Graduate Algebraic Geometry Seminar Fall 2017: Difference between revisions
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Abstract: Algebraic geometry and the prime spectrum arose from the study of subsets of C^n defined by polynomial equations. In many areas, we are often interested in subsets of R^n defined by polynomial equations and inequalities. This gives rise to real algebraic geometry and the real spectrum. We will introduce the concept of the real spectrum and how it differs from the prime spectrum, as well as some aspects of real commutative algebra. We will use these to discuss and prove Hilbert's 17th problem and move on to real algebraic geometry. If time permits, we will discuss the null-, positiv-, and nichtnegativ- stellensatzes and semi-algebraic geometry. Practically no background is assumed. | Abstract: Algebraic geometry and the prime spectrum arose from the study of subsets of C^n defined by polynomial equations. In many areas, we are often interested in subsets of R^n defined by polynomial equations and inequalities. This gives rise to real algebraic geometry and the real spectrum. We will introduce the concept of the real spectrum and how it differs from the prime spectrum, as well as some aspects of real commutative algebra. We will use these to discuss and prove Hilbert's 17th problem and move on to real algebraic geometry. If time permits, we will discuss the null-, positiv-, and nichtnegativ- stellensatzes and semi-algebraic geometry. Practically no background is assumed. | ||
"Abstract: In ancient times, before the coming of \mathbf{C}, there was another field. In the catacombs beneath Van Vleck, masked heresiarchs whisper eldrich conjectures about this shadowy object: that it was ordered, that it was not algebraically closed, and other perversions too horrible to name. | |||
But the ancients are dead, their cities are destroyed, and surely their secrets are lost to mankind forever....." ~Ed~ | |||
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Revision as of 20:53, 28 October 2015
When: Wednesdays 4:00pm
Where:Van Vleck B325
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
If there is a subject or a paper which you'd like to see someone give a talk on, add it to this list. If you want to give a talk and can't find a topic, try one from this list.
- Bondal and Orlov: semiorthogonal decompositions for algebraic varieties (Note: this is about cool stuff like Fourier-Mukai transforms)
- Braverman and Bezrukavnikov: geometric Langlands correspondence for D-modules in prime characteristic: the GL(n) case (Note: this title sounds tough but prime characteristic makes things easier)
- homological projective duality
- moment map and symplectic reduction
- the orbit method (for classifying representations of a Lie group)
- Kaledin: geometry and topology of symplectic resolutions
- Kashiwara: D-modules and representation theory of Lie groups (Note: Check out that diagram on page 2!)
- geometric complexity theory, maybe something like arXiv:1508.05788.
Fall 2015
| Date | Speaker | Title (click to see abstract) |
| September 2 | Ed Dewey | A^1 homotopy theory and rank-2 vector Bundles on smooth affine surfaces |
| September 9 | No one | No Talk |
| September 16 | Ed Dewey | A^1 homotopy theory and rank-2 vector Bundles on smooth affine surfaces (cont.) |
| September 23 | DJ Bruce | The Ring |
| September 30 | DJ Bruce | The Ring (cont). |
| October 7 | Zachary Charles | An Introduction to Real Algebraic Geometry and the Real Spectrum |
| October 14 | Zachary Charles | An Introduction to Real Algebraic Geometry and the Real Spectrum |
| October 21 | Eva Elduque | Symplectic Geometry I |
| October 28 | Moisies Heradon | Symplectic Geometry II |
| November 4 | Moisies Heradon | Symplectic Geometry III |
| November 11 | Nathan Clement | Moduli Spaces of Sheaves on Singular Curves |
| November 18 | Nathan Clement | Moduli Spaces of Sheaves on Singular Curves |
| November 25 | No Seminar Thanksgiving | TBD |
| December 2 | Jay Yang | TBD |
| December 9 | TBD | TBD |
| December 16 | TBD | TBD |
September 2
| Ed Dewey |
| Title: A^1 homotopy theory and rank-2 vector bundles on smooth affine surfaces |
|
Abstract: I will introduce the techniques used by Asok and Fasel to classify rank-2 vector bundles on a smooth affine 3-fold (arXiv:1204.0770). The problem itself is interesting, and the solution uses the A^1 homotopy category. My main goal is to make this category seem less bonkers. |
September 9
| No Talk |
| Title: N/A |
|
Abstract: There will be no GAG's talk this week as it conflicts with the computing workshop. |
September 16
| Ed Dewey |
| Title: A^1 homotopy theory and rank-2 vector bundles on smooth affine surfaces (cont). |
|
Abstract: I will introduce the techniques used by Asok and Fasel to classify rank-2 vector bundles on a smooth affine 3-fold (arXiv:1204.0770). The problem itself is interesting, and the solution uses the A^1 homotopy category. My main goal is to make this category seem less bonkers. |
