Martin Frankland: Realization problems in algebraic topology

Given a space X, its homotopy groups π_*X form a graded group equipped with the action of primary homotopy operations. Given a graded group with such operations, can it be realized as π_*X for some space X, and if so, can we classify all realizations? We will discuss some classification results and some non-realizability results, as well as obstruction-theoretic methods to tackle these problems. We will also survey realization problems for algebraic structures other than homotopy operations.

Matthias Kreck: An overlooked problem: Betti numbers of closed manifolds

Until rather recently nobody seems to have thought about the following very simple problem: Given a sequence of natural numbers b₀, ..., b_n, is there a closed n-dimensional manifold M with these Betti numbers? Work by Su, Su-Fowler, Su-Kennard and Kreck-Zagier shows that this is a very delicate problem which relates topology to non-trivial number theory. The role of bordism in attempts to solve this simple problem will be explained.

Marithania Silvero: Studying torsion in Khovanov homology

Khovanov homology of knots and links was introduced by Mikhail Khovanov at the end of last century. This link invariant, which categorifies Jones polynomial, was nicely reinterpreted by Viro in a purely combinatorial way in terms of Kauffman states. While conceptually simple, this definition becomes impractical when increasing the numbers of crossings of a link diagram. In this talk we present a new approach to extreme Khovanov homology introduced in [GMS].

With this point of view, we conjecture in [PS] that extreme Khovanov homology is torsion free and prove it for some particular families of links. We also present some advances on the study of the presence of torsion in Khovanov homology.

References:

[GMS] J. González-Meneses, P. M. G. Manchón and M. Silvero, A geometric description of the extreme Khovanov homology, Proceedings of the Royal Society of Edinburgh: Section A.

[PS] J. H. Przytycki and M. Silvero, Homotopy type of circle graphs complexes motivated by extreme Khovanov homology, arXiv:1608.03002.

Arturo Espinosa Baro: Topological complexity of subgroups of Artin's braid groups

Topological complexity, introduced by Farber during his study of the robot motion planning problem, is a numerical homotopy invariant, quite similar in spirit to the classic Lusternik-Schnirelmann category, with TC being significantly more difficult to compute. It is often possible to achieve such computation combining upper and lower cohomological bounds, but a class of spaces for which the computation of TC presents a particular challenge are the Eilenberg-MacLane spaces of torsion-free discrete groups. Farber requested a description of the TC of such spaces in terms of algebraic properties of the base group, and that objective seems to be out of reach at the present. In this context, M. Grant and D. Recio-Mitter, using a generalization of the notion of Poincaré duality, the Fadell-Neuwirth fibrations, and previous bounds of TC due to Grant and Grant-Lupton-Oprea, obtained bounds for certain subgroups of the braid groups, and, in particular, computed the TC of subgroups of the n strand braid groups which fixes any two strands. We will briefly introduce the notion of TC, and will proceed to describe the process followed by Grant and Recio-Mitter.

David Mendez: Homotopically rigid Sullivan algebras and their applications

The group of homotopy self-equivalences of a space is rarely trivial. Kahn was the first to obtain an example of one such space with non-trivial rational homology in the seventies. Later, Arkowitz and Lupton came across an example of a Sullivan algebra (equivalently, a rational homotopy type) with trivial homotopy self-equivalences. This algebra was used by Costoya and Viruel to solve Kahn's group realisability problem for finite groups, thus obtaining for any finite group G a rational space X whose group of homotopy self-equivalences is isomorphic to G. This construction also provide a way to obtain an infinite amount of homotopically rigid spaces. However, they all share their level of connectivity with the example of Arkowitz and Lupton.

The objective of this work is to illustrate that:

1. Homotopically rigid spaces are not as rare as they were though to be. We are able to obtain an infinite family of homotopically rigid spaces, showing a level of connectivity as high as desired.

2. Building blocks other than the example of Arkowitz and Lupton can be used to solve Kahn's realisability problem.

We can also apply the obtained results to differential geometry by enlarging the class of inflexible manifolds existing in literature and building new examples of strongly chiral manifolds.

References: C. Costoya, D. Méndez, A. Viruel, Homotopically rigid Sullivan algebras and their applications, arXiv:1701.03705 [math.AT].

