Graduate Algebraic Geometry Seminar Fall 2017: Difference between revisions

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


'''Where:'''Van Vleck TBD
'''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:
== Give a talk! ==
* Ask Questions Appropriately:
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.




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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.


===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.
* 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 Mumford "Picard Groups of Moduli Problems" (an early paper delving into the geometry of algebaric stacks)
===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.)
__NOTOC__
__NOTOC__


== Spring 2017 ==
== 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"| January 18
| bgcolor="#E0E0E0"| September 13
| bgcolor="#C6D46E"| Nathan Clement
| bgcolor="#C6D46E"| Moisés Herradón Cueto
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#January 18| TBD]]
| bgcolor="#BCE2FE"|[[#September 13| Vector bundles over the projective line]]
|-
|-
| bgcolor="#E0E0E0"| January 25
| bgcolor="#E0E0E0"| September 20
| bgcolor="#C6D46E"| Nathan Clement
| bgcolor="#C6D46E"| No Talk
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#January 25 | TBD]]
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar Fall 2017#September 20 | Reflecting on signing up for a talk]]
|-
|-
| bgcolor="#E0E0E0"| February 1
| bgcolor="#E0E0E0"| September 27
| bgcolor="#C6D46E"| TBD
| bgcolor="#C6D46E"| Moisés Herradón Cueto
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#February 1 | TBD]]
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar Fall 2017#September 27 | Vector bundles over an elliptic curve]]
|-
|-
| bgcolor="#E0E0E0"| February 8 
| bgcolor="#E0E0E0"| October 4
| bgcolor="#C6D46E"| TBD
| bgcolor="#C6D46E"| TBD
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#February 8 | TBD ]]  
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar Fall 2017#October 4 | TBD]]  
|-
|-
| bgcolor="#E0E0E0"| February 15
| bgcolor="#E0E0E0"| October 11
| bgcolor="#C6D46E"| TBD
| bgcolor="#C6D46E"| TBD
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#February 15 | TBD]]  
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar Fall 2017#October 11 | TBD]]  
|-
|-
| bgcolor="#E0E0E0"| February 22
| bgcolor="#E0E0E0"| October 18
| bgcolor="#C6D46E"| TBD
| bgcolor="#C6D46E"| TBD
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#February 22 | TBD]]  
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar Fall 2017#October 18 | TBD]]  
|-
|-
| bgcolor="#E0E0E0"| March 1
| bgcolor="#E0E0E0"| October 25
| bgcolor="#C6D46E"| TBD
| bgcolor="#C6D46E"| TBD
| bgcolor="#BCE2FE"| [[Graduate Algebraic Geometry Seminar#March 1 | TBD]]  
| bgcolor="#BCE2FE"| [[Graduate Algebraic Geometry Seminar Fall 2017#October 25 | TBD]]  
|-
|-
| bgcolor="#E0E0E0"| March 8
| bgcolor="#E0E0E0"| November 1
| bgcolor="#C6D46E"| TBD
| bgcolor="#C6D46E"| Michael Brown
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#March 8| TBD]]  
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar Fall 2017#November 1 | A Theorem of Orlov]]  
|-
|-
| bgcolor="#E0E0E0"| March 15
| bgcolor="#E0E0E0"| November 8
| bgcolor="#C6D46E"| TBD
| bgcolor="#C6D46E"| Michael Brown
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#March 15| TBD]]  
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar Fall 2017#November 8 | A Theorem or Orlov]]  
|-
|-
| bgcolor="#E0E0E0"| March 22
| bgcolor="#E0E0E0"| November 15
| bgcolor="#C6D46E"| Spring Break
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#March 22 | No Seminar. ]]
|-
| bgcolor="#E0E0E0"| March 29
| bgcolor="#C6D46E"| TBD
| bgcolor="#C6D46E"| TBD
| bgcolor="#BCE2FE"| [[Graduate Algebraic Geometry Seminar#March 29| TBD]]
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar Fall 2017#November 15 | TBD]]  
|-
|-
| bgcolor="#E0E0E0"| April 5
| bgcolor="#E0E0E0"| November 22
| bgcolor="#C6D46E"| TBD
| bgcolor="#C6D46E"| n/a
| bgcolor="#BCE2FE"| [[Graduate Algebraic Geometry Seminar#April 5| TBD]]   
| bgcolor="#BCE2FE"| [[Graduate Algebraic Geometry Seminar Fall 2017#November 22 | No Seminar]]   
|-
|-
| bgcolor="#E0E0E0"| April 12
| bgcolor="#E0E0E0"| November 29
| bgcolor="#C6D46E"| TBD
| bgcolor="#C6D46E"| TBD
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#April 12| TBD]]  
| bgcolor="#BCE2FE"| [[Graduate Algebraic Geometry Seminar Fall 2017#November 29 | TBD]]
|-
|-
| bgcolor="#E0E0E0"| April 19
| bgcolor="#E0E0E0"| December 6
| bgcolor="#C6D46E"| TBD
| bgcolor="#C6D46E"| TBD
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#April 19| TBD ]]  
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar Fall 2017#December 6 | What about stacks? ]]  
|-
|-
| bgcolor="#E0E0E0"| April 26
| bgcolor="#E0E0E0"| December 13
| bgcolor="#C6D46E"| TBD
| bgcolor="#C6D46E"| TBD
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#April 26| TBD ]]  
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar Fall 2017#December 14 | What about stacks? II ]]  
|}
|}
</center>
</center>


