Summer/Winter Schools

VSM Summer School 2025
September 14 to September 20, 2025 in Dienten am Hochkönig
The School takes place at the Universitätssport- und Seminarzentrum Dientnerhof.
The program will start on Monday, September 15 in the morning and finish on Friday, September 19 in the afternoon.
Participants are expected to arrive on Sunday, September 14 and to stay until Saturday, September 20. Registration will open in time and all student members of the VSM are welcome.

The confirmed speakers are:
David Damanik, Rice University – Furstenberg’s Theorem and the Anderson Model in the Non-Stationary Setting
After reviewing the classical Furstenberg theorem and its application to localization results for the one-dimensional Anderson model, we discuss recent extensions to the non-stationary case due to Anton Gorodetski and Victor Kleptsyn, as well as joint work with them on models with local correlations.

Ewelina Zatorska, University of Warwick – Mathematical Analysis of Multi-Phase Flow Models
This lecture series focuses on the mathematical analysis of multi-phase flows, with particular emphasis on compressible two-fluid systems in the isentropic and isothermal regimes. Such models are of considerable physical importance, appearing in a wide range of applications including geophysical and astrophysical flows, oil recovery, spray combustion, and coolant transport in nuclear reactors.

In contrast to compressible mixture models—where constituents interact at the microscopic level and are inseparably mixed—multi-phase flow models describe the macroscopic evolution of distinct, non-interacting compressible fluids that coexist in the same spatial domain. We will consider only macroscopic models, and discuss their derivation and connection to the single-phase compressible fluid models.

A central theme of the series will be the extension of well-posedness theory from single-phase to multi-phase systems. We will examine which results—such as the existence of local-in-time strong solutions and global-in-time weak solutions—can be generalized to two-fluid models. This part of the course will be based on a selection of recent results concerning both viscous and inviscid two-fluid systems with algebraic pressure closures. Some results also extend to models with PDE-based closures, such as the Baer–Nunziato system. If time permits, we will address the construction of relative entropy functionals and highlight the analytical advantages of PDE closures in this context.

In the final part of the series, we will explore the connection between compressible two-phase models and so-called constrained models, which have so far only been derived at a formal level. I will outline their main mathematical features and discuss emerging links with hydrodynamic models of interacting agents, a topic of growing interest in both applied mathematics and mathematical physics.

This course is designed for graduate students and researchers in mathematical fluid mechanics. Frequent references will be made to nowadays well-known results for the compressible Navier–Stokes equations. The problem sessions will introduce participants to analytical techniques originally developed for this system, which serve as a valuable foundation for the study of partial differential equations more broadly.

References:
1. D. Bresch, B. Desjardins, J.-M. Ghidaglia, E. Grenier, M. Hilliairet, Multifluid models including compressible fluids, in: Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, eds. Y. Giga, A. Novotný, Springer, Cham, pp. 1–52 (2018).
2. D. Bresch, P.B. Mucha, E. Zatorska, Finite-energy solutions for compressible two-fluid Stokes system, Archive for Rational Mechanics and Analysis, 232(2), 987–1029 (2019).
3. F. Bouchut, Y. Brenier, J. Cortes, J.-F. Ripoll, A hierarchy of models for two-phase flows, Journal of Nonlinear Science, 10, 639–660 (2000).
4. M. Ishii, T. Hibiki, Thermo-Fluid Dynamics of Two-Phase Flow, Springer, New York, (2006).
5. B.J. Jin, A. Novotný, Weak-strong uniqueness for a bi-fluid model for a mixture of non-interacting compressible fluids, Journal of Differential Equations, 268, 204–238 (2019).
6. Y. Li, E. Zatorska, On weak solutions to the compressible inviscid two-fluid model, Journal of Differential Equations, 299, 33–50 (2021).
7. Y. Li, E. Zatorska, Remarks on weak-strong uniqueness for two-fluid model, in: Geometric Potential Analysis, De Gruyter (2022).
8. Y. Li, Y. Sun, E. Zatorska, Large time behaviour for a compressible two-fluid model with algebraic pressure closure and large initial data, Nonlinearity, 33(8), 4075–4094 (2020).
9. A. Novotný, M. Pokorný, Weak solutions for some compressible multicomponent fluid models, Archive for Rational Mechanics and Analysis, 235, 355–403 (2020).
10. T. Piasecki, E. Zatorska, Maximal regularity for compressible two-fluid system, Journal of Mathematical Fluid Mechanics, 24(9), Article 96 (2022).
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Since the third planned speaker had to cancel his participation on short notice and it was not possible to find a replacement, each of the local professors participating in the school will give a 90 minutes lecture on a topic of broader interest.
These lectures are:

