## AP4: Exchanges of Experience and Joint Training for Young / Senior Researchers

- Cross-border training courses at the University of Szeged
- Cross-border training courses at the West University of Timisoara
- Research stages for senior members of the staff at the cross-border institution
- Stages of study for young members of the staff at the cross-border institution
- Cross-border training courses at the University of Szeged

On April 23, 2012, the LP (West University of Timisoara), appointed two experts to prepare and held training courses for the students in mathematics of the University of Szeged. After several discussions with our partners we decided the scientific content of these lectures ensuring that the students will obtain an optimal benefit. There were 45 participants at these training courses.

- Here are some photos from the lecture room:

- Here are some photos from the lecture room:
- The first lecture was held by L. Suciu. The scientific approach followed the content of some lectures presented at the 12th International Internet Seminar by B. Farkas, T. Eisner, M. Haase and R. Nagel

- The second lecture was held by A. Craciunescu

- Cross-border training courses at the West University of Timisoara

After the International Workshop on Functional Analysis, on October 15, 2012, we organized, at the West University of Timisoara, two cross-border training courses. Senior members of the Szeged implementing team (L. Kérchy and G. Röst) presented introductory courses on the applications of functional analysis, respectively differential equations.

- These short lectures, taking place in our Amphitheater A02, proved to be of a real interest among our students (the audience was of about 80 persons). Here are some photos from the lecture room:

- The first lecture was held by László Kérchy, Professor at the University of Szeged, the Head of the Department of Analysis and the Coordinator of the PP team. His presentation was, in fact, an introductory approach to various important problems in operator theory on Banach and Hilbert spaces (for example, the invariant subspace problem)

- The second lecture was held by Gergely Röst. It was a rigorous introduction to the theory of delay differential equations with many interesting applications (for example, the shower problem)

- These short lectures, taking place in our Amphitheater A02, proved to be of a real interest among our students (the audience was of about 80 persons). Here are some photos from the lecture room:
- Research stages for senior members of the staff at the cross-border institution

The Project Manager and the Coordinator of the LP team (members of the Timisoara team) visited their cross-border colleagues at the Univesity of Szeged (in the period February 15 - February 24, 2012) for a stage of research (and a management meeting).

Here the two researchers had the possibility to discuss on joint themes (related to the field of operator theory) with several members of the Department of Analysis. We should firstly mention Professor László Kérchy, the Head of the Department and the Editor-in-Chief of Acta Scientiarum Mathematicarum, who usually participates to the International Conference on Operator Theory (organized, every two years, in Timisoara). He is an experienced researcher and his advices certainly helped them to improve their theorems. In the same context their meeting with Béla Gábor Pusztai (his main research interests is in mathematical-physics) provided them the necessary tools to find good examples for their results (mainly of a theoretical nature). They also had interesting discussions with the two Ph.D. students in functional analysis: György Pál Gehér and Attila Péter Szalai. At the same time they were able to consult the library of the Bolyai Institute and to find new articles which are not available at their home institution.

During this stage of research the members of the LP team had the opportunity to attend the conference of Professor László Zsidó (University of Rome “Tor Vergata”, Italy) with the title: “Order structures in Hilbert spaces”. This famous mathematician received, in Szeged, the Béla Szőkefalvi-Nagy Medal of Honor.

- Here are some photos from this event:

In this period the two researchers gave the following presentations:

- Operators with Dense Ranges – held by D. Popovici (joint work with Z. Sebestyen)

- Dilatability of Linear Cellular Automata – held by A. Popovici (joint work with D. Popovici)

- Here are some photos from this event:

- Stages of study for young members of the staff at the cross-border institution

Two Ph.D. students (R. Muresan and C. Zaharia), members of the West University of Timisoara team, visited our partner colleagues of the University of Szeged for a stage of study in the period March 3 – March 10, 2012.

