Graduate Algebraic Geometry Seminar Spring 2024: Difference between revisions
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| bgcolor="#C6D46E" | | | bgcolor="#C6D46E" |Bella Finkel Holman | ||
| bgcolor="#BCE2FE" | | | bgcolor="#BCE2FE" |Lie Algebra Representations and Some Cute Applications | ||
| bgcolor="#BCE2FE" | | | bgcolor="#BCE2FE" |We'll introduce representations of Lie groups and Lie algebras and discuss some applications to quantum mechanics and quantum information theory. In particular, we will describe how the relationship between representations of a Lie group and its Lie algebra can elucidate the concept of "fractional spin." | ||
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| bgcolor="#E0E0E0" |03-20-2024 | | bgcolor="#E0E0E0" |03-20-2024 | ||
Revision as of 20:35, 6 February 2024
When: 2:30PM - 4:00PM every Wednesday starting January 31st, 2024. Talks are for 30 minutes - 1 hour with extra time for questions.
Where: Van Vleck B325
Who: All undergraduate and graduate students interested in algebraic geometry, abstract algebra, commutative algebra, representation theory, and related fields are welcome to attend.
Why: The purpose of this seminar is to learn algebraic geometry, commutative algebra, and broadly algebra itself, by giving and listening to talks in a informal setting. Sometimes people present an interesting topic or 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: 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.
Enrollment in Math 941: The correct section to enroll for Math 941 is is with primary instructor being Dima Arinkin. If you are signed up for this section, you are expected to give a talk to get a grade.
Organizers: John Cobb, Kevin Dao, Yu (Joey) Luo.
Feedback Form for Organizers: The form is anonymous. You can find it here.
Give a talk!
We need volunteers to give talks. Beginning graduate students are encouraged to give a talk. If you need ideas, see the wish list below, the main page, or talk to an organizer.
It is expected that people enrolled in Math 941: Seminar in Algebra must give a talk to get credit.
The sign-up for Math 941: Seminar in Algebra is a section with Dima Arinkin. Please make sure you signed up for this section if you want credit.
Sign up sheet: https://forms.gle/JofcgHVZyQmEKpcX7.
New Wishlist as of Spring 2024
The following is a list of topics that would be good to have in a graduate algebra and algebraic geometry seminar.
- 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.
- 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, 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".
- 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.
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 abstract algebra, algebraic geometry, representation theory, 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
- Save lengthy questions or highly technical questions for after the talk
Talks
| Date | Speaker | Title | Abstract |
| 01-31-2024 | Kevin Dao | Setting up GAGS + A Survival Guide to Sheaf Cohomology. | Discussion about GAGS expectations + getting list of speakers.
The short talk shall be about the basics of sheaf cohomology and all about telling the audience what they need to start computing things. The goal is to prove the genus-degree formula for smooth curves in the projective plane. |
| 02-07-2024 | Boyana Martinova | An Introduction to Cohen-Macaulay Rings | In this talk, I will introduce Cohen-Macaulay rings and discuss some key properties that make them a desirable class of rings to study. We'll explore some key techniques for determining whether a ring is Cohen-Macaulay and test various examples along the way. I plan to build all the dimension theory that is needed as we go, so this talk should be especially accessible to early graduate students. |
| 02-14-2024 | Caitlin Davis | Introduction to the Rational Normal Curve | The rational normal curve is an important example of many nice algebraic and geometric properties. I will discuss some of these properties, focusing on small concrete examples. This talk will aim to be accessible to grad students who have taken a semester or two of abstract algebra, and will not assume much (if any) algebraic geometry background. |
| 02-21-2024 | Jack Messina | Introducing Nonabelian Hodge Theory | |
| 02-28-2024 | Yiyu Wang | ||
| 03-06-2024 | Alex Mine | Gorenstein Rings and Duality | |
| 03-13-2024 | Bella Finkel Holman | Lie Algebra Representations and Some Cute Applications | We'll introduce representations of Lie groups and Lie algebras and discuss some applications to quantum mechanics and quantum information theory. In particular, we will describe how the relationship between representations of a Lie group and its Lie algebra can elucidate the concept of "fractional spin." |
| 03-20-2024 | Jacob Wood | ||
| 04-03-2024 | Ruocheng Yang | ||
| 04-10-2024 | Yaoxian Yang | ||
| 04-17-2024 | Ivan Aidun | ||
| 04-24-2024 | |||
| 05-01-2024 |