## SCV seminar 2017

Seminar organizer: Séverine Biard

### Mitja Nedic, Stockholm University: The convex combination problem for Herglotz-Nevanlinna functions in several variables, III

**Friday, 10. November, 11:40-13:00, Tgv227 in Tæknigarður. **

### Mitja Nedic, Stockholm University: The convex combination problem for Herglotz-Nevanlinna functions in several variables, II

**Friday, 10. November, 11:40-13:00, Tgv227 in Tæknigarður. **

### Mitja Nedic, Stockholm University: The convex combination problem for Herglotz-Nevanlinna functions in several variables, I

**Friday, 3. November, 11:40-13:00, Tgv227 in Tæknigarður. **

Abstract: Herglotz-Nevanlinna functions are holomorphic functions defined on the poly-upper half-plane having non-negative imaginary part. All such functions can be characterized via an integral representation formula as the sum of a real constant, a real-linear part and an integral involving a specific kernel function and a particular positive Borel measure. This constant, linear-part and measure uniquely correspond to a given function and are called the data of the function.

A common question concerning the integral representation of these functions asks to relate the data of one function to the date of another when the second function is constructed out of the first one. A particular problem of this type is the convex combination problem, where we build a Herglotz-Nevanlinna function of several variables out of a Herglotz-Nevanlinna function in one variable by replacing the argument of the one-variable function with a convex combination of independent variables.

In this talk, we present the solution to the convex combination problem for Herglotz-Nevanlinna functions. That is to say, we give explicit relations describing the data of the new function in terms of the data of the original function when the above construction is used.

### Örn Arnaldsson: The equivariant moving frames for Lie pseudo-groups

**Friday, 20. October, 11:40-13:00, Tgv227 in Tæknigarður. **

### Örn Arnaldsson: Infinite dimensional equivalence problems

**Friday, 13. October, 11:40-13:00, Tgv227 in Tæknigarður. **

### Örn Arnaldsson: Cartan’s solution to the finite dimensional equivalence problem

**Friday, 6. October, 11:40-13:00, Tgv227 in Tæknigarður. **

### Örn Arnaldsson: The signature of a coframe

**Friday, 29. September, 11:40-13:00, Tgv227 in Tæknigarður. **

### Örn Arnaldsson: Equivalnce problems for coframes

**Friday, 15. September, 11:40-13:00, Tgv227 in Tæknigarður. **

### Örn Arnaldsson: Introduction to pseduo-groups

**Friday, 8. September, 11:40-13:00, Tgv227 in Tæknigarður. **

### Örn Arnaldsson: Involutive moving frames

**Friday, 1. September, 11:40-13:00, Tgv227 in Tæknigarður. **

Abstract: In my recent thesis I showed that the equivariant moving frame for pseudo-groups and Cartan’s equivalence method are two sides of the same coin. This opened the door for a proof of termination of Cartan’s classic method as well as a new approach to equivalence problems that combines the two methods. In this talk I will survey the key ideas of the thesis and their applications.

### Masanori Adachi, Tokyo University of Science: Weighted Bergman spaces of domains with Levi-flat boundary: Two case studies

**Friday, 25. August, 11:00-12:00, VR2-157. **

Abstract: In contrast to bounded domains in Stein manifolds, it is not clear to what extent domains with Levi-flat boundary are capable of holomorphic function with slow growth. We shall answer this question in two cases, the space of geodesic segments and the maximal Grauert tube of a compact Riemann surface, which are realized as 1-convex domains with Levi-flat boundary. We describe the weighted Bergman spaces of these domains explicitly.

### Evgeny Poletsky, Syracuse University: Bounded psh functions on unbounded domains (joint work with N. Shcherbina)

**Friday, 30. June, 10:00-11:30, Tgv227 in Tæknigarður. **

Abstract: We will start with manifolds where all bounded psh functions are constants and present the criterion of Rosay for such manifolds. Then we will discuss S-manifolds introduced by Stoll. These manifolds have a psh exhaustion function that is maximal outside of a compact set and we will prove the result of Aytuna and Sadullaev that all bounded psh functions are constants on such manifolds.

