October 04-08, 2026
Podgorica
Podgorica
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We study a variant of the embedding functor \(Emb(M, N)\) that incorporates homotopical data from the frame bundle of the target manifold \(N\). Given a parallelized \(m\)-manifold \(M\) and an \(n\)-manifold \(N\) equipped with a section of its \(m\)-frame bundle, we define a modified embedding functor \(\widetilde{Emb}(M, N)\) that interpolates between the standard embedding and a reference framing. Using the manifold calculus of functors, we identify the Taylor tower of \(\widetilde{Emb}(M, N)\) with a mapping space of right modules over the Fulton–MacPherson operad. We prove a convergence theorem under a codimension condition, establishing a weak equivalence between \(\widetilde{Emb}(M, N)\) and its Taylor approximation. Finally, under rationalization, we describe the derived mapping space in terms of a combinatorial hairy graph complex, enabling computational access to the rational homotopy type of the space of embeddings.
A crystallization of a PL manifold is an edge-colored graph encoding a contracted triangulation of the manifold. The concept of regular genus generalizes the notions of surface genus and Heegaard genus for 3-manifolds to higher-dimensional closed PL manifolds. The regular genus of a PL manifold is a PL invariant. Determining the regular genus of a closed PL \(n\)-manifold remains a fundamental challenge in combinatorial topology. Small covers are closed \(n\)-manifolds admitting a locally standard \(\mathbb{Z}_2^n\)-action with orbit space homeomorphic to a simple convex polytope \(P^n\). In this talk, we construct the crystallizations of small covers over \(\Delta^{n-1} \times I\) and \(\Delta^2 \times \Delta^2\). We demonstrate that there are exactly \(1 + 2^{n-1}\) D-J equivalence classes of small covers over the prism \(\Delta^{n-1} \times I\), where \(n\geq 3\). For each \(\mathbb{Z}_2\)-characteristic function of \(\Delta^{n-1} \times I\), we construct a \(2^{n-1}(n+1)\)-vertex crystallization of the small cover \(M^n(\lambda)\) with regular genus \(1 + 2^{n-4}(n^2 - 2n - 3)\), where \(n\geq 4\). For the polytope \(P = \Delta^2 \times \Delta^2\), we classify all the small covers up to Davis-Januszkiewicz (D-J) equivalence and show that there are exactly seven such covers. Among these, one is \(\mathbb{RP}^2 \times \mathbb{RP}^2\), while the others are \(\mathbb{RP}^2\)-bundles over \(\mathbb{RP}^2\). Remarkably, each of these seven small covers has the regular genus 8.
We study axiomatization of degree theory for topological manifolds, similar to the one on Euclidean spaces, as elaborated in the book of Deimling, Nonlinear functional analysis. Our construction however differs, relying not on smooth approximations as Deimling, but cohomological methods. We study a generalization of the winding number concept. We also study a relevant product formula.
We prove that the moment-angle manifold Z_K is diffeomorphic to a connected sum of products of spheres when K is a starshaped (in particular, polytopal) 3-dimensional simplicial sphere with exactly two missing edges that are not adjacent to each other. For each moment-angle manifold in this family, one of the summands of the connected sum is a product of three spheres, as in the first example found by Fan, Chen, Ma and Wang.
For neighbourly starshaped simplicial spheres K of odd dimension, we prove the diffeomorphism Z_K = M_1 # ... # M_k, where each M_i is a product of two spheres; thus, the result previously obtained by Gitler and López de Medrano for polytopal spheres is extended to the case of starshaped spheres. We give an explicit description of the moment-angle manifolds corresponding to the non-polytopal Barnette and Brückner spheres.
This is a joint work with Victoria Kovyrshina, https://arxiv.org/abs/2609.19945.
In this lecture, we present recent results on universal simplicial complexes, their generalisation to the field \(\mathbf{F}_p\) for a prime \(p\), and the associated Buchstaber invariants. The talk is based on joint work with Aleš Vavpetič and Aleksandar Vučić.
Persistent homology is usually presented as a tool for extracting stable topological information from noisy data. In this talk, I will focus on how ideas and questions arising in topological data analysis lead to new results and perspectives in pure mathematics.
After recalling the basic structure and stability theory of one-parameter persistence, I will discuss three examples. The first concerns Vietoris–Rips complexes and Gromov hyperbolicity, where the perspective of persistent homology leads to refinements of classical results about the topology of Rips complexes and connects them to discrete Morse theory and computation. The second concerns Morse's functional topology, where persistence provides a rigorous framework for generalized Morse inequalities and for the classical argument of Morse and Tompkins concerning unstable minimal surfaces. Finally, I will indicate how persistence modules, viewed as representations of posets equipped with a metric notion of perturbation, lead to new questions about decomposition and stability in representation theory.
These examples illustrate how persistent homology, beyond its role in the analysis of data, provides new viewpoints and methods in geometry, topology, and algebra.
This paper systematically examines the analogue of the TD separation axiom for topological systems by comparing it with existing separation axioms. We discuss its properties and consequences for certain types of completely prime filters, as well as the internal structure of the frame associated with the topological system. Furthermore, we explore sobriety and weak sobriety of a topological system using the true-kernel and the false-kernel of a point of the frame associated to that topological system.
