# 2018/2019

### 2019 June 11th

**Konrad Lompert will give the lecture**

*Title:*
Bi-Hamiltonian structures induced by Invariant Nijenhuis
Tensors on homogeneous spaces

*Abstract:*
Studies of invariant Nijenhuis (1,1)-tensors on a homogeneous
space G/K of a reductive Lie group G allowed us to prove
Liouville integrability of the geodesic flow on two series
of homogeneous spaces of compact Lie groups for two kinds of
metrics: the normal metric and new classes of metrics related
to decomposition of G to two subgroups.
(A joint work with Andriy Panasyuk.)

In this talk I will overview used methods and discuss possible generalizations based on the use of weak Nijenhuis operators.

### 2019 June 4th

**Marcin Zubilewicz**

*Title:*
Curvature of unimodular webs (continuation)

### 2019 May 28th

**Marcin Zubilewicz**

*Title:*
Curvature of unimodular webs

*Abstract:*
A smooth k-web is a family of k distinct smooth foliations whose
leaves intersect generically. Objects of this kind have been
thoroughly studied in the setting of algebraic and symplectic
geometry. During the talk we will focus on the less-traveled
route: webs in the geometry of volume-preserving maps.

These unimodular webs possess nontrivial local structure, which manifests itself in the existence of curvature -- a symmetric 2-tensor covariant with respect to volume-preserving web-equivalences, named after its classical counterpart defined by Thomsen and Blaschke in the 1920s. This structure can also be described by means of a natural affine connection determined uniquely by the structure of a given web and a choice of the volume form on the ambient space.

Our main goal is to construct this connection, and to relate its curvature to the curvature of the web in order to establish another criterion for its triviality.

### 2019 March 12th

**Asahi Tsuchida**

*Title:*
On frontals for sub-Riemannian manifolds

### 2019 March 5th

**Mariusz Zając**

*Title:*
Homogeneous polynomials in some discrete problem

### 2019 January 15th

**Vincent Grandjean**

*Title:*
Equisingularity at infinity of real polynomials

*Abstract:*
In this work, jointly with N. Dutertre,
given a real polynomial function F in n variables,
we are looking at the regularity of the functions
"total curvature" and a "absolute total curvature"
attached to F. The function "total curvature" is defined
as the integral, over any given level of F, of the
Gauss-Kronecker curvature of the given level.
The function "absolute total curvature" is given by the
integral, over the given level, of the absolute value
of the Gauss-Kronecker curvature over the given level.
The Total absolute curvature of a Given Level is related
to Gauss-Bonnet formula; We show an equisingularity result
of this nature; both functions "total curvature" are continuous
at any regular value which satisfies Malgrange Condition.

### 2018 October 30th

**Wojciech Kryński** (IM PAN)

*Title:*
Geometric approach to the multipeakon solutions
to the Camassa-Holm equation

*Abstract:*
Multipeakons are special solutions to the Camassa-Holm equation.
They can be described by an integrable geodesic flow on a
Riemannian manifold. Singular points of the Riemannian metric
correspond to collisions of the multipeakons. We consider a
bi-Hamiltonian formulation of the system and exploit its first
integrals in order to analyse geodesics near a singular point
of the metric. We present a novel approach to the problem of the
dissipative prolongations of multipeakons after the collision time.

### 2018 October 16th (**at 4 pm**)

**Jun-Muk Hwang** (Korea Institute for Advanced Study, Seoul)

*Title:*
Rigidity of Legendrian singularities

*Abstract:*
Let (M, D) be a holomorphic contact manifold, i.e. a complex
manifold M of dimension 2m+1 equipped with a holomorphic contact
structure D.

An m-dimensional complex analytic subvariety V in M is called a Legendrian subvariety if the smooth locus of V is tangent to D. A Legendrian singularity means the germ of a Legendrian subvariety at a point.

We discuss conditions under which a Legendrian singularity becomes a cone singularity and explain how they are related to the geometry of Fano contact manifolds.

### 2018 October 2nd

**Shyuichi Izumiya** (Sapporo)

*Title:* Singularities of Cauchy horizons in Lorentz-Minkowski space