## Young Researchers Colloquium

### Jacek Krajczok, Filip Rupniewski, Jacopo Schino

Fridays **15:00 - 16:00**, room 403/Zoom.

__Next talk:__

__Previous talks:__

18.06.2021 Speaker: Fatima Zhora Goffi (Karlsruhe Institute of Technology, Institute for Analysis)

Title: Effective Characterization of Metamaterials

Abstract: When the period of unit-cells constituting metamaterials is no longer much smaller than the wavelength but only smaller, local material laws fail to describe the propagation of light in such composite media when considered at the effective level. Instead, nonlocal material laws are required. They have to be derived by approximating a general response function of the electric field in the metamaterial at the effective level. At first, we overcome this limitation by considering a Padé-type approximation of the response function of the electric field. It is shown that nine different constitutive relations can be derived. We present a checklist each of these constitutive relations has to pass in order to be physically and mathematically liable. It turned out that only one of these nine Padé approximations passes the checklist. Alternatively, we propose a general higher-order approximation for the same purpose. For such nonlocal constitutive relations, the classical interface conditions for the Maxwell equations are not sufficient to fix the amplitudes of all the propagating modes. Additional interface conditions are then required. We give a general formula of these interface conditions at any order of the constitutive relation.

**11.06.2021 Speaker:** Léonard Cadilhac (Paris-Saclay University, Department of Mathematics)

**Title:** Amenability and freeness

**Abstract:** In this talk, I will discuss amenability, a property of locally compact groups, and in particular its relationship with freeness. The property was introduced in the 1920s as an obstruction to paradoxical decompositions, in relation with the Banach-Tarski paradox. It admits numerous characterisations, among which a probabilistic one due to Kesten which will lead us to consider random walks on groups. In the last few minutes of the talk, I will present a generalisation of this result to the operator algebra setting.

**28.05.2021 Speaker:** Federico Bernini (University of Milano-Bicocca, Department of Mathematics and Applications)

**Title:** Semirelativistic Choquard equations with singular potentials and general nonlinearities arising from Hartree-Fock theory

**Abstract:** I am going to present recent work in collaboration with B. Bieganowski and S. Secchi, where a semirelativistic Choquard-type equation is considered. For this problem, we can show the existence or nonexistence of least energy solutions (called ground state solutions), though the presence of Hardy-type potential raises some problems. Moreover, a compactness result for the ground state solutions is proved.

Without entering in the details of the proof, I would like to explain the motivations that lie in the study of this problem, focusing in particular on the main difficulties we encountered and the techniques used to manage them.

21.05.2021 Speaker: Michał Godziszewski (University of Warsaw, Faculty of Mathematics, Informatics, and Mechanics)

Title: (Un)decidability of addition and multiplication in models of number theory

Abstract: One of the features of the arithmetical operations in natural numbers - addition and multiplication - is that they are almost trivially decidable: there is an easy algorithm for computing a sum or a product of two given natural numbers. From mathematical logic we know there exist mathematical structures that are non-isomorphic with the natural numbers, yet satisfy the first-order number theory. What is more, there are even uncountable models of number theory. All models of PA that are not isomorphic to the well-known standard model are called, surprise surprise, nonstandard. A somewhat natural question then is to ask if there are algorithms to compute the addition and multiplication of some nonstandard models? As Stanley Tennenbaum demonstrated in 1959, the answer is no: there is no nonstandard model of arithmetic, where addition or multiplication could be computable. The reason for that is that any nonstandard model of arithmetic codes information that is too complex to be accessed algorithmically.

During the talk, I will introduce the basic facts from mathematical logic, show how we know there are nonstandard models, sketch the argument for the Tennenbaum Theorem, and provide some rudiments on the computability-theoretic properties of the quotient structures of number theory and set theory.

**14.05.2021 Speaker:** Roberto Ognibene (University of Genoa, Department of Mechanical, Energetic, Industrial, and Transport Engineering)

**Title:** Spectral stability of the Laplacian under perturbation of the boundary conditions

**Abstract:** In this talk, I will discuss the behavior of the spectrum of the Laplacian on bounded domains, subject to moving mixed boundary conditions. More precisely, let us assume the boundary of the domain to be split into two parts, on which homogeneous Neumann and Dirichlet boundary conditions are respectively prescribed; let us then assume that, alternately, one of these regions “disappears” and the other one tends to cover the whole boundary. In this framework, I will first describe under which conditions the eigenvalues of the mixed problem converge to the ones of the limit problem (where a single kind of boundary condition is imposed); then, I will sharply quantify the rate of this convergence by providing an explicit first-order asymptotic expansion of the “perturbed” eigenvalues. These results have been obtained in collaboration with Veronica Felli and Benedetta Noris.

07.05.2021 Speaker: Alberto Franceschini (University of Trento, Department of Mathematics)

Title: Special varieties with a *C**-action of bandwidth less than or equal to *2*: (open) questions & (some) answers

Abstract: The idea of studying geometric spaces via group actions goes back to the Erlangen Program of Klein. When *X* is a projective variety, torus actions appear to be good candidates since the introduction of the GKM-theory in late 90s. The bandwidth of a *C**-action is an integer that measures, in some sense, the complexity of the *1*-dimensional torus action. Such a value is strongly linked to birational transformations of *X*, as pointed out by Occhetta, Romano, Solá Conde, and Wiśniewski. In this talk I will show who are the *C**-actions of minimal bandwidth for a good class of examples: rational homogeneous varieties with Picard number one. Moreover, I will discuss a characterization of varieties admitting a bandwidth one *C**-action and talk about (on-going) generalization in the case of bandwidth two.

**30.04.2021 Speaker:** Damian Orlef (Institute of Mathematics, Polish Academy of Sciences)

**Title:** From random groups to random propositional formulas

**Abstract:** Generally speaking, random groups are groups obtained by choosing a generating set and then introducing at random relations between the generators. Under appropriate conditions, this framework produces groups with interesting geometric properties, varying widely with the number of relations. In the talk I will describe in detail the so-called triangular binomial model of random groups (with relations of length 3) and describe its phase transitions. I will then show how the left-orderability of random triangular groups can be studied by considering satisfiability of random propositional formulas.

**23.04.2021 Speaker:** Sandra Lucente (University of Bari, Department of Physics)

**Title:** Making space for spaces

**Abstract:** In this talk, we summarize the various definitions of dimension. Our aim is to suggest a presentation of these ideas to a wide audience. For this reason, we follow a literary

approach, connecting every space dimension to an invisible city by the Italian writer Italo Calvino and a real European city.

**16.04.2021 Speaker:** Michał Łasica (Institute of Mathematics, Polish Academy of Sciences; University of Tokyo)

**Title:** The jagged landscape of fourth-order quasilinear parabolic systems

**Abstract:** I will briefly survey the existence theory of weak solutions to fourth-order quasilinear parabolic equations (and systems of equations), focusing on variational and monotonicity-based methods. I will try to explain how and why it's different from its better known second-order counterpart. Finally, I will present an existence result that I obtained recently in collaboration with Yoshikazu Giga.

**09.04.2021 Speaker:** Alexander Frei (University of Copenhagen)

**Title:** Relative Cuntz-Pimsner algebras: Gauge-invariant uniqueness theorem and the lattice of gauge-invariant ideals

**Abstract:** We start with an abstract definition of C*-correspondences comparing them to Fell bundles.

From here we go into a detailed analysis of covariances within the category of C*-correpondences.We obtain here a systematic reduction leading us to a parametrisation of relative Cuntz-Pimsner algebras.

With this at hand we arrive at the gauge-invariant uniqueness theorem, for all (arbitrary) gauge-equivariant representations at once.

From here we move on to the analysis part of the program. We study the covariances in the case of the Fock representation and its quotients. As a result we derive that the parametrisation of relative Cuntz-Pimsner algebras is classifying. In other words, we obtain a complete and intrinsic picture of the lattice of quotients, and equivalently of gauge-invariant ideals.

If time permits, we finish off with the next chapter on their induced Fell bundles, as already investigated by Schweizer.

**26.03.2021 Speaker:** Sanaz Pooya (Institute of Mathematics, Polish Academy of Sciences)

**Title:** Higher Kazhdan projections, L²-Betti numbers, and Baum-Connes conjectures

**Abstract:** The Baum-Connes conjecture suggests a link between operator algebras and topology/geometry. If it holds for a certain group, it provides topological tools to compute the K-theory of its reduced group C*-algebra. This conjecture has been confirmed for large classes of groups, such as amenable groups, but also for some Kazhdan's property (T) groups. Property (T) and its strengthening are driving forces in the search for potential counterexamples to the conjecture. Having property (T) for a group is characterised by the existence of a certain projection in the universal group C*-algebra of the group, known as the Kazhdan projection. It is this projection and its analogues in other completions of the group ring, which obstruct known methods of proof for the Baum-Connes conjecture.