We study the topology of spaces associated with both complex and real hyperplane arrangement \(J_{n+1,2}\). This arrangement arises naturally from the chamber decomposition of the hypersimplex \(\Delta_{n+1,2}\), the image of the moment map for the standard torus action on the complex Grassmannian \(G_{n+1,2}\).
In the complex case, we explicitly describe the cohomology of the associated space using Orlik-Solomon algebra. In the real case, we describe cellular structures on the one-point compactification of the associated space and investigate its homology using cellular and relative homology methods. Particular attention is given to the case \(J_{4,2}\) in connection with torus equivariant topology of the complex Grassmannians \(G_{4,2}\). This is a joint work with Svjetlana Terzić.
We observe and prove that the complexity of lasso for normalized data is smaller than for nonnormalized ones by relating this question to extremal combinatorics and algebraic graph theory. We employ a geometric approach to the lasso as a study of the tangency of the level sets of the least square objective function with the polyhedral boundary sets \(B(t)\) of the parameters in \(\mathbb R^p\) with the \(\ell_1\) norm equal to \(t\). We geometrically derive closed exact formulae for the solution of the lasso under the full rank assumption. We establish important general properties of the solutions of the lasso, which are known to be represented as a simple polygonal chain in \(\mathbb{R}^p\). Starting from \(p=2\) and \(p=3\), we show a striking difference in the maximal number of \(p\)-dimensional orthants a polygonal chain of a lasso solution can intersect in the case of normalized data vs. nonnormalized data. We prove that in the normalized case, the number \(h_{p,2}\) is a general upper bound for the number of segments of a lasso solution intersecting a \(p\)-dimensional orthant, where \(h_{p,r}\) is the maximal number of binary words of length \(p\) such that every two words match at least at \(r\) spots, \(r\le p\). We calculate \(h_{p,2}\) using spectral graph theory. It was known that for general data the sharp estimate for that number is \(2^{p-1}\), which we identify with \(h_{p,1}\). We also find an upper bound for the total number of segments of a lasso solution with normalized data in dimension \(p\), that is significantly less than \((3^p+1)/2\), a well-known sharp upper bound for the nonnormalized case. This is based on the joint work with Borislav Gajić.
In this study, the \(m\)-contiguity distance is introduced as an approximation of the contiguity distance between simplicial maps, and its fundamental properties are investigated. In addition, its relationships with the simplicial Lusternik--Schnirelmann category and discrete topological complexity are studied, together with its behavior under barycentric subdivision, strong homotopy equivalence, and in the context of aspherical spaces.
Toric topology studies spaces whose topology is encoded by combinatorial data such as simplicial complexes and simple polytopes. A fundamental question is to what extent this combinatorial information determines the topology of the resulting spaces. More generally, one may ask which topological invariants are sufficient to recover the homeomorphism type of a manifold arising in toric topology.
In this talk, we discuss several rigidity phenomena from this perspective. We begin with quasitoric and quoric manifolds, whose orbit-space descriptions lead to strong classification results. We then turn to moment--angle manifolds and explain how their topology can often be understood through associated principal bundles over quasitoric or quoric quotients. In the complex and quaternionic settings, this viewpoint leads to equivariant rigidity results, reducing classification problems to the study of characteristic classes of the corresponding kernel bundles.
We also discuss the real case, where flag simplicial complexes give rise to CAT(0) cubical geometry through the Davis complex. This provides a connection between toric topology and geometric rigidity phenomena related to the Farrell--Jones and Borel conjectures.
The aim of the talk is to illustrate how combinatorics, geometry, and topology interact in determining the homeomorphism type of spaces arising in toric topology.
I will explore the homotopy theory of Vietris-Rips complexes of hypercubes, focusing on the deep interplay between their topology and combinatorial structures. The goal is to understand key structural properties, such as higher connectivity, co-connectivity and the realisation of homotopy types, by introducing a new combinatorial-topological framework.
This is party joint work with Martin Bendersky and Salvatore Elia.
Operations on tempered cohomology theories (and complex-periodic cohomology theories more generally) are known to be intimately related to the isogenies of their associated algebraic groups. The Adams operations on K-theory, for example, are associated with the \(k\)th power map on \(\mathbb{G}_m\). Their analogues at height 2, Hecke operators on elliptic cohomology, are not as well-understood.
In this talk, I will describe a spectral algebreo-geometric approach to studying these operators. This begins with the spectral moduli stack of isogenies of oriented elliptic curves, which I show to be a (nonconnective) spectral Deligne-Mumford stack using a connected-étale decomposition and a generalization of Lurie's theorem on the representability of derived Hilbert spaces. We will then apply this to construct stacky Hecke operators in a manner similar to their classical moduli-theoretic construction—with the notable caveat that in the spectral case, Grothendieck duality introduces a nontrivial twist.
