北京雁栖湖应用数学研究院

Inspired by the GLMY path homology theory of directed graphs and its generalisations to quivers and marked categories, we associate a path chain complex to a marked simplicial set and define its path homology. In this talk, I will introduce a Borel-type construction for marked simplicial sets equipped with simplicial group actions and twisting functions. This construction is given by a marked version of the twisted Cartesian product using the box product. A classical theorem of Szczarba states that the twisted shuffle map induces a quasi-isomorphism between the chain complex of a twisted Cartesian product and an associated twisted tensor product. In the marked setting, we prove that this map restricts to an isomorphism of path chain complexes. As an application, digraphs with group actions admit a natural Borel construction as a special case of our framework. This leads to a notion of equivariant path homology, which can be computed via an explicit twisted tensor product. This is joint work with Prof. Shing-Tung Yau.

In joint works with Zhongjian Zhu, we obtained a criterion for a CW complex to be homotopy equivalent to the total spaces of $S^{2k-1}$-fibrations over $S^{2k}$ which was applied to give the homotopy classification of the total spaces of such fibrations and other related results.

We develop a homotopy theory for directed graphs using cubical homotopy groups, also known as A-groups or reduced GLMY homotopy groups. By localizing the category of directed graphs at morphisms that induce isomorphisms on these groups, we obtain an infinity-category of directed graphs. Our main result establishes that this infinity-category is equivalent to the infinity-category of spaces, thereby providing a discrete model for homotopy types in terms of directed graphs.

Topological features of material structures provide a new perspective for understanding material functions and properties, enabling more effective characterization of complex structural systems across different length scales. Unlike conventional geometric descriptors that often rely on local atomic environments, topology-based representations are capable of capturing global connectivity and intrinsic structural relationships in a more robust and generalized manner. This presentation introduces the application of topology-based approaches for property prediction and stability analysis of small molecular clusters, energy prediction and structural design of surface-phase crystals, and property prediction and structural screening of bulk-phase crystals. These studies are all based on feature extraction through topological data analysis combined with artificial intelligence-based prediction methods, providing a new research paradigm for computational materials science.

When studying spaces with an action of a finite group $G$, a powerful equivariant analogue of singular cohomology is $RO(G)$-graded Bredon cohomology, graded by real representations of $G$. Computations in this setting are often challenging and not well understood, even for $C_p$, the cyclic group of order $p$. In this talk, I will introduce $RO(G)$-graded cohomology in the context of $C_p$-equivariant surfaces, along with some structural results making computations easier.

Given a graded ring $A$, we want to know if it can be realized as the cohomology of a space $X_A$. We consider the case of a graded Stanley-Reisner ring $SR(K)$ corresponding to a simplicial complex $K$. When all the generators are in degree $2$ then we can construct $SR(K)$ as the cohomology of the Davis-Januszkiewicz space $DJ(K)$. In this talk we will consider some special cases when the degrees are higher, and relate the problem to graph colouring.

We give an overview of spectral moduli problems in the context of chromatic homotopy theory and power operations, leading to connections with Lubin–Tate and Drinfeld moduli towers, and the Jacquet–Langlands correspondence. Applications include homotopy fixed point spectral sequences which encode certain Galois cohomology groups and compute the homotopy groups of a chromatic localized sphere spectrum, as well as genuine equivariant spectra with respect to p-adic general linear groups. This is joint work with Xuecai Ma and Guozhen Wang.

The cohomological rigidity problem asks whether the cohomology ring of a manifold determines its diffeomorphism type. In this talk, I will discuss this problem for Bott manifolds. A Bott manifold is obtained as an iterated sequence of projective line bundles starting from a point, where each stage is the projectivization of the Whitney sum of two complex line bundles. I will sketch the proof that every isomorphism between the integral cohomology rings of two Bott manifolds is induced by a diffeomorphism. This is joint work with Suyoung Choi and Taekgyu Hwang.

The Vietoris-Rips complex $VR(n, r)$ of an $n$-hypercube graph at scale parameter $r$ is a simplicial complex whose topology has attracted growing attention in recent years. In particular, work of Aadamaszek. Adams and Feng has identified the homotopy type as a wedge of spheres in certain cases. Shukla conjectured that for $n \ge 2$, the reduced homology of $VR(n, r)$ is non-trivial if and only if the degree is $r + 1$ or $2^(r-1)$. In this talk, we use methods from graph theory in order to prove lower bounds on the connectivity of $VR(n, r)$ that contradict Shukla’s conjecture. This method will then be combined with a version of Alexander duality to prove upper bounds on the cohomological dimension. This is joint work with Martin Bendersky and Jelena Grbić.