Topološki seminar LJ-MB-ZG

Vodje seminarja: Petar Pavešić (Ljubljana), Iztok Banič (Maribor), Sonja Štimac (Zagreb)

Bodoča predavanja

Minula predavanja

Simetrična monoidalna kategorija indprofinVect

Predavateljica: Martina Stojic (Sveučilište u Zagrebu)

Povzetek: Govorit ću o kategoriji indprofinVect koja je ekvivalentna kategoriji striktnih ind-pro-objekata u kategoriji konačno-dimenzionalnih vektorskih prostora. Ta kategorija je konkretna kategorija vektorskih prostora s dodatnom strukturom i linearnih preslikavanja koja poštuju tu strukturu. Ona sadrži vektorske prostore i njihove duale, linearno kompaktne vektorske prostore, te ima prirodno definiran tenzorski produkt koji objedinjuje obični tenzorski produkt među vektorskim prostorima i upotpunjeni tenzorski produkt među linearno kompaktnim vektorskim prostorima. Kategorija je simetrična monoidalna i obuhvaća kategoriju lokalno linearno kompaktnih vektorskih prostora. U njoj se mogu opisati svi relevantni morfizmi koji kombiniraju ta dva svijeta (vektorske prostore, koji su prirodno objekti kategorije indfinVect, i njihove duale, objekte kategorije profinVect), kao što je naprimjer sparivanje beskonačno-dimenzionalnog vektorskog prostora i njegovog duala, te razna djelovanja duala na prostore i obratno. U svojoj disertaciji razvila sam teoriju za konkretnu kategoriju indproVect, koja je ekvivalentna kategoriji striktnih ind-pro-objekata prebrojive kofinalnosti u kategoriji vektorskih prostora. U tom slučaju mogla sam definirati i formalne baze i formalne sume u proVect te raditi i računati s njima kao s običnim bazama i konačnim sumama. Kategorija ima lijepa svojstva i moguće je u njoj interno definirati dosta kompleksne algebarske strukture. Pitanje je što se od toga sve može definirati u slučaju kad beskonačnost kontroliramo, umjesto prebrojivom kofinalnošću ind-objekata i pro-objekata kao u indproVect, konačnom dimenzionalnošću komponenti pro-objekata, kako je u indprofinVect.

Predavanje bo na FNM, Koroška cesta 160, v predavalnici 0/103.

Inhomogeneities in unimodal inverse limit spaces

Predavateljica: Ana Anušić (Sveučilište u Zagrebu)

Povzetek: Unimodal inverse limit space is a continuum obtained as an inverse limit on the unit interval with a single bonding map coming from the logistic family. Such continua are known to fail the homogeneity property, primarily because of the existence of points which are locally not homeomorphic to the Cantor set of arcs. We will investigate the structure of such points with respect to the dynamics of the bonding map.

Predavanje bo na FNM, Koroška cesta 160, v predavalnici 0/1.

Knot polynomials in the projective space

Predavatelj: Boštjan Gabrovšek (Univerza v Ljubljani)

Povzetek: Knot polynomials are invariants used to distinguish knots in the 3-dimensional Euclidean space.
We will generalize this concept for knots in the 3-dimensional projective space as well as other 3-manifolds.

Predavanje bo na FNM, Koroška cesta 160, v predavalnici 0/1.

Lozi-like maps

Predavateljica: Sonja Štimac (Sveučilište u Zagrebu)

Povzetek: : I will talk about Lozi-like family of maps, which is a generalization of the Lozi family of maps, and contains the Lozi family as a subfamily. The kneading theory developed for the Lozi maps holds in the same way for the Lozi-like maps: an itinerary of every point of the Lozi attractor is completely characterized by the set of kneading sequences.

While the situation may appear similar to what we see in one dimension for unimodal maps, there is a big difference. The analogue for the Lozi family is the family of the tent maps. There we have one parameter and one kneading sequence. For the Lozi family we have two parameters, but infinitely many kneading sequences. Thus, by using concrete formulas, we immensely restrict the possible sets of kneading sequences. It makes sense to conjecture that in a generic n-parameter subfamily of Lozilike maps, n kneading sequences determine all other kneading sequences. In fact, in an example at the end of my talk, we will see that two kneading sequences of the Lozi map determine the parameter values, and thus all kneading sequences.

This is joint work with Michal Misiurewicz.

Unimodal category

Predavatelj: Dejan Govc (Univerza v Ljubljani)

Povzetek: The concept of unimodal category [1] was first introduced by Baryshnikov and Ghrist in 2007. It is an invariant of functions $f:Xto[0,infty)$, similar in spirit to the Lyusternik-Schnirelmann category. I will give an overview of the basic definitions, known results and a monotonicity conjecture from [1]. Then I will discuss this conjecture in more detail. As is turns out, the conjecture holds for functions on $X=mathbb R$ as well as for certain well-behaved functions on $X=mathbb R^n$, but fails for more general functions.