The proposed talk will focus on the topology of non-singular complex algebraic surfaces, both affine and projective. Closely related is the topology of the universal of these spaces. In many situations, this universal cover turns out to be either an Eilenberg-MacLane space or a Moore space. I will discuss these properties and some related classification theorems. Special focus will be on surfaces fibered over a curve. For such surfaces one has a good information about its topology (and that of its covering) in terms of the knowledge of the base, the general fiber and the nature of the singular fibers.
There is defined the canonical action of the compact torus \(T^n\) on the complex Grassmann manifold \(G_{n,2}\). Our starting point is the \((U_n,p_n)\) model for \(X_n\) constructed by Buchstaber and Terzić (2022), where \(U_n = \Delta_{n,2}\times\mathcal F_n\) for the hypersimplex \(\Delta_{n,2}\) and the universal parameter space \(\mathcal F_n\) as defined in the works of Buchstaber and Terzić (2019, 2022). Buchstaber and Terzić (2023) proved that \(\mathcal F_n\) is diffeomorphic to the moduli space \(\mathcal M_{0,n}\) of stable rational curves of genus zero with \(n\) fixed ordered points. For the cases \(n = 5\) and \(n = 6\), we provide a description of the model \(U_n\) in terms of the hyperplane arrangement \(\mathcal{G}(n,2)\). We apply the resulting description to classify the Hassett spaces \(\mathcal{M}_{0,\mathcal{A}}\) for the case \(n = 5\). Based on joint work with Svjetlana Terzic.
In our previous work (arXiv:2511.10153), we introduced the notion of a locally standard T-pseudomanifold and obtained a classification under suitable assumptions. This class includes both quasitoric manifolds and projective toric varieties. In this talk, we introduce explicit descriptions of the equivariant intersection cohomology of certain locally standard T-pseudomanifolds. This is joint work with Yuya Koike.
Motivated by a work of Fu-So-Song, we associate a symmetric matrix A to a plane vector sequence v and give a formula to find the signature of A in terms of the sequence v. When A is nonsingular, we interpret the relation between A and A^{-1} from a topological viewpoint. In fact, one can associate an omnioriented quasitoric orbifold X of real dimension four to the sequence v and see that A^{-1} is the intersection matrix of the characteristic suborbifolds of X.
We introduce a new invariant associated to an action of a group \(G\) on a (possibly non-commutative) group \(K\). The invariant is based on the parametrized LS-category of the semi-direct product \(K\rtimes G\) over \(G\) and it also reflects a choice of a section, given by a derivation \(d\colon G\to K\). The invariant is denoted \(\mathrm{tc}(d)\) and takes integer values between \(\mathrm{cd}(K)\) and \(\mathrm{cd}(K)+\mathrm{cd}(G)\), reflecting the intricacies of the action of \(G\) on \(K\). We compute its values for semi-direct product decompositions of fundamental groups of mapping tori, of pure braid groups and of iterated semi-direct products of free groups. To achieve this, we relate \(\mathrm{tc}(d)\) to the cohomological dimension of its factors, to the sectional category of the inclusion of \(G\) in \(K\rtimes G\), and to the properties of torsors that are associated to the derivation \(d\).
This is a joint work with U. Ogrinec.
In this talk we will continue the exposition from the Pavešić's talk on Parametrized LS category and group actions and discuss two important classes of semi-direct product decompositions, arising from fundamental groups of mapping tori and from pure braid groups. To avoid repetition we will present examples that illustrate different approaches and computational methods.
Work of McGavran, Bosio-Meersseman and Gitler-Lopez de Medrano shows that a moment-angle manifold associated to a neighbourly triangulation of an odd dimensional sphere that is polytopal is diffeomorphic to a connected sum of products of two spheres. It has been an open problem for several years to give a homotopy theoretic proof that allows for an extension to non-polytopal cases. In this talk a solution to the problem is discussed. This is joint work with Amaranta Membrillo-Solis.
Bier spheres, named after Thomas Bier, have been studied from the viewpoint of combinatorics (simplicial complexes), topology (polyhedral products, toric manifolds), convex polytopes (generalized permutohedra, algorithmic Steinitz problem), game theory (cooperative games), experimental mathematics (nonpolytopal spheres), combinatorial optimization (submodular functions), algebraic statistics (convex rank tests), etc.
We present an overview of selected results from the project ``Bier-2016-2026'', as initiated at ICERM (2016), emphasizing the interplay of ideas from different mathematical fields. One of the cornerstone results is the following theorem, which links Bier spheres with braid arrangements, polymatroids, and generalized permutahedra. This result, together with its consequences and ramifications, paves the way for many applications, from polyhedral combinatorics to toric geometry and topology.
Theorem 1. Each Bier sphere admits a canonical star-shaped geometric realization whose radial fan coarsens the braid arrangement fan. Moreover, this fan is unimodular in the lattice \(A_n^\ast\), dual to the root lattice \(A_n\).
The talk is based on the past, present and future collaboration with participants and friends of Belgrade ``CGTA-project'' including: Siniša Vrećica, Duško Jojić, Gaiane Panina, Vladimir Grujić, Đorđe Baralić, Marinko Timotijević, Filip Jevtić, Ivan Limonchenko, Marija Jelić Milutinović, and many others.