FIRST DAY
Speaker: Mark Sioen (Vrije Universiteit Brussels, Brussels; Belgium)
Title: Zero-dimensionality and the Banaschewski compactification for approach spaces in its different forms
Abstract: The category App of approach spaces and contractions was introduced by R. Lowen in the 1980’s as an answer to the bad behaviour of metrizability of topological spaces w.r.t. taking products. As such, they represent the fragment of the numerical in-formation that can be saved when performing products, while maintaining compatibility with the usual topological product. Also App is covered within the extensive monoidal topology program initiated by M. M. Clementino, D. Hofmann and W. Tholen. In this talk we will focus on the notion of zero-dimensionality in approach theory and show that the full subcategory kZDApp2 of compact Hausdorff zero-dimensional objects is reflective within the category kZDApp2 of zero-dimensional Hausdorff approach spaces and contractions. We present several equivalent formulations for the corresponding reflector and show that it extends the well-known reflector from Hausdorff zero-dimensional topological spaces to compact Hausdorff zero-dimensional topological spaces,which is well-known under the name`Banaschewski-compactification’. This talk reports on joint work with E. Colebunders.
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Speaker: Maria Manuel Clementino (Universidade de Coimbra, Coimbra; Portugal)
Title: Categorical insights into ordered groups
Abstract: The goal of this talk is the study of (pre)ordered groups from a categorical perspective. In the firs part we will show that the category OrdGrp of (pre)ordered groups and monotone homomorphisms has both a topological and an algebraic flavour, providing immediate information on the behaviour of limits, colimits, factorization systems (as detailed in [2]). While in these respects it shares most of the properties with the category TopGrp of topological groups and continuous homomorphisms, split short exact sequences behave very differently in these two categories (as shown in [1,2,4]). This will be illustrated in the second part of the talk. If time permits, we will explain how this study can be extended so that it also includes metric groups, ultrametric groups, and probabilistic metric groups among other structures (as studied in [3]).
References
[1] M. M. Clementino, An invitation to topological semi-abelian algebras. In: New perspectives in Algebra, Topology and Categories, Coimbra Mathematical Texts, Vol.1, Springer Nature and University of Coimbra (2021), pp. 27-66.
[2] M. M. Clementino, N. Martins-Ferreira and A. Montoli, On the categorical behaviour of preordered groups, J. Pure Appl. Algebra 223 (2019), 4226-4245.
[3] M. M. Clementino and A. Montoli, On the categorical behaviour of V-groups, J. Pure Appl. Algebra 225 (2021) 106550.
[4] M. M. Clementino and C. Ruivo, On split extensions of preordered groups, Port. Math. (to appear).
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Speaker: Ando Razafindrakoto (University of the Western Cape, Bellville; South Africa)
Title: Monadic aspects of the ideal functor on the category of distributive lattices
Abstract: It is known that the construction of the frame of ideals from a distributive lattice induces a monad whose algebras are precisely the frames and frame homomorphisms. Using Jacobs’ results (see [1] below) on KZ-monad and Fakir’s construction of idempotent approximation of a monad (see [2] below), we show that successive iterations of the ideal functor on its algebras and coalgebras do not strictly lead to a new category.
References
[1] B. Jacobs, Bases as coalgebras, Logic Meth. Comp. Sci. 9 (2013),1-21.
[2] S. Fakir, Monade idempotente associée à une monade, C. R. Acad. Sci. Paris Sér. A 270 (1970), 99-101.
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Speaker: Sidney Morris (La Trobe University, Melbourne; Australia; and, Federation University Australia, Ballarat; Australia)
Title: Topology meets number theory
Abstract: Liouville proved the existence of a set ℒ of transcendental real numbers now known as Liouville numbers. Erdős proved that while ℒ is a small set in that its Lebesgue measure is zero, and even its s-dimensional Hausdorff measure, for each s > 0, equals zero, it has the Erdős property, that is, every real number is the sum of two numbers in ℒ. He proved ℒ is a dense Gδ - subset of ℝ and every dense Gδ - subset of ℝ has the Erdős property. While being a dense Gδ - subset of ℝ is a purely topological property, all such sets contain 𝔠 Liouville numbers. Each dense Gδ - subset of ℝ is homeomorphic to the product of a countably infinite number of copies of the discrete space ℕ of all natural numbers.