[1] Y. Baryshnikov and R. Ghrist, Unimodal category and topological statistics. Proceedings of NOLTA, 2011.

An Anderson-Choquet-type theorem and a characterization of weakly chainable continua

Predavatelj: Iztok Banič

We introduce the concept of proper convergence of a sequence of subspaces of a metric space and then prove that a continuum X is weakly chainable if there is a sequence of arcs converging properly to it. Also, we prove that a continuum X is weakly chainable if and only if there is a sequence of arcs in the Hilbert cube converging properly to an embedded copy of X. The proof is based on an Anderson-Choquet-type theorem (valid also for set-valued functions).

On a problem concerning products in the category of shape

Predavatelj: S. Mardešić

In 1977 Y. Kodama proved that the Cartesian product of an FANR and a paracompact space is a (direct) product in the shape category of topological spaces Sh(Top). Since metrizable movable continua generalize FANRs, it was natural to ask for products of such continua with other spaces. In the present paper we show that the Cartesian product of a metrizable movable continuum with a polyhedron need not be a product in Sh(Top). A counterexample is the Cartesian product of the Hawaiian earring and the polyhedron that is the pointed sum of a sequence of copies of circles.

Pro-Covering fibrations

Predavatelj: Nick Callor

We will discuss a certain class of inverse limits of covering spaces related to the Hawaiian Earring.

Predavanje bo na FNM, Koroška cesta 160, v predavalnici 0/103.

Substitution subshifts and renormalization for potential functions.

Predavatelj: Henk Bruin (University of Vienna)

Substitution subshift are a wide-spread way of creating aperiodic self-similar patterns. In this talk I want to present how to combine this with a type of renormalization in the space of potential functions. For standard examples, such as the Thue-Morse and the Fibonacci substitution subshift, this been worked out in great detail. The existence of fixed points of the renormalization operator is guaranteed for quite general substitutions, but the precise nature of these fixed points leaves open many questions, also related to thermodynamic questions in one-dimensional quasi-crystals and interval dynamics.
Predavanje bo na PMF Univerze v Zagrebu med 10. in 14. uro v predavalnici 104.

On the core version of the Ingram conjecture

Predavatelj: Jernej Činč (University of Vienna)

Ingram conjecture, stating that inverse limit spaces of two tent maps with different slopes in [1,2] are not homeomorphic, was proved recently by Barge, Bruin and Štimac. However, the proof that solves the Ingram conjecture, is based on observations on rays compactifying on the cores of the inverse limit spaces. The core version of the Ingram conjecture is thus still not proven in general. In this talk I will present a partial solution (concerning tent maps with non-recurrent critical points) to the core Ingram conjecture and discuss difficulties we encounter trying to prove the conjecture in general with our approach.
Predavanje bo na PMF Univerze v Zagrebu med 10. in 14. uro v predavalnici 104.

Plane embeddings of inverse limit spaces of tent maps

Predavateljica: Ana Anušić (Sveučilište u Zagrebu)

Inverse limits of tent maps are continua contained in the Hilbert cube. They often appear as global attractors of plane homeomorphisms. There is a well-known algorithm of embedding this continua in the plane using symbolic description of their arc-components. We will present the symbolic description and show how the current algorithm can be adapted to produce uncountably many different plane embeddings. For every arc-component there exists an embedding making it accessible from the complement.

Predavanje bo na Fakulteti za matematiko in fiziko Univerze v Ljubljani.

Inverse limits, inverse limit hulls and crossovers

Predavatelj: Iztok Banič (Univerza v Mariboru)

We give several characterizations of subspaces of products of countably many compact metric spaces that are inverse limits of inverse sequences. These characterizations are obtained both for inverse sequences with continuous single-valued bonding functions and for inverse sequences with upper semicontinuous set-valued bonding functions. The proofs are based on use of crossovers and inverse limit hulls.

Predavanje bo na Fakulteti za matematiko in fiziko Univerze v Ljubljani.

Categorification of twisting function

Predavatelj: Zoran Škoda (Sveučilište u Zagrebu)

By a construction of H. Cartan, a simplicial analogue of principal bundles, with simplicial group as the structure group, can be constructed as a twisted analogue of cartesian product
of the simplicial group and base simplicial set. The datum used for the twisting is the Cartan’s twisting function. Principal bundles are classified by the first nonabelian cohomology and can
carry connections in the sense of differential geometry. Higher nonabelian cocycles appear in many modern applications in geometry, topology, operator algebras and mathematical physics.
Using pseudosimplicial categories instead of simplicial sets we can categorify Cartan’s construction. After general introduction to the classical twisting function and modern motivation for higher cocycles, I will present a sketch of the categorification of twisting function in a joint work in progress with B. Jurčo. The coherences are rather elaborate, as we have calculated on the basis of Jardine’s supercoherence, hence they seem rather not practical for calculation. However, we expect to use them to aid a construction and further study of the expected categorification of universal simplicial principal bundle (classically obtained using delooping functor W bar of the simplicial group) in the pseudosimplicial language.