== September 14 ==
== 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%" | '''DJ Bruce'''
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Moisés Herradón Cueto'''
|-
|-
| bgcolor="#BCD2EE"  align="center" | Title: Vignettes In Algebraic Geometry
| bgcolor="#BCD2EE"  align="center" | Title: Vector Bundles over the projective line
|-
|-
| bgcolor="#BCD2EE"  |   
| bgcolor="#BCD2EE"  |   
Abstract:  
Abstract:  


Algebraic geometry is a massive forest, and it is often easy to become lost in the thicket of technical detail and seemingly endless abstraction. The goal of this talk is to take a step back out of these weeds, and return to our roots as algebraic geometers. By looking at three different classical problems we will explore various parts of algebraic geometry, and hopefully motivate the development of some of its larger machinery. Each problem will slowly build with no prerequisite assumed of the listener in the beginning.  
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 21 ==
== September 20 ==
<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%" | '''Moisés Herradón Cueto'''
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''No talk'''
|-
|-
| bgcolor="#BCD2EE"  align="center" | Title: Hilbert's 21 and The Riemann-Hilbert correspondence
| bgcolor="#BCD2EE"  align="center" | Title: You should sign up to give a talk
|-
|-
| bgcolor="#BCD2EE"  |   
| bgcolor="#BCD2EE"  |   
Abstract: Enough with the algebra! Away with the schemes and categories! Consider a differential equation with some singularities, such as y'=1/x. Analysis tells us that its solutions can be extended along paths on the complex plane, but when a path loops around the singular point, 0 in this case, the solution might change. This phenomenon is called monodromy. Hilbert's twenty-first problem asks about the possible inverse of the monodromy construction: if some monodromy is prescribed on the plane with some points removed, is there a nice (Fuchsian), linear differential equation whose solutions have this monodromy? Attempting to solve this problem will quickly take us back to our cozy algebraic geometry world of sheaves and vector bundles. For those of us to whom the word sheaves produces a cold sweat running down our backs, this topic is a great way to motivate and introduce sheaves, and will ultimately give us a reason to care about nontrivial vector bundles.
Abstract:  


No knowledge (or ignorance) of sheaves is required and the analysis in the talk will be contained in the tiny amount that I myself know.
TBD
|}                                                                         
|}                                                                         
</center>
</center>


