Seminarium: Geometria i topologia

I will talk about Reeb graphs of Morse functions and realization problems related to them. The Reeb graph of a function on a manifold is defined by contracting the connected components of its level sets. I will describe connections between Reeb graphs, epimorphisms onto free groups and systems of nonseparating 2-sided submanifolds. This allows us to study algebraic properties of epimorphisms onto free groups, such as the number of their equivalence classes or ranks of maximal epimorphisms.

24-04-19 | Wojciech Dybalski: Compton scattering in algebraic Quantum Field Theory

We consider Quantum Electrodynamics from the point of view of algebraic QFT. For a class of electrically charged representations of the algebra of observables, defined by Buchholz and Roberts, we construct scattering states describing one electron and photons. The main technical problem consists in showing that the single-electron states are vacua of the asymptotic photon fields. This is achieved by the mean ergodic theorem applied in a certain extended Hilbert space. (Joint work with Sabina Alazzawi).

06-03-19 | Karolina Paradysz: Zastosowanie Topologicznej Analizy Danych w ocenie jakości estymatorów klasy SMO na przykładzie danych z Narodowego Spisu Powszechnego

Narodowy Spis Powszechny w 2011 po raz pierwszy w Polsce przeprowadzony przy pomocy metody reprezentacyjnej co wiąże się z wyborem odpowiednio skonstruowanego estymatora, który na postawie próby najtrafniej będzie reprezentował badaną populację. W prezentacji skupiono się jedynie na jednej spośród wielu kategorii występujących w Spisie. Badaną kategorią jest niepełnosprawność, a odpowiedzi uzyskiwane od respondentów były deklaratywne, w związku z tym powstają zniekształcenia i braki odpowiedzi. Spośród wielu estymatorów pośrednich klasy SMO należy więc wybrać te, które będą spełniały kryteria formalne i merytoryczne (określają one pewien wzorzec, benchmark). W tym celu można wykorzystać także metody klastrowania, które pozwolą stworzyć wzorcowy układ odniesienia. Do stworzenia takiego benchmarku należałoby pójść o krok dalej niż zwykłe klastrowanie.

Rozwiązaniem zaproponowanym w prezentacji jest analiza kształtu chmury punktów danych (Topology Data Analysis) pozwalająca pogrupować je i znaleźć zależności między nimi. Metoda ta nie wymaga wcześniejszych założeń co do szukanych zależności oraz pozwala odkryć relacje pomiędzy punktami danych, które mogłyby przejść niewykryte w ramach standardowych wspólnych metod statystycznych i taksonomicznych.

Zdaniem G. Carlssona et al. (2005) metoda wykorzystuje homologię persystentną, której główną zaletą jest mała wrażliwość na zakłócenia, szum informacyjny, niedokładność danych. Niewielkie przesunięcia punktów danych nie są w stanie znacząco zmienić diagramu persystencji i płynących z niego wniosków.

30-01-19 | Thilo Kuessner: Parameters for module spaces of group representations

I will discuss the use of topology in the study of representations of finitely presented groups. In particular I will discuss Anosov representations of surface groups to PGL(3,C).

23-01-19 | Janusz Przewocki: Z-boundary for the discrete Heisenberg group

Z-structure is a generalization of Gromov boundary of a hyperbolic group or CAT(0) boundary for nonpositively-curved groups. In this talk we show how to construct Z-boundary for the discrete Heisenberg group. It occurs that it is a two-dimensional sphere.

12-12-18 | Bartosz Biadasiewicz: Bitcoin properties – analysis of users graph

I will describe project concerning analysis of a users graph of Bitcoin – a cryptocurrency. The way all the available history of transactions is stored makes it much easier to collect data and extend existing results by the new data in real time.

05-12-18 | Arturo Espinosa Baro: On the sectional category of subgroup inclusions and relative cohomologies

Let H and G be subgroups, with i a subgroup inclusion of H into G. The sectional category of i, secat(i), is the one associated to the fibration induced by i between the classifying spaces. We will extend a characterization of topological complexity due to Farber, Grant, Lupton and Oprea to the context of sectional category of subgroup inclusions. Also, we will employ a relative cohomology theory, the Adamson cohomology of G with respect to H, to study secat(i), describing a notion of canonical class analogous to the one developed by Bernstein, and generalizing a result of Costa and Farber to this sectional category. Finally, we will discuss some interesting conjectures related to the characterization of secat(i) and topological complexity of aspherical spaces.

14-11-18 | Zbigniew Błaszczyk: Topological complexity and efficiency of motion planning algorithms

I will discuss a variant of Farber’s topological complexity, defined for smooth compact Riemannian manifolds, which takes into account only motion planners with the lowest possible „average length” of the output paths. In particular, I will prove that it never differs from topological complexity by more than 1, thus showing that the latter invariant addresses the problem of the existence of motion planners which are „efficient”.

The talk is based on joint work with J. G. Carrasquel Vera.

07-11-18 | Marek Kaluba: Kazhdan Property (T) for SL_N(Z) and related groups

We present a completely new proof of Kazhdan Property (T) for the whole family of special linear groups defined over integers. The proof is strikingly simple and uses

1. the action of A_n on SL_N(Z) as the outer automorphism group preserving the (standard) generating set;
2. the known result that SL₃(Z) has property (T);
3. one computer-assisted computation in RSL₃(Z).