September 23
| DJ Bruce |
| Title: The Ring |
|
Abstract: The Grothendieck ring of varieties is an incredibly mysterious object that seems to capture a bunch of arithmetic, geometric, and topological data regarding algebraic varieties. We will explore some of these connections. For example, we will see how the Weil Conjectures are related to stable birational geometry. No background will be assumed and the speaker will try to keep things accessible to all. |
September 30
| DJ Bruce |
| Title: The Ring (cont.) |
|
Abstract: The Grothendieck ring of varieties is an incredibly mysterious object that seems to capture a bunch of arithmetic, geometric, and topological data regarding algebraic varieties. We will explore some of these connections. For example, we will see how the Weil Conjectures are related to stable birational geometry. No background will be assumed and the speaker will try to keep things accessible to all. |
October 7
| Zachary Charles |
| Title: An Introduction to Real Algebraic Geometry and the Real Spectrum |
|
Abstract: Algebraic geometry and the prime spectrum arose from the study of subsets of C^n defined by polynomial equations. In many areas, we are often interested in subsets of R^n defined by polynomial equations and inequalities. This gives rise to real algebraic geometry and the real spectrum. We will introduce the concept of the real spectrum and how it differs from the prime spectrum, as well as some aspects of real commutative algebra. We will use these to discuss and prove Hilbert's 17th problem and move on to real algebraic geometry. If time permits, we will discuss the null-, positiv-, and nichtnegativ- stellensatzes and semi-algebraic geometry. Practically no background is assumed. "Abstract: In ancient times, before the coming of \mathbf{C}, there was another field. In the catacombs beneath Van Vleck, masked heresiarchs whisper eldrich conjectures about this shadowy object: that it was ordered, that it was not algebraically closed, and other perversions too horrible to name. But the ancients are dead, their cities are destroyed, and surely their secrets are lost to mankind forever....." ~Ed~ |
October 14
| Zachary Charles |
| Title: An Introduction to Real Algebraic Geometry and the Real Spectrum |
|
Abstract: Algebraic geometry and the prime spectrum arose from the study of subsets of C^n defined by polynomial equations. In many areas, we are often interested in subsets of R^n defined by polynomial equations and inequalities. This gives rise to real algebraic geometry and the real spectrum. We will introduce the concept of the real spectrum and how it differs from the prime spectrum, as well as some aspects of real commutative algebra. We will use these to discuss and prove Hilbert's 17th problem and move on to real algebraic geometry. If time permits, we will discuss the null-, positiv-, and nichtnegativ- stellensatzes and semi-algebraic geometry. Practically no background is assumed. |
October 21
| Eva Elduque |
| Title: Symplectic Geometry I |
|
Abstract: "I seem to have committed myself to supplying abstracts. Unfortunately there is nothing remotely funny about symplectic geometry. I think I've never heard anything less intuitive than studying manifolds with symplectic 2-forms. Nonetheless it seems to be totally central to both enumerative geometry and geometric representation theory. Eva and Moises are going to take the bull by the horns and try to explain it to us. In order to give the bull a fighting chance, make sure not to let them get away with any "intuitive" remarks treating momenta as cotangent vectors. I'm pretty sure no one has actually understood that since Hamilton." ~Ed~ |
October 28
| Moisies Heradon |
| Title: Symplectic Geometry II |
|
Abstract: "When dynamics get hectic For reasons symplectic Don't sit and brute-force them all day. Find nice functions and list them - Integrable system! Let symmetries show you the way." ~Ed~ |
November 4
| Moisies Heradon |
| Title: Symplectic Geometry III |
|
Abstract: TBD |
November 11
| Nathan Clements |
| Title: Moduli Spaces of Sheaves on Singular Curves |
|
Abstract: I will explain some useful techniques for the study of sheaves on singular curves of arithmetic genus one. In particular, there are many isomorphisms between moduli spaces of different sorts of sheaves on a given curve coming from natural operations on sheaves. |
November 18
| Nathan Clements |
| Title: Moduli Spaces of Sheaves on Singular Curves |
|
Abstract: I will explain some useful techniques for the study of sheaves on singular curves of arithmetic genus one. In particular, there are many isomorphisms between moduli spaces of different sorts of sheaves on a given curve coming from natural operations on sheaves. |
November 25
| NO GAGS THIS WEEK |
| Title: No Talk Due to Thanksgiving |
|
Abstract: Enjoy the break! |
December 2
| Jay Yang |
| Title: TBD |
|
Abstract: TBD |
December 9
| TBD |
| Title: TBD |
|
Abstract: TBD |
December 16
| TBD |
| Title: TBD |
|
Abstract: TBD |