== September 28 ==
== 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"
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| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Moisés Herradón Cueto'''
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Moisés Herradón Cueto'''
|-
|-
| bgcolor="#BCD2EE"  align="center" | Title: Hilbert's 21 and The Riemann-Hilbert correspondence
| bgcolor="#BCD2EE"  align="center" | Title: Vector bundles over an elliptic curve
|-
|-
| bgcolor="#BCD2EE"  |   
| bgcolor="#BCD2EE"  |   
Abstract: Enough with the algebra! Away with the schemes and categories! Consider a differential equation with some singularities, such as y'=1/x. Analysis tells us that its solutions can be extended along paths on the complex plane, but when a path loops around the singular point, 0 in this case, the solution might change. This phenomenon is called monodromy. Hilbert's twenty-first problem asks about the possible inverse of the monodromy construction: if some monodromy is prescribed on the plane with some points removed, is there a nice (Fuchsian), linear differential equation whose solutions have this monodromy? Attempting to solve this problem will quickly take us back to our cozy algebraic geometry world of sheaves and vector bundles. For those of us to whom the word sheaves produces a cold sweat running down our backs, this topic is a great way to motivate and introduce sheaves, and will ultimately give us a reason to care about nontrivial vector bundles.
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.


No knowledge (or ignorance) of sheaves is required and the analysis in the talk will be contained in the tiny amount that I myself know.
|}                                                                         
|}                                                                         
</center>
</center>


== October 5 ==
== 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%" | '''No Talk This Week'''
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''TBD'''
|-
|-
| bgcolor="#BCD2EE"  align="center" | Title: Research Computing in Algebra
| bgcolor="#BCD2EE"  align="center" | Title: TBD
|-
|-
| bgcolor="#BCD2EE"  |   
| bgcolor="#BCD2EE"  |   
Abstract: This weeks seminar conflicts with the "Research Computing in Algebra" workshop, and so instead we will not be having seminar this week. Instead we encourage everyone -- but especially those with little computational experience -- to go and learn how computation plays a major role in the research of your algebra peers, and how you can begin to integrate computation into your own research. Contact Steve Goldstein for more information.
Abstract:  
 
TBD
 
|}                                                                         
|}                                                                         
</center>
</center>


== October 12 ==
== 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%" | '''Nathan Clement'''
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''TBD'''
|-
|-
| bgcolor="#BCD2EE"  align="center" | Title: Spectral Curves and Higgs Bundles
| bgcolor="#BCD2EE"  align="center" | Title: TBD
|-
|-
| bgcolor="#BCD2EE"  |   
| bgcolor="#BCD2EE"  |   
Abstract:  
Abstract:  
I will present some of the backround motivation for the study of Higgs Bundles, mainly pertaining to Nigel Hitchen's 1987 paper. I will then introduce the spectral curve associated to an operator and describe the relevant geometry.
 
TBD
 
|}                                                                         
|}                                                                         
</center>
</center>


== October 19 ==
== 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%" | '''Nathan Clement'''
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''TBD'''
|-
|-
| bgcolor="#BCD2EE"  align="center" | Title: Spectral Curves and Blowups
| bgcolor="#BCD2EE"  align="center" | Title: TBD
|-
|-
| bgcolor="#BCD2EE"  |   
| bgcolor="#BCD2EE"  |   
Abstract:  
Abstract:  
Continuing on from last time, I will now take a closer look at the geometry of the spectral curve.  The main construction will be the lifting of a spectral curve to a blow up of the ambient surface, and the main tool for studying the geometry of this new spectral curve will be intersection theory in a surface.
 
TBD
 
|}                                                                         
|}                                                                         
</center>
</center>


== October 26 ==
== 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%" | '''Andrei Caldararu'''
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''TBD'''
|-
|-
| bgcolor="#BCD2EE"  align="center" | Title: What is Mirror Symmetry?
| bgcolor="#BCD2EE"  align="center" | Title: TBD
|-
|-
| bgcolor="#BCD2EE"  |   
| bgcolor="#BCD2EE"  |   
Abstract: Mirror Symmetry is a surprising discovery made in physics around 1992. Its main initial statement was the conjecture that one can calculate certain enumerative invariants (curve counts) on a Calabi-Yau threefolds by carying out an apparently unrelated calculation (solving a differential equation) related to a very different Calabi-Yau threefold. Later, two mathematical explanations of mirror symmetry were proposed, one algebraic by Maxim Kontsevich (Homological Mirror Symmetry) and one geometric by Strominger-Yau-Zaslow.
Abstract:  
 