They had several scientific conversations with Professor G. Röst. The topics discussed include delay differential equations, bifurcation theory, nonlinear dynamics, mathematical epidemiology. The two students also had the opportunity to collaborate with Professors Janos Karsai and Maoxing Liu. They visited the mathematical library which is situated in the building of the Bolyai Institute and has a large collection of mathematical books and journals. Also they participated to the graduate course “Nonlinear Dynamics” taughed by Professor Tibor Krisztin (the Head of the Bolyai Institute). Other course attended was “Functional Differential Equations” which was held as a part of the Mathematical Epidemiology Research Seminar. There they met the members of the EPIDELAY Research Group (Delay Differential Equations and Mathematical Biology): Gergely Röst, Diána Knipl, Kyeongah Nah, Maoxing Liu and Attila Denes. On the last day, before departure, our Ph.D. students attended the presentation of the lecturer Maoxing Liu entitled “Networks and epidemic models”.

In the period November 27-29, 2012 Gergely Röst, senior member of the Project Partner (University of Szeged) visited the cross-border institution (West University of Timisoara) for a stage of research.

Useful scientific discussions were engaged on this occasion with some members of the LP differential equations group (Raluca Mureşan, Cristina Praţa, Petre Preda and Claudia Zaharia). The main subject was the applicability of delay differential equations in various models simulating the propagation of infectious diseases.

On November 28, 2012, Professor Gergely Röst was the main speaker at the Scientific Seminar “Differential Equations and Nonlinear Analysis” – SEDAN (coordinated by Professor Petru Jebelean). His interesting presentation, entitled

“From simple dynamics to chaos via non-monotone delayed feedback”, has been followed, besides the two implementing teams, by members of the teaching staff of the West and Technical Universities of Timisoara and by two PhD students from Szeged, Diána Knipl and Kyeongah Nah.

- Here are some photos from their stage:

In this period the two researchers gave the following presentations:

- On-uniform dichotomy of evolutionary processes on the real line – held by R. Muresan

- Generalized Ulam – Hyers stability for the additive functional equation in various spaces – held by C. Zaharia

In the period June 27 - 30, 2012, three young researchers (Raluca Mureşan, Cristina Praţa and Claudia Zaharia), members of the Lead Partner (West University of Timisoara) implementing team visited the cross-border partner institution (University of Szeged) for a stage of study.

On the first day (June 27, 2012) they met professor Gergely Röst and the EPIDELAY (Delay Differential Equations and Mathematical Biology) research group. Several discussions were engaged on their recent research work. More precisely they studied models for the human influenza in terms of some asymptotic properties of evolution equations. The members of the EPIDELAY group expressed their interest for a posible future joint collaboration with the Timisoara team on the asymptotic properties of differentiable systems with applications in mathematical biology.

On June 28, 2012, our three Ph.D. students visited the university library, where they spent several hours documenting on current research results related to their theses. They studied several articles and the famous books of L. Barreira, C. Valls - Ordinary Differential Equations: Qualitative Theory (Graduate Studies in Mathematics 137, American Mathematical Society, 2012), respectively L. Barreira - Dimension and Recurrence in Hyperbolic Dynamics (Progress in Mathematics 272, Birkhäuser, 2008). On the same day they attended the weekly scientific seminar of the EPIDELAY group.

During this period our young researchers had the opportunity to participate at the international meeting "CSM - The Second Conference of PhD Students in Mathematics" (http://www.math.u-szeged.hu/CSM/). The following talks were given:

- Raluca Mureşan - Uniform exponential stability for evolution families that are not exponentially bounded

- Cristina Praţa - A version of a theorem of A. M. Lyapunov for the stability and instability of evolution families in Banach spaces

- Claudia Zaharia - A characterization of the completeness of random normed spaces

In the period November 26-30, 2012 three young researchers (Ábel Garab, György Gehér and Attila Szalai), members of the Project Partner (University of Szeged), visited their cross-border institution (West University of Timisoara) for stages of study.

Ábel Garab, together with his thesis coordinator (Gergely Röst), participated to several discussions with the differential equations group. The two young members of the functional analysis group from Szeged (György Gehér and Attila Szalai; PhD students of Professor László Kérchy) studied, under the coordination of A. Popovici, D. Popovici and N. Suciu, various problems related to Hilbert C*-modules (Banach spaces endowed with a complete inner product having values in a C*-algebra), Wold-type decompositions and dilation results on these objects, as well as some applications to prediction and K-theory.

- Here are some photos from their stage: