IMPAN Colloquium

Wednesdays at 14:15 in Room 321

Organizers: Piotr Achinger, Marcin Lara, Mateusz Wasilewski, Aneta Wróblewska-Kamińska

UPCOMING MEETINGS

FEBRUARY 19

Jan Rozendaal (IMPAN Warszawa)
The local smoothing conjecture

The local smoothing conjecture for the Euclidean wave equation is one of the main open problems in harmonic analysis. It is concerned with determining, in a quantitative sense, the extent to which waves can collide and focus in regions of flat space. This conjecture is known to imply several other major and seemingly unrelated problems in harmonic and geometric analysis, all of which have been open for decades, and work on the conjecture has also led to advances in areas such as analytic number theory. In this talk I will discuss the local smoothing conjecture, versions of the conjecture on smooth manifolds, and some recent work on manifolds with rough metrics. 

PAST MEETINGS (2024/2025)

OCTOBER 2

Janusz Grabowski (IMPAN Warszawa)
Multiplication by reals on manifolds

The playground for differential geometry is formed by manifolds. They are geometric structures that locally are like Euclidean spaces, e.g., spheres, tori, or projective spaces. What is crucial is that we can develop a differential calculus on manifolds like on Rn.

Particularly important, also in applications, are linear structures generalizing vector spaces and called vector bundles. They are locally products U x Rn, where the base U is an open subset of a Euclidean space and the fibers {p} x Rn carry vector space structures. The multiplication by reals is defined globally on every vector bundle, while the vector addition is defined on each fiber separately.

The main message is that one can forget about the addition, since every vector bundle is completely determined by its multiplication by reals. This approach, originated by Mikołaj Rotkiewicz and me, simplifies radically big parts of differential geometry. In particular, the concept of the compatibility of a vector bundle structure with other geometric structures on the manifold is then very easy, as we can use only the multiplication by reals and forget the other parts of the vector bundle structure. For instance, two vector bundle structures are compatible if the corresponding multiplications by reals commute.

We can define also abstract multiplications by reals on manifolds. Unexpectedly, such structures are very rigid and have been fully described locally by our tandem with Rotkiewicz. Canonical examples, that are not vector bundles,  are higher tangent bundles, i.e. bundles of jets of curves on manifolds.

OCTOBER 9

Artem Dudko (IMPAN Warszawa)
On spectrum of self-similar groups and graphs

Studying spectra of self-similar groups and related questions led to many important discoveries. Self-similar groups contain examples of groups solving various important open questions, including the  Grigorchuk group (the first example of finitely generated groups of intermediate growth), the Basilica group (first example of amenable but not subexponentially amenable groups), and the Lamplighter group (a counterexample to the strong Atiyah conjecture). In this talk I will give an introduction to self-similar groups, discuss methods of studying their spectral properties and spectral properties of associated graphs, and present some recent results on this topic (joint with Rostislav Grigorchuk).

OCTOBER 16

Martin Bridson (University of Oxford & Clay Mathematics Institute)
Chasing finite shadows of infinite groups through geometry

There are many situations in geometry and group theory where it is natural, convenient or necessary to explore infinite groups via their actions on finite objects – i.e. via the finite quotients of the group. But how much understanding can one really gain about an infinite group by examining its finite images? Which properties of the group can one recognise, and when does the set of finite images determine the group completely? How hard is it to decide what the finite images of an infinite group are?

In this talk, I will sketch some of the rich history of these problems and present results that illustrate how the subject has been transformed in recent years by input from low-dimensional topology and the study of non-positively curved spaces.

OCTOBER 23

Masha Vlasenko (Kyiv School of Economics & IMPAN Warszawa)
Varieties in the mirror and arithmetic

In physics, string theory considers elementary particles as one-dimensional objects called strings. In this theory spacetime has more than four dimensions, and the extra dimensions are described by complex manifolds of a special type, the so-called Calabi-Yau manifolds. Mirror symmetry is a phenomenon when two Calabi-Yau manifolds look very different geometrically, but are nevertheless equivalent when used as extra dimensions to describe interactions of particles. I plan to tell a story from the beginnings of mirror symmetry, when Candelas, de la Ossa, Green and Parkes discovered that difficult problems in enumerative geometry could be solved quickly ''in the mirror'' using differential equations. Physicists also noticed that the so-called instanton numbers observed on the other side of the mirror turned out to be integers. Together with Frits Beukers we were able to verify this conjecture in several key examples of mirror symmetry.

