Format of the event:

This one-week long collaborative workshop (25-29 January 2027) will give the opportunity to 5 groups of participants to work under the supervision of leader on a research project. There will be also colloquium style talks and round tables to give the opportunity to all participants to meet with many researchers in a friendly atmosphere.

What will participants get out of it?

1. an in-depth knowledge of the concepts, notions and perceptions of a research topic
2. experience with several of the most commonly used research methods and skills for conducting research in the area of a research topic
3. working in groups on research level questions
4. ability to report research findings in writing and orally at an academic level
5. start a research network

Organization committee:

1. Jérémy Blanc (University of Neuchâtel)
2. Gloire Grâce Bockondas (University of Neuchâtel and Université Marien Ngouabi)
3. Elisa Gorla (University of Neuchâtel)
4. Christophe Ritzenthaler (Université de Rennes 1 and Université Côte d'Azur, France)
5. Felix Schlenk (University of Neuchâtel)

Group leaders

1. Jean Bertoin (University of Zürich): Yule process, random recursive tree, and some applications
2. Jérémy Blanc (University of Neuchâtel), Christian Urech (ETH Zürich) and Susanna Zimmermann (University of Bâle), : birational Geometry and Cremona groups
3. Annalisa Buffa (EPFL): Reduced and full-order models for hemodynamics: numerical simulation, comparison, and analysis
4. Felix Schlenck (University of Neuchâtel): Geometric problems arising in classical mechanics

Group leader: Jean Bertoin (probability theory, population model, combinatorics)

Title: Yule process, random recursive tree, and some applications

Abstract: The Yule process is a fundamental stochastic process in continuous time which has been introduce more than 100 years ago. It can be viewed as a population model in which individuals beget children at constant rate, independently one from the others. The Yule process has deep connections to a variety of random structures, such as random recursive trees, random permutations, Polya urn, preferential attachment, etc., and continues to be of crucial importance in many areas of research.
The project concerns the situation where individuals beget twins instead of a single child. One needs to adapt the classical techniques and then analyze how the main results of the theory are transformed.

Prerequisites: Students should have a good background in probability theory, notably Markov chains in discrete and in continuous time, martingales.

References: Yule GU. 1925 II.—A mathematical theory of evolution, based on the conclusions of Dr. J.C. Willis, F.R.S. Phil. Trans. R. Soc. Lond. B, 213, 21.
Simon HA. 1955 On a class of skew distribution functions. Biometrika, 42, 425.
H.~M. Mahmoud, {\it Evolution of random search trees}, Wiley-Interscience Series in Discrete Mathematics and Optimization A Wiley-Interscience Publication, , Wiley, New York, 1992; MR1140708
J.~W. Pitman, {\it Combinatorial stochastic processes}, Lecture Notes in Mathematics, 1875, Springer, Berlin, 2006; MR2245368
Amaury Lambert; Ages, sizes and (trees within) trees of taxa and of urns, from Yule to today. Philos Trans R Soc Lond B Biol Sci 13 February 2025; 380 (1919): 20230305.

Group leaders: Jérémy Blanc, Christian Urech and Suzanna Zimmerman (algebraic geometry)

Title: Birational Geometry and Cremona Groups

Abstract: The Cremona group is the group of birational transformations of projective space. It has a long history and was extensively studied by mathematicians in the late 19th century, most notably by Luigi Cremona, whose name it bears. Research on the group has continued ever since, and powerful new methods have been developed in recent years. In dimension 2, in particular, we now have a detailed understanding of base-points, the structure of birational transformations, generators of the group, and group theoretical properties, especially over algebraically closed fields. In this project, we will begin with some of the foundations of birational geometry before turning to concrete questions about the Cremona group that can be approached through explicit computations, carried out either by hand or with computer assistance. Participants will be guided by Jérémy Blanc, Christian Urech, and Susanna Zimmermann, who will introduce concrete constructions, help develop a solid understanding of the relevant techniques, and support participants in working toward original answers to open questions.

Prerequisites: The project draws mainly on algebra (polynomials, ideals, compositions, group theory etc.) and geometry (base loci, curves, intersections, etc.).

Group leader: Annalisa Buffa and Francesco Sala (numerical analysis)

Title : Reduced and full-order models for hemodynamics: numerical simulation, comparison, and analysis

Abstract: Mathematical and computational models of blood flow are increasingly used to study cardiovascular function and to support the analysis of clinically relevant hemodynamic quantities. Depending on the level of detail required, these models may range from three-dimensional fluid dynamics simulations to simplified one-dimensional or lumped-parameter descriptions. While detailed models can provide rich information, reduced models are computationally cheaper, but their reliability depends on the assumptions and parameters used in their construction. The goal of this project is to investigate how different levels of modeling complexity can be used to describe blood flow in vascular vessels. After introducing the main modeling ideas behind full-order and reduced-order descriptions, participants will work on the numerical implementation and analysis of a simplified hemodynamic model. The project will include simulations in simplified vascular configurations, comparison between models of different complexity, and the study of how selected parameters influence relevant quantities of interest. Depending on the background of the participants, the project may focus more on mathematical modeling, numerical discretization, computational implementation, or sensitivity analysis. The practical work will mainly rely on Python.

Prerequistes: numerical analysis, some basis in partial differential equations and their numerical approximation.

Group leader: Felix Schlenk

Title: A geometric problem arising in classical mechanics

Abstract : In symplectic geometry, it is interesting to compare sets up to symplectomorphism. These are the diffeomorphisms that leave Hamilton's equations invariant. A systems with antipodal symmetry on the 2-sphere is described in the cotangent bundle of real projective space RP^2. Almost toric fibrations provide a very convenient way to represent many 4-dimensional symplectic manifolds by 2-dimensional pictures, see [2]. This base diagram for T^*RP^2 is the infinite wedge generated by the vectors (0,1) and (4,1). It was recently shown in [1] that triangles in this wedge are symplectomorphic to codisc-bundles of Randers metrics (namely: special Finsler) metrics, as in the metrics modeling equatorial wind on the round 2-sphere described by Katok [4]. The ECH capacities (a famous sequence of symplectic capacities constructed in [3]) were computed for these triangles T(a,b) in [5]. We start with teams studying parts of the above material, and explaining them to each other. We then try to use the ECH capacities and computer codes to understand when the ellipsoid over T(a,b) symplectically embeds into the one over T(a',b') for a,b,a',b'>0.

References [1] N. Adaloglou and J. Hauber. Quantitative symplectic topology of Katok's examples and equivariant symplectic embeddings soon on arXive
[2] J. Evans. Lectures on Lagrangian torus fibrations. London Mathematical Society Student Texts, 105. Cambridge University Press, Cambridge, 2023.
[3] M. Hutchings. Quantitative embedded contact homology. J. Differential Geom. 88 (2011) 231–266.
[4] A. B. Katok. Ergodic perturbations of degenerate integrable Hamiltonian systems. Math. USSR, Izv. 7 (1974) 535–571. doi: 10.1070/IM1973v007n03ABEH001958.
[5] J. Trejos. ECH capacities of concave singular toric domains. arXiv:2509.23429