Young Researchers Colloquium

dr Tristan Bice (IMPAN),dr Michał Gaczkowski (IMPAN), dr Tatiana Shulman (IMPAN)

Fridays 15:00 - 16:00, room 403.

After colloquium we have cheese and wine meeting.


Next seminar:

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.


Previous seminars:

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 $Lp(\Omega), \Omega \subset \Rn$, 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.

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