Masterclass: Seiberg-Witten Invariants

Lin Jianfeng (Tsinghua University)

In this course, we present the definition of the family Seiberg–Witten invariants and the Bauer–Furuta invariants. We apply this to diffeomorphism groups, discussing exotic diffeomorphisms, homological instability and infinite generation of mapping class groups. In the end, we cover some interactions with symplectic and algebraic geometry such as the symplectic Torelli group, exoctic symplectomorphisms on K3 surfaces and the monodromy of Milnor fibrations.

Thomas Kragh (Uppsala University)

I will start by explaining how a TQFT structure can be used as a tool for computing Bauer-Furuta invariants. In particular, this motivates constructing certain stable homotopy types associated to spin-c 3-manifolds. I will discuss how to do this in a sequence of increasingly general cases.
The main part will start with me outlining the basic ideas of Conley index theory, as it is central to all the constructions. Next, I will go through a simplified version of Manolescu's first construction in the case where the first Betti number vanishes. Then I will discuss a simplified version of the fully unfolded theory of Khandhawit--Lin--Sasahira. Finally, I will sketch how to twist the theory to obtain a construction for general 3-manifolds. The final part is joint work with Behrens and Hedenlund.

Tomasz Mrowka (MIT)

This lecture series covers the basics of Seiberg-Witten equations including spin-c structures, Dirac operators and motivations for the equations. It also covers some vanishing theorems, including a proof of the adjunction inequalities, and some calculations for the Seiberg-Witten invariants for Kähler manifolds. Finally, we will present a stable homotopy refinement by extracting stable homotopy classes from Fredholm maps and discussing S^1 and the Pin(2)-symmetry, Bauer-Furuta invariants and the 10/8 theorem.

András Stipsicz (Alfréd Rényi Institute of Mathematics)

In the five lectures we plan to introduce basic notions and constructions from the theory of smooth four-manifolds. Lecture 1 will focus on the algebraic topology of 4-manifolds, and recalls the fundamental theorems of Freedman and Donaldson, while discussing intersections forms and also reflects on the 11/8-conjecture, the most important question regarding realizability. Lecture 2 shows basic properties of the Seiberg-Witten invariants, discusses genus function of four-manifolds, and using the blow-up procedure, provides  several interesting examples, while keeping track of surfaces in the resulting four-manifolds. In Lecture 3 we showcase some further constructions, namely knot surgeries and rational blow-downs. Lecture 4 returns to earlier examples: we discuss elliptic surfaces and Lefschetz fibrations. Finally in Lecture 5 we show further exotica, now  with fundamental group Z_2.

The lectures on Monday, Tuesday and Thursday take place in Auditorium 6. On Wednesday morning, we are in Auditorium 1. On Wednesday afternoon, we are in Auditorium 4. On Friday, we are in Auditorium 5.

The conference dinner will begin at 6:45 PM at Foodclub Nørrebro. For anyone interested, we can meet at 6:15 PM in front of the building and walk to the restaurant all together.

The conference/masterclass will take place at the Department of Mathematical Sciences, University of Copenhagen. See detailed instructions on how to reach Copenhagen and the conference venue.

Tickets and passes for public transportation can be bought at the Copenhagen Airport and every train or metro station. You can find the DSB ticket office on your right-hand side as soon as you come out of the arrival area of the airport. Alternatively, you can use the "Rejsekort" app to book a ticket for public transit.

A journey planner in English is available.

More information on the "find us" webpage.

We kindly ask the participants to arrange their own accommodation.

We recommend Hotel 9 Små Hjem, which is pleasant and inexpensive and offers rooms with a kitchen. Other inexpensive alternatives are Steel House Copenhagen (close to city centre), and CabInn, which has several locations in Copenhagen: the Hotel City (close to Tivoli), Hotel Scandinavia (Frederiksberg, close to the lakes), and Hotel Express (Frederiksberg) are the most convenient locations; the latter two are 2.5-3 km from the math department. Somewhat more expensive – and still recommended – options are Hotel Nora and  Ibsen's Hotel.

An additional option is to combine a stay at the CabInn Metro Hotel with a pass for Copenhagen public transportation (efficient and reliable). 

Registration is closed.

In case you need a confirmation of registration for visa purposes please contact Jan Tapdrup.

Organisers:

Jonathan Clivio (jocl "at" math.ku.dk)

Lucas Larsen (lukas.larsen "at" math.ku.dk)

Fadi Mezher (fadi.mezher "at" kit.edu)

Nathalie Wahl (wahl "at" math.ku.dk)

Admin:

Jan Tapdrup (jt "at" math.ku.dk)