Predavanje bo na FNM, Koroška cesta 160, v predavalnici 0/103.

On Topological Complexity and Robot Motion Planning

Predavateljica: Aleksandra Franc (Univerza v Ljubljani)

In the last 15 years several attempts have been made to use topology to analyse configuration spaces and solve the problem of robot motion planning. We will look at a few of the results and work through some interesting examples.

Predavanje bo na FNM, Koroška cesta 160, v predavalnici 0/103.

Barbara di Fabio (University of Bologna)

Predavatelj: A stable combinatorial distance for reeb graphs of surfaces

The shape similarity problem has been studied since long by the computer vision community for dealing with shape classi?cation and retrieval tasks. Shape properties, such as curvature, are encoded in compact representations of shapes, namely, shape descriptors. In this framework, shape similarity can be measured by de?ning an appropriate distance on the set of the chosen shape descriptors. A question that deserves attention is the choice of the distance used to compare shape descriptors. Indeed, it is clear that any data acquisition is subject to perturbations, noise and approximation errors and, without stability, distinct computational investigations of the same object could produce completely different results. So a major problem in shape comparison concerns the stability against data perturbations.
In this seminar we focus on the Reeb graph shape descriptor. Reeb graphs provide a method to combinatorially describe the shape of a manifold endowed with a Morse function.
We define a combinatorial metric for Reeb graphs of orientable surfaces in terms of the cost necessary to transform one graph into another by edit operations. The main contributions we present are the stability property and the optimality of this edit distance. More precisely, the stability result states that changes in the functions, measured by the maximum norm, imply not greater changes in the corresponding Reeb graphs, measured by the edit distance. The optimality result states that our edit distance discriminates Reeb graphs better than any other metric for Reeb graphs of surfaces satisfying the stability property.

Predavanje bo potekalo v predavalnici 104 na PMF Univerze v Zagrebu.

Cardinality of inverse limits

Predavatelj: Niko Tratnik (Univerza v Mariboru)

We explore the cardinality of generalised inverse limits. Among other things, we show that, for any n in {aleph_0, c, 1,2,3,4,…}, there is an upper semicontinuous function with the inverse limit having exactly n points. We also prove that if f is an upper semicontinuous function whose graph is a continuum, then the cardinality of the corresponding inverse limit is either 1, aleph_0 or c.

Predavanje bo potekalo v predavalnici 104 na PMF Univerze v Zagrebu.

Computability of compact manifolds

Predavatelj: Zvonko Iljazović (Zagreb)

If S is a semi-computable subset of Euclidean space, then S
need not be computable. However, under certain topological
assumptions on S, we can conclude that S is computable. In
this talk the following result will be presented: if S is a
compact manifold with boundary and if the boundary of S is
computable, then S is computable.

Predavanje bo potekalo v Ljubljani, soba 2.02, Jadranska 21, ob 10. uri

Generalizations of Markov interval functions and inverse limits

Predavateljica: Tjaša Lunder (Maribor)

We generalize Markov interval functions in two different ways. First, we generalize them to set-valued functions, defined with respect to a finite set of points on the interval; we call these functions generalized Markov interval functions. Second, we generalize them to set valued functions, defined with respect to a countable set of points on the interval; we call them countably Markov interval functions. We show that any two inverse limits with generalized Markov interval bonding functions with the same pattern are homeomorphic. We also show that any two inverse limits with countably Markov interval bonding functions with the same pattern are homeomorphic. Both cases generalize the result from S. Holte, Inverse limits of Markov interval maps, Topology Appl. 123, 2002, 421–427.

Predavanje bo potekalo v Ljubljani, soba 2.02, Jadranska 21, ob 10. uri

Ważewski’s universal dendrite as an inverse limit with one set-valued bonding function

Predavateljica: Tina Sovič

Abstract: We construct a family of upper semi-continuous set-valued functions f (belonging to the class of so-called comb functions), such that for each of them the inverse limit of the inverse sequence of intervals [0, 1] and f as the only bonding function is homeomorphic to Ważewski’s universal dendrite. Among other results we also present a complete characterization of comb functions for which the inverse limits of the above type are dendrites.

Predavanje bo na PMF Univerze v Zagrebu med 10. in 14. uro v predavalnici 104.