Homepage of Martin Klimeš

Centre of Mathematics of the University of Porto,
Rua do Campo Alegre 687, 4169-007 Porto, Portugal.


About me:

I am currently a postdoc at the Centre of Mathematics of University of Porto. Previously I've been a postdoc in Vienna (2017-18), Montreal (2016-17), Strasbourg (2014-15) and and an ATER in Dijon (2015-16). I finished my PhD at Université de Montréal in 2014 under supervision of Christiane Rousseau.

I work in the area of analytic ordinary differential equations, a domain in the intersection of analysis and complex geometry and with many connections to mathematical physics.
One recurring and very essential topic in my research is that of the relation between the (continuous) dynamics of a given differential system and the (discrete) dynamics of its monodromy/holonomy/return map, and how does this relation persist under deformations or limit situations. Namely, I've been interested in the problems of confluence of singularities in complex ODEs, both linear and non-linear, in analytic classification of parametric families of linear differential systems, in isomonodromic deformations of meromorphic connections on Riemann surfaces, and in Painlevé equations. My latest interest is in dynamical approach to the study of singularities in CR geometry.

My research interests:

Analytic theory of ordinary differential equations, singularities of foliations, normalization of vector fields, Stokes phenomenon, confluence of singularities,
meromorphic linear systems and their isomonodromic deformations, holomorphic dynamics, exact WKB analysis, singularities in CR geometry, Painlevé equations.

Articles and preprints:

M. Klimes, Confluence of singularities in hypergeometric systems, (to appear in Funkcialaj Ekvacioj (2018)), arXiv

M. Klimes, C. Rousseau, Generic 2-parameter perturbations of parabolic singular points of vector fields in C, Conform. Geom. Dyn. 22 (2018)  arXiv

M. Klimes, Stokes phenomenon and confluence in non-autonomous Hamiltonian systems, Qual. Theory Dyn. Syst. 17 (2018)  arXiv

M. Klimes, Wild monodromy of the fifth Painleve equation and its action on the wild character variety: approach of confluence, preprint 2018.  arXiv

M. Klimes, Analytic classification of families of linear differential systems unfolding a resonant irregular singularity. preprint 2014.  arxiv

M. Klimes, Confluence of Singularities of Non-linear Differential Equations via Borel-Laplace Transformations, J. Dyn. J. Dyn. Control Syst. 22 (2016)  arxiv


M. Klimes, Unfolded singularities of analytic differential equations. PhD thesis under the direction of Christiane Rousseau, 2014. TEL

L. Klimes, M. Klimes, Perturbation expansions of complex-valued travel-time along real-valued reference rays. Geophys. J. Int. 186 (2011), 751-759.

Presentation slides:

Confluence of singularities in non-autonomous Hamiltonian systems and non-linear Stokes phenomenon. Holomorphic foliations in Alsace, Paris 2017

Non-linear Stokes phenomenon in the fifth Painlevé equation. Irregular connections, character varieties and physics, Paris 2017

On the center manifold of unfolded complex saddle-node singularities. AAGADE, Bedlevo 2015