Let (M, ⍵) be a rational ruled symplectic 4-manifold endowed with an effective Hamiltonian S1-action, and let ℱ denote a connected component of the fixed point set of the action. Consider EmbS1(M, ℱ, c), the space of S1-equivariant embeddings of standard symplectic balls of capacity c centered at ℱ, equipped with the C∞ topology. In this talk, we will show that EmbS1(M, ℱ, c) is path-connected, which is equivalent to the uniqueness of equivariant symplectic blowup of size c at ℱ up to equivariant symplectic isotopy. Moreover, we will discuss the homotopy type of the space of images of equivariant embeddings of symplectic balls of capacity c centered at ℱ. This is based on a joint work in progress with Pranav Chakravarthy, Liat Kessler, and Martin Pinsonnault.
Szczarba’s twisted shuffle provides a quasi-isomorphism between the chain complex of a twisted Cartesian product and the corresponding twisted tensor product. We extend this construction to marked simplicial sets, where certain edges are marked and simplicial group actions are required to preserve these markings. The homology of a marked simplicial set is defined using its path chains, motivated by path homology theories for digraphs, quivers, and marked categories. We show that, in this marked setting, the classical chain homotopy equivalence given by Szczarba’s twisted shuffle strengthens to a chain isomorphism when the marked twisted Cartesian product is defined via the box product. As an application, this approach yields a natural Borel construction for digraphs with group actions and a Borel-type equivariant path homology, together with an explicit twisted chain complex computing this homology.
Finding the minimal number of fixed points of a periodic Hamiltonian flow on a compact manifold is, in general, an open problem. We will see how one can obtain lower bounds for the number of fixed points of a circle action on an almost complex manifold, by retrieving information from a special Chern number.
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 joint work with Martin Bendersky.
Hessenberg varieties are closed subvarieties of flag varieties introduced by De Mari-Procesi-Shayman. Their geometry and topology are related to other research areas such as quantum cohomology of flag varieties, logarithmic derivation modules in hyperplene arrangements, and Stanley's chromatic symmetric functions in graph theory. These connections are discovered in regular nilpotent and regular semisimple Hessenberg varieties. In order to give an explicit presentation of the cohomology rings of regular nilpotent Hessenberg varieties, Abe-Harada-Horiguchi-Masuda introduced polynomials fij. In this talk we quantize the polynomials fij by a method of Fomin-Gelfand-Postnikov, and we talk about a relation between their quantizations fij and the coordinate rings of regular semisimple Hessenberg varieties. In particular, we give a connection between the coordinate rings of regular semisimple Hessenberg varieties and the cohomology rings of regular nilpotent Hessenberg varieties.
The flag variety Flag(ℂn) has a structure of a homogeneous space SL(n, ℂ)/B, where B is the subgroup consists of all upper triangular matrices in SL(n, ℂ)/B. Let U be the commutator subgroup of B. By choosing a suitable character B → ℂ, we obtain an ample line bundle SL(n, ℂ) ×B ℂ, where the action of B on ℂ is given by the chosen character. Let H be the algebraic torus consists of all diagonal matrices in SL(n, ℂ). Then, B can be decomposed into the semi-direct product as B = H ⋉ U. Therefore we may regard Flag(ℂn) as a quotient of SL(n, ℂ)/U by an action of the algebraic torus (ℂ∗)n-1. On the other hand, the algebraic torus H × H ≅ (ℂ∗)2n-2 acts on SL(n, ℂ)/U by "double-sided" multiplications. We consider quotients by actions of H × H on SL(n, ℂ)/U, restricted to an (n-1)-dimensional algebraic subtorus (ℂ∗)n-1 → H × H and a linear character 𝜒: (ℂ∗)n-1 → ℂ∗. We aim to find conditions for 𝜒 so that this quotient is well-behaved.
We restrict our attention to the case n = 3. We study the double-sided actions of (ℂ∗)2 on SL(3, ℂ)/U and the associated quotients. We give an explicit necessary and sufficient condition for SL(3, ℂ)/U to agree with the 𝜒-stable locus in its affine closure. This is joint work with Yoshinori Hashimoto and Hisashi Kasuya.
