# 2018/2019

#### 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