Bogdan Batko: Weak index pairs and the Conley index for discrete multivalued dynamical systems

Motivation to revisit the Conley index theory for discrete multivalued dynamical systems [T. Kaczynski and M. Mrozek, Topology Appl. 65 (1995), pp. 83–96] stems from the needs of broader real applications, in particular in the problem of reconstructing dynamics from samples or in combinatorial dynamics. We introduce a new, less restrictive definition of the isolating neighborhood. It turns out that then the main tool for the construction of the index, i.e., the index pair, is no longer useful. In order to overcome this obstacle we use the concept of weak index pairs. We prove that each isolating neighbourhood admits weak index pairs. Moreover, it is sufficient to well-pose the definition of the index.

Wojciech Politarczyk: Twisted Alexander polynomials

I will start with a short review of homology with twisted coefficients. Then I will show how can we extract classical Alexander polynomial of a knot from twisted homology of its complement. Next I will describe construction of twisted Alexander polynomials and sketch some of its applications.

Marek Kaluba: The computational aspects of property (T)

Kazhdan's Property (T) is a well known concept in the theory of group actions. Its numerous applications include finite generation of lattices, fixed-point properties of isometric actions, constructions of expanding graphs and product replacement algorithm. However a complicated notion requires a serious fire-power to be established. Indeed to prove that a group has property (T) requires a non-trivial effort even in the case of most classical examples, such as SL(3,Z).

We hope to ease the effort by drawing from the field of semi-definite programming and cone-optimisation. Using the Positivestellensatz and following the work of Ozawa and Netzer & Thom we will show how to translate property (T) into a semi-definite optimisation problem. Given an explicit generating set S of a finitely presented group G, this will (possibly) allow us to produce a “witness” for the property (T) and simultaneously estimate the Kazhdan's constant for (G,S).

Petar Pavesic: A generalization of covering spaces

In Spanier's classical textbook on algebraic topology much of the exposition on covering spaces is done within a more general setting of Hurewicz fibration with unique path-lifting property. Since then the concept was not very widely used, but it has reappeared recently in relation to some problems in “wild topology”.

Petar Pavesic: Topological complexity of a fibration

Topological complexity of a map was devised in order to study the manipulation complexity of a mechanical device with a given forward kinematic map. Since kinematic maps that appear in robotics normally have singularities, the resulting concept is not homotopy invariant. However, if the map under examination is indeed a fibration, then the theory assumes a more homotopy-theoretic flavor.

Jose Carrasquel: Rational homotopy theory for non-simply connected spaces

We will discuss several new and not-so-new approaches to rational homotopy theory for non-simply connected spaces.

Masafumi Sugimura: Inverse limits of Burnside rings for p-groups

Let G be a finite group and H a subgroup of G. Also let V be an R[G]-module. Denote by V^• the one point compactification of V. Moreover, let F be a set of subgroups of G and {f_H}_H∈F a family of H-maps f_H: V^• -> V^•. Then we have the following problem: does there exist a G-map f_G: V^• -> V^• such that f_G is H-homotopy equivalent to f_H for any F∈G?

We denote by A(G) the Burnside ring of G. By evaluating coker[res^G: A(G) -> ∏_H∈F A(H)], we can understand the difficulty of the above problem. Denote by L(G,F) the inverse limit of A(G) and by B(G,F) the image of res^G. Then it is known that coker[res^G] can be decomposed into the direct sum of L(G,F)/B(G,F) and P(G,F)/B(G,F). Define Q(G,F) = L(G,F)/B(G,F). Morimoto showed that Q(G,F) is isomorphic to the product of Q(G/G^p, F_G/G^p) for primes p which divide k_G, where k_G is Oliver's number and G^p is the smallest normal subgroup of G with |G/G^p| a p-power. Because G/G^p is a p-group, it is important to calculate Q(G,F) when G is p-group. Hara and Morimoto calculated Q(G,F) in the case G = A₄, the alternating group of degree 4. They also calculated Q(G,F) when G = C_p, C_p × C_p, C_pⁿ and C_p × C_q as p-group, where p and q are distinct primes and n is a positive integer. However, little is known about Q(G,F) in the case of more complicated groups G. In this talk, we will calculate Q(G,F) for G = C_p^m × C_pⁿ for n=1, 2.

