Benjamin McKay, Cork University. Chern-Simons forms and rolling surfaces on surfaces

Martedì 20 Maggio 2025 ore 16:00 aula D'Antoni

Abstract: 

Rolling a surface along another surface is described by an underdetermined ODE system. A nice open problem: if you have two surfaces, and you construct the associated ODEs, and I have two surfaces, and I construct the associated ODEs, when is your ODE system isomorphic to mine? There are examples where the surfaces are nowhere locally isometric. The local invariants of these ODE systems were studied by Cartan in 1910. He found a surprising relationship to a flag variety of G2.  We consider holomorphic ODE systems of the same type, and look for equations on the Chern classes and Chern-Simons classes of the complex manifolds they live on.

Dmitry Yakubovich, Universidad Autonoma de Madrid. The similarity to a normal operator, resolvent growth and a problem concerning subharmonic functions

Martedì 6 Maggio 2025 ore 16:00 aula D'Antoni

Josias Reppekus, University of Amsterdam. On analytic discs in pseudoconvex C^0 boundaries

Martedì 29 Aprile 2025 ore 16:00 aula D'Antoni

Abstract:

I will discuss results on the existence of analytic discs in C^0 boundaries of pseudoconvex domains, that is, pseudoconvex domains whose boundary is locally the graph of a continuous (real-valued) function. The central tool is the solution to the Dirichlet problem for the Levi form with continuous boundary data by Nikolay Shcherbina.

In complex dimension 2, I conclude that a disk in the closure of such a domain D will always be completely contained inside the domain or completely contained in its boundary. Moreover, a boundary point lies on a boundary disk if and only if one can delete a neighbourhood of the point from the closure of D and recover the boundary point in the plurisubharmonic hull of this set. 

The first result is motivated by limit maps of recurrent Fatou components. The latter result has been proven by David Catlin in the case of smooth boundaries in dimension 2 and generalised to higher dimensions by Sahutoglu and Straube under strict pseudoconvexity conditions. We conjecture a different more natural generalisation holds also in higher dimensions in the continuous case.

This is joint work with Nikolay Shcherbina and Tobias Harz.

Sergei Yakovenko, Weizmann Institute of Science. The Rolle models in analysis and geometry

Martedì 11 Febbraio 2025 ore 16:00 aula D'Antoni

Abstract:

The Rolle theorem from the first year calculus course asserts that between any two roots of a differential real function of one real variable there must be a root of its variable. This can be translated into an obvious inequality between the number of isolated zeros N(f) of a smooth function and that of its derivative N(f'). I will try to present a number of generalizations of this result for the cases where the straightforward reformulation fails, e.g. for an entire function exp nz -1 which may have plenty of isolated zeros and its derivative n exp nz that is non-vanishing.  

These results have numerous applications for the complexity of sets defined by ordinary and Pfaffian differential equations in the real and complex domain. The talk will be a brief survey of many recent results obtained with Gal Binyamini and Dmitry Novikov, largely inspired by Askold Khovanskii. 

Carsten Lunde Petersen, University of Copenaghen. Convergence of critical points of sequences of polynomials, where roots are Scarce.

Martedì 29 Ottobre 2024 ore 16:00 aula D'Antoni

Abstract:

There are many results on the statistical properties of the root locus of various families of polynomials. Examples are Brolin’s theorem for the family of iterates of any fixed polynomial of degree at least $2$.

The monograph by Stahl and Totik on the sequence of monic polynomials orthogonal with respect to a fixed probabibility meaure $\mu$ on $\C$ with compact non-polar support. And more generally the sequence of polynomials which are extremal in $L^p(\mu)$ for any $p\geq 1$ including the $L^\infty$ case, where the extremal polynomials are called Chebychev polynomials.

Recently attention has shifted to the more general case of a sequence of polynomials $q_k$, where the root distribution converges to a limit probability measure.

Assuming this condition I shall in this talk address the question of the distribution of critical points, i.e. the zeros of the derivatives $q_k’$ in areas where the roots of $q_k$ are scarce, where the root distribution converges to a limit probability measure.

For the extremal polynomials in $L^p(\mu)$, which are mentioned above, it is well known that the number of roots of $q_k$ are uniformly boundedon any compact set outside the polynomial convex hull of the support of $\mu$.

A similar property also holds in dynamics. This shows that scarceness of the roots of the sequence of polynomials $q_k$ is a common property for many interesting sequences of polynomials.

This is joint work with Christian Henriksen, The Technical University of Denmark and Eva Uhre, Roskilde University.

Nefton Pali, University of Paris-Sud (Orsay). On maximal totally real embeddings

Martedì 15 Ottobre 2024 ore 16:00 aula D'Antoni

Abstract:

It has been 64 years since the existence of a complex structures on Grauert Tubes was proven for the first time by Bruhat-Whitney. Still, up to now, the explicit form of such structure has remained quite mysterious to the community of experts in the field. 

This is finally clarifed in a recent joint article with Bruno Salvy, where we consider complex structures with totally real zero section of the tangent bundle.  We assume that the complex structure tensor is real analytic on the fibers of the tangent bundle. This hypothesis is quite natural in view of the existence result of Bruhat-Whitney.

For any torsion-free connection acting on the real analytic sections of the tangent bundle of a real analytic manifold, we provide a simple and explicit expression of the coefficients of the Taylor expansion on the fiber of the associated canonical complex structure.

An explicit global expression for the above coefficients is important for applications to analytic micro local analysis over manifolds, as it allows an explicit global construction of the complex extension of a given global Fourier integral operator defined on a real analytic manifold.