Graduate Algebraic Geometry Seminar: Difference between revisions

Latest revision as of 21:30, 7 October 2024

When? Where?: Link to current semester

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 an informal setting. Sometimes people present an interesting paper they find. Other times people give a prep talk for the Algebraic Geometry Seminar or present techniques motivated by the Applied Algebra seminar. Other times people give a series of talks on a topic they have been studying in-depth. Regardless the goal of GAGS is to provide a supportive and inclusive place for all to learn more about algebraic geometry and commutative algebra.

How: If you want to get emails regarding time, place, and talk topics (which are often assigned quite last minute) add yourself to the gags mailing list: gags@g-groups.wisc.edu by sending an email to gags+subscribe@g-groups.wisc.edu. If you prefer (and are logged in under your wisc google account) the list registration page is here.

Current Organizers: Kevin Dao, Yu (Joey) Luo, and Boyana Martinova.

Give a talk!

We need volunteers to give talks this semester. If you're interested, follow the link above to the current semester. 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:

New Wish List as of Fall 2024

This wishlist is based on requests from graduate students (new and old). Don't be intimidated by the list (especially as a new graduate student), a lot of the topics here are advanced. You are always welcome to give a talk on a topic that does not appear on this list. If you are looking for a topic and none of the ones listed below sound compelling to you, you can always reach out to one of the organizers for more ideas!

Topics in Representation Theory. There are many topics one can discussion: explaining Lie algebra representations via Fulton-Harris's book (Lecture 7-9), Brauer theory, the Stone-von Neumann theorem, classification and determination of unitary representations, the Harish-Chandra isomorphism, Borel-Bott-Weil, historical results such as Frobenius determinants. Quiver representations are another topic; there is a well-written book by Ralf Schiffler you could look at for this topic.
The Riemann-Roch Theorem, its generalizations: Grothendieck-Riemann-Roch, Hirzebruch-Riemann-Roch, and applications.
GAGA Theorems and how to use them. Some ideas on important results to talk about can be found here. For some references to look at: the appendix in Hartshorne's Algebraic Geometry, Serre's original GAGA paper, and Neeman's book Algebraic and Analytic Geometry.
Cohen-Macaulay rings and schemes and variants of this type. A useful topic for those working with "mild singularities". The standard reference for this stuff is the book by Brunz and Herzog, but Eisenbud's Commutative Algebra book also has a lot of things to say about CM rings.
Hodge Theory for the working Algebraic Geometer. What is the Hodge decomposition? What is the Hard Lefschetz Theorem? What is the statement of the Hodge conjecture? Dolbeault cohomology?
Algebraic Curves via Hartshorne Chapter IV. What can be said projective curves of degree d and genus g? How do (did) people study algebraic curves? What are the important facts about curves a working algebraic geometer should know?
Algebraic Suraces via Hartshorne Chapter V and Beauville's Complex Algebraic Surfaces. What does the birational classification of complex algebraic surfaces look like? How should we classify objects?
Vector Bundles on P^n. A good reference for this is "Vector Bundles on Complex Projective Spaces" by Christian Okonek. Interesting points of discussion could inclued any of: Horrock's Criterion for vector bundles, Beilinson's Theorem, splitting of uniform bundles of rank r<n, moduli of stable 2-bundles, constructions of vector bundles on P^n for low values of n, Serre's construction of rank 2 bundles, proof of the Grothendieck-Birkhoff Theorem, and etc. These are all very classical problems / theorems in algebraic geometry and a talk on these topics would make a great expository talk.
Basics of Moduli: functor of points, representable functors, moduli of curves M_g, moduli of Abelian varieties of dimension g, and why do we care? A reference is Harris and Morrison, but there is the now growing textbook by Jarod Alper titled "Stacks and Moduli". Lots of lots of examples are encouraged.
What is a syzygy? Compute some minimal free resolutions and tell people about how syzygies can tell you a lot about a curve. The Geometry of Syzygies by David Eisenbud is also a good reference and introduction to this topic.
Derived categories and the Fourier-Mukai Transform. Introduce derived categories and explain their importance in algebraic geometry e.g. through the Fourier-Mukai transform. The book "Fourier-Mukai Transforms in Algebraic Geometry" by Daniel Huybrechts is a good reference for this stuff, but there is also the notes by Andrei Căldăraru on the Arxiv which are more to the point.
Introduction to Algebraic Stacks: there are a number of references for this e.g. Alper's notes on Moduli, "Algebraic Stacks" by Tomas L. Gomez, the original paper of Deligne and Mumford titled "The Irreducibility of the Space of Curves of Given Genus", Martin Olsson's book "Algebraic Spaces and Stacks", and so on. Examples would be strongly encouraged over technical details and Alper's notes and/or Gomez's article are the best for this.
There are many many classes of varieties out there that people are interested -- pick one and it could very well be a talk on its own! Here are a few examples; abelian varieties, secant varieties, tangent varieties, Kazdan-Lutszig varieties, toric varieties, flag varieties, Fano varieties, Prym varieties, and beyond.