A GKM manifold is an almost complex manifold endowed with a torus action with isolated fixed points, whose equivariant cohomology is described by its GKM graph, a combinatorial object associated in terms of data on the fixed point set. During this talk, we discuss 6-dimensional GKM manifolds with 4 fixed points. We discuss possible GKM graphs, and for each type of graph we discuss the existence of manifold. We show that six types occur.
(P1) complex projective space ℂℙ3 with standard complex structure
(P2) blow up of S6 at a fixed point, diffeomorphic to ℂℙ3
(P3) ℂℙ3 with non-standard almost complex structure
(Q1) complex quadric Q3 with standard complex structure
(Q2) blow up of S6 along isotropy 2-sphere, diffeomorphic to Q3
(S) S2 × S4, obtained as equivariant gluing along orbits of two S6's
This is joint with Shintaro Kuroki, Mikiya Masuda, and Takashi Sato.
The wedge operation preserves PL~spheres and their Buchstaber numbers. A seed is a PL sphere that cannot be obtained as a wedge of another PL sphere. If a seed has maximal Buchstaber number p, then its dimension is at most 2p - p - 2. In this talk, we discuss the tightness of this bound. This is joint work with Suyoung Choi.
We investigate the global deformation rigidity problem of rational homogeneous manifolds of Picard number one which were developed by Hwang, Mok and others. In particular, we focus on the role of varieties of minimal rational tangents. Starting with similar ideas, we introduce some recent global deformation rigidity results of some quasi-homogeneous varieties, symmetric varieties and horospherical varieties, with Picard number one.
In this talk, I will introduce the notion of a locally standard T-pseudomanifold over a convex polytope and present a classification theorem for these spaces up to equivariant homeomorphism. This work generalizes the classification theorem of quasitoric manifolds over simple convex polytopes due to Davis-Januszkiewicz to arbitrary convex polytopes. If time permits, I will also introduce locally standard T-pseudomanifolds over topologically stratified pseudomanifolds under certain conditions, and present their classification (arXiv:2511.10153). This is joint work with Yuya Koike.
It is known by the work of Klyachko that the Chern classes of the permutohedral variety XAn are represented by elementary symmetric polynomials. In this talk, we focus on the product of Chern classes ckcn-k on XAn. By employing purely combinatorial methods related to the permutohedral fan 𝚫An, we present an explicit closed formula expressing this product as a multiple of the top Chern class cn in the rational cohomology ring. As an application, we provide a concrete computation of the corresponding Chern numbers <ckcn-k , [XAn]>.
A Bier sphere is defined as deleted join of a simplicial complex different from the entire simplex and its Alexander dual complex. This simple and beautiful construction yields a class of triangulated spheres that have starshaped realizations due to the result of Jevtić, Timotijević and Živaljević.
In this talk, for an arbitrary simplicial complex K ≠ 𝛥[m] on [m] = { 1, 2, ..., m }, we will introduce the canonical complete regular fan ΣK whose underlying simplicial complex is isomorphic to the (m-2)-dimensional Bier sphere of K. We will then discuss some important geometric and topological properties of the canonical toric manifold XK of complex dimension (m-1) and its real (m-1)-dimensional counterpart XKℝ associated with the fan ΣK. Finally, we will consider their applications in geometrical combinatorics, toric topology, and bordism theory.
The talk is based on joint works with Vladimir Grujić, Marinko Timotijević, and Rade Živaljević.
I will present a recent preprint with the same title. Recall that a smooth complex Fano variety is called prime if the Picard group has rank 1 or equivalently the second Betti number is equal to 1. Moreover, recall that an action of a group G is called semi-free if the stabilizer of every point is G or {1G}. The main result classifies smooth prime complex Fano 4-folds having a semi-free holomorphic ℂ∗-action into four deformation families, distinguished by their Fano index, one possibility for each integer 𝜄 satisfying 2 ≤ 𝜄 ≤ 5. Three of the families are unique up to isomorphism and were known to have semi-free algebraic torus actions. For the remaining example with Fano index 2, for which each member is realized as a linear section of the Grassmanian Gr(2,6), certain elements were shown to have semi-torus actions recently by Alexander Kuznetsov. The proof of the main classification proceeds by studying the action of the compact torus S1 ⊂ ℂ∗ which is Hamiltonian with respect to certain Kähler forms.