In this talk, after providing background on the topic, I will introduce a generalisation of Kazhdan projections. Employing these projections we provide a link between the surjectivity of various versions of the Baum-Connes map and the L²-Betti numbers of the group. This is based on joint work with Kang Li and Piotr Nowak.

**19.03.2021 Speaker:** Artem Dudko (Institute of Mathematics, Polish Academy of Sciences)

**Title:** Finding order within the chaos: a brief introduction to Julia sets

**Abstract:** Informally speaking, the Julia set *J(p)* of a polynomial *p(z)* of a complex variable *z* is the set of points near which the iterates of *p(z)* behave chaotically. Despite being associated with chaos, Julia sets have a lot of beautiful structures. In this talk, I will give a brief introduction to Julia sets of polynomials and will try to explain why there is so much order within them.

**12.03.2021 Speaker:** Piotr Antoni Kozarzewski (Military University of Technology)

**Title:** Parametrised measures — examples & applications

**Abstract:** The aim of this talk is to introduce the classical notions of parametrised measures — Young Measures and DiPerna–Majda measures like in [2] or [1]. Thepossiilities given by this notions will be illustrated on Euler's equation of incompressible fluids, as well as on several elementary examples. I will also sketch several modern approaches to DiPerna–Majda measures following [3] and explain the topological obstacles lying in the background, solved in [4].

[1] J.–J. Alibert and G. Bouchitté, *Non-uniform integrability and generalized Young measures*, Journal of Convex Analysis, 4 (1997), 129–147.

[2] R. J. DiPerna and A. J. Majda, *Oscillations and ioncentrations in weak solutions of the incompressible fluid equations*, Communications in Mathematical Physics, 108 (1987), 667–689.

[3] A. Kałamajska, *On Young measures controlling discontinuous functions*, Journal of Convex Analysis, 13 (2006), 177–192.

[4] P. A. Kozarzewski, *On certain compactification of an arbitrary subset of R^m and its applications to DiPerna–Majda measures theory*, submitted, (2020).

**05.03.2021 Speaker:** Sven Raum (University of Stockholm, Department of Mathematics)

**Title:** Superrigidity for group operator algebras

**Abstract:** It is a classical problem to recover a discrete group from various rings or algebras associated with it, such as the integral group ring. By analogy, in an operator algebraic framework, we want to recover torsion-free groups from certain topological completions of the complex group ring, such as the reduced group C*-algebra. Groups for which this is possible are called C*-superrigid.

I will start this talk with a discussion of how a group can be recovered from its group rings, before I introduce the reduced group C*-algebras and describe the state-of-the-art in C*-superrigidity. I will end with a short account on other kinds of superrigidity for group operator algebras putting the subject into a bigger perspective.

**26.02.2021 Speaker:** Hanieh Keneshlou (Institute of Mathematics, Polish Academy of Sciences)

**Title:** The birational geometry of Hurwitz spaces

**Abstract:** The Hurwitz space parametrizes *d*-sheeted simply branched covers of the projective line by smooth curves of genus *g*. The study of Hurwitz spaces plays an important role in shedding light on the geometry of the moduli spaces of curves *M_g*. In this talk, I will survey our knowledge on the birational geometry of Hurwitz spaces and I will present some results in this direction.

**22.01.2021 Speaker:** Oskar Stachowiak (University of Warsaw, Faculty of Physics)

**Title:** Counting paths in directed graphs

**Abstract:** Graph theory is considered one of the oldest and most accessible branches of combinatorics, and has numerous natural connections to other areas of mathematics. In particular, directed graphs, or quivers, are fundamental tools in representation theory and in noncommutative topology. In my talk, I will focus on the specific problem in the combinatorics of finite directed graphs: how to maximize the number of all paths of a fixed length in certain classes of graphs. Based on joint work with Piotr M. Hajac.

**15.01.2021 Speaker:** Alessandra De Luca (University of Milano-Bicocca, Department of Mathematics and Applications)

**Title:** From the Monotonicity Formula to the Unique Continuation Property for elliptic problems

**Abstract:** In my talk, I will focus on the study of the unique continuation property for second-order elliptic equations. To this aim, an important tool is the monotonicity formula for the so-called *Almgren's frequency function* associated with the solution of the problem, which can be derived using a suitable Pohozaev-type identity. I will give you an idea at first dealing with the cases of harmonic functions and also of very general perturbed elliptic problems.

After that, I will present an approximation argument which I developed in order to prove the unique continuation property when the domain is highly non-smooth due to the presence of a crack and, as a consequence of this construction, I will show some results related to problems where the fractional laplacian is involved.

**08.01.2021 Speaker:** Marzena Śniegowska (Nicolaus Copernicus Astronomical Center & Center for Theoretical Physics, Polish Academy of Sciences)

**Title**: Black holes in a nutshell

**Abstract**: I will briefly mention about this year's Nobel prize in Physics and about amazing result of EHT collaboration which is the first direct visual evidence of a supermassive black hole's silhouette. However, most of my talk will be focused on black holes in a more general and rather simplified way. I will explain, from an observational astronomer's point of view, how we can explore this part of astrophysics.

**18.12.2020** Speaker: Marco Gallo (University of Bari, Department of Mathematics)

**Title**: Climbing a mountain can really make my day? Some insights on variational methods and the search for normalized solutions

**Abstract**: The goal of the talk is to give some ideas of the variational methods used to solve partial differential equations, focusing on the tool of the Mountain Pass theorem. We will then move to the case of normalized solutions for nonlinear Schrödinger equations, giving a picture of some recent techniques involving both the geometry and the compactness for the Lagrangian formulation of the problem. Finally, we will highlight how these tools well suit the case of fractional nonlocal operators.

**11.12.2020 Speaker:** Mateusz Wasilewski (KU Leuven)

**Title:** Random quantum graphs

**Abstract:** The study of quantum graphs emerged from quantum information theory. One way to define them is to replace the space of functions on a vertex set of a classical graph with a noncommutative algebra and find a satisfactory counterpart of an adjacency matrix in this context. Another approach is to view undirected graphs as symmetric, reflexive relations and "quantize" the notion of a relation on a set. In this case, quantum graphs are operator systems and the definitions are equivalent. Doing this has some consequences already for classical graphs; viewing them as operator systems of a special type has already led to introducing a few new "quantum" invariants.

Motivated by developing the general theory of quantum graphs, I will take a look at random quantum graphs, having in mind that the study of random classical graphs is very fruitful. I will show how having multiple perspectives on the notion of a quantum graph is useful in determining the symmetries of these objects.

Joint work with Alexandru Chirvasitu.

**4.12.2020 Speaker:** Michele Zaccaron (University of Padova, Department of Mathematics)

**Title:** Domain perturbation theory: results in spectral shape optimization via a functional analytic approach

**Abstract:** In this talk, we consider eigenvalue problems for two second-order differential operators: the former is important in linear elasticity and it involves the Laplacian, one of the most known and studied operators. The latter arises in the theory of electromagnetism and it is deeply related to Maxwell's equations, involving the so-called "curl curl" operator.

Addressing the classical question "Can one hear the shape of a drum?", we will focus on the study of the dependence of the eigenvalues upon shape perturbation, i.e. the perturbation of the domain in which the PDE is set.

The talk will contain a first introductory part in which I will try to explain the possible motivations and issues that arise when dealing with such eigenvalue problems, presenting some known results in spectral shape optimization for the Dirichlet/Neumann/Steklov Laplacian.

Then a more technical part will follow showing some of the tools and techniques used in the functional analytic study of these differential problems in order to give the audience a flavour of the ideas and instruments in my area of research. Here we will make use of the curl curl operator as an example, showing also recent results regarding optimization of the eigenvalues under suitable constraints (fixed volume or perimeter).

**27.11.2020 Speaker:** Javier de Lucas Araujo (KMMF UW)

**Title:** Classification of finite-dimensional Lie algebras of Hamiltonian vector fields on the plane and applications

**Abstract:** In this talk, I will start by describing the fundamental properties of Lie systems. A Lie system is a system of non-autonomous differential equations whose general solution can be written as a function of a finite generic family of particular solutions and some constants. Sophus Lie laid down the fundamental results on the theory of Lie systems. In particular, I will explain the celebrated Lie--Scheffers theorem, which states that every Lie system amounts to a curve in a finite-dimensional Lie algebra of vector fields. Lie classified all finite-dimensional Lie algebras of vector fields on the real line and on the plane satisfying that their vector fields span a regular distribution. His results were not fully explained, which led to the posterior appearance of several false claims in the literature. This problem was finally fixed by Artemio Gonzalez, Niki Kamran, and Peter J. Olver towards the end of the XX century.

