MAY 15

Paweł Dłotko (IMPAN)
What is the shape of things?

In this talk, I will lay down an introduction to methods of computational and applied topology developed in my Dioscuri Centre for Topological Data Analysis. We will start with standard topologically inspired data characteristics, like persistent homology and subsequently move to Euler curves and profiles, as well as various Reeb-graph-like characteristics, that can be computed for finite data samples. We will discuss how the development of this methodology is driven by concrete problems in applied sciences and discuss a number of successful cases where they were used.

MAY 22

Adam Kanigowski (Uniwersytet Jagielloński / University of Maryland)
Ergodic and statistical properties of smooth systems

MAY 29

Adam Nowak (IMPAN)

JUNE 5

JUNE 12

OCTOBER 4

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.

OCTOBER 11

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.

OCTOBER 18

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. 

OCTOBER 25

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.

NOVEMBER 8

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. 

NOVEMBER 15

Boban Velic̆ković (CNRS/Université Paris Cité)
Higher derived limits

NOVEMBER 22
Barbara and Jaroslav Zemanek Prize Award Ceremony

14:15–15:15   Introductory lecture

Eric Ricard (CNRS/Caen)
Weak type inequalities in non commutative analysis

We will introduce the field of non commutative analysis, where one replaces function spaces by operators algebras and related objects. There have been quite a lot of developments in the last 20 years thanks to operator spaces theory and especially works by Pisier and Junge. As in harmonic analysis, most of the results  are obtained from end point estimates but people considered BMO spaces  as much easier than weak L_1 until Leonard Cadilhac came up with new ideas in his thesis.  

15:45–16:45    Lecture of the laureate

Léonard Cadilhac (Sorbonne Université)
Non-commutative maximal functions and ergodic theory

In non-commutative analysis, maximal inequalities were first formulated in the 70's but their systematic study only began 30 years later in works of Pisier, Junge and Xu. Since a family of positive operators does not necessarily admit a supremum, maximal functions offer many challenges when trying to generalize their theory from standard measure spaces to non-commutative ones. This is seen in their very definition, in their interpolation properties, and in the techniques employed to prove them. In this talk, I will illustrate this facts, mainly focusing on a maximal function coming from the ergodic theory of group actions. 

NOVEMBER 29

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.

DECEMBER 6

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.

DECEMBER 13

Przemysław Wojtaszczyk (IMPAN Warszawa)
Nonlinear widths; mix of topology, Banach spaces and numerical algorithms

The primary concern of approximation theory is the development of methods to approximate general functions f by simpler, easier to compute functions.  It has as its origin the study of the approximation of functions by n dimensional linear spaces of algebraic or trigonometric polynomials.  Over the last decades, motivated by various theoretical and practical problems approximation has evolved into using what are commonly  referred to as nonlinear approximation methods.  Especially in numerical analysis we see the growing interest in the use of non-linear approximation procedures (algorithms). While it is well known  that nonlinear methods of approximation can often perform dramatically better than linear methods, there are still questions on how to measure the optimal performance possible for such methods.  More precisely: we want to estimate how well we can approximate the set K using a specified class of approximation methods. This is expressed by widths. The width is a sequence of numbers, say (E_n(K)_X), where n means the number of parameters used.

In order to quantify how well we can approximate functions f,  we must specify at least

1. how to measure the accuracy of the approximation. Typical candidates here are L_p norms.

2. what we know about the function f,  to be approximated.  The natural description here is to specify that f is an element of a certain compact subset K of the Banach space X. We often refer to such a compact set K as a model class.

3. what approximation procedures we want to use

4. how many parameters we use.

 In my talk I plan

• To present more  precisely the general approximation schemes.

• To present historically first non-linear widths and the topology they are based on.

• To present recently introduced  stable  non-linear widths and their connections with topology and functional analysis and numerical algorithms.

• To show some relations between some widths.

DECEMBER 20

Jakub Witaszek (Princeton University)
The interplay between complex and arithmetic singularities

JANUARY 10

Mateusz Kwaśnicki (Wrocław University of Science and Technology)

Liouville's theorems for Lévy operators

Classical Liouville's theorem states that all bounded harmonic functions are constant. A more general variant only requires one-sided bound, and yet another version asserts that harmonic functions bounded by a polynomial are polynomials.