Janusz Przewocki: Z-struktury dla rozszerzeń grup

Temat tego referatu dotyczy problemów, nad którymi obecnie pracuję z D. Osajdą. Zostaną wprowadzone podstawowe pojęcia związane z brzegami grup oraz ich istotne zastosowania. Następnie omówimy uogólnienie pojęcia brzegu za pomocą Z-struktur wprowadzonych przez M. Bestvinę, a także ich podstawowe własności. Ostatnią część referatu będą stanowić otwarte problemy nad którymi pracujemy i szkice dowodów.

Aleksandra Franc: Algorithmic approach to topological complexity

Our goal is to produce an algorithm that for a given simplicial complex X constructs an optimal motion planner on X × X. We will show a few examples of the results the algorithm produces on spheres, torus, Klein bottle, projective plane and other 2-dimensional surfaces.

Andrzej Weber: Bazy Auerbacha i topologia przestrzeni flag

Niech V będzie przestrzenią Banacha skończonego wymiaru. Mówimy, że a₁, a₂, ..., a_n ∈ V jest bazą Auerbacha, gdy |a_i|=|a_i^*|= 1 dla każdego i = 1, 2, ..., n. Istnienie takiej bazy dla dowolnej przestrzeni Banacha skończonego wymiaru zostało wykazane przez Auerbacha w 1930 roku. Własności baz Auerbacha były badane przez wielu autorów. W roku 2005 Plichko wykazał, że muszą istnieć co najmniej dwie istotnie różne bazy Auerbacha. Następnie Pełczyński postawił hipotezę, że w przestrzeni wymiaru n istnieje co najmniej n baz Auerbacha. Wykażemy, że w dowolnej przestrzeni Banacha wymiaru n musi być co najmniej n(n-1)/2+1 baz Auerbacha. Dowód jest topologiczny i korzysta z obliczenia kategorii Lusternika-Schnirelmanna dla przestrzeni flag. Dla przestrzeni Banacha ogólnego typu otrzymujemy oszacowanie lepsze stosując teorię Morse'a.

Maria Marchwicka: Wybrane zagadnienia z topologii DNA

Tematem seminarium będą wybrane zagadnienia z zastosowań topologii do analizy struktury oraz reakcji enzymatycznych cząsteczek DNA. Przedstawię podstawowe pojęcia z teorii węzłów znajdujące zastosowanie w genetyce molekularnej. W szczególności skupię się na wielomianie Jonesa i HOMFLYPT. Zaprezentuję również model wstęgi oraz twierdzenie Călugăreanu opisujące rozkład liczby opleceń w cząsteczce na liczbę skrętów i zwojów. Omówię przemiany w węzłach i splotach DNA zachodzące pod wpływem działania topoizomeraz oraz miejscowo specyficznej rekombinazy.

Jose Gabriel Carrasquel-Vera: Rational homotopy theory for Schwarz's sectional category

We will give an introduction to rational homotopy theory and discuss how these techniques can be applied to give algebraic lower bounds for Schwarz's sectional category and, in particular, for Farber's topological complexity. We will also give a generalisation of the Félix-Halperin theorem for rational Lusternik-Schnirelmann category in the more general context of sectional category.

Antoni Pierzchalski: Stare i nowe o gradientach - nieredukowalnych składnikach pochodnej kowariantnej

Opowiem o gradientach Steina Weissa. W szczególności, o kompletnym rozwiązaniu problemu eliptyczności operatorów rzędu drugiego powstałych ze złożeń gradientów. Opowiem także o najnowszych wynikach dotyczących zachowania na brzegu, w szczególności o metodzie konstrukcji naturalnych warunków brzegowych i ich eliptyczności w wiązkach form skośnych lub symetrycznych.

Bogusław Hajduk: Konstrukcje rozmaitości symplektycznych i kontaktowych

Omówię klasyczne i nieklasyczne wykonywalne na rozmaitościach symplektycznych i kontaktowych: chirurgia kontaktowa; symplektyczna suma Gompfa (sklejanie rozmaitości symplektycznych); konstrukcje struktur symplektycznych na wiązkach; chirurgia podrozmaitości w rozmaitościach kontaktowych.