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Summer School: Algebraic Stacks and Related Topics

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This is a Summer School of the SFB/TRR 45 Bonn-Essen-Mainz financed by the DFG (Deutsche Forschungsgemeinschaft). It will take place from August 31st until September 4th, 2015 at the University of Mainz (Germany).

What
  • Summer School
When Aug 31, 2015 09:00 AM to
Sep 04, 2015 01:00 PM
Where Mainz, 03-428 and 05-432
Contact Name
Contact Phone +49-6131-3921295
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This summer school is intended for advanced master students, PhD students, and Young Researchers in Algebra, Number Theory and Geometry. 

Its aim is to provide an introduction to the theory of algebraic stacks and its applications to algebraic and arithmetic geometry.

Here is the list of participants.

We are very grateful to Nivedita Bhaskhar, Raymond van Bommel, Daniel Bragg, David Holmes, Siddharth Mathur,  and Arne Smeets for providing typed notes of the talks.

Some pictures of the event can be found here.

 

Invited Speakers:

 

Jarod Alper, Australian National University (Australia)

Martin Olsson, University of California (USA)

Max Lieblich, University of Washington (USA)

Lenny Taelman, University of Amsterdam (Netherlands)

Angelo Vistoli, Scuola Normale Superiore di Pisa (Italy)

 

There will be four lectures of 4x1 hour by Alper, Olsson, Lieblich, and Vistoli, and an extra 2 hours lecture by Taelman.

 

Abstracts and titles


Jarod Alper

Title:  Artin algebraization and the local quotient structure of algebraic stacks (notes typed by Jarod himself)

Abstract: These lectures will begin with a discussion of Artin approximation and algebraization followed with several applications.  I will then develop an equivariant analogue of Artin approximation and algebraization which will be applied to prove a local structure theorem for algebraic stacks.  Namely, the main goal of these lectures is to establish the following theorem:  every algebraic stack, locally of finite type over an algebraically closed field with affine stabilizers, is etale-locally a quotient stack in a neighborhood of a point with linearly reductive stabilizer group.  Numerous applications of this theorem will also be discussed.

Recommended literature:

M. Artin, Algebraic approximation of structures over complete local rings, Inst. Hautes Etudes Sci. Publ. Math. (1969), no. 36, 23-58.

M. Artin, Algebraization of formal moduli. I, Global Analysis (Papers in Honor of K. Kodaira), Univ. Tokyo Press (1969), 21-71.

B. Conrad and J. de Jong, Approximation of versal deformations, http://math.stanford.edu/~conrad/papers/approx.pdf

J. Alper, J. Hall, and D. Rydh, A Luna etale slice theorem for algebraic stacks, http://arxiv.org/abs/1504.06467


Martin Olsson 

Title: Algebraic stacks and log geometry (notes typed by Nivedita Bhaskhar and Siddharth Mathur, favorite example of Martin is here)

Abstract: In these lectures I will discuss the connection between algebraic stacks and logarithmic geometry in the sense of Fontaine, Illusie, and Kato.  This connection provides many examples for the theory of stacks and offers a new perspective on log geometry, and I will discuss applications in both directions.

Recommended literature:

Logarithmic geometry and Moduli http://arxiv.org/abs/1006.5870

Logarithmic geometry and algebraic stacks https://math.berkeley.edu/~molsson/lvs3.pdf

 


Max Lieblich

Title: The unreasonable effectiveness of twisted sheaves (notes typed by Daniel Bragg)

Abstract: I will discuss twisted sheaves and their surprisingly broad applications to abstract algebra, number theory, and the geometry of moduli spaces. As I hope to explain, the key to unlocking the power of twisted sheaves lies in embracing their stacky origins.


 

Lenny Taelman

Title: The stack of curves of genus one

Abstract: In these lectures we will study one example of an algebraic stack: the stack M_1 that classifies curves of genus one (without a chosen base point). In particular, we will compute its cohomology, count its number of points over finite fields, and study the invariants (`characteristic classes') of genus one curves coming from cohomology classes of the stack M_1.


 

Angelo Vistoli

Title: Gerbes (notes typed by David Holmes)

Abstract: Gerbes are 2-categorical analogues of principal bundles. In the first part of the course I will discuss the abstract theory of gerbes, in particular the cohomological interpretation of gerbes with abelian bands. In the second part I will give some applications, possibly to twisted sheaves, essential dimension and fundamental gerbes.

