Local Organiser : Ludmil Katzarkov

Organisers : Benoît Bertrand, Erwan Brugallé, Ilia Itenberg, Grigory Mikhalkin

Vienna 14th-15th 2012

Dec. 14, Hoersaal 2, UZA2

14:30 G. Mikhalkin (Geneva)

Immersed real and tropical plane curves

We look at topological classification of tropical and real rational planar curves, in particular those of degree 5. (Joint work in progress with Ilia Itenberg and Johannes Rau.)

15:30 Coffee

16:00 I. Itenberg (Paris)

Quantum enumeration of tropical curves.

Recently, Florian Block and Lothar Goettsche introduced new polynomial multiplicities for plane tropical curves. We show that these multiplicities give rise to a new invariant way to enumerate plane tropical curves. This enumeration can be interpreted as a certain refinement of Mikhalkin's tropical enumeration of complex curves and has applications concerning enumeration of real curves. (Joint work with Grigory Mikhalkin.)

19:30 Wine and Cheese at Garnisongasse 3, 2nd floor

Dec. 15, C 209, UZA4

09:00 Coffee

09:30 M. Polyak (Technion)

Knot and 3-manifold invariants via counting graphs and surfaces

I will start with a short overview of the so-called perturbative invariants of links and 3-manifolds. While these invariants were intensively studied in the last two decades, many questions, inconsistencies and problems persist. The reason is that the construction is based on some complicated Feynman integrals (involving uni/trivalent graphs) in the perturbative Chern-Simons theory and a lot of technicalities are involved. In the main part of the talk I will describe an alternative elementary combinatorial construction of these invariants. It involves counting trivalent graphs in a link diagram. This approach immediately extends to invariants of 3-manifolds. Our construction can be restated in terms of counting certain surfaces ending on a link diagram. The latter description seem to be a combinatorial counterpart of a so-called "large N duality conjecture" by Gupakomar-Vafa, relating Chern-Simons theory to open strings.

10:45 H. Markwig (Saarbruecken)

Tropical Hurwitz cycles

(joint work with Aaron Bertram and Renzo Cavalieri) Double Hurwitz numbers count genus g degree d covers of the projective line with fixed ramification profile over zero and infinity and only simple ramification otherwise. They are piece-wise polynomial in the entries of the two special ramification profiles. The wall-crossing formulas can be expressed in terms of "smaller" Hurwitz numbers. Tropical analogues of double Hurwitz numbers have been helpful to discover some of their interesing features. We can also understand a double Hurwitz numbers as a zero-dimensional cycle in an appropriate moduli space of covers, resp. Ist push-forward into the moduli space of n-marked stable curves. We generalize this point of view by allowing higher-dimensional cycles corresponding to covers where we do not fix all simple ramification points. We start by restricting to the case of genus zero. We consider both the tropical and algebraic version of these generalized Hurwitz cycles and study their connection, their piece-wise polynomial structure and wall-crossing behavious. In this talk, we will concentrate on tropical Hurwitz cycles.

12:00 L. Katzarkov (Vienna)

TSC and phantoms

We will look at theory of phantoms from categorical and tropical prospective.