TBD
 
|}                                                                         
|}                                                                         
</center>
</center>


== November 2 ==
== 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%" | '''Daniel Erman'''
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Michael Brown'''
|-
|-
| bgcolor="#BCD2EE"  align="center" | Title: Deformation Theory
| bgcolor="#BCD2EE"  align="center" | Title: A theorem of Orlov
|-
|-
| bgcolor="#BCD2EE"  |   
| bgcolor="#BCD2EE"  |   
Abstract: Deformation Theory, What does it know? Does it know things? Let's find out!
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 9 ==
== 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%" | '''Brandon Boggess'''
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Michael Brown'''
|-
|-
| bgcolor="#BCD2EE"  align="center" | Title: Quasicoherent Sheaves and Saturation
| bgcolor="#BCD2EE"  align="center" | Title: A Theorem of Orlov
|-
|-
| bgcolor="#BCD2EE"  |   
| bgcolor="#BCD2EE"  |   
Abstract: Given a module, one can form a quasicoherent sheaf on an affine scheme. In much the same way, we can get a quasicoherent sheaf on a projective scheme from any graded module. Unlike in the affine case, this construction fails to give an equivalence of categories. We will examine this construction and explore how saturation can fix this problem.
Abstract: This will be a continuation of the previous talk.
 
|}                                                                         
|}                                                                         
</center>
</center>


== November 16 ==
== 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%" | '''Wanlin Li'''
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''TBD'''
|-
|-
| bgcolor="#BCD2EE"  align="center" | Title: Gonality of modular curves in characteristic p
| bgcolor="#BCD2EE"  align="center" | Title: TBD
|-
|-
| bgcolor="#BCD2EE"  |   
| bgcolor="#BCD2EE"  |   
Abstract: My talk is based on Bjorn Poonen's paper with this title. He gave a proof of given a bound on gonality, there are only finitely many modular curves in characteristic p. The same result for characteristic 0 was given by Abramovich in 1966. I will sketch the proof in this talk. This paper used Technics from both number theory and algebraic geometry.
Abstract:  
 
TBD
 
|}                                                                         
|}                                                                         
</center>
</center>


== November 23 ==
== 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"
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</center>
</center>


== November 30 ==
== 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"
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|-
|-
| bgcolor="#BCD2EE"  |   
| bgcolor="#BCD2EE"  |   
Abstract: TBD
Abstract:  
 
TBD
|}                                                                         
|}                                                                         
</center>
</center>


== December 7 ==
== 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%" | '''David Wagner'''
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Nathan Clement'''
|-
|-
| bgcolor="#BCD2EE"  align="center" | Title: Generic Freeness and the Dimension of Fibres
| bgcolor="#BCD2EE"  align="center" | Title: What about stacks?
|-
|-
| bgcolor="#BCD2EE"  |   
| bgcolor="#BCD2EE"  |   
Abstract: The fact that the image of a projective variety is closed was known in some special cases as early as Newton, who gave ingenious methods for computing equations of the image (by hand!!). There is no need, though, to ask only about the set of positive-dimensional fibres; somewhat more generally, and under very modest assumptions about the schemes in question, the dimension of fibres is semi-continuous on the source (i.e. only jumps up). Guided carefully by David Eisenbud, we begin by proving the generic freeness lemma of Grothendieck and then pass on to the thoroughly lovely Chevalley's Theorem. After accepting a few basic facts about dimension (plus more theorems), our pastoral traipse through the domain of commutative algebra will be basically self-contained.
Abstract:
 
TBD
 
|}                                                                         
|}                                                                         
</center>
</center>


== December 14 ==
== 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]
Line 300: Line 363:


== Past Semesters ==
== 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_Fall_2016 Fall 2016]


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"