Graduate Algebraic Geometry Seminar Fall 2017: Difference between revisions

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'''
'''
'''When:''' Wednesdays 4:00pm
'''When:''' Wednesdays 3:30pm


'''Where:'''Van Vleck B325
'''Where:'''Van Vleck B321 (Fall 2017)
[[Image:cat.jpg|thumb|220px| | Lizzie the OFFICIAL mascot of GAGS!!]]
[[Image:cat.jpg|thumb|220px| | Lizzie the OFFICIAL mascot of GAGS!!]]


'''Who:''' YOU!!
'''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 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.
'''Why:''' The purpose of this seminar is to learn algebraic geometry and commutative algebra 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. 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@lists.wisc.edu. The list registration page is [https://admin.lists.wisc.edu/index.php?p=11&l=gags here].
'''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 [https://admin.lists.wisc.edu/index.php?p=11&l=gags here].
'''
'''


== Give a talk! ==
We need volunteers to give talks this semester. If you're interested contact [mailto:juliette.bruce@math.wisc.edu Juliette], 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.
== 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:
== 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.


===Specifically Vague 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===


== Give a talk! ==
===Interesting Papers & Books===
We need volunteers to give talks this semester. If you're interested contact [mailto:djbruce@math.wisc.edu 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.
* ''Symplectic structure of the moduli space of sheaves on an abelian or K3 surface'' - Shigeru Mukai.
 
* ''Residues and Duality'' - Robin Hatshorne.
** Have you heard of Serre Duality? Would you like to really understand the nuts and bolts of it and its generalizations? If so this book is for you. (You wouldn't need to read the whole book to give a talk ;).)
 
* ''Coherent sheaves on P^n and problems in linear algebra'' - A. A. Beilinson.
** In this two page paper constructs the semi-orthogonal decomposition of the derived category of coherent sheaves on projective space. (This topic is very important, and there are a ton of other resources for this result and the general theory of derived categories.)


* ''Frobenius splitting and cohomology vanishing for Schubert varieties'' - V.B. Mehta and A. Ramanathan.
** In characteristic p the fact that (x+y)^p=x^p+y^p means that one has the Frobenius morphism, which sends f to f^p. In this paper the authors introduce the notion of what it means for a variety to be Frobenius split, and use this to prove certain cohomologcal vanishing results for Schubert varieties. Since then Frobenius splitting -- and its related cousins (F-regularity, strong F-regularity, F-purity, etc.) have played large roles in geometry and algebra in characteristic p. This is a good place to get a sense for what kicked all this stuff off!


== Wish List ==
* ''Schubert Calculus'' - S. L. Kleiman and Dan Laksov.
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.
** An introduction to Schubert calculus suitable for those of all ages. I am told the paper essentially only uses linear algebra!


* Bondal and Orlov: semiorthogonal decompositions for algebraic varieties (Note: this is about cool stuff like Fourier-Mukai transforms)
* ''Rational Isogenies of Prime Degree'' - Barry Mazur.
** In this paper Mazur classifies all isogenies of rational elliptic curves of prime order. As a result of this he deduces his famous result that the torsion subgroup of an elliptic curve (over Q) is one of 15 abelian groups. This definitely stares into the land of number theory, but certainly would still be of interest to many.


* 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'')
* ''Esquisse d’une programme'' - Alexander Grothendieck.
** Originating from a grant proposal in the mid 1980's this famous paper outlines a tantalizing research program, which seeks to tie numerous different areas of math (algebraic geometry, Teichmuller theory, Galois theory, etc.) together. This is where Grothendieck introduced his famous Lego game and dessin d'enfant. While just a research proposal this paper has seemingly inspired a ton of cool math, and will allow you to "blow peoples’ minds". (The original paper is in French, but there are English translations out there.)


* homological projective duality
* ''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.) (The original paper is in French, but there are English translations out there.)


* moment map and symplectic reduction
* ''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.


* the orbit method (for classifying representations of a Lie group)
* ''Picard Groups of Moduli Problems'' - David Mumford.
** This paper is essentially the origin of algebraic stacks.


* Kaledin: geometry and topology of symplectic resolutions
* ''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.


* Kashiwara: D-modules and representation theory of Lie groups (Note: Check out that diagram on page 2!)
* ''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?''.


* geometric complexity theory, maybe something like arXiv:1508.05788.
* ''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.


