Difference between revisions of "Graduate Algebraic Geometry Seminar Fall 2017"
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Here are the topics we're '''DYING''' to learn about! Please consider looking into one of these topics and giving one or two GAGS talks. | Here are the topics we're '''DYING''' to learn about! Please consider looking into one of these topics and giving one or two GAGS talks. | ||
+ | ===Specific Topics=== | ||
* D-modules 101: basics of D-modules, equivalence between left and right D-modules, pullbacks, pushforwards, maybe the Gauss-Manin Connection. Claude Sabbah's introduction to the subject could be a good place to start. | * D-modules 101: basics of D-modules, equivalence between left and right D-modules, pullbacks, pushforwards, maybe the Gauss-Manin Connection. Claude Sabbah's introduction to the subject could be a good place to start. | ||
* Sheaf operations on D-modules (the point is that then you can get a Fourier-Mukai transform between certain O-modules and certain D-modules, which is more or less how geometric Langlands is supposed to work) | * Sheaf operations on D-modules (the point is that then you can get a Fourier-Mukai transform between certain O-modules and certain D-modules, which is more or less how geometric Langlands is supposed to work) | ||
− | * David | + | ===Famous Theorems=== |
+ | |||
+ | ===Interesting Papers=== | ||
+ | |||
+ | * ''Géométrie algébraique et géométrie analytique'' - J.P. Serre. | ||
+ | ** A projective variety X over the complex numbers has two lives, an algebraic and an analytic, depending on which topology one wishes to work with. That is one can think about X as a complex manifold and work with holomorphic functions or as an algebraic variety and work with regular functions. Hence to any complex projective variety we have two sheaf theories and as a result two cohomology theories. In this famous paper Serre compares these two and shows they are in fact the same. (''Note: This is a super fundamental result that is used all the time; normally in the following way: Uhh... What do you mean by cohomology? Well by GAGA or something it doesn't really mater.) | ||
+ | |||
+ | * ''Limit linear series: Basic theory''- David Eisenbud and Joe Harris. | ||
+ | ** One of the more profitable tools -- especially when studying moduli spaces -- in a geometers tool box is the theory of degenerations. However, sometimes we care about more than just the variety we are degenerating and want to keep track of things like vector/line bundles. In this paper Eisenbud and Harris develop the theory of degenerating a curve together with a linear series. From this they prove a ton of cool results: M_g is of general type for g>24, Brill-Noether theory, etc. | ||
+ | |||
+ | * ''Picard Groups of Moduli Problems'' - David Mumford. | ||
+ | ** This paper is essentially the origin of algebraic stacks. | ||
+ | |||
+ | * ''The Structure of Algebraic Threefolds: An Introduction to Mori's Program'' - Janos Kollar | ||
+ | ** This paper is an introduction to Mori's famous ``minimal model'' program, which is a far reaching program seeking to understand the birational geometry of higher dimensional varieties. | ||
+ | |||
+ | * ''Cayley-Bacharach Formulas'' - Qingchun Ren, Jürgen Richter-Gebert, Bernd Sturmfels. | ||
+ | ** A classical result we all learn in a first semester of algebraic geometry is that 5 points in the plane (in general position) determine a unique plane conic. One can similarly show that 9 (general) points in the plane determine a unique plane cubic curve. This paper tries to answer the question: ``What is equation for this cubic curve?''. | ||
+ | |||
+ | * ''On Varieties of Minimal Degree (A Centennial Approach)'' - David Eisenbud and Joe Harris. | ||
+ | ** Suppose X is a projective variety embedded in projective space so that X is not contained in any hyperplane. By projecting from general points one can see that the degree of X is at least codim(X)+1. This paper discusses the classification of varieties that achieve this lower degree bound i.e. varieties of minimal degree. This topic is quite classical and the paper seems to contain a nice mixture of classical and modern geometry. | ||
__NOTOC__ | __NOTOC__ |
Revision as of 10:05, 28 December 2016
When: Wednesdays 4:00pm
Where:Van Vleck TBD
Who: YOU!!
Why: The purpose of this seminar is to learn algebraic geometry by giving and listening to talks in a informal setting. Talks are typically accessible to beginning graduate students and take many different forms. Sometimes people present an interesting paper they find. Other times people give a prep talk for the Friday Algebraic Geometry Seminar. Other times people give a series of talks on a topic they have been studying in-depth.
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@lists.wisc.edu. The list registration page is here.
Give a talk!
We need volunteers to give talks this semester. If you're interested contact DJ, or just add yourself to the list (though in that case we might move your talk later without your permission). Beginning graduate students are particularly encouraged to give a talk, since it's a great way to get your feet wet with the material.