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Speaker: Linus Kramer (Universität Münster, Münster; Germany)
Title: Old and new results about automatic continuity
Abstract: Suppose that G and H are topological groups and that f : G → H is a group homomorphism. Then one can consider conditions on G, H, f that ensure that f is continuous and open. A classical result by E. Cartan says that this is the case if G is a compact semisimple Lie group and if H is a finite-dimensional unitary group. I will survey recent results of this type. This is based on joint work with Braun, Hofmann and Varghese.
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Speaker: Helge Glöckner (Universität Paderborn, Paderborn; Germany)
Title: Totally disconnected contraction groups
Abstract: An automorphism α of a topological group G is called contractive if all α-orbits converge to the neutral element. A pair (G, α) of a locally compact group and a contractive automorphism of G is called a locally compact contraction group. As shown by Siebert, every locally compact contraction group is a direct product of a simply connected nilpotent Lie group and a totally disconnected, locally compact contraction group. In the talk, I shall describe results concerning the latter groups, obtained in joint work with George A. Willis. The most recent results concern the structure of contraction groups which are torsion groups.
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Speaker: Kang Li (FAU Erlangen-Nürnberg, Erlangen; Germany)
Title: Kirillov's orbit method to the Baum-Connes conjecture for algebraic groups
Abstract: The orbit method for the Baum-Connes conjecture was first developed by Chabert and Echterhoff in the study of permanence properties for the Baum-Connes conjecture. Together with Nest they were able to apply the orbit method to verify the conjecture for almost connected groups and p-adic groups. In this talk, we will discuss how to prove the Baum-Connes conjecture for linear algebraic groups over local fields of positive characteristic along the same idea. It turns out that the unitary representation theory of unipotent groups plays an essential role in the proof. As an example, we will concentrate on the Jacobi group, which is the semi-direct product of the symplectic group with the Heisenberg group. It is well-known that the Jacobi group has Kazhdan’s property (T), which is an obstacle to prove the Baum-Connes conjecture. If time permits, we will also discuss my recent joint work with Maarten Solleveld about quasi-reductive groups.
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SECOND DAY
Speaker: Dikran Dikranjan (University of Udine, Udine; Italy)
Title: Characterizing Lie groups by their zero-dimensional subgroups
Abstract: Lie groups are often characterized via restraints imposed on their closed subgroups, as the classical criterion of Glushkov characterizing the Lie groups as the locally compact groups having no small subgroups. In this vein is the characterization [3] of the locally compact groups (the so called ``exotic Lie groups’’) containing no copies of the discrete integers ℤ and the compact groups ℤp of p-adic integers. The compact abelian exotic Lie groups were characterized much earlier in [1] under the name ``exotic tori’’. The exotic Lie groups have many nice properties, yet they need not be Lie groups, i.e., these more relaxed (compared to Glushkov’s Theorem) conditions do not characterize the Lie groups in the class of all locally compact groups. The aim of this talk will be to discuss stronger restrictions on the closed zero-dimensional subgroups of a compact-like group L to ensure that L is a Lie group. The compactness properties we consider include (local) compactness,(local) ω-boundedness, (local) countable compactness, (local) precompactness, (local) minimality and sequential completeness. Here is a sample of our characterizations:
(i). A topological group is a Lie group if and only if it is locally compact and has no infinite compact zero-dimensional subgroups.
(ii). An infinite topological group is a compact Lie group if and only if it is sequentially complete, precompact, locally minimal, contains a non-empty open connected subset and has no infinite compact zero-dimensional subgroups.
(iii). An abelian topological group G is a Lie group if and only if G is locally minimal, locally precompact and all closed zero-dimensional subgroups of G are discrete.
(iv). An abelian topological group is a compact Lie group if and only if it is minimal and has no infinite closed zero-dimensional subgroups.
These results were obtained jointly with D. Shakhmatov [2].
References
[1] D. Dikranjan and I. Prodanov, A class of compact abelian groups, Annuaire Univ. Sofia, Fac. Math. Méc. 70 (1975/76), 191–206.
[2] D. Dikranjan and D. Shakhmatov, Characterizing Lie groups by controlling their zero-dimensional subgroups, Forum Math. 30(2) (2018), 295–320.
[3] D. Dikranjan and L. Stoyanov, Compact groups with totally dense torsion part, Baku International Topological Conference (Baku, 1987), “Elm”, Baku, (1989), 231–238.