* ''The Gromov-Witten potential associated to a TCFT'' - Kevin J. Costello.
** This seems incredibly interesting, but fairing warning this paper has been described as ''highly technical'', which considering it uses A-infinity algebras and the derived category of a  Calabi-Yau seems like a reasonable description. (This paper may be covered in Caldararu's Spring 2017 topics course.)
__NOTOC__
__NOTOC__


== Fall 2015 ==
== Fall 2017 ==


<center>
<center>
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| bgcolor="#BCD2EE" width="300" align="center"|'''Title (click to see abstract)'''
| bgcolor="#BCD2EE" width="300" align="center"|'''Title (click to see abstract)'''
|-
|-
| bgcolor="#E0E0E0"| September 2
| bgcolor="#E0E0E0"| September 13
| bgcolor="#C6D46E"| Ed Dewey
| bgcolor="#C6D46E"| Moisés Herradón Cueto
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#September 2| A^1 homotopy theory and rank-2 vector Bundles on smooth affine surfaces]]
| bgcolor="#BCE2FE"|[[#September 13| Vector bundles over the projective line]]
|-
|-
| bgcolor="#E0E0E0"| September 9
| bgcolor="#E0E0E0"| September 20
| bgcolor="#C6D46E"| No one
| bgcolor="#C6D46E"| No Talk
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#September 9| No Talk ]]
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar Fall 2017#September 20 | Reflecting on signing up for a talk]]
|-
|-
| bgcolor="#E0E0E0"| September 16
| bgcolor="#E0E0E0"| September 27
| bgcolor="#C6D46E"| Ed Dewey
| bgcolor="#C6D46E"| Moisés Herradón Cueto
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#September 9| A^1 homotopy theory and rank-2 vector Bundles on smooth affine surfaces (cont.) ]]
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar Fall 2017#September 27 | Vector bundles over an elliptic curve]]
|-
|-
| bgcolor="#E0E0E0"| September 23 
| bgcolor="#E0E0E0"| October 4
| bgcolor="#C6D46E"| DJ Bruce
| bgcolor="#C6D46E"| TBD
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#September 16| The Ring ]]  
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar Fall 2017#October 4 | TBD]]  
|-
|-
| bgcolor="#E0E0E0"| September 30
| bgcolor="#E0E0E0"| October 11
| bgcolor="#C6D46E"| DJ Bruce
| bgcolor="#C6D46E"| TBD
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#September 23| The Ring (cont). ]]  
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar Fall 2017#October 11 | TBD]]  
|-
|-
| bgcolor="#E0E0E0"| October 7
| bgcolor="#E0E0E0"| October 18
| bgcolor="#C6D46E"| Zachary Charles
| bgcolor="#C6D46E"| TBD
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#October 7| An Introduction to Real Algebraic Geometry and the Real Spectrum]]  
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar Fall 2017#October 18 | TBD]]  
|-
|-
| bgcolor="#E0E0E0"| October 14
| bgcolor="#E0E0E0"| October 25
| bgcolor="#C6D46E"| Zachary Charles
| bgcolor="#C6D46E"| TBD
| bgcolor="#BCE2FE"| [[Graduate Algebraic Geometry Seminar#October 14| An Introduction to Real Algebraic Geometry and the Real Spectrum]]  
| bgcolor="#BCE2FE"| [[Graduate Algebraic Geometry Seminar Fall 2017#October 25 | TBD]]  
|-
|-
| bgcolor="#E0E0E0"| October 21
| bgcolor="#E0E0E0"| November 1
| bgcolor="#C6D46E"| Eva Elduque
| bgcolor="#C6D46E"| Michael Brown
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#October 21| Symplectic Geometry I]]  
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar Fall 2017#November 1 | A Theorem of Orlov]]  
|-
|-
| bgcolor="#E0E0E0"| October 28
| bgcolor="#E0E0E0"| November 8
| bgcolor="#C6D46E"| Moisies Heradon
| bgcolor="#C6D46E"| Michael Brown
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#October 28| Symplectic Geometry II]]  
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar Fall 2017#November 8 | A Theorem or Orlov]]  
|-
|-
| bgcolor="#E0E0E0"| November 4
| bgcolor="#E0E0E0"| November 15
| bgcolor="#C6D46E"| Moisies Heradon
| bgcolor="#C6D46E"| TBD
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#November 4| Symplectic Geometry III]]  
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar Fall 2017#November 15 | TBD]]  
|-
|-
| bgcolor="#E0E0E0"| November 11
| bgcolor="#E0E0E0"| November 22
| bgcolor="#C6D46E"| Nathan Clement
| bgcolor="#C6D46E"| n/a
| bgcolor="#BCE2FE"| [[Graduate Algebraic Geometry Seminar#November 11| Moduli Spaces of Sheaves on Singular Curves]]   
| bgcolor="#BCE2FE"| [[Graduate Algebraic Geometry Seminar Fall 2017#November 22 | No Seminar]]   
|-
|-
| bgcolor="#E0E0E0"| November 18
| bgcolor="#E0E0E0"| November 29
| bgcolor="#C6D46E"| Nathan Clement
| bgcolor="#C6D46E"| TBD
| bgcolor="#BCE2FE"| [[Graduate Algebraic Geometry Seminar#November 18| Moduli Spaces of Sheaves on Singular Curves]]   
| bgcolor="#BCE2FE"| [[Graduate Algebraic Geometry Seminar Fall 2017#November 29 | TBD]]   
|-
| bgcolor="#E0E0E0"| November 25
| bgcolor="#C6D46E"| No Seminar Thanksgiving
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#November 25| TBD ]]
|-
| bgcolor="#E0E0E0"| December 2
| bgcolor="#C6D46E"| Jay Yang
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#December 2| TBD ]]
|-
|-
| bgcolor="#E0E0E0"| December 9
| bgcolor="#E0E0E0"| December 6
| bgcolor="#C6D46E"| TBD
| bgcolor="#C6D46E"| TBD
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#December 9| TBD ]]  
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar Fall 2017#December 6 | What about stacks? ]]  
|-
|-
| bgcolor="#E0E0E0"| December 16
| bgcolor="#E0E0E0"| December 13
| bgcolor="#C6D46E"| TBD
| bgcolor="#C6D46E"| TBD
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#December 16| TBD ]]  
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar Fall 2017#December 14 | What about stacks? II ]]  
|}
|}
</center>
</center>


