Dawid Kielak (University of Oxford)
Fibring in group theory and topology
I will introduce the topological concept of fibring over the circle, and its algebraic counterpart. I will then discuss the role fibring played in Thurston's programme of understanding 3-manifolds, and how the algebraic approach can play a similar role in understanding low dimensional groups.
Maciej Dołęga (IMPAN Kraków)
Enumerative combinatorics – how enumeration helped to discover new bridges in mathematics
In my talk I am going to describe enumerative combinatorics – the area that studies problems that were natural and interesting already for ancient mathematicians but has seen a very quick development in recent years. I will focus on these recent developments and I will describe two examples how enumerative combinatorics helped to build new bridges and fascinating directions of research in probability, representation theory and enumerative geometry.
Grigor Sargsyan (IMPAN Gdańsk)
Forcing Axioms and Determinacy Axioms: towards a unified theory of infinity
Forcing Axioms are axioms of infinity generalizing the Baire Category Theorem. They have been proposed to mitigate the effect of forcing, and since then have been very useful in solving wide range of problems in many areas of mathematics. Determinacy Axioms are game theoretic axioms asserting the determinacy of infinite two player games. They resolve classical questions from analysis. They are fundamentally rooted in different mathematical ideas, depict radically different pictures of the universe, solve mathematical problems according to the prevailing intuition of the practitioners and are logically incompatible, creating deep ambiguities in our understanding of infinity. In this talk, we will describe an approach to unify them.
Šárka Nečasová (Czech Academy of Sciences)
On the motion of fluid in a moving domain, applications to fluid structure, questions of uniqueness, regularity, and collisions
Problems of fluid flow inside a moving domain deserve a lot of interest as they appear in many practical applications. Such problems can also be seen as a preparation step for research of fluid-structure interaction problems. Research of the compressible version of the Navier-Stokes system dates back to the nineties when the groundbreaking result of the existence of the global weak solutions to the compressible barotropic Navier–Stokes system on a fixed domain was proved by P. L. Lions and, later, by E. Feireisl and collaborators who extended the existence result to more physically relevant state equations. After that the theory of weak solutions was extended to the problem of fluid flow inside a moving domain. Such existing theory was applied to more complicated problem e.g. to the interaction between system of heat conducting fluid with a shell of Koiter type, or into the case of two compressible mutually noninteracting fluids and a shell of Koiter type encompassing a time dependent 3D domain filled by the fluids. Moreover, the theory of compressible fluids filling the smooth bounded domain where inside of a domain rigid body/bodies is/are moving and the motion satisfies the conservation of force and momentum was studied, and some questions arise in these problems e.g. uniqueness, regularity. Even, in the case of an incompressible fluid with moving a rigid body the problem cannot be treated easily since the physical domain is time-dependent. Another very important question is a problem of collision. What we can show? It is possible to prove collision or to show no-collision result.
Jakub Skrzeczkowski (University of Oxford)
Nonlocal PDEs everywhere!
Nonlocal PDEs (partial differential equations) are a class of equations combining local (say, a function or its derivative evaluated at the given point) and global quantities (say, an integral of the function). I will discuss several areas where such equations arise, including mathematical physics (derivations of equations), numerical analysis (particle-type method for diffusion) and statistics (analysis of the recently proposed Stein variational gradient descent method for sampling). Several open problems will be discussed and some partial solutions will be proposed.
Boban Velic̆ković (CNRS/Université Paris Cité)
Higher derived limits
Barbara and Jaroslav Zemanek Prize Award Ceremony
14:15–15:15 Introductory lecture
Eric Ricard (CNRS/Caen)
Weak type inequalities in non commutative analysis
Léonard Cadilhac (Sorbonne Université)
Non-commutative maximal functions and ergodic theory
Jarosław Buczyński (IMPAN Warszawa)
Three stories of Riemannian and holomorphic manifolds
On Wednesday afternoon you are going to hear a bunch of stories about manifolds, focusing on two main characters: a compact holomorphic manifold and a Riemannian manifold. The talk consists of three seemingly independent parts. In the first part, the main character is going to be a compact holomorphic manifold, and as in every story, there will be some action going on. This time we act with the group of invertible complex numbers, or even better, with several copies of those. The spirit of late Andrzej Białynicki-Birula until this day helps us to understand what is going on. The second part is a tale of holonomies, it begins with "a long time ago,..." and concludes with "... and the last missing piece of this mystery is undiscovered til this day". The main character here is a Riemannian manifold, but the legacy of Marcel Berger is the background all the time. In the third part we meet legendary distributions, which are subbundles in the tangent bundle of one of our main characters. Among others, distributions can be foliations, or contact distributions, which like yin and yang live are the opposite sides of the world, yet they strongly interact with one another. Ferdinand Georg Frobenius is supervising this third part. Finally, in the epilogue, all the threads and characters so far connect in an exquisite theorem on classification of low dimensional contact manifolds. In any dimension the analogous classification is conjectured by Claude LeBrun and Simon Salamon, while in low dimensions it is proved by Jarosław Wiśniewski, Andrzej Weber, in a joint work with the narrator.
Mikołaj Frączyk (Uniwersytet Jagielloński)
Large subgroups in higher rank
Let G be a higher-rank semisimple Lie group (for example, SL_n(R), n > 2). The famous Margulis' arithmeticity theorem allows us to more or less classify the lattices of G. On the other hand, our understanding of infinite covolume discrete subgroups of G is far from complete. It seems that in higher rank, it is hard to find examples of "large" discrete subgroups other than lattices. It is natural to wonder whether there are new rigidity statements that would explain this situation. In my talk, I'll make this question more precise and present several instances of such rigidity phenomena, for example, my joint work with Gelander on confined discrete subgroups and the joint work with Minju Lee on discrete subgroups with finite Bowen-Margulis-Sullivan measure.
Przemysław Wojtaszczyk (IMPAN Warszawa)
Nonlinear widths; mix of topology, Banach spaces and numerical algorithms
Jakub Witaszek (Princeton University)
The interplay between complex and arithmetic singularities
Mateusz Kwaśnicki (Wrocław University of Science and Technology)
Piotr Miłoś (IMPAN Warszawa / IDEAS NCBR)