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Speaker: Bernardo Rodrigues (University of Pretoria, Pretoria; South Africa)
Title: On some projective divisible codes from the Chevalley group F4(2)
Abstract: This relates to joint work with Amin Saeidi, University of Limpopo, South Africa. A linear [n,k,d]-code C is called projective if no two columns of a generator matrix are linearly dependent, i.e. the columns represent pairwise distinct points in a projective (k-1)-dimensional space. Moreover, a code C over 𝔽p is said to be ∆-divisible if the Hamming weight wt(c) of every codeword c in C is divisible by an integer ∆ >1. In particular, q-ary ∆-divisible codes were studied in [1] with some applications given there. The study that we present in this talk forms part of a programme in which we examine relevant properties of ∆-divisible projective codes of small dimension (in particular irreducible modules of small dimension) and few-weight codes on which finite non-abelian simple groups act faithfully and irreducibly as permutation groups of automorphisms. For illustrative purposes in the talk we consider G = F4(2) to be the adjoint Chevalley group of type F4 over a finite field 𝔽2. Using a representation theoretic approach we present the codes as submodules of the permutation modules over 𝔽2 and examine them in connection with unital algebras, in particular exceptional Jordan algebras. The codes in question are binary projective ∆-divisible codes of dimension 26 that admit the group G as a permutation group of automorphisms acting transitively on the coordinate positions. The specific codes have parameters: [69615,26,32640]2, [69888,26,32768]2, [15667200,26,7831552]2, [16707600,26,8323200]2, [16773120,26,8384512]2 and [17821440,26,8880128]2, respectively. We obtain the main parameters of the codes, describe the full automorphism groups of these structures, and give a geometric description and also examine the nature of the classes of non-zero codewords of the codes.
References
[1] M. Kermaier and S. Kurz. On the lengths of divisible codes. IEEE Trans. Inform. Theory 66 (7) (2020), 4051–4060.
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Speaker: Dharmanand Baboolal (University of KwaZulu-Natal, Durban; South Africa)
Title: The Roelcke uniformity on a localic group
Abstract: Roelcke [6] showed that the infimum of the left and the right uniformity is a compatible uniformity on any topological group. The infimum of any two compatible uniformities on a completely regular topological space is known generally not be compatible, so this a special feature for topological groups. The purpose of this talk is to see what happens in the context of localic groups, that is, group objects in the category of locales.
References
[1] B. Banaschewski and J.C. Vermeulen, On the completeness of localic groups, Comment. Math. Univ. Carolinae 40,2 (1999), 293 – 307.
[2] J. Picado, Weil uniformities for frames, Comment. Math. Univ. Carolinae 36 (1995), 357 – 370.
[3] J. Picado, Structured frames by Weil entourages, Appl. Categ. Struct. 8 (2000), 351–366.
[4] J. Picado and A. Pultr, Frames and locales: Topology without points, Frontiers in Mathematics, vol. 28, Springer, Basel, 2012.
[5] J. Picado and A. Pultr, Entourages, covers and localic groups, Appl. Categ. Struct. 21 (2013), 49 – 66.
[6] W. Roelcke and S. Dierolf, Uniform structures on topological groups and their quotients, McGraw-Hill, New York,1981.
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Speaker: Ana Belén Avilez García (University of the Western Cape, Bellville; South Africa)
Title: Farness for sublocales and an insertion theorem for uniform frames
Abstract: In classical topology an insertion theorem was proved for uniform spaces in [2]. That is, a result where given two functions, under necessary and sufficient conditions, one can show the existence of a uniformly continuous function in-between. Here we will present the point-free version of this result, i.e. an insertion theorem for uniform frames. We will first recall some of the background regarding uniform frames and real-valued functions. We will discuss the farness relation between sublocales of a locale, and define uniform continuity for general, not necessarily continuous, real-valued functions on a frame. The notion of farness is not only interesting in itself, but it also allows us to characterize uniform continuity and to present an insertion theorem for uniform (in fact, preuniform) frames. Due to its complexity, the proof of this result will not be discussed in this talk, but we will present some of its applications. Finally, we will compare this theorem with another insertion result for frames. The content of this talk is joint work with Igor Arrieta, and it is based on the paper [1].
References
[1] I. Arrieta and A. B. Avilez. A general insertion theorem for uniform locales. J. Pure Appl. Algebra, 227 (2023), Art. No. 107329.
[2] D. Preiss and J. Vilimovský. In-between theorems in uniform spaces Trans. Amer. Math. Soc. 261 (1980), 483–501.