== September 2 ==
== September 13 ==
<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%" | '''Ed Dewey'''
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Moisés Herradón Cueto'''
|-
|-
| bgcolor="#BCD2EE"  align="center" | Title: A^1 homotopy theory and rank-2 vector bundles on smooth affine surfaces
| bgcolor="#BCD2EE"  align="center" | Title: Vector Bundles over the projective line
|-
|-
| bgcolor="#BCD2EE"  |   
| bgcolor="#BCD2EE"  |   
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.
Abstract:  
 
Next week I will do an overview of Atiyah's classification of bundles on an elliptic curve. Today, I will talk about the tools needed to do this: cohomology of vector bundles. My goal is to keep a loose, islander, Ibizan pace where I will not define anything very rigorously, yet we will get our hands dirty with some computations, not all of which you have sat down and done before (if you have, what is your life? Why am I the one giving this talk?). Our aimless drift will hopefully get us to the much easier classification of vector bundles on the projective line, and we will have achieved the feat of using cohomology to prove a statement that doesn't contain the word cohomology! Flowery crowns are optional.
|}                                                                         
|}                                                                         
</center>
</center>


== September 9 ==
== September 20 ==
<center>
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
|-
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''No Talk'''
|-
| bgcolor="#BCD2EE"  align="center" | Title: N/A
|-
| bgcolor="#BCD2EE"  | 
Abstract: There will be no GAG's talk this week as it conflicts with the computing workshop.
|}                                                                       
</center>
== September 16 ==
<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%" | '''Ed Dewey'''
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''No talk'''
|-
|-
| bgcolor="#BCD2EE"  align="center" | Title: A^1 homotopy theory and rank-2 vector bundles on smooth affine surfaces (cont).
| bgcolor="#BCD2EE"  align="center" | Title: You should sign up to give a talk
|-
|-
| bgcolor="#BCD2EE"  |   
| bgcolor="#BCD2EE"  |   
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.
Abstract:  
|}
</center>


== September 23 ==
TBD
<center>
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
|-
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''DJ Bruce'''
|-
| bgcolor="#BCD2EE"  align="center" | Title: The Ring
|-
| bgcolor="#BCD2EE"  | 
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.
|}                                                                         
|}                                                                         
</center>
</center>
== September 30 ==
 
== September 27 ==
<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%" | '''DJ Bruce'''
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Moisés Herradón Cueto'''
|-
|-
| bgcolor="#BCD2EE"  align="center" | Title: The Ring (cont.)
| bgcolor="#BCD2EE"  align="center" | Title: Vector bundles over an elliptic curve
|-
|-
| bgcolor="#BCD2EE"  |   
| bgcolor="#BCD2EE"  |   
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.  
Abstract:  
 