Based on the Gonzalez, Kamran, and Olver classification of finite-dimensional Lie algebras of vector fields on the plane, which I will present and analyse, I will describe which of such Lie algebras can be considered as Lie algebras of Hamiltonian vector fields relative to a symplectic structure. We shall also show some of their relevant physical applications and their use in the theory of Lie systems. Finally, we will provide easily implementable algebraic and geometric methods to check whether a finite-dimensional Lie algebra of vector fields can be considered as a Lie algebra of Hamiltonian vector fields relative to a symplectic structure.

**20.11.2020 Speaker: **Maciej Gałązka (MIM UW)

**Title: **Examples of ranks of polynomials on toric varieties.

**Abstract: **We introduce the notion of rank of a polynomial for the Veronese embedding. We generalize it to some toric varieties. We investigate some lower bounds for rank and check when they can determine it. We examine some examples.

**13.11.2020 Speaker:** Caterina Sportelli (Università degli Studi di Bari Aldo Moro)

**Title: **Gradient-type quasilinear elliptic systems: existence and multiplicity results via a variational approach

**Abstract: **The first traces of variational methods date back to the 17th century but one hundred more years were necessary for the formal statement of the new theory Calculus of Variations, namely the pioneer works of Euler and Lagrange, which studies the existence of minimum points.

Nowadays, a variational approach is indispensable for dealing with many nonlinear phenomena and suitable advanced techniques allow one to look also for critical levels of minimax type.

In this talk, I will discuss the existence of solutions for a class of coupled quasilinear elliptic systems of gradient type with coefficients which depend also on the solution itself.

Although classical variational approaches fail, I will discuss how to overcome the difficulties that arise by introducing a suitable Banach space and a generalized version of the classical Ambrosetti-

Rabinowitz Mountain Pass Theorem.

Finally, I will show that, under assumptions of symmetry, a multiplicity result can be stated, too.

**06.11.2020 Speaker:** Zofia Grochulska (MIM UW)

**Title: **Finding a replacement for diffeomorphisms.

**Abstract: **I will talk about a part of analysis which has a nonempty intersection with topology as it deals with homeomorphisms equipped with some notion of differentiability. We will try to learn if they behave like diffeomorphisms or not. In particular, I will talk about what is known about the so-called Ball-Evans question regarding the approximation of homeomorphisms by diffeomorphisms. The answer turns out to be non-trivial and important from the point of view of applications.

**23.10.2020 Speaker:** Francesco Esposito (University of Calabria, Department of Mathematics and Computer Science)

**Title: **The moving planes method of Aleksandrov-Serrin for some semilinear elliptic problems.

**Abstract: **The moving planes method is one of the most important techniques that have been used in recent years to establish some qualitative properties of positive solutions of nonlinear elliptic equations, like symmetry and monotonicity. In particular, the aim of this talk is to discuss the application of this technique to positive solutions of some semilinear elliptic problems under zero Dirichlet boundary conditions.

The first part of the talk is focused on the study of symmetry and monotonicity properties of classical solutions via the moving planes method of Aleksandrov and Serrin. The second part is dedicated to the description of a nice variant of this technique in the case of singular solutions.

**16.10.2020 Speaker:** Marcin Napiórkowski (KMMF UW)

**Title: **Hot topics in cold gases

**Abstract:** Since the first experimental realization of Bose-Einstein condensation in cold atomic gases in 1995 there has been a surge of activity in this field. Ingenious experiments have allowed us to probe matter close to zero temperature and reveal some of the fascinating effects quantum mechanics has bestowed on nature. It is a challenge for mathematical physicists to understand these various phenomena from first principles, that is, starting from the underlying many-body Schrödinger equation. In my talk, I shall explain some of the problems and results mathematicians (including myself) are working on.

**09.10.2020 Speaker:** Haonan Zhang (IST Austria)

**Title: **Convexity and concavity of trace functionals and why they matter

**Abstract:** In this talk I will give a brief introduction to the convexity and concavity of trace functionals involving trace and matrices. They play an important role in quantum information theory. Since Lieb's celebrated work in 1973 resolving the conjecture of Wigner-Yanase-Dyson, this topic has seen great progress. As an example, I will introduce a conjecture in quantum information theory of Audenaert-Datta (and a stronger one of Carlen-Frank-Lieb) in recent years, explain the connection with convexity/concavity of trace functionals, and show how to solve them in a simple way.

**02.10.2020 Speaker:** Eleonora Romano (University of Trento)

**Title: **An introduction to the Minimal Model Program and the case of surfaces

**Abstract:** The aim of this seminar is to give an introduction to the main problem in Birational Geometry, which is the birational classification of complex smooth projective varieties. To this end, we first introduce essential objects as divisors, cone of curves, and extremal contractions. Then, we will focus on the Minimal Model Program (MMP), by discussing the case of surfaces.

**19.06.2020 Speaker:** Kang Li (IMPAN)

**Title:** Kirillov's orbit method for the Baum-Connes conjecture for algebraic groups

**Abstract:** The orbit method for the Baum-Connes conjecture was first developed by Chabert and Echterhoff in the study of permanence properties for the Baum-Connes conjecture. Together with Nest they were able to apply the orbit method to verify the conjecture for almost connected groups and p-adic groups. In this talk, we will discuss how to prove the Baum-Connes conjecture for linear algebraic groups over local fields of positive characteristic along the same idea. It turns out that the unitary representation theory of unipotent groups plays an essential role in the proof. As an example, we will concentrate on the Jacobi group, which is the semi-direct product of the symplectic group with the Heisenberg group. It is well-known that the Jacobi group has Kazhdan's property (T), which is an obstacle to prove the Baum-Connes conjecture.

**12.06.2020 Speaker:** Arturo Martínez-Celis (IMPAN)

**Title:** The Lindelöf property in products

**Abstract:** A topological space is Lindelöf if every open cover has a countable subcover. Unlike the property of being compact, the Lindelöf property is not preserved under products; one can easily show that the Sorgenfrey Line (the real numbers with the topology generated by the half-open intervals [a,b)) is Lindelöf but the product with itself is not. On the other hand, compact spaces always have a Lindelöf product with any Lindelöf space. Spaces such that their product with every Lindelöf space is Lindelöf are called productively Lindelöf. In this talk we will discuss these properties, mainly in the metrizable setting, see some examples and counterexamples, and their relation with some old questions in Topology.

**05.06.2020 Speaker:** Jacek Krajczok (IMPAN)

**Title:** Tomita-Takesaki theory and locally compact quantum groups

**Abstract:** During the talk I will recall the notions of a von Neumann algebra and a weight. Later on, I will state the main results of the Tomita-Takesaki theory and present it in a couple of examples. In the second part of my talk I will show how this theory is used in the theory of locally compact quantum groups. In particular, I will discuss a relation between traciality of the Haar integrals and unimodularity of the dual quantum group.

**29.05.2020 Speaker:** Michał Miśkiewicz (MiM UW)

**Title:** Geometric PDEs – how standard things get hard

**Abstract:** Starting with examples of geometrically motivated partial differential equations, I will illustrate Hadamard's notion of a well-posed problem. Then I will discuss the harmonic map flow, which is a simple generalization of the classical heat equation to the setting of maps taking values in a given manifold. We will see how this geometric restriction causes fundamental problems in the analysis of solutions.

**22.05.2020 Speaker:** Jakub Siemianowski (IM PAN)

**Title:** Topological approach to elliptic PDEs

**Abstract:** I begin with recalling some topological tools and giving us an intuition to them. In particular, I present one way of generalizing Bolzano's intermediate value theorem in higher dimensions and its connections with other existential theorems like Brouwer's fixed point theorem. Finally, I show how to use this tools to solve systems of elliptic partial differential equations.

**13.03.2020 Speaker:** Jacopo Schino (IMPAN)

**Title:** Distributions or: how I learned to stop worrying and pretend they're functions

**Abstract:** Distributions are often known as "a generalization of functions", but what does it mean exactly? In this talk, I will define formally what a distribution is, with a particular emphasis on how some operations with distributions (e.g. differentiation or convolution) are somehow induced by, and therefore generalize, the same operations with functions (whence the concept of distributions as generalized functions). I will also show possible applications (mostly to PDE's) of the theory of distributions.

**06.03.2020 Speaker:** Klaudiusz Czudek (IM PAN)

**Title:** Random walks on the interval

**Abstract:** Fix two increasing homeomorphisms of the interval into itself and assign to them some probabilities. Pick a starting point from the interior of the interval and consider a random walk in which we decide where to go in the next step by selecting randomly a homeomorphism according to the assigned distribution. I will describe what is the statistical behavior of this random walk.

