Spring School: Irregular Hodge Structures and Stability Conditions

Classical Variations of Hodge structures give rise to bundles with connections that have logarithmic poles. Connections with higher order poles lead to a more general notion of Hodge structure which is called "irregular" or "non-commutative". The notion of irregularity refers to work of Deligne where he introduces exponential periods that indicate a non-integral, even irrational degree of their Hodge filtration, e.g. a period of \sqrt{\pi} indicates a Hodge degree of 1/2. Deligne also relates such periods to the cohomology of the twisted de Rham complex, i.e. the usual de Rham differential d is replaced by d+df for f a regular function. The more recent terminology "non-commutative" refers to the circumstance that such Hodge structures should appear as the natural Hodge structures to associate to noncommutative spaces. As suggested by Katzarkov-Kontsevich-Pantev, such spaces are represented by dg- or A_\infty-categories which appear for instance in homological mirror symmetry as Fukaya(-Seidel)-categories or categories of matrix factorisations. The purpose of this school is to provide the participants with working level skills of dealing with irregular Hodge structures and to generate familiarity with applications of these in homological mirror symmetry and enumerative geometry. Stability conditions on the associated categories can be thought of as an analogy of what polarisations do in complex geometry. They enable counting objects and such counts reappear in the Stokes data of the irregular connection of the associated Hodge structure - closing the circle that connects these concepts.

Speakers:

Tentative Schedule:

Registration: Monday, Feb 29, 9:00 - 9:30, Room 05-432 (Hilbertraum), 5th floor

Course material

• Claus Hertling: course description and exercises pdf, solutions to exercise 2
• Claude Sabbah: lecture notes and exercises pdf
• Paolo Stellari: course description and exercises pdf
• Victoria Hoskins: exercises with hints pdf
• Jacopo Stoppa: course plan and exercises pdf

Not far from Mainz is the famous Kloster Eberbach, site of the movie casting for "The Name of the Rose" after the well-known novel by recently deceased Umberto Eco. We booked for you a "strolling wine tasting tour" of the monastry on Wednesday afternoon. This is a combination of a guided tour with a wine tasting of six different classy regional Rieslings. We collect the symbolic contribution of 10 EUR from those interested in the event on Monday morning. A bus will pick us up at 2pm on Wednesday. The tour at the Kloster begins at 3pm and takes about 90 minutes. The bus will leave again from the monastry at 5pm to bring you back to the hotel.

References:

• "The Space of Stability Conditions on Abelian Threefolds, and on some Calabi-Yau Threefolds" by Arend Bayer, Emanuele Macrì, Paolo Stellari: http://arxiv.org/abs/1410.1585
• "Stability conditions on triangulated categories" by Tom Bridgeland: http://arxiv.org/abs/math/0212237
• "Stability conditions and Stokes factors" by Tom Bridgeland, Valerio Toledano-Laredo: http://arxiv.org/abs/0801.3974
• "Stability data, irregular connections and tropical curves" by Sara Angela Filippini, Mario Garcia-Fernandez, Jacopo Stoppa:  http://arxiv.org/abs/1403.7404
  
• "Examples of non-commutative Hodge structures" by Claus Hertling, Claude Sabbah: http://arxiv.org/abs/0912.2754
• "Stratifications for moduli of sheaves and moduli of quiver representations", Victoria Hoskins: http://arxiv.org/abs/1407.4057
• "Non-commutative Hodge structures" by Claude Sabbah: http://arxiv.org/abs/1107.5890

Participants:

Organizers:

• Helge Ruddat (Mainz)
• Duco van Straten (Mainz)
• Stefan Müller-Stach (Mainz)

If you have any questions related to the funding or the accommodation, please write to Jutta Gonska

Registration and Financial Support:

 
Full financial support is available for members of the SFB/TRR 45. There will be limited support for other participants, too.

Registration is possible until January 31, 2016, at the latest.

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 "Spring 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 spring 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 05-426 (fifth floor).

Coffee and registration will be in room 05-432 (Hilbertraum).

Miscellaneous: 

• Map of Campus
• Location of Campus within the City of Mainz
• Weather in Mainz
• Deutsche Bahn website