We will regain our continental composture and discuss Atiyah's classification of bundles on an elliptic curve. There will be a ton of preliminary stuff, some lemmas, some theorems and some sketchy proofs. The sun will rise on the east and set on the west, and in the mean time we will learn all the isomorphism classes of vector bundles on an elliptic curve over any field.
 
|}                                                                         
|}                                                                         
</center>
</center>
== October 7 ==
 
== October 4 ==
<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%" | '''Zachary Charles'''
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''TBD'''
|-
|-
| bgcolor="#BCD2EE"  align="center" | Title: An Introduction to Real Algebraic Geometry and the Real Spectrum
| bgcolor="#BCD2EE"  align="center" | Title: TBD
|-
|-
| bgcolor="#BCD2EE"  |   
| bgcolor="#BCD2EE"  |   
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:  
 
TBD
 
|}                                                                         
|}                                                                         
</center>
</center>


== October 14 ==
== October 11 ==
<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%" | '''Zachary Charles'''
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''TBD'''
|-
|-
| bgcolor="#BCD2EE"  align="center" | Title: An Introduction to Real Algebraic Geometry and the Real Spectrum
| bgcolor="#BCD2EE"  align="center" | Title: TBD
|-
|-
| bgcolor="#BCD2EE"  |   
| bgcolor="#BCD2EE"  |   
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:  
 
TBD
 
|}                                                                         
|}                                                                         
</center>
</center>


== October 21 ==
== October 18 ==
<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%" | '''Eva Elduque'''
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''TBD'''
|-
|-
| bgcolor="#BCD2EE"  align="center" | Title: TBD
| bgcolor="#BCD2EE"  align="center" | Title: TBD
|-
|-
| bgcolor="#BCD2EE"  |   
| bgcolor="#BCD2EE"  |   
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. 
Abstract:  


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."
TBD
 
~Ed~


|}                                                                         
|}                                                                         
</center>
</center>


== October 28 ==
== October 25 ==
<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%" | '''Moisies Heradon'''
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''TBD'''
|-
|-
| bgcolor="#BCD2EE"  align="center" | Title: Symplectic Geometry II
| bgcolor="#BCD2EE"  align="center" | Title: TBD
|-
|-
| bgcolor="#BCD2EE"  |   
| bgcolor="#BCD2EE"  |   
Abstract:
Abstract:  


"When dynamics get hectic
TBD
 
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~


|}                                                                         
|}                                                                         
</center>
</center>


== November 4 ==
== November 1 ==
<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%" | '''Moisies Heradon'''
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Michael Brown'''
|-
|-
| bgcolor="#BCD2EE"  align="center" | Title: Symplectic Geometry III
| bgcolor="#BCD2EE"  align="center" | Title: A theorem of Orlov
|-
|-
| bgcolor="#BCD2EE"  |   
| bgcolor="#BCD2EE"  |   
Abstract: TBD
Abstract: I will discuss the main theorem of Orlov's "Derived Categories of Coherent Sheaves and Triangulated Categories of Singularities". This very powerful theorem provides a comparison between the derived category of coherent sheaves on certain schemes and a related gadget called the "singularity category". Orlov's theorem recovers Beilinson's semiorthogonal decomposition of the bounded derived category of projective space as a special case.
 
 
|}                                                                         
|}                                                                         
</center>
</center>


== November 11 ==
== November 8 ==
<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%" | '''Nathan  Clements'''
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Michael Brown'''
|-
|-
| bgcolor="#BCD2EE"  align="center" | Title: Moduli Spaces of Sheaves on Singular Curves
| bgcolor="#BCD2EE"  align="center" | Title: A Theorem of Orlov
|-
|-
| bgcolor="#BCD2EE"  |   
| bgcolor="#BCD2EE"  |   
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.
Abstract: This will be a continuation of the previous talk.
 
|}                                                                         
|}                                                                         
</center>
</center>
== November 18 ==
 
== November 15 ==
<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%" | '''Nathan Clements'''
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''TBD'''
|-
|-
| bgcolor="#BCD2EE"  align="center" | Title: Moduli Spaces of Sheaves on Singular Curves
| bgcolor="#BCD2EE"  align="center" | Title: TBD
|-
|-
| bgcolor="#BCD2EE"  |   
| bgcolor="#BCD2EE"  |   
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.
Abstract:  
 
TBD
 
|}                                                                         
|}                                                                         
</center>
</center>
== November 25 ==
 