Wishlists from Days of Yore

Wishlists from past years can now be found here.

Semesters

Toby the OFFICIAL mascot emeritus of GAGS!!

@@ Line 1: / Line 1: @@
-'''When? Where?:''' [https://wiki.math.wisc.edu/index.php/Graduate_Algebraic_Geometry_Seminar_Spring_2024 Link to current semester]
+'''When? Where?:''' [https://wiki.math.wisc.edu/index.php/Graduate_Algebraic_Geometry_Seminar_Fall_2024 Link to current semester]
 '''Who:''' All undergraduate and graduate students interested in algebraic geometry, commutative algebra, and related fields are welcome to attend.
@@ Line 17: / Line 17: @@
 * Ask Questions Appropriately:
-==The List of Topics that we Made February 2018==
+== New Wish List as of Fall 2024 ==
+This wishlist is based on requests from graduate students (new and old). Don't be intimidated by the list (especially as a new graduate student), a lot of the topics here are advanced. You are always welcome to give a talk on a topic that does not appear on this list. If you are looking for a topic and none of the ones listed below sound compelling to you, you can always reach out to one of the organizers for more ideas!
+*Topics in Representation Theory. There are many topics one can discussion: explaining Lie algebra representations via Fulton-Harris's book (Lecture 7-9), Brauer theory, the Stone-von Neumann theorem, classification and determination of unitary representations, the Harish-Chandra isomorphism, Borel-Bott-Weil, historical results such as Frobenius determinants. Quiver representations are another topic; there is a well-written book by Ralf Schiffler you could look at for this topic.
+*The Riemann-Roch Theorem, its generalizations: Grothendieck-Riemann-Roch, Hirzebruch-Riemann-Roch, and applications.
+*GAGA Theorems and how to use them. Some ideas on important results to talk about can be found [https://en.wikipedia.org/wiki/Algebraic_geometry_and_analytic_geometry#Important_results here]. For some references to look at: the appendix in Hartshorne's Algebraic Geometry, Serre's original GAGA paper, and Neeman's book Algebraic and Analytic Geometry.
+*Cohen-Macaulay rings and schemes and variants of this type. A useful topic for those working with "mild singularities". The standard reference for this stuff is the book by Brunz and Herzog, but Eisenbud's Commutative Algebra book also has a lot of things to say about CM rings.
+*Hodge Theory for the working Algebraic Geometer. What is the Hodge decomposition? What is the Hard Lefschetz Theorem? What is the statement of the Hodge conjecture? Dolbeault cohomology?
+* Algebraic Curves via Hartshorne Chapter IV. What can be said projective curves of degree d and genus g? How do (did) people study algebraic curves? What are the important facts about curves a working algebraic geometer should know?
+*Algebraic Suraces via Hartshorne Chapter V and Beauville's Complex Algebraic Surfaces. What does the birational classification of complex algebraic surfaces look like? How ''should'' we classify objects?
+*Vector Bundles on P^n. A good reference for this is "Vector Bundles on Complex Projective Spaces" by Christian Okonek. Interesting points of discussion could inclued any of: Horrock's Criterion for vector bundles, Beilinson's Theorem, splitting of uniform bundles of rank r<n, moduli of stable 2-bundles, constructions of vector bundles on P^n for low values of n, Serre's construction of rank 2 bundles, proof of the Grothendieck-Birkhoff Theorem, and etc. These are all very classical problems / theorems in algebraic geometry and a talk on these topics would make a great expository talk.
+*Basics of Moduli: functor of points, representable functors, moduli of curves M_g, moduli of Abelian varieties of dimension g, and why do we care? A reference is Harris and Morrison, but there is the now growing textbook by Jarod Alper titled "Stacks and Moduli". Lots of lots of examples are encouraged.