Based on the result we give much better estimates for the so called Kazhdan constants, which are very close to the existing upper bounds.

We will discuss how the proof may extend to the other families of groups with symmetric generating sets.

24-10-18 | Piotr Mizerka: Nonexistence of smooth one fixed point actions of finite groups on low-dimensional spheres

In this talk we focus on smooth one fixed point actions of finite groups on low-dimensional spheres. The algebraic characterization of groups admitting such actions is already known – these are so called Oliver groups. We show nonexistence of one fixed point actions of several such groups for spheres of particular dimensions. For obtaining the results, the GAP software was used. Joint work with Agnieszka Borowiecka.

18-10-18 | Rade T. Živaljević: Topology and combinatorics of unavoidable complexes

The partition number \pi(K) of a simplicial complex K\subseteq 2^{[m]} is the minimum integer \nu such that for each partition A_1\uplus\ldots\uplus A_\nu = [m] of [m] at least one of the sets A_i is in K. A complex K is r-unavoidable if \pi(K)\leq r. We say that a complex K is almost r-non-embeddable in R^d if for each continuous map f: |K| \to R^d there exist r vertex disjoint faces \sigma_1,\ldots, \sigma_r of |K| such that f(\sigma_1)\cap\ldots\cap f(\sigma_r)\neq\emptyset. Motivated by problems of Tverberg-Van Kampen-Flores type we review several results (obtained in collaboration with Duško Jojić, Wacław Marzantowicz and Siniša Vrećica) which link together the combinatorics and topology of these two classes of complexes. One of our central observations (see Theorem 4.6 in https://arxiv.org/abs/1603.08472), summarizing and extending results of G. Schild, B. Grünbaum and many others, is that interesting examples of (almost) r-non-embeddable complexes can be found among the joins K = K_1 * … * K_s of r-unavoidable complexes. From a broader perspective unavoidable complexes are relevant for other mathematical fields as well, which will be illustrated by examples from cooperative game theory and polyhedral combinatorics.

17-10-18 | Siniša Vrećica: Chessboard complexes

A crucial and one of the most significant properties of Algebraic topology is the applicability of its results in different areas of Mathematics. Many important results are established in this way, starting with The fundamental theorem of algebra, Brouwer fixed point theorem, the ham sandwich theorem. In some cases a quite unexpected application of topological result provided a solution to the long-standing conjecture (such as L. Lovász proof of Kneser conjecture). The principal example of such topological result is famous Borsuk-Ulam theorem.

We illustrate the applicability of topological methods and results by presenting an important configuration space – chessboard complex, and by showing how its properties could be used in solving the problems in other areas of Mathematics. Chessboard complex appears in different ways: as a coset complex of the symmetric group by some of its subgroups (stabilizing some elements), as a matching complex of a complete bipartite graph, as a complex of partial injective functions from one finite set to the other, as a deleted join of a finite set (or a repeated deleted join of a point).

Actually we define several versions of this complex and show how each of them is motivated by some mathematical question. For example, we show how a cycle-free chessboard complex appears in establishing the symmetric analogue of the cyclic homology of algebras, and how generalized and symmetrized versions appear in establishing the generalizations of van Kampen-Flores theorem and Tverberg-type theorems.

Our dominant interest is in the connectivity properties of a chessboard complex (which reduces to determining its homology groups), but we consider some other properties as well.

The talk is based on joint papers with R. Živaljević, and some with D. Jojić.

27-09-18 | Gregory Lupton: Homotopy Theory in the Digital Topology Setting

I will present a progress report on some work joint with John Oprea and Nick Scoville. An n-dimensional digital image is a finite subset of the integer lattice in R^n, together with an adjacency relation. For instance, a 2-dimensional digital image is an abstraction of an actual digital image consisting of pixels. Our work consists of developing notions and techniques from homotopy theory in the setting of digital images.

In an extensive literature, a number of authors have introduced concepts from topology into the study of digital images. But some of these notions, as they appear in the literature, do not seem satisfactory from a homotopy point of view. Indeed, some of the constructs most useful in homotopy theory, such as cofibrations and path spaces, are absent from the literature. Working in the digital setting, we develop some basic ideas of homotopy theory, including cofibrations and path fibrations, in a way that seems more suited to homotopy theory. We illustrate how our approach may be used, for example, to study Lusternik-Schnirelmann category and topological complexity in a digital setting. One future goal is to develop a characterization of a „homotopy circle” (in the digital setting) using the notion of topological complexity. This is with a view towards recognizing circles, and perhaps other features, using these ideas. This talk will include a survey of the basics on topological notions in the setting of digital images.

24-09-18 | Ai Guan: Gauge equivalence for complete L-infinity algebras

Maurer-Cartan elements in differential graded Lie algebras, and more generally L-infinity algebras, appear in many areas of mathematics, such as homotopical algebra, differential geometry and deformation theory. In this talk we will show how gauge equivalence of Maurer-Cartan elements can be characterized as a left homotopy in a model category sense. We will then discuss some of the consequences of this characterization: an entirely homotopical proof to Schlessinger-Stasheff’s theorem, a short formula for gauge equivalence, and a strong homotopy generalisation of T. Voronov’s non-abelian Poincare lemma.