## Young Researchers Colloquium

### Adam Abrams, Joachim Jelisiejew, Jacopo Schino

Fridays **14:45 - 15:45**, room 403.

After colloquium we have cheese and wine meeting.

__Next seminar:__

**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.

__Previous seminars:__

**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.