** **

** ****WINTER SCHOOL IN CONVEX SYMPLECTIC GEOMETRY**

**WINTER SCHOOL IN CONVEX SYMPLECTIC GEOMETRY**

**FEBRUARY 20-24, 2017 (BOCHUM) **

**Aims of the School**

This winter school is intended for graduate students and postdocs who are interested in the relationships between convex geometry and symplectic geometry. Its goal is not only to educate participants on recent developments in convex symplectic geometry but also to trigger new collaborations. There will be a precourse on basic symplectic geometry.

This winter school is funded by the Bochum-Cologne SFB collaboration program CRC/TRR 191 "Symplectic Structures in Geometry, Algebra and Dynamics”.

**To participate**

The school will cover travel and lodging expenses for a limited number of participants. Who wishes to participate should send an email to Frau Dzwigoll before January 15, 2017. A social dinner will take place on wednesday evening.

Speakers

Alberto Abbondandolo (Bochum)

Juan Carlos Álvarez Paiva (Lille)

Umberto Hryniewicz (Rio de Janeiro)

Michael Hutchings (Berkeley)

Jungsoo Kang (Münster)

Yaron Ostrover (Tel Aviv)

Daniel Rosen (Tel Aviv)

**Directions**

General directions and maps can be found here.

In the morning of Monday 20.2 we will have pre-courses by **Jungsoo Kang**, in which he will cover some prerequisites from symplectic geometry.

All lectures will take place in NA 01/99. Coffee breaks will take place in NA 4/24.

8:00 - 9:00 Registration (NA 5/34)

9:15 - 10:30 Kang

11:15 - 12:30 Kang

14:15 - 15:30 Ostrover

16:15 - 17:30 Abbondandolo

**Tuesday 21**

9:15 - 10:30 Rosen

11:15 - 12:30 Alvarez Paiva

14:15 - 15:30 Abbondandolo

16:15 - 17:30 Hryniewicz

**Wednesday 22**

9:15 - 10:30 Ostrover

11:15 - 12:30 Hutchings

19:00 Social dinner

**Thursday 23**

9:15 - 10:30 Hryniewicz

11:15 - 12:30 Alvarez Paiva

14:15 - 15:30 Hutchings

16:15 - 17:30 Abbondandolo

**Friday 24**

9:15 - 10:30 Hryniewicz

11:15 - 12:30 Hutchings

14:15 - 15:30 Alvarez Paiva

**TITLES AND ABSTRACTS**

**A. Abbondandolo**: *“A fixed point theorem for disc maps and its consequences in convex and Finsler geometry”
*

Abstract: I will discuss a fixed point theorem for area preserving diffeomorphisms of the disc relating the action of a fixed point to the Calabi invariant of the map. This theorem has consequences on the minimal action of closed characteristics on the boundary of convex bodies in the 4-dimensional symplectic space and on shortest closed geodesics on the 2-dimensional sphere.

**J. C. Alvarez Paiva**:

*"Contact-geometric techniques in systolic and convex geometry”*

**Abstract**: These talks will be an introduction to the systolic geometry of contact and Finsler manifolds spiced with applications to convex geometry. Specifically, the following two topics will be covered:

1. Ivanov's beautiful generalization of Pu's systolic inequality to reversible Finsler metrics and its applications to the Viterbo conjecture, Mahler's inequality, and the seventh Busemann-Petty problem on estimating the area of the unit sphere in a three-dimensional normed space.

2. The work of Alvarez Paiva and Balacheff on the systolic criticality of Zoll Finsler manifolds and regular contact forms.

**U. Hryniewicz**:

*"Action, index and Calabi"*

**Abstract**: I will discuss the relations between action and index of periodic points and the Calabi invariant of area-preserving diffeomorphisms on a surface. Then I will explain how to use these relations to achieve two goals: (1) study Reeb flows in dimension three and construct contact forms with high systolic ratio on a general oriented 3-manifold; and (2) construct dynamically convex contact forms on the 3-sphere with systolic ratio arbitrarily close to two.

**M. Hutchings**: *"Computing symplectic capacities of convex toric domains"*

**Abstract**: “Convex toric domains” are convex domains in C^n that are invariant under the action of T^n. We explain how to compute various symplectic capacities for these domains via combinatorial formulas. These computations have applications to deciding when one convex toric domain can be symplectically embedded into another. The capacities that we will compute are (1) analogues of the Ekeland-Hofer capacities, defined from positive S^1-equivariant symplectic homology, and (2) ECH capacities, defined using embedded contact homology in the case n=2. Part (1) is joint work with Jean Gutt, and part (2) is based on joint work with Keon Choi, Dan Cristofaro-Gardiner, David Frenkel, and Vinicius Ramos.

**Y. Ostrover** and **D. Rosen**: *"What we talk about when we talk about symplectic measurements" *

**Abstract**: We shall discuss several topics regarding symplectic measurements of convex domains in the classical phase space.

In particular: Viterbo's volume-capacity conjecture (current status and related open questions), the relation with Mahler conjecture from convex geometry, computational complexity of estimating capacities, and more.