OCTOBER 30
Barbara and Jaroslav Zemanek Prize Award Ceremony

14:15–15:15   Introductory lecture 

Stuart White (University of Oxford)
Introduction to the classification of simple nuclear C*-algebras

In this talk I'll give an introduction to simple nuclear C*-algebras and their classification. This will be illustrated by examples coming from group actions. My aim will be to describe at a high level the paradigm shift from using internal structure to classify, to obtaining internal structure from classification that Chris Schafhauser's work made possible.  No prior knowledge of operator algebras will be assumed.


15:45–16:45    Lecture of the laureate

Christopher Schafhauser (University of Nebraska–Lincoln)
Lifting problems in C*-algebras and applications

A classical problem of Halmos asks which essentially normal operators (those commuting with their adjoint modulo a compact operator) on Hilbert space are compact perturbations of normal operators (those commuting with their adjoint).  A complete solution was obtained by Brown, Douglas, and Fillmore in the early 70s, and their solution led to the introduction of algebraic topological methods in operator algebras.  In particular, for a compact metric space X, they considered all embeddings of C(X) into the quotient B(H)/K(H) of bounded operators on a Hilbert space modulo the compact operators and showed that homotopy classes of such embeddings form an abelian group K_1(X), which is the degree one term for a generalized homology theory dual to topological K-theory.  Building on this, Kasparov developed a much more general extension theory, studying lifting problems along more general quotient maps up to a stabilized notion of homotopy.  I will discuss some recent progress in `non-stable’ extension theory with applications to embedding problems and classification problems for simple nuclear C*-algebras.

NOVEMBER 6

Maciej Borodzik (IMPAN Warszawa)
Slice knots

Understanding knot concordance is one of the most difficult and most intricate problem in low-dimensional topology. It has connections with deformation of singularities, geometry of plane curves on the one side, and the smooth Poincare conjecture in dimension four, on the other. In this talk I will present classical and modern tools used for studying knot concordance.

NOVEMBER 13

Iwona Chlebicka (MIMUW)
Fast p-Laplace evolution

Being a natural nonlinear counterpart of the heat equation, p-Laplace evolution equation is studied since 1960s. It's difficult to say whether the significant interest it enjoys is driven more by its numerous applications or by its intricate and profound intrinsic mathematical properties, which vary with the exponent p. For p=2 one deals with the linear equation, which is well-understood. In the interval 1<p<2, numerous thresholds emerge, governing the dynamics. My talk will focus on the relaxation towards self-similarity of nonnegative solutions and its rate across various subranges of small p. It is based on a joint project arXiv:2405.05405 with Matteo Bonforte (UAM, Madrid) and Nikita Simonov (Sorbonne University, Paris).

NOVEMBER 20

Joanna Kułaga-Przymus (Nicolaus Copernicus  University)
Randomness of the Möbius function and dynamics

The Möbius function, central to analytic number theory, is conjectured to exhibit randomness similar to independent sequences. During my talk I will concentrate on the interplay between number theory and dynamics, presenting two main strategies towards proving Sarnak's conjecture on the absence of correlations between the Möbius function and deterministic dynamical systems. I will include results on Veech's conjecture, joint with A. Kanigowski, M. Lemańczyk and T. de la Rue and provide insights into the current obstacles in establishing Sarnak's conjecture.

NOVEMBER 27

Błażej Wróbel (IMPAN Wrocław/University of Wrocław)
Three discrete maximal functions in high dimensions - Gaussians, balls and spheres

High-dimensional phenomena constitute an important research theme that has been present in harmonic analysis for several decades. It was initiated by E. M. Stein in 1980s  yet still a number of open problems remain unresolved. The main task is usually to obtain dimension-free estimates for norms of various operators as the dimension goes to infinity. The study of dimension-free estimates in the discrete setting was initiated several years ago by J. Bourgain, E. M. Stein, M. Mirek and myself. It is an example of discrete analogues in harmonic analysis - a vibrant area of modern harmonic analysis.


During my talk I will discuss discrete maximal functions, for Gaussians, balls, and spheres. The study of dimension-free estimates for these maximal functions requires bounding certain multiplier symbols, which are highly symmetric exponential sums. This requires interdisciplinary tools from analytic number theory (Warring's problem) and combinatorics. A preliminary step in the case of balls and spheres is a count of lattice points in these sets which has to be performed in a uniform (dimension-free) manner. I will report on recent progress with M. Mirek and T.Z. Szarek (spheres and Gaussians), and with J. Niksiński (balls and spheres).