**28.02.2020 Speaker:** Boulos El Hilany (IM PAN)

**Title:** On polynomial maps having maximally-dimensional preimages

**Abstract:** The coordinates of points in the codomain of a complex polynomial map are polynomials in the coordinates of points in its domain. Given such a map (under some mild assumptions), the complex dimension of a generic point's preimage depends solely on the dimensions of the source and target spaces. This, however, does not prevent some polynomial maps bringing about a preimage whose dimension is higher than expected. Points in the target space producing such preimages that have the maximal dimension form the maximality set of a map. I aim to give a glimpse at the possible topological properties that a complex polynomial map can have and their relation to its discrete invariants, such as the supports, that is, the sets of exponent vectors of the monomials in the respective underlying polynomials appearing with non-zero coefficients. Given a complex polynomial map, I will present a description for the (non-)emptiness of its maximality set using the corresponding supports' configuration.

**24.01.2020 Speaker:** Asahi Tsuchida (IM PAN)

**Title:** Introduction to catastrophe theory

**Abstract:** The goal of this talk is to prove Thom’s elementary catastrophe theorem.

Starting from physical phenomenon, we look over singularity theory of smooth maps and their stability.

Several comments on related topics are given if it is possible.

**17.01.2020 Speaker:** Zofia Michalik (MIM UW)

**Title:** The issues with volatility in asset price models

**Abstract:** The famous Black-Scholes formula for option pricing, published in 1973, was a breakthrough in financial mathematics. Since then it has been widely used by practitioners, mainly due to its simpleness. One of the main drawbacks of the model is that it assumes that the volatility of the asset price process is deterministic, which does not reflect the reality. This motivates the need for a better model, in which the volatility of the asset price is itself stochastic. In my talk I will take a Black-Scholes model as a starting point and then present some of the stochastic volatility models and the mathematics behind them.

**10.01.2020 Speaker:** Fulgencio Lopez Serrano (IM PAN)

**Title: **Maximal equilateral sets

**Abstract:** A set is equilateral if every two points are the same distance apart. In the plane the vertices of an equilateral triangle are an equilateral set. We will present results about maximal equialteral sets in the euclidean space. If possible we present results for general Banach spaces.

**13.12.2019 Speaker:** Reza Mohadmmadpour (IM PAN)

**Title:** Approximation of the maximal Lyapunov exponent

**Abstract:** Let T : X → X be a discrete dynamical system. A skew product over T is a dynamical system F acting on the product space X × Y such that π ⚬ F = T, where π: X × Y → X is the projection on the first coordinate. We are interested in skew products that act linearly on the second coordinate, so Y needs to be a vector space. Such skew products are called linear cocycles.

In this talk, we will focus on the Lyapunov exponents of linear cocycles. In particular, we will show that the maximal Lyapunov exponent can be approximated by periodic points.

**6.12.2019 Speaker:** Maksymilian Grab (MIM UW)

**Title: **Invariant Theory for pedestrians

**Abstract:** As already pointed out by Felix Klein in the nineteenth century, group actions are ubiquitous in geometry. Modern complex algebraic geometry is no exception to this slogan: homogeneous spaces, toric geometry, constructions of moduli spaces -- to name just a few topics closely related to algebraic group actions and their quotients. On the other hand, the construction of a quotient of an affine group action in algebraic category is more subtle than just taking the evident space parametrizing orbits of the action.

My goal in this talk will be to motivate through examples and present the definition of a quotient in sense of Geometric Invariant Theory (GIT) for a reductive algebraic group action on an affine variety over the field of complex numbers. Time permits, the audience will be exposed to a simple example unveiling a bit of non-obvious relation between GIT and the birational classification of algebraic varieties. Along the way we may meet a Hilbert problem.

**29.11.2019 Speaker:** Michał Godziszewski (MIM UW)

**Title:** Forcing, large cardinals, and the multiverse of models of set theory

**Abstract:** Gödel's discovery of the incompleteness phenomenon and Cohen's proof that the the failure of the Continuum Hypothesis is consistent with ZFC showed that there exist many incompatible structures satisfying the basic axioms of set theory. Sophisticated techniques developed by set theorists over the course of the 20th century, notably Cohen's method of forcing, showed that many interesting properties of sets are not decided by ZFC, so that we can equally well consider both ZFC augmented by such a property or by its negation. In parallel various statements unprovable in ZFC have been considered as candidates for new axioms for mathematics.

An important class of the above are Large Cardinal Axioms (LC) that (usually) posit existence of infinite cardinal numbers that: are uncountable, share certain combinatorial properties with ω, reflect (to a certain degree) the structure of the entire universe of sets and can be regarded as strengthenings of ZFC-principles for generating new sets: usually interpreted in allowing for maximizing the universe's height. However, as it was first shown by A. Levy and R. Solovay, certain important and natural combinatorial problems, such as CH are independent of ZFC + LC. The method of forcing that was invented by P. Cohen to demonstrate the independence of CH from ZFC led eventually to another class of axiom-candidates (but it was by no means the purpose of the method).

The so-called Forcing Axioms (FA) are certain generalizations of the Baire Category Theorem. The first example of FA was isolated by D. Martin (hence its name: Martin's Axiom) from the study of the use of iterated forcing in R. Solovay's and S. Tennebaum's proof of consistency of Suslin's Hypothesis (SH)—arguably second most important (after CH) problem of set theory in the first half of the 20th century. Related ideas led to isolation of the notion of properness of forcing by S. Shelah in his study of Jensen's Forcing, leading to proving consistency of SH with GCH. What was the role of these axioms? FAs allowed for transforming forcing from a method of showing independence into a way of proving conditional theorems. Noteworthily, most of them settle the value of the continuum to be the second uncountable cardinal number, and some of FAs have been successfully used since then also outside set theory (e.g., in combinatorics, topology, measure theory, real analysis, and functional analysis). However, the main intrinsic reason for why they can be considered as candidates for axioms is that they can be interpreted in allowing for maximizing the width of the set-theoretic universe (in a certain technical sense), and hence are presumably closely related to the iterative conception of set, and they express certain absoluteness principles (called generic absoluteness).

If we believe that to each of these various axiomatic systems corresponds a mathematical universe satisfying them, then there is a multiverse of mathematical universes, each with its own version of mathematics. These different mathematical universes agree to a significant degree, especially with respect to the mathematics of the finite, but give different answers to questions of the higher infinite. The purpose of the talk is to introduce the basic tools of contemporary axiomatic set theory and illustrate the multiverse perspective on foundations of mathematics with some recent results of my joint work with V. Gitman, T. Meadows, and K. Williams, concerning the collection of countable, recursively saturated models of set theory.

**22.11.2019 Speaker:** Masha Vlasenko (IM PAN)

**Title:** What is a period?

**Abstract:** Periods are numbers arising as integrals of algebraic functions over domains described by polynomial equations or inequalities with

rational coefficients. They include all algebraic numbers but also transcendental numbers, such as π and many other important constants

of mathematics and physics. Representations of periods as integrals help to notice relations and prove identities among them. That is why

one can think of periods as a class of "handy" numbers.

An algebraic number naturally comes with the set of its Galois conjugates. These are the other solutions of its minimal polynomial

equation with rational coefficients. Can one generalize the notion of conjugate numbers to periods? Although the definition of periods looks

so simple, it will allow us to have a glance at some central ideas of modern arithmetic geometry.

**15.11.2019 Speaker:** Artem Dudko (IM PAN)

**Title:** How to sum up divergent series and why this is useful

**Abstract****:** By definition, a divergent series is... divergent, so does not sum up in a usual sense. In the first part of the talk starting with simple examples I will show a few approaches to summing up divergent series. In the second part of the talk I will explain the Borel-Laplace summation procedure for divergent power series and will show some applications for solving difference and differential equations and for studying dynamical systems.

**8.11.2019 Speaker:** Adam Abrams (IM PAN)

**Title:** Mathematics in digital communication

**Abstract:** How can two people have a completely public conversation and both learn information that an eavesdropper does not? How can Gmail deliver messages securely, Netflix stream video efficiently, and Bitcoin transfer money reliably?

I will present several applications of mathematics to modern communication, including (depending on time) encryption, error correction, and compression. I will also discuss the main ideas behind distributed blockchain currencies.

**25.10.2019 Speaker:** Jacopo Schino (IM PAN)

**Title:** Partial differential equations: an overview of the classical theory

**Abstract:** Do you know why the surface of water keeps making circles even after the stone has sunk, but when you're talking to somebody you don't just hear his/her voice again and again? Ever wondered why a drop of milk spreads so fast in a glass of water, or why they bothered giving harmonic functions one specific name?

This Friday I'll try to answer these questions looking closely at some important PDE's: I'll explain their physical derivation, highlight analogies and differences among them, emphasizing their physical meanings, and show the diverse methods used to build solutions.