Classifying equivariant smooth closed manifolds up to equivariant bordism is one of fundamental problems in topology. We will mainly focus on the case of equivariant geometric unoriented bordism of G-actions fixing isolated points where G = ℤ2k, which can directly be associated with G-representation theory. In this talk, I will introduce some new progresses, especially for the homology description and the dimension formulae of equivariant geometric bordism groups, which involve a connection with the universal complexes of DJ theory, and the uses of matroid theory and spectral sequence etc.
Motivated by Fu-So-Song [1], one can associate a symmetric matrix with a plane vector sequence. Those matrices are related to tridiagonal matrices through inverse. I will discuss this correspondence. Half of my talk will be what I talked at the 7th KTTW and the other half will be about further observations.
[1] X. Fu, T. So, and J. Song, The integral cohomology ring of four-dimensional toric orbifolds, arXiv:2304.03936.
[2] X. Fu, T. So, and J. Song, Cohomology bases of toric surfaces, Topology Appl. 369 (2025), Paper No. 109392, 20 pp.
[3] M. Masuda, Symmetric matrices defined by plane vector sequences, arXiv:2503.06836.
This talk is based on the joint work arXiv:2410.02105 with Raymond Chou and Brendon Rhoades. The topic includes the computation of the equivariant cohomology ring of the space Xn,k,k of n-tuples of lines in ℂk that span d-dimensional subspaces. Our work is a continuation of the earlier work by Pawlowski and Rhoades, which introduced and studied the k=d case, Xn,k,k, to realize a certain generalized covariant ring appearing in the Delta conjecture as a cohomology ring. I will explain our main technique, Orbit Harmonics, to compute the (equivariant) cohomology ring.
Moment-angle manifolds provide a wide class of examples of non-Kähler compact complex manifolds with a holomorphic torus action. A complex structure on a moment-angle manifold Z is defined by a complete simplicial fan. When the fan is rational, the manifold Z is the total space of a holomorphic bundle over a toric variety with fibres compact complex tori. In this case, the invariants of the complex structure of Z, such Dolbeault cohomology and the Hodge numbers, can be analysed using the Borel spectral sequence of the holomorphic bundle.
In general, a complex moment-angle manifold Z is equipped with a canonical holomorphic foliation ℱ which is equivariant with respect to the algebraic torus action. Examples of moment-angle manifolds include the Hopf manifolds, Calabi-Eckmann manifolds, and their deformations. The holomorphic foliated manifolds (Z, ℱ) are models for irrational toric varieties.
We describe the basic de Rham and Dolbeault cohomology algebras of the canonical holomorphic foliation on a moment-angle manifold, LVMB-manifold or any complex manifold with a maximal holomorphic torus action. Namely, we show that the basic cohomology has a description similar to the cohomology algebra of a complete simplicial toric variety due to Danilov and Jurkiewicz. The Hodge decomposition for the basic Dolbeault cohomology is proved by reducing to the transversely Kähler (equivalently, polytopal) case using a foliated analogue of toric blow-up. We also provide a DGA model for the ordinary Dolbeault cohomology algebra of Z.
The game of SET is a popular card game of pattern finding, which corresponds to finding a line in a finite affine space. In this talk, we explain the game and introduce related games designed mathematically similar ways based on linear and affine relations of finite vector spaces. This talk is intended to be light, easy to follow, and fun. I apologize that the topic is not related to toric topology in any way.
There are exactly 16 Gorenstein Fano toric surfaces and they correspond to 16 equivalence classes of reflexive lattice polygons in the plane. As a natural generalization of toric varieties, we can consider normal equivariant compactifications of reductive algebraic groups. By the Luna-Vust theory, the normal equivariant embeddings of a redective algebraic group G are classified by combinatorial objects called colored fans, which generalize the fans appearing in the classification of toric varieties. Furthermore, many geometric properties of a Fano group compactification can be described from its algebraic moment polytope which encodes the structure of representation of G on the spaces of sections of tensor powers of the anticanonical line bundle. In this talk, I will give a classification of Gorenstein Fano equivariant compactifications of semisimple complex Lie groups with rank two, and determine which of them are equivariant K-stable and admit (singular) Kähler-Einstein metrics.
The circle actions on low-dimensional manifolds have been studied in a few articles. In this talk, I'll introduce several classes of manifolds which are the orbit spaces resulting from circle actions on some manifolds. Then, I'll discuss some properties of these new manifolds.
