem. | ||
|} | |} | ||
</center> | </center> |
Latest revision as of 17:57, 21 November 2021
When: 5:00-6:00 PM Thursdays
Where: TBD
Who: All undergraduate and graduate students interested in algebraic geometry, commutative algebra, and related fields are welcome to attend.
Why: The purpose of this seminar is to learn algebraic geometry and commutative algebra by giving and listening to talks in a informal setting. Sometimes people present an interesting paper they find. Other times people give a prep talk for the Algebraic Geometry Seminar. Other times people give a series of talks on a topic they have been studying in-depth. Regardless the goal of GAGS is to provide a supportive and inclusive place for all to learn more about algebraic geometry and commutative algebra.
How: If you want to get emails regarding time, place, and talk topics (which are often assigned quite last minute) add yourself to the gags mailing list: gags@g-groups.wisc.edu by sending an email to gags+subscribe@g-groups.wisc.edu. If you prefer (and are logged in under your wisc google account) the list registration page is here.
Organizers: John Cobb, Colin Crowley.
Contents
Give a talk!
We need volunteers to give talks this semester. If you're interested, please fill out this form. Beginning graduate students are particularly encouraged to give a talk, since it's a great way to get your feet wet with the material. If you would like some talk ideas, see the list on the main page.
Fall 2021 Topic Wish List
This was assembled using input from an interest form at the beginning of the semester. Choose one and you will have the rare guarantee of having one interested audience member. Feel free to add your own.
- Stacks for Kindergarteners
- Motives for Kindergarteners
- Applications of Beilinson resolution of the diagonal, Fourier Mukai transforms in general
- Wth did June Huh do and what is combinatorial hodge theory?
- Computing things about Toric varieties
- Reductive groups and flag varieties
- Introduction to arithmetic geometry -- what are some big picture ideas of what "goes wrong" when not over an algebraically closed field?
- Geothendieck '66, "On the de Rham Cohomology of Algebraic Varieties"
- Going from line bundles and divisors to vector bundles and chern classes
- A History of the Weil Conjectures
- Mumford & Bayer, "What can be computed in Algebraic Geometry?"
- A pre talk for any other upcoming talk
Being an audience member
The goal of GAGS is to create a safe and comfortable space inclusive of all who wish to expand their knowledge of algebraic geometry and commutative algebra. In order to promote such an environment in addition to the standard expectations of respect/kindness all participants are asked to following the following guidelines:
- Do Not Speak For/Over the Speaker
- Ask Questions Appropriately
Talks
Date | Speaker | Title (click to see abstract) |
September 30 | Yifan Wei | On Chow groups and K groups |
October 7 | Owen Goff | Roguish Noncommutativity and the Onsager Algebra |
October 14 | Peter YI WEI | Pathologies in Algebraic Geometry |
October 21 | Asvin G | Introduction to Arithmetic Schemes |
October 28 | Caitlyn Booms | Classifying Varieties of Minimal Degree |
November 4 | John Cobb | Syzygies and Koszul Cohomology |
November 11 | Colin Crowley | Introduction to Geometric Invariant Theory |
November 23 | Connor Simpson | Combinatorial Hodge Theory |
December 2 | Alex Mine | Fourier-Mukai Transforms |
December 9 | Yu Luo | Stacks for Kindergarteners |
September 30
Yifan Wei |
Title: On Chow groups and K groups |
Abstract: We define Chow groups and K groups for non-singular varieties, illustrate some basic properties, and explain how intersection theory is done using K groups (on a smooth surface). Then we proceed to compute the K group of a non-singular curve. On higher dimensions there might be some issues, if time permits we will show how these issues can be mitigated, and why Grothendieck-Riemann-Roch is one of the greatest theorems in algebraic geometry (in my humble opinion). |
October 7
Owen Goff |
Title: Roguish Noncommutativity and the Onsager Algebra |
Abstract: While throughout algebraic geometry and many other fields we like commutative rings, we often wonder what happens if our ring is not commutative. Say, for instance, you have A^2, but instead of xy=yx you have a relation xy = qyx for some constant q. In this talk I will discuss the consequences of this relation and how it relates to an object of combinatorial nature called the q-Onsager algebra. |
October 14
Peter YI WEI |
Title: Pathologies in Algebraic Geometry |
Abstract: This talk serves as a brief discussion on pathologies in algebraic geometry, inspired by a short thread of Daniel Litt’s twitter. No hard preliminaries! :) |
October 21
Asvin G |
Title: Introduction to Arithmetic Schemes |
Abstract: Many of us are comfortable working with varieties over the complex numbers (or other fields) but part of the magic is that it's almost as easy to consider varieties over more exotic rings like the integers or the p-adics. I'll explain how to think about such varieties and then use them to prove the birational invariance of Hodge numbers for Calabi-Yau's over the complex numbers using results from finite fields and p-adic analysis! TBD |
October 28
Caitlyn Booms |
Title: Classifying Varieties of Minimal Degree |
Abstract: The degree of a variety embedded in projective space is a well-defined invariant, and there is a sense in which some varieties have minimal degree. Long ago, Del Pezzo and Bertini classified geometrically all possible projective varieties of minimal degree. More recently, Eisenbud and Goto gave an algebraic notion that classifies such varieties. In this talk, we will introduce the necessary background and explore these two theorems and the ways they are connected. |
November 4
John Cobb |
Title: Syzygies and Koszul Cohomology |
Abstract: Early on in the history of algebraic geometry it was recognized that many properties/invariants of projective varieties could be deduced by looking at their hyperplane sections. Starting in the 1950s, this classical picture was gradually refined into general theory by people like Serre and Kodaira — many hard-earned numbers could now be obtained by more brainless methods. I hope to motivate a few ideas introduced in the 1980’s as a continuation of this story beginning from Serre’s vanishing theorem. |
November 11
Colin Crowley |
Title: Introduction to Geometric Invariant Theory |
Abstract: Given a group action on a variety, is there a quotient variety? How do you construct it? Geometric invariant theory gives partial answers to these questions for projective varieties and a particular class of groups (reductive groups). I’ll give an overview of how GIT quotients work, which will be in the language of Hartshorne chapter one and does not require any knowledge of schemes. (Although I may need to talk a little about ample line bundles. I haven't decided yet.) With the remaining time I'll sketch how these ideas are used in constructing (coarse) moduli spaces of semistable vector bundles, and mention which areas of math use these ideas today. |
November 23
Connor Simpson |
Title: Combinatorial Hodge Theory |
Abstract: We'll talk about log-concave sequences of natural numbers and the evolution of methods for proving log-concavity over the past decade. In particular, we'll talk about "what June Huh did", then talk about alternative roads to similar results, including recent work by Chan & Pak. |
December 2
Alex Mine |
Title: Fourier-Mukai Transforms |
Abstract: I'll say a few things about derived category of sheaves and talk about Fourier-Mukai transforms, which are certain functors between the derived categories of sheaves on two schemes. In particular, I will try to elucidate what is so "Fourier" about them. |
December 9
Yu Luo |
Title: Stacks for Kindergarteners |
Abstract: After struggling for a while, a kindergartner manages to build a LEGO Death Star. One day, while our kindergartner is at school, their father manages to break it. He hurriedly buys a new set, builds it, and secretly replaces the broken Death Star. Even though our kindergartner does not know it, we know that two Death Stars are not the same. That is, even though the two Death Stars are isomorphic, they are not canonically isomorphic. Motivated by this, we define the stacks in algebraic geometry to study the moduli problem. |