# Michigan Derived Algebraic Geometry RTG Learning Workshop

## May 17-19, 2012

East Hall, University of Michigan
Organized by Bhargav Bhatt and Daniel Erman

## Goal of workshop

The goal of this workshop is for the participants to obtain a rudimentary understanding of the motivation and uses of derived algebraic geometry. Since this is a learning workshop, the talks will be given by non-experts, and the assumed background will be minimal. For instance, we assume that most of the audience (and most of the speakers!) will have little or no previous exposure to derived algebraic geometry.

This workshop is being partially funded by an NSF Grant for Michigan's RTG in Algebraic Geometry. We are grateful to the NSF and to the Michigan Department of Mathematics for making this workshop possible.

## Plan of lectures

We plan to have nine 50-minute lectures over the course of three days. Note that we are very open to suggestions from interested participants about how to improve this plan.
1. Motivation: Bezout's Theorem . Derived algebraic geometry gives a new perspective on Bezout's theorem, particularly in the case of degenerate intersections (e.g. interesting a line with itself in the projective plane.) The material will be drawn from the introduction to Lurie's thesis.
2. Simplicial sets and higher categories. We will introduce some of the category theory notions that appear in derived algebraic geometry.
3. Simplicial commutative rings, I. We introduce the notion of a simplicial commutative rings and include some examples.
4. Simplicial commutative rings, II. Further discussion of examples and properties of simplicial rings.
5. Cotangent complex, I. The cotangent complex is essential to the study of deformation theory. We outline these ideas in the "classical" format in which they were discovered by Quillen, Andre, and Illusie.
6. Cotangent complex, II. Derived algebraic geometry provides a clarifying perspective on the cotangent complex.
7. Derived schemes. This will be a discussion of to add a derived structure onto a scheme.
8. Application to obstruction theories. Obstruction theories, and the construction of a virtual fundamental class, can be seen in terms of the introduction of a derived structure. The focus will be on examples, such as M_g(X,\beta).
9. Application to loop spaces. Loop spaces provide interesting examples of derived schemes.

## References

There are lots of possible references for this workshop, and we have collected some of our favorites here. (And thanks to everyone who emailed us with their own suggestions! Further suggetions are very welcome.)

Background Material
We are asking all participants to do a bit of preparation before the workshop. In particular, we would like everyone to look at the following topics (possible references are given in each bullet point). Note that we do not expect you to be an expert on these topics! We just want you to have some familiarity with the following definitions and examples.
1. Simplicial sets and the Dold-Kan correspondence. We suggest notes by Akhil Matthew, available here.
2. Model Categories.. Many possible references here, including Wikipedia, a survey by Dwyer and Spalinski available here, or the book "Model Categories" by Hovey.
3. Spend an hour or two reading at least one of the "Introductory References" from below.
Introductory References on Derived Algebraic Geometry
There is a huge list of potential references for this workshop. We begin with a few introductory references.
1. "What is a Derived Stack" by Vezzosi. We strongly encourage all participants to read this 4-page introduction to some of the ideas in derived algebraic geometry. (PDF version )
2. "A note on the cotangent complex in derived algebraic geometry" by Vezzosi. This short note is quite readable, though it does assume some previous knowledge of deformation theory
3. "Simplicial presheaves and derived algebraic geometry" by Toen. ( PDF version ).
4. The introduction of Lurie's Thesis. This introduction provides motivation for the study of derived algebraic geometry. (PDF version).
Further References
Here are some further references that may be of interest.
1. "Higher and derived stacks: a global overview" by Toen (arXiv version).
2. "Andre-Quillen homology of commutative algebras" by Iyengar. (arXiv version).
3. Quillen's MIT notes "Homology of Commutative Rings".
4. Quillen's paper "On the (co-)homology of commutative rings".
5. Lurie's Higher Algebra, especially chapter 8.

Funding
The deadline for applying for funding has passed. Unfortunately, we do not expect to be able to any additional applicants at this point.

## Schedule

 Date and time Location Speaker Title Notes Thursday, May 17 9:00-10:10 Math Atrium (East Hall) Coffee and registration 10:10-11:00 260 Dennison Hall Daniel Erman Motivation: Bezout's Theorem Notes 11:10-12:00 260 Dennison Hall Philsang Yoo Simplicial sets and higher categories Notes 2:10-3:00 260 Dennison Hall Akhil Matthew Simplicial Commutative Rings, I Notes or LaTeX version of notes . 3:00-3:30 Math Atrium (East Hall) Tea 3:30-?? Math Atrium (East Hall) Informal Q&A 5:00-6:00 Dominic's Conference Happy Hour Friday, May 18 9:00-10:10 Math Atrium (East Hall) Coffee 10:10-11:00 296 Dennison Hall George Boxer Simplicial Commutative Rings, II Notes 11:10-12:00 296 Dennison Hall Daniel Litt Cotangent Complex, I Notes 2:10-3:00 296 Dennison Hall Alex Perry Cotangent Complex, II Notes 3:00-3:30 Math Atrium (East Hall) Tea 3:30-?? Math Atrium (East Hall) Informal Q&A Saturday, May 19 9:00-10:10 Math Atrium (East Hall) Coffee 10:10-11:00 296 Dennison Hall Bhargav Bhatt Derived Schemes Notes 11:10-12:00 296 Dennison Hall Jonathan Wise Applications to Obstruction Theories Notes 2:10-3:00 296 Dennison Hall Anatoly Preygel Applications to Loop Spaces Notes

## Participant List

Here is a partial list of participants. Please let us know about any misspellings.
1. Atanas Atanasov, Harvard
2. Bhargav Bhatt, Michigan
3. George Boxer , Harvard
4. Jonathan Campbell , Stanford
5. Jinwon Choi , UIUC
6. Atoshi Chowdhury , Stanford
7. Philip Egger , Northwestern
8. Chris Elliott , Northwestern
9. Daniel Erman, Michigan
10. Giovanni Faonte , Yale
11. Alberto Garcia Raboso , UPenn
12. Christopher Gomes , UIC
13. Francois Greer , Stanford
14. Nathan Grigg , UW
15. Wei Ho , Columbia
16. Xin Jin , Northwestern
17. Clemens Koppensteier , Northwestern
18. Raju Krishnamoorthy , Columbia
19. Binglin Li , UC Davis
20. Daniel Litt , Stanford
21. Luigi Lombardi , UIC
22. Akhil Mathew , Harvard
23. Evgeny Mayanski , PSU
24. Yusuf Mustopa , Michigan
25. Andrew Niles , Berkeley
26. Vivek Pal , Columbia
27. Xuanyu Pan , Columbia
28. Alex Perry , Harvard
29. Anatoly Preygel , MIT
30. Marius Radulescu , Chicago State University
31. Daniel Ross ,
32. Matt Satriano, Michigan
33. Daniel Schultheis , UCSD
34. Michel van Garrell , Caltech
35. Matt Wechter , UIC
36. Jonathan Wise ,
37. Matt Woolf , Harvard
38. Philsang Yoo , NW
39. Naizhen Zhang , UC Davis
40. Yi Zhu , Stonybrook
41. Runpo Zong , Princeton
42. Pal Zsamboki , UW