An orbifold is a topological space that is locally modelled on the quotient of a Euclidean space by a finite group action. One way in which orbifolds arise in symplectic geometry is as symplectic reductions of Hamiltonian actions at regular values. Therefore, even to understand symmetries of symplectic manifolds, we are thrown back to the question of studying (symmetries of) symplectic orbifolds. In this talk we consider Hamiltonian S1-actions on closed four dimensional symplectic orbifolds under the assumptions in the title. To such an action we attach a labelled multigraph that completely determines the action up isomorphism, generalizing the construction by Karshon for manifolds. Time permitting, we may discuss the analogue of minimal spaces in this context, i.e. spaces on which no equivariant blow-down can be performed. This is joint work with Leonor Godinho and Grace Mwakyoma-Oliveira.
In toric topology, the cohomological rigidity problem asks whether two toric spaces with isomorphic cohomology rings have the same homotopy / homeomorphism / diffeomorphism type. In this talk, I will discuss results on the cohomological rigidity of toric orbifolds with respect to homotopy types.
This talk is based on joint work with Xin Fu, Jongbaek Song and Stephen Theriault.
We study the homotopy theory of polyhedral products associated to a combinatorial generalisation of a manifold known as a pseudomanifold. As special cases, we show that loop spaces of moment-angle manifolds associated to any triangulation of S2 and S3, or neighbourly triangulations of S2n+1, decompose as a product of spheres and loops on spheres. This is joint work with Lewis Stanton.
We introduce a systematic method for generating set-theoretic operads through iterated application of the power set functor. Our key observation is that this functor preserves the operadic structure: the power set of any set operad yields another set operad, enabling the construction of infinite families of operads from any initial one. Applying this framework to the permutative operad introduced by Chapoton [3], we uncover a striking hierarchy. The first iteration recovers the commutative triassociative operad, introduced by Vallette [4]. The second produces both operads on simplicial complexes described by Ayzenberg [2] and Abramyan-Panov [1], and we show that they are infinitely generated. This reveals these seemingly disparate structures as part of a unified operadic tower. We leverage this perspective to establish that the arrows of any (small) cocontinuous cocomplete symmetric monoidal category carry a natural algebra structure over these simplicial complex operads. This operadic unification provides a conceptual framework that simultaneously explains multiple variants of polyhedral products—each arising from different symmetric monoidal structures on topological spaces. Going further, we construct a new operad on pairs of simplicial complexes based on the polyhedral join, introduced by Ayzenberg [2]. This operad contains the two simplicial complex operads mentioned above as suboperads and supports a generalized notion of algebra that extends our earlier constructions.
[1] Semyon A. Abramyan and Taras E. Panov. Higher whitehead products in moment—angle complexes and substitution of simplicial complexes. Proceedings of the Steklov Institute of Mathematics, 305(1):1–21, May 2019.
[2] A. A. Ayzenberg. Substitutions of polytopes and of simplicial complexes, and multigraded betti numbers. Transactions of the Moscow Mathematical Society, 74:175–202, April 2014.
[3] Frédéric Chapoton. Un endofoncteur de la catégorie des opérades, volume 1763 of Lecture Notes in Mathematics, page 105–110. Springer Berlin Heidelberg, Berlin, Heidelberg, 2001.
[4] Bruno Vallette. Homology of generalized partition posets. Journal of Pure and Applied Algebra, 208(2):699–725, February 2007.
Bier spheres form a notable subclass of simplicial spheres in that they contain infinitely many non-polytopal examples. Recently, it has been shown that each Bier sphere can be realized as a smooth complete fan, which further strengthens its connection to toric geometry. In this talk, we investigate how the homotopy types of all full subcomplexes of Bier spheres can be classified and computed. We also discuss several applications of these results, including consequences for invariants arising in toric topology. This is joint work with Suyoung Choi and Seonghyeon Yu.
In this talk, we discuss a morphism from the Peterson variety to the toric orbifold associated to Cartan matrices. As an application, we calculate the totally nonnegative part of the Peterson variety in arbitrary Lie type. This confirms a conjecture of Rietsch for all Lie types. This is joint work with Hiraku Abe and Tao Gui.