Over the last few years similar results have been studied for harmonic functions for Lévy operators. That is, the equation Δf = 0 is replaced by the non-local PDE Lf = 0, where L is a Lévy operator, that is, a translation-invariant integro-differential operator satisfying the positive maximum principle. These results have applications in regularity theory for non-local PDEs.

Alibaud, del Teso, Endal and Jakobsen (DOI:10.1016/j.matpur.2020.08.008) proved that bounded L-harmonic functions are constant. This result was soon extended by Berger and Schilling (DOI:10.7146/math.scand.a-132068), and by Berger, Schilling and Shargorodsky (arXiv:2211.08929). Further papers study particular Lévy operators or more narrow classes of such operators. For example, a complete description of functions harmonic with respect to the fractional Laplace operator was given by Fall (DOI:10.1090/proc/13021), and by Chen, D'Ambrosio and Li (DOI:10.1016/j.na.2014.11.003).

In a recent paper Liouville's theorems for Lévy operators (arXiv:2301.08540), together with Tomasz Grzywny we prove three results. The first one is Liouville's theorem for arbitrary Lévy operators which only requires a one-sided bound on L-harmonic functions. The second one describes polynomially bounded L-harmonic functions, under suitable assumptions. Finally, our third result proves that these assumptions cannot be omitted: we construct an unusual non-polynomial L-harmonic function for a suitable Lévy operator L. Our results include and extend all previously known Liouville's theorems for Lévy operators.

FEBRUARY 7

Piotr Miłoś (IMPAN Warszawa / IDEAS NCBR) & Łukasz Kuciński (IMPAN Warszawa)

When mathematics meets AI

In our talk, we will shortly present a landscape of AI developments related to solving math problems. In recent years, we have witnessed a surge of work based on large language models, as well as massive formalization efforts, which now touch upon mainstream developments in mathematics. We will also present our work in these directions.

FEBRUARY 14

Marcin Bownik (IMPAN Gdańsk)

Beyond the Kadison-Singer problem

The aim of this talk is to give an overview of the solution of the Kadison-Singer problem (1959) by Marcus, Spielman, and Srivastava (2015). This problem was known to be equivalent to a large number of problems in analysis such as Anderson paving conjecture (1979), Bourgain-Tzafriri restricted invertibility conjecture (1991), Weaver’s conjecture (2004), and Feichtinger’s conjecture (2005). The amazing solution of this problem uses methods which are very far from analyst's toolbox such as real stable polynomials, interlacing families of polynomials, or multivariable barrier method. It involves a key concept of a mixed characteristic polynomial, which is an multilinear analogue of the usual characteristic polynomial. At the same time, the Kadison-Singer problem shows the unity of mathematics as it connects a large number of areas: operator algebras (pure states), set theory (ultrafilters), operator theory (paving), random matrix theory, linear and multilinear algebra, algebraic combinatorics (real stable polynomials), functional analysis (frame theory), and harmonic analysis (exponential frames).

In the last part of the talk we discuss several developments beyond the Kadison-Singer problem. This includes: the conjecture of Akemann and Weaver showing Lyapunov-type theorem for trace class operators, the solution of discretization problem for continuous frames by Freeman and Speegle, and the existence of syndetic Riesz sequences of exponentials.

FEBRUARY 21

Yonatan Gutman (IMPAN Warszawa)
Optimal representation of dynamical systems 

The field of dynamical systems started more than one hundred years ago with the appearance of Poincaré’s "Les méthodes nouvelles de la Mécanique Céleste". In the last century the  discipline has undergone a considerable expansion with deep contributions by renowned mathematicians.  Notwithstanding, in essence, a dynamical system is a simple object, consisting of a phase space X and a transformation T:X—>X. However when performing an experiment involving a dynamical system one is often interested in a convenient representation. This can be achieved with the help of observables f:X—>R and their associated time-delayed measurements, e.g., f(x), f(Tx), f(T^2 x),… 

After a general introduction to dynamical systems, I will describe several problems related to optimal - in a precise manner to be defined - representations of dynamical systems.