**18.10.2019 Speaker:** David Martí-Pete (IM PAN)

**Title:** Wandering domains in transcendental dynamics

**Abstract:** We will start by introducing the main concepts in the iteration of holomorphic functions in the complex plane, such as the Fatou and Julia sets. We will focus on the iteration of transcendental entire functions (you can think of the exponential function). Wandering domains are components of the Fatou set that are not eventually periodic. Such components are specific to the iteration of transcendental functions in one complex variable: polynomials and rational maps do not have wandering domains. We will discuss how to classify wandering domains, their relationship with the singular values of the function, and the different ways of constructing them that exist up to now.

**11.10.2019 Speaker:** Mariusz Tobolski (IM PAN)

**Title:** A (locally) trivial talk

**Abstract:** I will discuss the main ideas of noncommutative topology and use the definition of the local-triviality dimension to illustrate how do they work in practice. The local-triviality dimension generalizes the concept of a locally trivial principal bundle, which is pivotal in algebraic topology and fundamental in gauge field theories in physics.

**4.10.2019 Speaker:** Rami Ayoush (IM PAN)

**Title:** Fourier-analytic approach to the 2-wave cone condition

**Abstract:** During the talk I will discuss recent developments in the problem of estimating the Hausdorff dimension of measures satisfying PDEs as well as its connections to the theory of Fourier multipliers.

I will show how to prove a dimension estimate corresponding to a certain more restrictive — in the structural sense — version of the 2-wave cone condition (of Arroyo-Rabasa, De Philippis, Hirsch, and Rindler) in more general Fourier-analytic setting. The sketch of the proof will be based on the example of gradients from BV(ℝ³).

This is joint work with M. Wojciechowski.

**14.06 Speaker:** Alessandro Sisto (ETH)

**Title:** Studying groups using geometry

**Abstract:** Given a finitely generated group G, and a choice of finite generating set S, one can associate to (G, S) a metric space, called Cayley graph. The choice of S does not matter up to an equivalence relation called "quasi-isometry". A large part of geometric group theory is dedicated to studying the (large-scale) geometric properties of Cayley graphs and how they relate to the algebraic properties of groups. I will introduce all the relevant notions, and then focus on one of the main large-scale properties in geometric group theory, namely Gromov-hyperbolicity.

**7.06.19 Speaker:** Marcin Małogrosz (MiNI PW)

**Title:** Can one hear the shape of a quadrilateral?

**Abstract:** Although it is possible to hear the volume (1911 Weyl) and the measure of the boundary (1980 Ivrii) of a given domain, in general one can’t hear its shape (1964 Milnor for manifolds; 1992 Gordon, Webb and Wolpert for polygons). However if it is known in advance that the domain is a triangle (1990 Durso) or that is has certain symmetries and an analytic boundary (2010 Zelditch) then the sound allows to identify the domain up to isometries! During the talk I will shed some light on these issues (or make things much more obscure depending on your mathematical taste) and (hopefully) answer the title question.

**31.05.19 Speaker: **Adam Abrams (IM PAN)

**Title:** Coding sequences and the security problem for square billiards

**Abstract:** This talk will explore two problems related to billiards on a square table: (1) what sequences can arise from recording which side is involved in each successive bounce of a billiard ball, and (2) how easily can one prevent an “assassin” at one location from firing a billiard to hit a target elsewhere on the table? These problems display some surprising connections to computer science, to other dynamical systems, and to topology.

**24.05.19 Speaker:** Arkadiusz Mecel (MIM UW)

**Title: **Gröbner bases and the automaton property of Hecke-Kiselman algebras.

**Abstract: **Many algebraic structures are defined by generators and relations, starting from the symmetric group whose presentation is based on elementary transpositions and braid-type relations. Such constructions often involve graphs, as in the case of Coxeter groups. One benefit of this approach is that in the algebraic work we intuitively tend to work in the language of words, i.e. the elements of a free algebras, be it monoids or groups, or rings. In 1965 Buchberger considered the Gröbner basis as a set of multivariate polynomials that have desirable algorithmic properties. In the commutative setting, every set of polynomials can be transformed into a finite Gröbner basis. This process generalizes three familiar techniques: Gaussian elimination for solving linear systems of equations, the Euclidean algorithm for computing the greatest common divisor of two univariate polynomials, and the Simplex Algorithm for linear programming. In the world of non-commutative polynomials, however, things get complicated, as the process of obtaining the Grobner basis requires the choice of generators and an ordering on monomials, and may not terminate. In order to overcome this difficulty we often turn to structures based on automatons.

In the work with J. Okninski and M. Wiertel (both from University of Warsaw) we consider Hecke-Kiselman monoids introduced as a generalization of 0-Hecke algebras of Coxeter groups. These are semigroup algebras constructed in a very natural way from graphs. The combinatorics of these algebras is fairly simple to work with, yet quite a few elementary formulated problems appear as supringly difficult.

**17.05.19 Speaker:** Maria Donten-Bury (MIM UW)

**Title:** Resolution of singularities

**Abstract:** The aim of the talk is to show how one can deal with singularities of algebraic varieties. I will focus on the class of quotient singularities, i.e. singularities of quotients of smooth algebraic varieties by finite group actions, which provide a lot of interesting examples.

**10.05.2019 Speaker: **Carlos Perez-Sanchez (FUW)

**Title: **An introduction to tensor models

**Abstract: **The interest in generalizing random matrix models --- seen as 2D-random

geometry framework --- to higher dimensions led Ambjørn, Durhuus, and

Jonsson to introduce tensor models by the end of last century. With this

decade's findings on the missing "1/N-expansion", Gurau further propelled

tensor models. This talk is an introduction to their combinatorial,

topological, and, if time allows, physical aspects.

**26.04.2019 Speaker: **Jacopo Schino (IM PAN)

**Title: **A further step into variational methods

**Abstract: **Last year I showed how to use (the infinite-dimension version of) the Weierstrass Theorem to find a weak solution of a certain linear elliptic (i.e., time-independent) problem. For some nonlinearities, though, this method is ineffective. In this talk I will show how to use other tools to solve an elliptic problem whose nonlinearity is of power type, with the exponent being in a certain range.

**12.04.2019 Speaker: **Mitsuru Wilson (IM PAN)

**Title: **Quantum Groups in Action

**Abstract: **Quantum groups are a generalization of groups whose algebras of (continuous) functions are generalized to Hopf algebras. Important group theoretic notions such as group action extend to the Hopf algebra setting. The goal of my talk is to serve as an introduction to quantum groups, actions by quantum groups, and some of my recent results.

**05.04.2019 Speaker: **Welington Cordeiro (IM PAN)

**Title: **Chaos theory: expansivity and sensitivity to initial conditions

**Abstract: **The sensitivity of chaotic systems to initial conditions is sometimes called the Butterfly Effect. The idea is that a butterfly flapping its wings in a South American rainforest could, in principle, affect the weather in Texas. This idea was first publicized by meteorologist Edward Lorenz, who constructed a very crude model of the convection of the atmosphere when it is heated from below. A property stronger than sensitivity is expansivity. The dynamics of expansive systems may be very complicated, but it is quite well understood. In this talk, we will explore the dynamics of systems with intermediate

behavior between sensitivity to initial conditions and expansivity.

**29.03.2019 Speaker: **Adam Śpiewak (MiM UW)

**Title: **Fractal measures in dynamical systems

**Abstract: **Fractal measures are probability distributions having properties analogous to fractal sets (self-similarity/complicated structure at arbitrary small scales/non-integer dimension). Even very simple models are not yet fully understood, with major open problems being an area of active research. During the talk I will present several examples of such measures and discuss their properties. Emphasis will be placed on examples originating from dynamical systems, where they arise naturally as invariant or stationary measures.

**22.03.2019 Speaker: **Tomasz Dębiec

**Title: **Conserved quantities and regularity in fluid dynamics.

**Abstract: **Conserved or dissipated quantities, like energy or entropy, are at the heart of the study of many classes of time-dependent PDEs in connection with fluid mechanics. This is the case, for instance, for the Euler and Navier-Stokes equations, for systems of conservation laws, and for transport equations. In all these cases, a formally conserved quantity may no longer be constant in time for a weak solution at low regularity. In this talk we discuss the interplay between regularity and conservation of energy in the realm of ideal incompressible fluids.