DECEMBER 4

Feliks Przytycki (IMPAN Warszawa)
Geometric pressure and periodic orbits for iteration of quadratic polynomials in the complex plane

Hyperbolic Hausdorff dimension of the Julia set J(f) for a rational mapping f (ratio of two polynomials) of degree at least 2 on the Riemann sphere is defined as the supremum of the Hausdorff dimensions of its invariant hyperbolic subsets (usually it is just the Hausdorff dimension of the Julia set itself). It is the first zero t = t0 of the geometric pressure function P(f, −tlog|f′|). There are various equivalent definitions of this pressure, e.g. variational. I will sketch the proof that it can be expressed via periodic trajectories in the case of quadratic polynomials, partially answering an old problem. The method I use is to show that there are not many periodic trajectories going in bunches, using Milnor’s Orbit Portraits for external rays and their arguments.

A special case is the question of how many (at most) periodic trajectories of period n can be entirely in the disc B(x, rn) for rn small enough, for Cremer’s fixed point x (or periodic trajectory). Cremer’s means that the linear part of f at x is multiplication by exp(2πiα) with α real irrational and f is not linearizable at x (that happens if α is fast approximated by its continued fraction rational convergents pn/qn). I can give an answer for quadratic polynomials and rn ≤ exp(−δn): at most one periodic trajectory for any δ > 0 and all n large enough.

DECEMBER 18

Jacek Jendrej (Institut de Mathématiques de Jussieu)
Dispersion, solitons, their stability and emergence

Dispersive partial differential equations are evolution equations (that is, involving the time variable) whose solutions preserve the energy, but can still decay in large time due to the fact that various frequencies propagate with distinct velocities. In some cases, there exist non-trivial special solutions called solitons, which do not change their shape as time passes. Usually they are not stable in the usual Lyapunov sense, and the study of their (appropriately defined) stability is a major challenge.

I will discuss these issues in the case of a nonlinear wave equation known as the φ4 model. I will introduce the solitons for this model and explain why the question of their stability is a difficult one. A special case was solved a few years ago by Kowalczyk, Martel and Muñoz. I will also mention related results on similar models.

JANUARY 8

Justyna Signerska-Rynkowska (Dioscuri Center in Topological Data Analysis, Gdansk University of  Technology)
Low-dimensional dynamics enhancing modeling of neuronal activity


The talk will start with a brief introduction to the history of neuron modelling, including models of Lapicque, Hodgkin-Huxley and FitzHugh-Nagumo. These pioneering models were proposed decades ago and since then many other models arose, including hybrid systems combining continuous dynamics with discrete resets accounting for neuronal spiking. 


Due to the variety of these models, it is important to indicate the most universal and reliable tools allowing understanding dynamical mechanisms shaping their properties, related to diversity of neuronal activity and biological relevance. We will see how the theory of low dimensional dynamics (e.g. rotation theory, S-unimodal maps, phase-space methods) can be effectively applied in the analysis of map-based and hybrid neuron models, commonly used in computational and theoretical neuroscience. I will also mention recent tools adapted from nonlinear time series analysis and computational topology facilitating, among others, classification of spike-trains and phase-portraits in neuron models.

JANUARY 15

Piotr Nowak (IMPAN Warsaw)
Rigid groups and spectral gaps

The goal of this talk is to discuss Kazhdan’s property (T), a classical, powerful rigidity property of groups, and a new approach to proving it via algebraic spectral gaps for the Laplacian in the group ring. These new methods grew out of a new description of property (T) in terms of noncommutative sums of square due to Ozawa, and in particular I will present how it was used to prove property (T) for Aut(F_n), the automorphism groups of free groups, for n at least 5. I will also discuss some applications and generalizations to higher cohomology and higher index theory. 

JANUARY 22

Jacinta Torres (Jagiellonian University)
Algebraic combinatorics and geometry in representation theory

The first part of this talk will be a brief introduction to the representation theory of finite groups, Lie algebras and groups, and associative algebras, mostly over the field of complex numbers. In the second part of the talk I will explain how objects in representation theory can be modelled using combinatorics and geometry. Finally, I will highlight some of the landmark advances in the field and present some of my own contributions as well as various open problems.