FEBRUARY 28

Piotr Gwiazda (IMPAN Warszawa)
From compressible Euler equation to porous media equation

One of the ways to understand various phenomena in the real world is to describe them by mathematical models. Which in our case are systems of partial differential equations. But having various such models it is also important to find relations between them. We will concentrate on the so-called high-friction limit for systems arising in fluid mechanics and we will study a combined system of Euler, Euler-Korteweg and Euler-Poisson equations with friction and exponential pressure with exponent γ>1. We rigorously derive the scalar diffusive equation as a limit of the Euler-like equation using the relative entropy method.

The proof is formulated in a frame of dissipative measure-valued solutions  (which are "weaker" than the weak one) of the Euler-like equation which are known to exist on arbitrary intervals of time.

MARCH 6

Łukasz Grabowski (Leipzig University)

Brief overview of the cost of groups and equivalence relations

I will give a brief overview of the so-called cost - an invariant of groups and equivalence relations introduced by Levitt and Gaboriau, which gives insight into various conjectures in algebra and topology. I will concentrate on presenting a result of

Hutchcroft and Pete on the cost of groups with property (T), and the generalisation of this result to equivalence relations (this last result is a joint work with Hector Jardon Sanchez and Sam Mellick).

MARCH 20

Ewelina Zatorska (University of Warwick)
Dissipative Aw-Rascle system: various notions of solutions

Abstract: During my talk I will introduce the Aw-Rascle model of one line vehicular traffic and then it’s dissipative version in multi-dimensions. I will explain connections with other models of mathematical fluid mechanics and kinetic theory and introduce definitions of suitable weak solutions. I will then discuss an interesting problem of singular limit leading to hard-congestion model, and present the proof of this result in one-dimensional setting.

MARCH 27

Christophe Eyral (IMPAN)

Topology of complex hypersurface singularities

The aim of this talk which is intended for general audience is to introduce classical tools of singularity theory for the study of isolated complex hypersurface singularities. In particular, this includes the conic structure theorem, the Milnor fibration theorem, and the presentation of the Milnor number. My intention is to reach, at the end of the talk, the statement of the Lê-Ramanujam theorem: "In a family of complex hypersurfaces with isolated singularities, the invariance of the Milnor number implies the invariance of the embedded topological type."

APRIL 10

Gábor Szabó (KU Leuven)

On classification of C*-algebras and their dynamics

I will start this talk by motivating C*-algebras as the intuitive concept of a noncommutative topological space. From this point of view, many interesting invariants such as topological K-theory extend to the category of C*-algebras. I will give a brief glimpse into the classification program for simple C*-algebras by K-theoretical data. For certain classes of examples, an important consequence is homotopy rigidity, which means that homotopy equivalence implies isomorphism, resembling the spirit of classical phenomena such as Mostow's theorem in geometry or the Borel conjecture. I will then outline ongoing research effort to unravel such rigidity phenomena at the level of dynamical systems on C*-algebras.

APRIL 17

Borys Kuca (Uniwersytet Jagielloński)
The Szemerédi theorem and beyond

Looking for patterns in sets of numbers is among the oldest and most fundamental mathematical endeavors. A quintessential result in this direction is the Szemerédi theorem which asserts that each subset of integers of positive density contains an arithmetic progression of arbitrary (finite) length. Often viewed as an example of a "deep" mathematics due to its elaborate and diverse proofs, the Szemerédi theorem has stimulated far-reaching developments in areas as diverse as combinatorics, number theory, harmonic analysis, ergodic theory and model theory. In this talk, I will survey some of the recent progress on the Szemerédi theorem and its generalisations.

APRIL 24

Adam Skalski (IMPAN)
On certain Hecke algebras arising as deformations of group algebras of Coxeter groups

Operator algebras associated with discrete groups have played a significant role in the theory of operator algebras since its inception over 80 years ago, and remain a central theme of research still now.  I will recall fundamental questions related to this class and present certain operator algebras which can be viewed as deformations of algebras of (right-angled) Coxeter groups: the so-called q-Hecke operator algebras.

We shall see how the algebras in question arise in various natural ways, in particular related to groups acting on buildings, and later characterise some of their properties in terms of the deformation parameters.