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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. Sometimes people present an interesting paper they find. Other times people give a prep talk for the 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 by sending an email to firstname.lastname@example.org. If you prefer (and are logged in under your wisc google account) the list registration page is here.
Organizers: John Cobb, Colin Crowley.
Give a talk!
We need volunteers to give talks this semester. If you're interested, please fill out this form. Beginning graduate students are particularly encouraged to give a talk, since it's a great way to get your feet wet with the material. If you would like some talk ideas, see the list on the main page.
Spring 2022 Topic Wish List
This was assembled using input from an interest form at the beginning of the semester. Choose one and you will have the rare guarantee of having one interested audience member. Feel free to add your own.
- Computing things about Toric varieties
- Reductive groups and flag varieties
- Geothendieck '66, "On the de Rham Cohomology of Algebraic Varieties"
- Going from line bundles and divisors to vector bundles and chern classes
- A History of the Weil Conjectures
- Mumford & Bayer, "What can be computed in Algebraic Geometry?"
- A pre talk for any other upcoming talk
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
|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.