Graduate Algebraic Geometry Seminar Spring 2020
When: Thursday 5:00-6:00 PM CST
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: email@example.com. The list registration page is here.
Organizers: David Wagner, Colin Crowley.
Give a talk!
We need volunteers to give talks this semester. If you're interested contact Colin or David, 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:
|Date||Speaker||Title (click to see abstract)|
|January 29||Colin Crowley||Lefschetz hyperplane section theorem via Morse theory|
|February 5||Asvin Gothandaraman||An Introduction to Unirationality|
|February 12||Qiao He||Title|
|February 19||Dima Arinkin||Blowing down, blowing up: surface geometry|
|February 26||Connor Simpson||Intro to toric varieties|
|March 4||Peter||An introduction to Grothendieck-Riemann-Roch Theorem|
|March 11||Caitlyn Booms||Intro to Stanley-Reisner Theory|
|March 25||Steven He||Braid group action on derived categories|
|April 1||Vlad Sotirov||Title|
|April 8||Maya Banks||Title|
|April 15||Alex Hof||Embrace the Singularity: An Introduction to Stratified Morse Theory|
|April 22||Ruofan||Birational geometry: existence of rational curves|
|April 29||John Cobb||Title|
|Title: Lefschetz hyperplane section theorem via Morse theory|
|Abstract: Morse theory allows you to learn about the topology of a manifold by studying the critical points of a nice function on the manifold. This perspective produces a nice proof of the theorem in the title, which concerns the homology of smooth projective varieties over C. I'll explain what the theorem says, say something about what Morse theory is and why it's related, and then finish with a neat example. I'm aiming to make this understandable to someone who's taken algebraic geometry 1 and topology 1.|
|Title: An introduction to unirationality|
|Abstract: I will introduce the notion of unirationality and show that cubic hypersurfaces are unirational (following Kollar). If time permits, I will also show that unirational varieties are simply connected.|
|Title: Blowing down, blowing up: surface geometry|
|Abstract:A big question in algebraic geometry is how much one can change a variety without affecting it `generically'. More precisely, if two varieties are birational, how far can they be from being isomorphic?
The question is trivial for (smooth projective) curves: they are birational if and only if they are isomorphic. In higher dimension, the situation is much more interesting. The most fundamental operation are the `blowup', which is a kind of alteration of a variety within its birational isomorphism class, and its opposite, the blowdown.
In my talk, I will introduce blowups and discuss their properties. Then (time permitting) I would like to look deeper at the case of surfaces, where the combination of blowups and intersection theory provides a complete and beautiful picture. (If we do get to this point, I won't assume any knowledge of intersection theory: to an extent, this talk is my excuse to introduce it.)
|Title: Intro to Toric Varieties|
|Abstract: A brief introduction to toric varieties: how to build them, formulas for computing topological data, toric blow-ups, and more.|
|Title: An introduction to Grothendieck-Riemann-Roch Theorem|
|Abstract: The classical Riemann-Roch theorem tells you about how topological (genus) and analytical (through line bundle) properties on compact Riemann surface (i.e. smooth projective curve) relate to each other. Moreover, this theorem can be generalized to any vector bundles (or coherent sheaves) over any smooth projective varieties. Eventually, Grothendieck “relativized” this theorem as a property of a morphism between two projective varieties. In this talk I will introduce basic notions to formulate this theorem. If time permitting, enough examples will be given appropriately.|
|Title: Intro to Stanley-Reisner Theory|
|Abstract: Stanley-Reisner theory gives a dictionary between combinatorial objects (simplicial complexes) and algebraic objects (Stanley-Reisner rings). In this talk, I will introduce the main objects of study in this theory, describe this dictionary with several examples, and discuss how Stanley-Reisner theory can help us investigate algebra-geometric questions.|
|Title: Braid group action on derived category|
|Abstract: In this talk, I will define spherical object and A_m-configuration in derived category of coherent sheaves, and say a few words about the motivation coming from the homological mirror symmetry.|
|Title: Embrace the Singularity: An Introduction to Stratified Morse Theory|
|Abstract: Early on in the semester, Colin told us a bit about Morse
Theory, and how it lets us get a handle on the (classical) topology of smooth complex varieties. As we all know, however, not everything in life goes smoothly, and so too in algebraic geometry. Singular varieties, when given the classical topology, are not manifolds, but they can be described in terms of manifolds by means of something called a Whitney stratification. This allows us to develop a version of Morse Theory that applies to singular spaces (and also, with a bit of work, to smooth spaces that fail to be nice in other ways, like non-compact manifolds!), called Stratified Morse Theory. After going through the appropriate definitions and briefly reviewing the results of classical Morse Theory, we'll discuss the so-called Main Theorem of Stratified Morse Theory and survey some of its consequences.
|Title: Birational geometry: existence of rational curves|
|Abstract: Rational curves on a variety control its birational geometry. It thus is important to determine whether they exist. People didn’t know how to do this systematically, before Mori discovered a deformation lemma which detect their existence, and bound their degree if they exist. I will briefly introduce Mori’s insight.|
The List of Topics that We Made February 2018
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:
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.
- 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? 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
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.)