JANUARY 29

Marcin Napiórkowski (University of Warsaw)
Bose-Einstein condensation: an ongoing mathematical challenge

Proving Bose-Einstein condensation in the thermodynamic limit remains a major open problem in mathematical physics. In my talk, I will explain the content of the conjecture and review recent progress in the study of bosonic many-body systems. 

PAST MEETINGS (2023/24)

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

 In my talk I plan

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

 

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.

MAY 8

Tomasz Komorowski (IMPAN)
On the Conversion of Work into Heat: Microscopic Models and Macroscopic Equations


Nature has a hierarchical structure with macroscopic behavior arising from the dynamics of atoms and molecules. The connection between different levels of the hierarchy is however not always straightforward, as seen in the emergent phenomena, such as phase transition and heat convection. Establishing in a mathematical precise way the connection between the different levels is the central problem of rigorous statistical mechanics. One of the methods leading to  such results is to introduce some stochasticity inside the system. We summarize  some of the results obtained recently concerning the  derivation of the macroscopic heat equation from the  microscopic  behavior of   a  harmonic chain with a stochastic perturbation. We focus our attention on the emergence of macroscopic boundary conditions.

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

One of the central discoveries in the theory of dynamical systems was that differentiable (or smooth) systems can display strongly chaotic behavior and in many ways behave like a sequence of random coin tosses. In this talk we will describe appearance and interactions of chaotic properties in smooth dynamics. We will highlight main developments, describe the state of the art and discuss some open problems in the field.

MAY 29

Adam Nowak (IMPAN)
Genuinely sharp heat kernel estimates in various contexts


The talk will concern the problem of finding truly sharp description of the behavior of heat kernels in various classical contexts. We will briefly overview the state of the knowledge and give an account of recent results within this topic, including for instance genuinely sharp heat kernel bounds on the Euclidean spheres.

JUNE 5

Tomasz Pełka (MIM UW)
Monodromy at radius zero and its applications

Consider a family of smooth complex algebraic manifolds over a punctured disc. Going once around the origin one gets a self-diffeomorphism of a fiber, called the monodromy. It is only well defined up to an isotopy. In my talk, I will explain how to choose a representative which is a symplectomorphism with particularly simple dynamics. The idea is to extend the smooth family from the punctured disc to the annulus, and look at the monodromy over the inner circle ("at radius zero"), where all the choices become irrelevant. For isolated hypersurface singularities, the resulting dynamics can be used to recover the multiplicity, and prove its invariance under μ-constant deformations. For maximal Calabi-Yau degenerations, a large part of a "radius-zero" fiber is covered by Lagrangian tori, which can be used to approximate the ones postulated by Mirror Symmetry.

JUNE 12

Rafał Latała (MIM UW)
On the work of Michel Talagrand – the 2024 Abel Prize laureate 

The 2024 Abel Prize was awarded to Michel Talagrand "for his groundbreaking contributions to probability theory and functional analysis, with outstanding applications in mathematical physics and statistics". During the talk I will briefly discuss some of Talagrand's scientific achievements in three fields, honored by the Norwegian Academy of Science and Letters: suprema of stochastic processes, concentration of measure and  spin glasses.

JUNE 19

Rostislav Grigorchuk (Texas A&M)
Fractal, liftable and scale groups


Scale  groups  are  closed subgroups of the group of isometries of a regular tree that fix an end of the tree and are vertex-transitive.  They  play  an  important  role  in the study  of  locally  compact  totally  disconnected  groups as  was  recently observed  by  P-E.Caprace and  G.Willis.  In  the  past  they  were  studied   in  the  context  of abstract  harmonic  analysis,  random  walks   and  amenability. It  is a  miracle  that  they  are  closely  related  to  fractal groups,  a  special subclass  of  self-similar  groups.

In my  talk  I  will discuss  two ways  of  building  scale  groups.  One  is  based  on  the  use  of  scale-invariant  groups studied  by  V. Nekrashevych  and  G. Pete,  and  a second is  based  on  the  use  of  liftable  fractal groups.  The  examples  based on both  approaches  will  be  demonstrated  using  such groups  as  Lamplighter, Basilica,  Hanoi Tower Group, Group  of  Intermediate  Growth (between  polynomial  and  exponential) constructed  by the speaker  in  1980,  and  GGS-groups. Additionally,  the group  of  isometries  of  the  ring  of  integer  p-adics  and group of  dilations  of  the  field  of  p-adics  will  be  mentioned  in  the  relation  with the  discussed  topics. A  joint  work  with  Dmytro  Savchuk.