After that we will move to the recent results of Harz, Shcherbina and Tomassini who were interested when smooth bounded psh function cannot separate point on a domain \(\Omega\) in \(\mathbb C^n\)$. For this they introduced the notion of the core that is the set of points \(z\) in \(\Omega\) where there is no a smooth bounded psh function strictly psh near \(z\). As it happens all bounded psh functions are constant on this set but it can be very wild.

The rest of the talk will be devoted to the HShchT-problem for general psh functions. There the core is related to the Green function and some other objects. You will not be bored.

### Séverine Biard: Estimates for the complex Green operator on CR submanifolds of hypersurface type: Symmetry and Interpolation.

**Friday, 16. June, 10:00-11:30, Tgv227 in Tæknigarður. **

Abstract: Although the complex Green operator on CR submanifolds of hypersurface type is naturally compared to the $\bar\partial$-Neumann operator on pseudoconvex domains,

some of its properties differ. Mainly, compactness estimates hold for forms of symmetric bidegrees but those estimates do not percolate up the tangential \(\bar\partial\)-complex. However, in a joint work with E. Straube, we prove a result of interpolation of compactness estimates for the complex Green operator, giving an alternative to the percolation. We also show that Sobolev estimates hold for forms of symmetric bidegrees.

### Jón Ingólfur Magnússon: Intersection Theory on Complex Manifolds

**Tuesday, 11. April, 15:00-16:00, Tgv227 in Tæknigarður. **

### Jón Ingólfur Magnússon: Intersection Theory on Complex Manifolds

**Wednesday, 5. April, 10:00-11:30, Tgv227 in Tæknigarður. **

### Jón Ingólfur Magnússon: Intersection Theory on Complex Manifolds

**Wednesday, 29. March, 10:00-11:30, Tgv227 in Tæknigarður. **

### Jón Ingólfur Magnússon: Intersection Theory on Complex Manifolds

**Wednesday, 22. March, 10:00-11:30, Tgv227 in Tæknigarður. **

### Jón Ingólfur Magnússon: Intersection Theory on Complex Manifolds

**Wednesday, 15. March, 10:00-11:30, Tgv227 in Tæknigarður. **

### Jón Ingólfur Magnússon: Intersection Theory on Complex Manifolds

**Wednesday, 8. March, 10:00-11:30, Tgv227 in Tæknigarður. **

### Jón Ingólfur Magnússon: Intersection Theory on Complex Manifolds

**Wednesday, 1. March, 11:40-12:40, Tgv227 in Tæknigarður. **

### Jón Ingólfur Magnússon: Intersection Theory on Complex Manifolds

**Wednesday, 22. February, 11:40-12:40, Tgv227 in Tæknigarður. **

### Jón Ingólfur Magnússon: Intersection Theory on Complex Manifolds

**Friday, 17. February, 14:00-15:30, Tgv227 in Tæknigarður. **

### Severine Biard: Compactness estimates for the complex Green operator

**Friday, 13. January, 10:00-11:30, Tgv227 in Tæknigarður. **

Abstract: I will talk about a sufficient condition, denoted \((P_q)\) for the compactness of the complex Green operator of (for smooth pseudoconvex compact CR submanifolds of hypersurface type. I will give the idea of the proof as well.

### Sylvain Arguillère: Sub-riemannian geometry and connections with CR geometry

**Friday, 6. January, 10:00-11:30, Tgv227 in Tæknigarður. **

Abstract: Sub-Riemannian geometry is the study of non-holonomic constraints in sub-Riemannian manifolds. I will give a brief overview of the main results of sub-Riemannian geometry, while establishing links with CR manifolds of hyper surface type and the \(\overline \partial_b\) operator.