**15.03.2019 Speaker: **Zuzanna Szymańska.

**Title: **Mathematical Modelling in Biology and Medicine. Can we cure cancer with calculus?

**Abstract: **Over the last 30 years there has been an intensive development of mathematical modelling in the biomedical sciences. Models developed in collaboration with biologists and physicians have repeatedly enabled the verification of existing and emerging new research hypotheses as well as facilitated the design of new experiments. There are many differences between both normal and cancer cells and between healthy and cancerous tissue. Some of these key differences concern properties of individual cells and how quickly they divide, migrate or even evade the normal process of cell death. Other properties are concerned with how a solid tumour spreads to secondary parts of the body through the processes called invasion and metastasis. At some point in the development of a solid tumour, cancer cells from the primary cancerous mass of cells migrate and invade the local tissue surrounding the tumour. This initial invasion of the local tissue is the first stage in the complex process of secondary spread where the cancer cells travel to other locations in the individual and set up new tumours called metastases. These secondary tumours are responsible for around 90% of all (human) deaths from cancer. Knowing precisely how cancer cells invade the local tissue would enable better treatment protocols to be developed and consequently better individualised patient care. Cancer invasion and spread is, by its nature, a complicated phenomenon involving many inter-related processes across a wide range of spatial and temporal scales. Therefore, the theoretical support from mathematical modelling, analysis, computational simulation and systems biology in understanding these processes is extremely necessary. The last decades have witnessed enormous advances in our understanding of the molecular basis of cell structure and function. Scientists have made impressive advances in elucidating the mechanisms mediating cell-signalling and its consequences for the control of gene expression, cell proliferation and cell motility. With the rapid development of experimental methods, huge amounts of genetic, proteomic, biochemical and visual data become available. At the same time, the development of mathematical and computational models of various aspects of cancer growth has significantly contributed to the "theoretical side" - mathematical and computational models constructed in collaboration with biologists and clinicians have repeatedly enabled the verification of existing and formulation of new research hypotheses, and also facilitated the design of new experiments. In my talk I will give a short overview of current state-of-the-art in mathematical modelling of cancer disease.

**8.03.2019 Speaker: **Janusz Czyż.

**Title: **What is an aftermath of the Polish Mathematical School?

**Abstract: **The Polish Mathematical School flourished between the world wars, with

Tarski and Sierpinski works on Cantor-Godel arithmetic, Banach's functional

analysis, Marian Smoluchowski's works in statistical physics, among others.

While the Second World War struck with destructive pover at the Polish

Mathematical School both figuratively and literally (Józef Marcinkiewicz was

the youngest victim among this group), some of the scientists and much of

their ideas survived and are still either classical (topology, Banach spaces

and algebras) or inspiring new developments (Łukasiewicz's 3-value logic,

Mycielski-Steihaus axiom of determination). This was commemorated by

conducting the International Congress of Mathematicians in Warsaw in 1983,

which was preceded by a problem session for Warsaw mathematicians, led by

Michael F. Atiyah.

**1.03.2019 Speaker: **Artem Dudko

**Title:** On the spectrum of the Grigorchuk group.

**Abstract:** The Grigorchuk group *G* is a group which was constructed in 1980 and solved several open problems in group theory. The spectrum of a group is a certain set containing important information about the group. In my talk, I will explain the notion of the spectrum of a group, introduce the Grigorchuk group *G*, and present a joint result with R. Grigorchuk, namely, the calculation of the spectrum of *G.* The talk is intended for non-specialists. I will give all necessary definitions and provide some examples.

**25.01.2019 Speaker: **Ignacio Vergara

**Title:** Unitarizable groups and the Dixmier problem.

**Abstract: **This will be an introductory talk on group representations on Hilbert spaces and the notion of unitarizability. A group is said to be unitarisable if every uniformly bounded representation is "similar" to a unitary representation. In 1950, Day and Dixmier proved (independently) that every amenable group is unitarizable. The converse remains open and it is known nowadays as the Dixmier problem.

My goal in this talk is to explain the previous paragraph in detail, without assuming any knowledge of group representations or amenability. If time allows, I will give a complete proof of the Day–Dixmier theorem.

**11.01.2019 Speaker: **Adam Abrams

**Title:** Introduction to Dynamical Systems

**Abstract: **Dynamical systems studies changes to systems over time. With such a broad description, almost anything can be presented as a dynamical system, and indeed mathematical dynamics relates a wide array of topics. Fortunately, there are some important definitions and results that can be made accessible to a wide range of mathematicians. This talk will cover several important notions in dynamics, such as minimality, mixing, and ergodicity. We will address these concepts mainly through instructive examples rather than through detailed proofs.

**14.12.2018 Speaker:** Janusz Czyż

**Title:** From Copernicus to the Polish School of Mathematics and Logic

**Abstract:** The Copernicean Revolution was an event both in astrophysics or physics and in

mathematics. Namely, Copernicus' pupil Rheticus in Cracov did initiate

counting multidigital trigonometric tables for which a logarithmic calculus

was needed. Also Copernicean Mercuroid defined in the ``de Revolutionibus...''

was a masterpiece in the art of approximation and the most advanced geometric

curve.

**7.12.2018 Speaker:** Tomasz Pełka

**Title:** Classifying planar rational cuspidal curves

**Abstract: **Let E be a closed algebraic curve on a complex projective plane; and assume that E is homeomorphic to a projective line. The classification of such curves, up to a projective equivalence, is a classical open problem with interesting counterparts in topology and symplectic geometry. The Coolidge-Nagata conjecture ('59), proved recently by Koras and Palka, asserts that every such curve is obtained from a line by some Cremona transformation.

The known curves can in fact be constructed inductively. I will sketch this nice picture during my talk, including, as an example, two newly discovered families of curves. But to prove this conjecture, one needs to study the complement of such curve. The simple topology of this affine surface makes the algebraic subtlety clearly visible. In the most difficult case when it is of log general type, the classical tools, such as the logarithmic Minimal Model Program applied to the minimal smooth completion (X,D), are not sufficient. The idea of Palka is to study the pair (X,(1/2)D) instead. This leads to the Negativity Conjecture, which asserts that the log Kodaira dimension of K+(1/2)D is negative. I will explain why this is the natural extension both of Coolidge-Nagata and some other, open problems concerning rigidity of (X,D). I will also report on our recent classification, up to a projective equivalence, of rational cuspidal curves satisfying this conjecture. They turn out to share certain unexpected properties. The aim of this talk is, on one hand, to advertise the log MMP modifications as a modern tool applicable in a much broader context, and on the other hand, to give new, elementary, but interesting examples of the rich geometry of planar curves and Cremona maps.

**23.11.2018 Speaker: **Piotr Hajac

**Title: **Operator algebras that one can see

**Abstract:** Operator algebras are the language of quantum mechanics just as much as differential geometry is the language of general relativity. Reconciling these two fundamental theories of physics is one of the biggest scientific dreams. It is a driving force behind efforts to geometrize operator algebras and to quantize differential geometry. One of these endeavours is noncommutatvive geometry, whose starting point is natural equivalence between commutative operator algebras (C*-algebras) and locally compact Hausdorff spaces. Thus noncommutative C*-algebras are thought of as quantum topological spaces, and are researched from this perspective. However, such C*-algebras can enjoy features impossible for commutative C*-algebras, forcing one to abandon the algebraic-topology based intuition. Nevertheless, there is a class of operator algebras for which one can develop new ("quantum") intuition. These are graph algebras, C*-algebras determined by oriented graphs (quivers). Due to their tangible hands-on nature, graphs are extremely efficient in unraveling the structure and K-theory of graph algebras. We will exemplify this phenomenon by showing a CW-complex structure of the Vaksman-Soibelman quantum complex projective spaces, and how it explains its K-theory.

(in the academic year 2018-19 the organizers were dr Tristan Bice,dr Michał Gaczkowski, dr Tatiana Shulman)

**15.06.2018 Speaker: **Jacopo Schino

**Title:** An introduction to variational methods.

**Abstract: **In this talk I'm going to present an introduction to variational methods, by which we mean looking for (weak) solutions to a certain PDE problem as critical points of a proper functional.

We'll go through a simple linear problem: I'll recall the new notion of solution and show how we get there, then I'll prove the existence of a solution by the infinite-dimensional version of Weierstrass Theorem and, if I have time, the uniqueness of such solution.

**8.06.2018 Speaker: **Reza Mohammadpour

**Title:** Lyapunov exponents Of cocycles.

**Abstract: **Lyapunov exponents tell us the rate of divergence of nearby trajectories – a key component of chaotic dynamics. For one

dimensional maps Lyapunov exponent at a point is a Birkhoff average of $log|f'|$ along the trajectory of this point. For a

typical point for an ergodic invariant measure it is equal to the average of $log|f'|$ with respect to this measure.

In this talk,we will give an introduction about Lyapunov exponents of cocycles,Moreover we will give a few examples which are related

to PDE,Smooth Dynamics and Probability.We deal with Lyapunov exponents of products of random i.i.d. 2x2 matrices of determinant

$\pm1$.We will see how lyapunov exponent gives information about the norm of growth of the matrices $A^{n}(x)$,Finally we will

discuss conditions in which the Lyapunov exponents are always positive.

