BrainMap

Cornelia Vizman

prof. dr. habil.

Professor - UNIVERSITATEA DE VEST TIMISOARA

Researcher | Teaching staff

Curriculum Vitae (10/01/2019)

Expertise & keywords

Geometry and dynamics of singular vorticities in ideal fluids Call name: P 4 - Proiecte de Cercetare Exploratorie, 2020
PN-III-P4-ID-PCE-2020-2888 2021 - 2023

Role in this project:

Coordinating institution: UNIVERSITATEA DE VEST TIMISOARA

Project partners: UNIVERSITATEA DE VEST TIMISOARA (RO)

Affiliation:

Project website: http://quasar.physics.uvt.ro/~vizman/idei_cv

Abstract:

The project proposes the study the geometry and dynamics of singular vorticities in an ideal fluid, with emphasis on those with codimension one vorticity support, i.e. vortex sheets.

Multiple solutions for systems with singular ϕ-Laplacian operator Call name: P 1 - SP 1.1 - Proiecte de cercetare Postdoctorală
PN-III-P1-1.1-PD-2016-0040 2018 - 2020

Role in this project:

Coordinating institution: UNIVERSITATEA DE VEST TIMISOARA

Project partners: UNIVERSITATEA DE VEST TIMISOARA (RO)

Affiliation: UNIVERSITATEA DE VEST TIMISOARA (RO)

Project website: https://sites.google.com/site/sysphilaplaciancserban

Abstract:

This research project focus on boundary value problems for partial differential systems, as well as for differential systems with singular ϕ-Laplacians. The prototype for such Laplacians is the mean curvature operator in Minkowski space which has significant importance in differential geometry and special relativity. We intend to obtain multiplicity of solutions for singular ϕ-Laplacians systems subjected to Dirichlet, periodic or Neumann boundary conditions. In order to prove the expected results one make use of variational and/or topological methods.

Nonlinear Grassmannians and diffeomorphism groups Call name: P 4 - Proiecte de Cercetare Exploratorie
PN-III-P4-ID-PCE-2016-0778 2017 - 2019

Role in this project:

Coordinating institution: UNIVERSITATEA DE VEST TIMISOARA

Project partners: UNIVERSITATEA DE VEST TIMISOARA (RO)

Affiliation: UNIVERSITATEA DE VEST TIMISOARA (RO)

Project website: https://physics.uvt.ro/~vizman/pce_cv/results.html

Abstract:

The nonlinear Grassmannians are spaces of unparameterized curves or higher dimensional submanifolds, called like this by analogy with the Grassmannian of linear subspaces. They can be viewed as homogeneous spaces for diffeomorphism groups of the ambient manifold or as spaces of parameterized submanifolds that differ by reparametrization. Nonlinear Grassmannians are the smooth versions of shape spaces used in shape analysis and computer vision. Our issue is on one hand to find nice geometric structures on them and on the other hand to study geometric flows on them. We consider more general "adorned" nonlinear Grassmannians, i.e. shape spaces, each shape endowed with a geometric structure (e.g. differential form). Being homogeneous spaces of diffeomorphisms groups, nolinear Grassmannians are good candidates for coadjoint orbits of groups of diffeomorphisms or central extensions of them. The groups of diffeomorphisms that we plan to focus on are those preserving a volume form, a symplectic form, a differential character, a contact and an Engel structure.

Kirillov's orbit method attaches group representations to its coadjoint orbits. For projective representations one has to replace the group with a central extension. These are richer in coadjoint orbits than the groups themselves. We plan to find coadjoint orbits of central extensions of diffeomorphism groups in the realm of nonlinear Grassmannians. In the case of volume preserving diffeomorphisms these can shed light on vortex dynamics.

Putting a Riemannian metric on shaoe spaces, one can define distance between shapes. One uses weighted Sobolev metrics. We plan to consider this type of metrics on adorned nonlinear Grassmannians and address questions related to their geodesic equation, induced geodesic distance, completeness, as well as properties of the curvature.

Diffeomorphism groups, fluid dynamics and related geometric structures Call name: Exploratory Research Projects - PCE-2011 call
PN-II-ID-PCE-2011-3-0921 2011 - 2016

Role in this project:

Coordinating institution: Universitatea de Vest din Timisoara

Project partners: Universitatea de Vest din Timisoara (RO)

Affiliation: Universitatea de Vest din Timisoara (RO)

Project website: http://www.physics.uvt.ro/~vizman/idei_cv/

Abstract:

The dual pair concept in Poisson geometry was introduced by Weinstein in a famous paper from 1983. In the same year, another remarkable paper due to Marsden and Weinstein appeared, where these ideas merged with those of Arnold and Marsden on the geometric underpinning of the Euler equations. One of the main objects of this paper was a general construction of dual pair of momentum maps for the Euler equations: a geometric tool that explains both Clebsch variables and point vortex solutions. The 2011 paper of Gay-Balmaz and Vizman gives the rigorous answer, using central extensions of the groups of Hamiltonian and volume preserving diffeomorphisms. It carries out a similar program for the n-dimensional Camassa-Holm equation, whose dual pair of momentum maps was introduced by Holm and Marsden in 2004 in the context of singular solutions, but without the necessary transitivity results.

The symmetry/conservation duality, is a feature of such dual pairs of momentum maps. Our plan is to carry out a deeper and more systematic study of (infinite dimensional) dual pairs, especially in connection with dynamical systems, to find new dual pairs inspired by fluid dynamics, to better understand vortex dynamics by using special non-linear Grassmannians and reduction, to exploit the link to non-commutative integrable Hamiltonian systems, to find new properties and new generalizations of dual pairs (eventually related to contact geometry or generalized complex structures).

Geometric structures applied in stability and control problems for dissipative dynamical systems Call name: Projects for Young Research Teams - TE-2011 call
PN-II-RU-TE-2011-3-0006 2011 - 2014

Role in this project:

Coordinating institution: Universitatea de Vest din Timisoara

Project partners: Universitatea de Vest din Timisoara (RO)

Affiliation: Universitatea de Vest din Timisoara (RO)

Project website: https://sites.google.com/a/e-uvt.ro/sdd/

Abstract:

The study of dissipation is very important in real life situations. The conservative models are a good approximation of phenomena that are encountered in nature, but experience teaches us that there are always imperfections that lead to dissipation. From this perspective we consider that it is very important to understand the dynamics of dissipative systems. Such systems are found in satellite orientation problems, magnetization problems, plasma physics – just to name a few fields.

The project objective is to study the geometry underlying the dissipative forces in order to understand their effect on the dynamics of an initial conservative system. Our study is based on a large set of open problems that appear from mathematical physics, such as: the general form of Morrison dissipation on an arbitrary dimensional phase space; the Riemannian metric underlying this generalized Morrison dissipation and its relation to the well known double bracket dissipation; the possible generalization of the results on dissipative N-point-vortex models in the plane to the N-point-vortex models on the sphere. The purpose of our research is two folded: to understand the finite dimensional case and also to study its infinite dimensional counterpart.

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List of research grants as project coordinator or partner team leader
Significant R&D projects for enterprises, as project manager
R&D activities in enterprises
Peer-review activity for international programs/projects