Recommended literature:

The Nori fundamental gerbe of a fibered category http://arxiv.org/abs/1204.1260

Essential dimension and algebraic stacks http://arxiv.org/abs/math/0701903

Brauer groups and quotient stacks http://arxiv.org/abs/math/9905049


 

Schedule:

 

  Monday Tuesday Wednesday Thursday Friday
9:00-10:00 Registration
Alper
Lieblich
Olsson
Taelman
10:00-10:30
Coffee break
Coffee break Coffee break Coffee break Coffee break
10:30-11:30
Vistoli
Lieblich
Alper
Vistoli
Taelman
11:45-12:45
Alper
Olsson
Vistoli
Alper
Olsson
13:00-15:00
Lunch Lunch Lunch Lunch Lunch 
15:00-16:00
Lieblich
Vistoli
Free afternoon
Lieblich

16:00-16:30
Coffee break Coffee break Free afternoon
Coffee break
16:30-17:30
Olsson
Questions
Free afternoon
Questions

 

Registration will be in room 05-432 (5th floor).

All lectures will take place in room 03-428 (3rd floor).

 

Organizers:

 

Ariyan Javanpeykar (Mainz)

Ronan Terpereau (Mainz)

If you have any questions related to the funding or the accomodation, please write to Frau Gonska: gonska@mathematik.uni-mainz.de

 

Registration and Financial Support:

 
Full financial support is available for members of the SFB/TRR 45.

Registration is now closed.


Due to the large number of participants it is unfortunately not possible to admit any more people. This, unfortunately, also applies for SFB members.


Accommodation:


We have reserved a block of rooms at Hotel Hammer and Cityhotel Neubrunnenhof  and Advena Hotel Mainz, all hotels are located close to the main train station in Mainz. Participants are asked to please contact one of the hotels directly by stating the key word "Summer School University of Mainz". Please note, however, that the number of rooms is limited and rooms are allocated on a "first come - first serve" basis so early reservation is recommended.

Of course, it is possible to make reservations in other hotels as well. If you prefer that, we refer to the home page of the City of Mainz where you will find a detailed list of hotels in and around Mainz.

 

Travel information:

Closest airport is located at Frankfurt/Main. From there, there are frequent trains to Mainz central station (Hauptbahnhof).

The fastest way to the Institut fur Mathematik from the Hauptbahnhof (Mainz main railway station) is to take Bus line 69 which departs from platform F. Get off at "Duesbergweg", the Mathematics building is to your right behind the parking lot. (The walk is about 2 minutes.) The timetable for the Bus line 69 is here (please pay attention to the fact that the summer school takes place during a vorlesungsfrei period, i.e. you have to watch the last column).  

Alternatively,  you may take bus lines 9, 54, 55, 58, 68; the bus stop is "Friedrich-von-Pfeiffer-Weg". From there, walk over the pedestrian bridge to the university campus and follow the street to the right. After about 100 meters there is a left curve. After passing the Mensa (campus cafeteria) you see the Mathematics building right in front of you. (The walk is about 10 minutes.)

 

All lectures will take place at the Institute of Mathematics of the University of Mainz, Staudinger Weg 9, in room 03-428.


Suggestions for the evenings and Wednesday afternoon:

 

  • There will be a diner in restaurant organized on Wednesday (probably at Heiliggeist, to be confirmed). We will meet at 19.00 inside the restaurant (so that we do not block the street waiting altogether outside). If you are interested to come, then please inform the organizers by Tuesday evening.
  • For Wednesday afternoon, if you want to visit the city we recommend you to visit the Gutenberg museum, the Mainz cathedral, and the St. Stephan church. However, if you prefer to walk outside the city, you can either hike along the Rhein (see here for more information) or else here is a pleasant walk that you can do in three/four hours.
  • Here is a list of some nice places where you can eat in Mainz: Heiliggeist, Mosch Mosch, Lomo, Bully's burgers, Pomp, Sausalitos, Santiago, Buddha.
  • If you fancy wine there is a wine salon on Tuesday 1st September. Moreover, there is also a wine market from Thursday 3rd till Sunday 6th.

Miscellaneous:

 

 


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