Andreas Cap, Universität Wien – Non-involutive distributions 
Consider an open subset $U\subset\mathbb R^n$ that describes the possible states of a mechanical system. Assume further, that in each point $x\in U$, the directions in which the system can actually move form a $k$-dimensional linear subspace $E_x\subset \mathbb R^n$. These subspaces are said to define a distribution $E=\{E_x:x\in U\}$ on $U$ (not to be confused with distributions in the sense of generalized functions).  For some choices of $E$, this can lead to constants of the motions, which restrict the movement and for sufficiently many such constants (the case that $E$ is involutive), one arrives at integrable systems. For other choices, the so-called Chow-Rasehevskii theorem shows  no such restrictions arise and indeed, the system can reach any point in $U$. 

It turns out that simple examples of systems, like a cartwheel or a car moving on (an open subset of) a plane, lead to the latter behavior. Trying to associate a length or an energy to the allowed paths of such a system then leads to the concept of a sub-Riemannian metric, and the study of such metrics is of interest both in pure and applied mathematics.

A variant of the Chow-Rasehevskii theorem then shows that the arclength of allowed curves gives rise to a metric on $U$, which induces the usual topology on $U$. 

In my talk, I will outline the basic properties and discuss in detail the examples of a cartwheel and a car, which lead to a contact distribution (with $k=2$) in dimension 3 and a so-called Engel distribution (also with $k=2$) in dimension 4. Both these types of distributions have no local invariants and infinite dimensional groups of automorphisms. In the end of the talk, I will briefly discuss higher dimensional cases, in particular an example with $k=2$ in dimension two, which describes to balls rolling on each other without twisting or slipping. Surprisingly, classical work of E. Cartan shows that in this case the distribution $E$ itself endows $U$ with a „geometry“. It has local invariants and finite dimensional automorphisms related to an exceptional Lie algebra of type $G_2$.

Michael Drmota, TU Wien – The Moebius Randomness Principle
The Moebius Randomness Principle says that the sum \sum_{n\le x} x_n \mu(n) = o(N) for all „reasonable sequences“ x_n, where \mu(n) denotes the Moebius function
(defined by \m(n) = (-1)^k if n = p_1p_2… p_k is the product of k different prime numbers
and \m(n) = 0 otherwise). For example, this principle holds for x_n = 1 and is then equivalent to the prime number theorem. It also holds for periodic sequences x_n and is then equivalent the the Dirichlet prime number theorem.
The notion „reasonable sequence“ was formalized by Sarnak who conjectured that the Moebius randomness principle holds for all so-called deterministic sequences, that is, sequences that are of the form x_n = f(T^n y_0), where (Y,T) is a zero-entropy dynamical system and f is a continuous function. This conjecture is still widely open (nevertheless it holds, for example, for nil-sequences and automatic sequences). This conjecture is also very close to the Chowla conjecture that formalizes the asymptotic independence between addition and multiplication of integers. Actually Sarnak showed that the Chowla conjecture implies his conjecture. The purpose of this talk is to give some background information on this subject and to present also some ideas how sums that involve prime numbers can be analytically handled.

Gerald Teschl, Universität Wien – Post Quantum Cryptography on one foot 
Our entire modern communication relies on Public Key Crytography. The security of the classical algorithms (RSA and Diffie-Hellmann) currently in use rely on the hardness of the factorization and discrete logarithm problems. Both can be effectively solved on a (sufficiently powerful) quantum computer. This looming threat has 2016 lead to an international competition, under the governance of the NIST, for new quantum safe algorithms. Recently the first algorithms have been standardized and are currently being deployed worldwide. 

In contrary to the classical algorithms, which only require division with remainder for a basic understanding, the new algorithms are based on more sophisticated mathematics. In this talk I’ll try to convey some basic ideas. Given the time constraints the talk will mainly focus on lattice-based cryptography, which is a fascinating area on its own. A bit of algebra (polynomial rings, cyclic groups, finite fields) will be sufficient to follow the talk.

Christian Krattenthaler, Universität Wien – Bijections
Bijections are an important „workhorse“ in (enumerative and algebraic) combinatorics. In this presentation, I will explain *why* they are important, and discuss some instructive examples. Finally, I will present several notorious (hopeless?) open problems of finding bijections between certain concrete sets of combinatorial objects, some of which are open for a long time.

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