**25.05.2018 Speaker: **Monika Szczepanowska

**Title:** KNOT ENERGIES, SURFACE ENERGIES AND GENUS BOUNDS.

**Abstract: **Energies of submanifolds are an analytic tool for studying topological and geometric properties of embedded submanifolds. Interest in energies was rekindled in the nineties by pioneering works of Freedman, of He and Wang and of O'Hara, who used energies of curves to study knottedness of links in R^3. One of the most useful energies is the Menger curvature defined originally for curves in R^3, and which was studied in detail by many authors. The Menger curvature can be generalized to surfaces and to higher-dimensional submanifolds. The intuition behind the energy is that, if a submanifold has complicated topological behavior (for example the curve is knotted), then the energy should be large. In other words, small energy

should imply simple topology. We want to look at some basic definitions and intuitions in this field. For instance,

we shall review some properties of energies, and discuss difficulties and unexpected traps one encounters while trying to extend the definition of energy to higher dimensions. At the end, we will sketch a proof that there exists a positive constant C such that, for every smooth, closed and connected surface S in R^3 of total area 1, the genus g(S) and the energy E(S) of the surface S satisfy the inequality: g(S) less or equal to C times E(S).

**11.05.2018 Speaker: **Samuel Evington

**Title:** The C*-Algebra Petting Zoo.

**Abstract: **On a warm sunny Friday afternoon in Warsaw, we shall take a close look at some of the key examples of C*-algebras. In the process, we will learn a bit about the general theory of C*-algebras and why they are worth studying. I will then speak briefly about the dramatic recent progress in the classification of C*-algebras and my research in this area.

**13.04.2018 Speaker: **Marcin Małogrosz

**Title:** Stability analysis of stationary solutions of reaction-diffusion systems via the linearisation principle and spectral analysis of Schrodinger operators.

**Abstract: **Roughly speaking a steady state of a dynamical system is called stable if it attracts every trajectory starting from its sufficiently small neighbourhood. In the finite dimensional case (system of ordinary differential equations in the Euclidean space) the linearisation theorem states that stability of an equilibrium is equivalent to the stability of the Jacobian matrix at that equilibrium if the spectrum of the matrix is separated from the imaginary axis. On the other hand the Routh-Hurwitz theorem gives equivalent conditions for the stability of a matrix in terms of the signs of certain polynomial expressions of its entries. These two theorems make the finite dimensional case somewhat understood and allow to perform the stability analysis of steady states of concrete systems with the use of numerical computations. The situation is more complicated in the case of infinitely dimensional dynamical systems, specifically systems of partial-differential equations of reaction diffusion type. Although the linearisation principle extends with the Jacobian matrix being replaced by a Schrodinger operator, there is no counterpart of the Routh-Hurwitz theorem and only partial results concerning stability of such operators are known. We will discuss these matters in detail during the talk.

**23.03.2018 Speaker: **Antoni Kijowski

**Title:** On functions with the mean value property.

**Abstract: **I will discuss functions possessing the mean value property. Such class has been introduced in the metric measure spaces as one of possible definitions of harmonic functions in this setting. During the talk I will focus on the consequences which the property supports and present a large collection of examples to complete the analysis.

**16.03.2018 Speaker: **Damian Orlef

**Title:** How to study generic groups and why?

**Abstract: **I will talk about the Gromov density model, in which a group is obtained by specifying generators and then imposing a random set of relations between them.

The model is simple, yet produces groups with interesting geometric properties and provides testing grounds for various methods and conjectures. I will describe

some of the landscape, focusing later on left-orderability and property (T) (which leads to a connection with random graphs). No previous knowledge of the topic(s) is assumed.

**26.01.2018 Speaker**: Vasiliki Evdoridou (IM PAN)

**Title**: The escaping set of transcendental entire functions

**Abstract:** In this talk we will give an introduction to the iteration of transcendental entire functions focusing on the escaping set. The escaping set consists of points that go to infinity under iteration and it plays an important role in the area. After giving the definition and discussing some of its properties, we will look at the structure of the escaping set. In particular, we will present two specific examples of transcendental entire functions for which the structure of the escaping set differs significantly.

**19.01.2018** Speaker: Konrad Aguilar (University of Denver).

**Title:** Topologies for the ideal space of C*-algebraic inductive limits.

**Abstract:** C*-algebras can be viewed as operator norm-closed

self-adjoint subalgebras of bounded operators on Hilbert spaces.

Because of this, the theory of C*-algebras benefits from a rich

representation theory. As the kernel of a representation is a norm

closed two-sided ideal of a C*-algebra, a major tool in this theory

comes in the form of topologies on ideals, where ideals are seen as

points. This begins with the Jacobson topology on primitive ideals,

which are ideals formed by the kernels of a non-zero irreducible

representations. From this, J.M.G. Fell developed a topology on all

norm closed two-sided ideals of a C*-algebra. Motivated in part by

this, we developed a new topology on the ideal spaces of C*-algebras

formed by inductive limits (C*-inductive limits). We then compare this

topology to the other topologies discussed. We also present that our

topology does agree with Fell's topology for the particular

C*-inductive limits called approximately finite-dimensional

C*-algebras (AF-algebras). As an application, we first provide a

metric topology on certain quotients of AF-algebras using the tools of

Noncommutative Metric Geometry, in particular M.A. Rieffel's compact

quantum metric spaces and F. Latremoliere's quantum Gromov-Hausdorff

propinquity. Next, we introduce sufficient conditions for when our

topology on ideals produces a continuous map from certain sets of

ideals to the associated space of quotients. An example of such a

continuous map is given by the Boca-Mundici AF-algebras. This shows

that the act of taking a quotient can be seen to be continuous at the

level of viewing ideals and quotients as points of topological spaces.

**15.12.2017 Speaker: **Tomasz Kochanek

**Title:** The Szlenk index and asymptotic geometry of Banach spaces.

**Abstract: **The notion of Szlenk index was introduced in 1968 by W. Szlenk in order to show that there is no universal Banach space in the class of all separable reflexive Banach spaces. Its origins stem from the Cantor-Bendixson index which is well-known in topology. Since the pioneering paper by Szlenk, his index and several similar ordinal indices have proven to be extremely useful tools in Banach space theory. During the talk, we will discuss some intuitions behind the Szlenk index and the Szlenk power type, and some of the most recent results on this topic. These involve connections with asymptotic geometry of Banach spaces and the theory of asymptotic structures, which is quite fundamental for understanding the current state of knowledge in the structural theory of Banach spaces.

**8.12.2017 Speaker: **Krzysztof Ziemiański

**Title:** Directed Algebraic Topology.

**Abstract: **I will talk about directed spaces; these are topological spaces with some additional structure, which can be used for modelling concurrent programs. I will define directed counterparts of classical topological invariants and present main problems which are investigated in this area.

**1.12.2017 Speaker: **Piotr Achinger

**Title:** Around monodromy.

**Abstract: **Do you feel that you are going around in circles and not getting anywhere? Things may not be as bad as they seem. You might be getting somewhere, but not realizing it because you aren't aware of your personal monodromy*.

In this lecture, I will provide a gentle overview of the concept of monodromy in the context of algebraic geometry, algebraic topology, differential equations, and number theory.

* (c) Nick Katz

**24.11.2017 Speaker: **Masha Vlasenko

**Title:** Formal groups and congruences.

**Abstract:** I will give a friendly introduction to the theory of formal group laws focusing on arithmetic questions such as integrality and local invariants.

**10.11.2017 Speaker:** Tristan Bice (IM PAN).

**Title:** <<-Increasing Approximate Units in C*-Algebras (joint work with Piotr Koszmider).

**Abstract:** It is well known that every C*-algebra has an increasing approximate unit w.r.t. the usual partial order on the positive unit ball. We consider the strict order << instead, where a << b means a = ab. Here again it is well known that every separable or sigma-unital C*-algebra has a <<-increasing approximate unit, but the general case remained unresolved. In this talk we outline our recent work showing that this extends to omega_1-unital C*-algebras but not, in general, to omega_2-unital C*-algebras. In particular, we consider C*-algebras defined from Kurepa/Canadian trees which are scattered and hence LF but not AF in the sense of Farah and Katsura. It follows that whether all separably representable LF-algebras are AF is independent of ZFC.