Wish List
Here are the topics we're DYING to learn about! Please consider looking into one of these topics and giving one or two GAGS talks.
Specific Topics
- D-modules 101: basics of D-modules, equivalence between left and right D-modules, pullbacks, pushforwards, maybe the Gauss-Manin Connection. Claude Sabbah's introduction to the subject could be a good place to start.
- Sheaf operations on D-modules (the point is that then you can get a Fourier-Mukai transform between certain O-modules and certain D-modules, which is more or less how geometric Langlands is supposed to work)
Famous Theorems
Interesting Papers
- Géométrie algébraique et géométrie analytique - J.P. Serre.
- A projective variety X over the complex numbers has two lives, an algebraic and an analytic, depending on which topology one wishes to work with. That is one can think about X as a complex manifold and work with holomorphic functions or as an algebraic variety and work with regular functions. Hence to any complex projective variety we have two sheaf theories and as a result two cohomology theories. In this famous paper Serre compares these two and shows they are in fact the same. (Note: This is a super fundamental result that is used all the time; normally in the following way: Uhh... What do you mean by cohomology? Well by GAGA or something it doesn't really mater.)
- Limit linear series: Basic theory- David Eisenbud and Joe Harris.
- One of the more profitable tools -- especially when studying moduli spaces -- in a geometers tool box is the theory of degenerations. However, sometimes we care about more than just the variety we are degenerating and want to keep track of things like vector/line bundles. In this paper Eisenbud and Harris develop the theory of degenerating a curve together with a linear series. From this they prove a ton of cool results: M_g is of general type for g>24, Brill-Noether theory, etc.
- Picard Groups of Moduli Problems - David Mumford.
- This paper is essentially the origin of algebraic stacks.
- The Structure of Algebraic Threefolds: An Introduction to Mori's Program - Janos Kollar
- This paper is an introduction to Mori's famous ``minimal model program, which is a far reaching program seeking to understand the birational geometry of higher dimensional varieties.
- Cayley-Bacharach Formulas - Qingchun Ren, Jürgen Richter-Gebert, Bernd Sturmfels.
- A classical result we all learn in a first semester of algebraic geometry is that 5 points in the plane (in general position) determine a unique plane conic. One can similarly show that 9 (general) points in the plane determine a unique plane cubic curve. This paper tries to answer the question: ``What is equation for this cubic curve?.
- On Varieties of Minimal Degree (A Centennial Approach) - David Eisenbud and Joe Harris.
- Suppose X is a projective variety embedded in projective space so that X is not contained in any hyperplane. By projecting from general points one can see that the degree of X is at least codim(X)+1. This paper discusses the classification of varieties that achieve this lower degree bound i.e. varieties of minimal degree. This topic is quite classical and the paper seems to contain a nice mixture of classical and modern geometry.
Spring 2017
Date | Speaker | Title (click to see abstract) |
January 18 | Nathan Clement | TBD |
January 25 | Nathan Clement | TBD |
February 1 | TBD | TBD |
February 8 | TBD | TBD |
February 15 | TBD | TBD |
February 22 | TBD | TBD |
March 1 | TBD | TBD |
March 8 | TBD | TBD |
March 15 | TBD | TBD |
March 22 | Spring Break | No Seminar. |
March 29 | TBD | TBD |
April 5 | TBD | TBD |
April 12 | TBD | TBD |
April 19 | TBD | TBD |
April 26 | TBD | TBD |
January 18
Nathan Clement |
Title: TBD |
Abstract: TBD |
January 25
Nathan Clement |
Title: TBD |
Abstract: TBD |
February 1
TBA |
Title: TBD |
Abstract: TBD |
February 8
TBA |
Title: TBD |
Abstract: TBD |
February 15
TBA |
Title: TBD |
Abstract: TBD |
February 22
TBA |
Title: TBD |
Abstract: TBD |
March 1
TBA |
Title: TBD |
Abstract: TBD |
March 8
TBA |
Title: TBD |
Abstract: TBD |
March 15
TBA |
Title: TBD |
Abstract: TBD |
March 22
Spring Break |
Title: No Seminar. |
Abstract: n/a |
March 29
TBA |
Title: TBD |
Abstract: TBD |
April 5
TBA |
Title: TBD |
Abstract: TBD |
April 12
TBA |
Title: TBD |
Abstract: TBD |
April 19
TBA |
Title: TBD |
Abstract: TBD |
April 26
TBA |
Title: TBD |
Abstract: TBD |