== November 22 ==
<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%" | ''' NO GAGS THIS WEEK '''
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''No Seminar This Week'''
|-
|-
| bgcolor="#BCD2EE"  align="center" | Title: No Talk Due to Thanksgiving
| bgcolor="#BCD2EE"  align="center" | Title: Enjoy Thanksgiving!
|-
|-
| bgcolor="#BCD2EE"  |   
| bgcolor="#BCD2EE"  |   
Abstract: Enjoy the break!
Abstract: n/a
|}                                                                         
|}                                                                         
</center>
</center>
== December 2 ==
 
== November 29 ==
<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%" | '''Jay Yang'''
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''TBD'''
|-
|-
| bgcolor="#BCD2EE"  align="center" | Title: TBD
| bgcolor="#BCD2EE"  align="center" | Title: TBD
|-
|-
| bgcolor="#BCD2EE"  |   
| bgcolor="#BCD2EE"  |   
Abstract: TBD
Abstract:  
 
TBD
|}                                                                         
|}                                                                         
</center>
</center>
== December 9 ==
 
== December 6 ==
<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%" | '''TBD'''
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Nathan Clement'''
|-
|-
| bgcolor="#BCD2EE"  align="center" | Title: TBD
| bgcolor="#BCD2EE"  align="center" | Title: What about stacks?
|-
|-
| bgcolor="#BCD2EE"  |   
| bgcolor="#BCD2EE"  |   
Abstract: TBD
Abstract:
 
TBD
 
|}                                                                         
|}                                                                         
</center>
</center>
== December 16 ==
 
== December 13 ==
<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%" | '''TBD'''
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Nathan Clement'''
|-
|-
| bgcolor="#BCD2EE"  align="center" | Title: TBD
| bgcolor="#BCD2EE"  align="center" | Title: What about stacks? II
|-
|-
| bgcolor="#BCD2EE"  |   
| bgcolor="#BCD2EE"  |   
Abstract: TBD
Abstract:  
|}                                                                      
 
TBD
|}                    
</center>
</center>


== Organizers' Contact Info ==
== Organizers' Contact Info ==
[http://www.math.wisc.edu/~djbruce DJ Bruce]
[http://www.math.wisc.edu/~juliettebruce Juliette Bruce]


[http://www.math.wisc.edu/~clement Nathan Clement]
[http://www.math.wisc.edu/~clement Nathan Clement]


[http://www.math.wisc.edu/~dewey/ Ed Dewey]
[https://www.math.wisc.edu/~moises Moisés Herradón Cueto]
 
== Past Semesters ==
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Spring_2017 Spring 2017]
 
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Fall_2016 Fall 2016]
 
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Spring_2016 Spring 2016]
 
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_(Fall_2015) Fall 2015]

Latest revision as of 00:25, 28 February 2019

When: Wednesdays 3:30pm

Where:Van Vleck B321 (Fall 2017)

Lizzie the OFFICIAL mascot of GAGS!!

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. 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. 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@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 Juliette, 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.

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:


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.

Specifically Vague 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 & Books

  • Symplectic structure of the moduli space of sheaves on an abelian or K3 surface - Shigeru Mukai.
  • Residues and Duality - Robin Hatshorne.
    • Have you heard of Serre Duality? Would you like to really understand the nuts and bolts of it and its generalizations? If so this book is for you. (You wouldn't need to read the whole book to give a talk ;).)
  • Coherent sheaves on P^n and problems in linear algebra - A. A. Beilinson.
    • In this two page paper constructs the semi-orthogonal decomposition of the derived category of coherent sheaves on projective space. (This topic is very important, and there are a ton of other resources for this result and the general theory of derived categories.)
  • Frobenius splitting and cohomology vanishing for Schubert varieties - V.B. Mehta and A. Ramanathan.
    • In characteristic p the fact that (x+y)^p=x^p+y^p means that one has the Frobenius morphism, which sends f to f^p. In this paper the authors introduce the notion of what it means for a variety to be Frobenius split, and use this to prove certain cohomologcal vanishing results for Schubert varieties. Since then Frobenius splitting -- and its related cousins (F-regularity, strong F-regularity, F-purity, etc.) have played large roles in geometry and algebra in characteristic p. This is a good place to get a sense for what kicked all this stuff off!
  • Schubert Calculus - S. L. Kleiman and Dan Laksov.
    • An introduction to Schubert calculus suitable for those of all ages. I am told the paper essentially only uses linear algebra!
  • Rational Isogenies of Prime Degree - Barry Mazur.
    • In this paper Mazur classifies all isogenies of rational elliptic curves of prime order. As a result of this he deduces his famous result that the torsion subgroup of an elliptic curve (over Q) is one of 15 abelian groups. This definitely stares into the land of number theory, but certainly would still be of interest to many.
  • Esquisse d’une programme - Alexander Grothendieck.
    • Originating from a grant proposal in the mid 1980's this famous paper outlines a tantalizing research program, which seeks to tie numerous different areas of math (algebraic geometry, Teichmuller theory, Galois theory, etc.) together. This is where Grothendieck introduced his famous Lego game and dessin d'enfant. While just a research proposal this paper has seemingly inspired a ton of cool math, and will allow you to "blow peoples’ minds". (The original paper is in French, but there are English translations out there.)
  • 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.) (The original paper is in French, but there are English translations out there.)
  • 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.
  • The Gromov-Witten potential associated to a TCFT - Kevin J. Costello.
    • This seems incredibly interesting, but fairing warning this paper has been described as highly technical, which considering it uses A-infinity algebras and the derived category of a Calabi-Yau seems like a reasonable description. (This paper may be covered in Caldararu's Spring 2017 topics course.)