**3.11.2017 Speaker:** Safoura Zadeh (IM PAN).

**Title:** Isomorphisms between the left uniform compactification of locally compact groups.

**Abstract:** For a locally compact group $G$, let $C_{b}(G)$ be the space of all complex-valued, continuous and bounded functions on $G$ equipped with the sup-norm, and $LUC(G)$ be the subspace of $C_{b}(G)$ consisting of all functions $f$ such that the map $G\to C_b(G);x\mapsto l_xf$ is continuous, where $l_xf$ is the function defined by $l_xf(y)=f(xy)$, for each $y\in G$. The subspace $LUC(G)$ forms a unital commutative C*-algebra. We can induce a multiplication on the Gelfand spectrum of $LUC(G)$, $G^{LUC}$, with which $G^{LUC}$ forms a semigroup. In this talk, I study some properties of $G^{LUC}$, the so called right topological semigroup compactification of $G$. I also discuss the question of when the corona, $G^{LUC}\setminus G$, determines the underlying topological group $G$.

**20.10.2017 Speaker:** Mateusz Wasilewski (IM PAN).

**Title:** Non-commutative techniques in classical probability

**Abstract:** I will discuss two classical probabilistic objects -- random walks and birth-death processes -- using the language of operator theory. We will see how commutation relations between certain operators (related to the generators of the aforementioned processes) allow us to perform explicit computations; the combinatorial tools from free probability, such as non-crossing partitions, will appear naturally. If time permits, I will show how to make a transition from classical probability to quantum probability.

P.S. I may present an example involving Darth Vader and stormtroopers.

**9.06.2017 Speaker:** Olli Toivanen (IM PAN)

**Title:** Regularity in generalized Orlicz spaces

**Abstract: **A Lebesgue space $L^{p}(\Omega), \Omega \subset \Rn$, is the space of those functions f for which $\int_\Omega |f|^p\,dx < \infty$. This sort of an integrability condition can be generalized in various ways, such as saying "instead of integrating a power p, let's consider some other function of f", or "instead of a fixed power p, let's allow p to be p(x), or, a function of the point x \in \Omega". These approaches lead, respectively, to Orlicz spaces and to variable exponent Lebesgue and Sobolev spaces.

It is an interesting question whether these and other generalizations can be brought together and covered by a "super-generalization", and whether (say) the minimizers of that integral would have regularity (say Hölder continuity) with assumptions even remotely as good as those of the individual cases.

Somewhat surprisingly, this seems to be the case. I will speak on these generalized Orlicz or Musielak-Orlicz spaces, and on the various recent results in building a regularity theory on them. I'll start from "Hölder regularity of quasiminimizers under generalized growth conditions", by Harjulehto, Hästö, and myself; Calc. Var. PDEs, 56 (2), 2017.

**2.06.2017 Speaker:** Andrey Krutov (IMPAN)

**Title:** Introduction to the geometry of partial differential equations

**Abstract: **We will consider the basic material on the geometric approach to partial differential equations and symmetries, including an introductory part on the geometry of jet spaces.

**26.05.2017 Speaker:** Saeed Ghasemi (IMPAN)

**Title:** SAW*-algebras and sub-Stonean spaces

**Abstract: ** SAW*-algebras are C*-algebras which are noncommutative analogous of sub-Stonean spaces (F-spaces) in topology i.e., spaces for which two disjoint open, σ-compact sets have disjoint closures. Many properties of sub-Stonean spaces were generalized to general SAW*-algebras. For example Pedersen showed that the corona algebras of sigma-unital C*-algebras are SAW*, which generalizes the fact that Cech-Stone remainders of locally compact σ-compact Hausdorff spaces are sub-Stonean spaces. I will talk about the continuous maps from products of compact spaces into sub-Stonean spaces. In particular, it is well-known that the are no injective continuous maps from the product of infinite compact spaces into a sub-Stonean space. I will present a generalization of this result to SAW*-algebras i.e., there are no surjective maps from SAW*-algebras onto C*-tensor products of two infinite-dimensional C*-algebras. This in particular answers a question of Simon Wassermann who conjectured that the Calkin algebra is essentially non-factorizable.

**19.05.2017 Speaker:** Karen Strung (IMPAN)

**Title:** Smale spaces and C*-algebras

**Abstract:** In this talk, I will describe the hyperbolic dynamical systems known as Smale spaces. These includes such well-known examples as the shifts of finite type, hyperbolic toral automorphisms and Anosov diffeomorphisms. Each Smale space gives rise to topological equivalence relations coming from the stable, unstable, and homoclinic relations. I will show how these can be used to construct C*-algebras and describe some of their structural properties.

**12.05.2017 Speaker:** Jarosław Mederski (IMPAN)

**Title: **Nonlinear Maxwell equations

**Abstract:** Our aim is to solve the system of Maxwell equations in the presence of nonlinear polarization in a bounded domain. A solution of the system describes propagation of electromagnetic fields in a nonlinear medium. We show that the problem leads to a semilinear equation involving the curl-curl operator. In the talk we present the functional setting and variational methods which allow to deal with the curl-curl operator and find ground state solutions. We recall the classical mountain pass theorem as well as recent generalized Nehari manifold approaches. At the end of my talk we discuss new directions of research and some open problems which seem to be important from the physical point of view and challenging from the mathematical side.

**5.05.2017 Speaker:** Iwona Skrzypczak (IMPAN)

**Title:** Approximation in anisotropic and non-reflexive Musielak-Orlicz spaces

**Abstract:** The talk will concern the generalization of the Sobolev spaces, namely the general Musielak-Orlicz spaces, where the norm is

governed by an integral of a general convex function, depending not only on the function, but also on the spacial variable.

The highly challenging part of analysis in the general Musielak-Orlicz spaces is giving a relevant structural condition implying approximation properties of the space. However, we are equipped not only with the weak-* and strong topology of the gradients, but also with the intermediate one, namely - the modular topology.

The brief presentation of the setting and ideas are planned to be clear even for those who are not used to the nonlinear analysis.

**21.04.2017 Speaker:** Lashi Bandara (University of Gothenburg)

**Title: **Functional calculus for bisectorial operators and applications to geometry

**Abstract:** The bounded holomorphic functional calculus for bisectorial operators can be thought of as an implicit Fourier theory in settings where the transform cannot be defined. It has been particularly useful in low-regularity situations such as Euclidean domains and in obtaining non-smooth perturbation estimates. The power of the tool lies in the fact that its boundedness can often be obtained via real-variable harmonic analysis methods. In this talk, I'll give an introduction to these operators, the functional calculus, its connection to harmonic analysis and more recent applications to geometry.

**7.04.2017 Speaker:** Paweł Józiak (IMPAN)

**Title:** Algebra meets probability.

**Abstract:** Many mathematical disciplines can have direct connections to other, but we are sometimes convinced that some of them are separated and the connection is indirect and distant. One such example could be the theory of probability and pure algebra -- the aim of the talk would be to convince that in fact algebra and probability are conversant to each other. During the talk, I'd like to present a purely algebraic/combinatorial proof of the central limit theorem (CLT), one of the building blocks of the modern probability theory. If time permits, I will also smuggle a bit of a modern branch of operator algebras, called the free probability theory (with its own CLT). No algebraic nor probabilistic prerequisites will be necessary, the talk is aimed to be understandable by general audience (e.g. undergraduate mathematics or physics students).

**24.03.2017 Speaker:** Marithania Silvero (IMPAN)

**Title: **A combinatorial approach to Khovanov homology

**Abstract:** Khovanov homology is a link invariant introduced in 2000 by Mikhail

Khovanov. This bigraded homology categorifies Jones polynomial and it has been

proved to detect the unknot. In this talk we present a new approach to extreme

Khovanov homology in terms of a specific graph constructed from the link

diagram. With this point of view, we pose a conjecture related to the

existence of torsion in extreme Khovanov homology and show some examples where

the conjecture holds.

21.05.2021 Speaker: Michał Godziszewski (University of Warsaw)

Title: (Un)decidability of addition and multiplication in models of number theory

Abstract: One of the features of the arithmetical operations in natural numbers - addition and multiplication - is that they are almost trivially decidable: there is an easy algorithm for computing a sum or a product of two given natural numbers. From mathematical logic we know there exist mathematical structures that are non-isomorphic with the natural numbers, yet satisfy the first-order number theory. What is more, there are even uncountable models of number theory. All models of PA that are not isomorphic to the well-known standard model are called, surprise surprise, nonstandard. A somewhat natural question then is to ask if there are algorithms to compute the addition and multiplication of some nonstandard models? As Stanley Tennenbaum demonstrated in 1959, the answer is no: there is no nonstandard model of arithmetic, where addition or multiplication could be computable. The reason for that is that any nonstandard model of arithmetic codes information that is too complex to be accessed algorithmically.

During the talk, I will introduce the basic facts from mathematical logic, show how we know there are nonstandard models, sketch the argument for the Tennenbaum Theorem, and provide some rudiments on the computability-theoretic properties of the quotient structures of number theory and set theory.