+*What is a syzygy? Compute some minimal free resolutions and tell people about how syzygies can tell you a lot about a curve. The Geometry of Syzygies by David Eisenbud is also a good reference and introduction to this topic.
+* Derived categories and the Fourier-Mukai Transform. Introduce derived categories and explain their importance in algebraic geometry e.g. through the Fourier-Mukai transform. The book "Fourier-Mukai Transforms in Algebraic Geometry" by Daniel Huybrechts is a good reference for this stuff, but there is also the notes by Andrei Căldăraru on the Arxiv which are more to the point.
+* Introduction to Algebraic Stacks: there are a number of references for this e.g. Alper's notes on Moduli, "Algebraic Stacks" by Tomas L. Gomez, the original paper of Deligne and Mumford titled "The Irreducibility of the Space of Curves of Given Genus", Martin Olsson's book "Algebraic Spaces and Stacks", and so on. Examples would be strongly encouraged over technical details and Alper's notes and/or Gomez's article are the best for this.
+*There are many many classes of varieties out there that people are interested -- pick one and it could very well be a talk on its own! Here are a few examples; abelian varieties, secant varieties, tangent varieties, Kazdan-Lutszig varieties, toric varieties, flag varieties, Fano varieties, Prym varieties, and beyond.__NOTOC__
-On February 21st of the Month of February of The 2018th Year of the Seventh Age of The Sun, the People Present at GAGS Compiled Ye Followinge Liste of Topics They Wished to Hear Aboute:
+=== Wishlists from Days of Yore ===
+Wishlists from past years can now be found [[Old GAGS Wish Lists|here]].
-Feel free to edit the list and/or add references to learn this stuff from. Since then, we've succeeded in talking about some of these, which doesn't mean there shouldn't be another talk. Ask around or look at old semester's websites.
+== Semesters ==
-* Schubert Calculus, aka how many lines intersect four given lines in three-dimensional space? The answer to this question is prettiest when you think about it as a problem of intersecting subvarieties in the Grassmanian. ''What is the Grassmanian, you say?'' That's probably a talk we should have every year, so you should give it!
-* Kindergarten GAGA. GAGA stands for Algebraic Geometry - Analytic Geometry. Serre wrote a famous paper explaining how the two are related, and you could give an exposition suitable to kindergardeners.
-* Katz and Mazur explanation of what a modular form is. What is it?
-* Kindergarten moduli of curves.
-* What is a dualizing sheaf? What is a dualizing complex? What is Serre duality? What is local duality? Can local duality help us understand Serre duality?
-* Generalizations of Riemann - Roch. (Grothendieck - Riemann - Roch? Hirzebruch - Riemann - Roch?)
-* Hodge theory for babies
-* What is a Néron model?
-* What is a crystal? What does it have to do with D-modules? [http://www.math.harvard.edu/~gaitsgde/grad_2009/SeminarNotes/Nov17-19(Crystals).pdf Here's an encouragingly short set of notes on it].
-* What and why is a dessin d'enfants?
-* DG Schemes.
-==Ed Dewey's Wish List Of Olde==__NOTOC__
-Back in the day Ed and Nathan made this list of topics they wanted to hear. They all sound super duper cool, but it's also true that they had many years of AG behind their backs, so this list might not be very representative of what the GAGS audience wants to hear bout.
-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)
-===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.)
+[[Image:newcat.jpg|thumb|220px| Toby the OFFICIAL mascot emeritus of GAGS!!]]
-== Semesters ==
+[https://wiki.math.wisc.edu/index.php/Graduate_Algebraic_Geometry_Seminar_Fall_2024 Fall 2024]
-[https://wiki.math.wisc.edu/index.php/Graduate_Algebraic_Geometry_Seminar_Fall_2025 Fall 2025]
 [https://wiki.math.wisc.edu/index.php/Graduate_Algebraic_Geometry_Seminar_Spring_2024 Spring 2024]