Fall 2017

Date Speaker Title (click to see abstract)
September 13 Moisés Herradón Cueto Vector bundles over the projective line
September 20 No Talk Reflecting on signing up for a talk
September 27 Moisés Herradón Cueto Vector bundles over an elliptic curve
October 4 TBD TBD
October 11 TBD TBD
October 18 TBD TBD
October 25 TBD TBD
November 1 Michael Brown A Theorem of Orlov
November 8 Michael Brown A Theorem or Orlov
November 15 TBD TBD
November 22 n/a No Seminar
November 29 TBD TBD
December 6 TBD What about stacks?
December 13 TBD What about stacks? II

September 13

Moisés Herradón Cueto
Title: Vector Bundles over the projective line

Abstract:

Next week I will do an overview of Atiyah's classification of bundles on an elliptic curve. Today, I will talk about the tools needed to do this: cohomology of vector bundles. My goal is to keep a loose, islander, Ibizan pace where I will not define anything very rigorously, yet we will get our hands dirty with some computations, not all of which you have sat down and done before (if you have, what is your life? Why am I the one giving this talk?). Our aimless drift will hopefully get us to the much easier classification of vector bundles on the projective line, and we will have achieved the feat of using cohomology to prove a statement that doesn't contain the word cohomology! Flowery crowns are optional.

September 20

No talk
Title: You should sign up to give a talk

Abstract:

TBD

September 27

Moisés Herradón Cueto
Title: Vector bundles over an elliptic curve

Abstract:

We will regain our continental composture and discuss Atiyah's classification of bundles on an elliptic curve. There will be a ton of preliminary stuff, some lemmas, some theorems and some sketchy proofs. The sun will rise on the east and set on the west, and in the mean time we will learn all the isomorphism classes of vector bundles on an elliptic curve over any field.

October 4

TBD
Title: TBD

Abstract:

TBD

October 11

TBD
Title: TBD

Abstract:

TBD

October 18

TBD
Title: TBD

Abstract:

TBD

October 25

TBD
Title: TBD

Abstract:

TBD

November 1

Michael Brown
Title: A theorem of Orlov

Abstract: I will discuss the main theorem of Orlov's "Derived Categories of Coherent Sheaves and Triangulated Categories of Singularities". This very powerful theorem provides a comparison between the derived category of coherent sheaves on certain schemes and a related gadget called the "singularity category". Orlov's theorem recovers Beilinson's semiorthogonal decomposition of the bounded derived category of projective space as a special case.


November 8

Michael Brown
Title: A Theorem of Orlov

Abstract: This will be a continuation of the previous talk.

November 15

TBD
Title: TBD

Abstract:

TBD

November 22

No Seminar This Week
Title: Enjoy Thanksgiving!

Abstract: n/a

November 29

TBD
Title: TBD

Abstract:

TBD

December 6

Nathan Clement
Title: What about stacks?

Abstract:

TBD

December 13

Nathan Clement
Title: What about stacks? II

Abstract:

TBD

Organizers' Contact Info

Juliette Bruce

Nathan Clement

Moisés Herradón Cueto

Past Semesters

Spring 2017

Fall 2016

Spring 2016

Fall 2015

Retrieved from "https://wiki-dev.math.wisc.edu/index.php?title=Graduate_Algebraic_Geometry_Seminar_Fall_2017&oldid=17059"