Abstracts

Yunhyung Cho (Sungkyunkwan University)

Title : Classification of semifree monotone Hamiltonian $S^1$ manifolds

Abstract : In this talk, we consider a six dimensional monotone symplectic manifold $M$ and show that if $M$ admits a semifree Hamiltonian circle action and has an isolated minimum, then $M$ is Fano.

Alastair Darby (Xi’an Jiaotong–Liverpool University)

Title: Equivariant cohomology of torus orbifolds

Abstract: We calculate the integral cohomology, in terms of generators and relations, of locally standard torus orbifolds whose odd degree cohomology vanishes. This involves extending the notion of a GKM-graph to the orbifold case and considering certain integrality conditions. This is joint work with Shintaro Kuroki and Jongbaek Song.

Hiroaki Ishida (Kagoshima University)

Title: Towards transverse toric geometry

Abstract: We say that an effective action of a compact torus $G$ on a connected smooth manifold $M$ is \textit{maximal} if there exists a point $x \in M$ such that $\dim G + \dim G_x = \dim M$. On the other hand, a compact connected complex manifold equipped with a compact torus action has a holomorphic foliation coming from the torus action. In this talk, we discuss a classification of compact connected complex manifolds with maximal torus actions up to transversely equivalence. If time permits, we also discuss the basic cohomology and basic Dolbeault cohomology of such manifolds.

This talk is based on a joint work with Roman Krutovsky and Taras Panov.

Eunjeong Lee (IBS-CGP)

Title: Grossberg--Karshon twisted cubes and connected walks

Abstract: Let $G$ be a complex semisimple simply connected algebraic group of rank $r$. Let $\mathbf{i} = (i_1,i_2,\dots,i_n) \in [r]^n$ be a word decomposition, and let $\ell=(\ell_1,\dots,\ell_n)$ be a sequence of non-negative integers. Grossberg and Karshon introduced a virtual lattice polytope associated to $\mathbf{i}$ and $\ell$, called a \textbf{twisted cube}, whose lattice points encode characters of representations of $B$. More precisely, counted lattice points in twisted cube with sign according to a density function, one get the character of the generalized Demazure module associated to $\mathbf{i}$ and $\ell$. We introduce the notion of \textbf{hesitant connected $\ell$-walks} and then prove that the associated Grossberg--Karshon twisted cube is a closed convex polytope precisely when $\mathbf i$ is a hesitant-connected-$\ell$-walk-avoiding.

Hao Li (Fudan University)

Title: $\alpha$-invariant of submanifolds of generalised Bott towers

Abstract: Stolz theorem states that for a $n$-dimensional simple connected spin manifold where $n\geq 5$, it admits a positive scalar curvature if and only if the $\alpha$-invariant is zero. Weiping Zhang gave an equation to compute the $\alpha$-invariant of spin complex hypersurfaces. In this talk, I use these tools to construct a family of submanifolds of generalised Bott towers which admit positive scalar curvatures.

Zhi Lü (Fudan University)

Title: On orbit braid groups

Abstract: Let $M$ be a (topological) connected manifold with an action of a finite group $G$. We establish the notion of (pure) orbit braid groups for the $G$-manifold. We will see that this notion is not only the generalization to the classical (pure) braid groups defined by the fundamental groups $\pi_1(F(M,n))$ and $\pi_1(F(M,n)/\Sigma)$ of the configuration space $F(M,n)$ of $M$, but also the classical (pure) braid groups can be regarded as subgroups of the orbit braid group defined by us. In addition, we also study the relation between the orbit braid group and the fundamental group of the orbit configuration space $F_G(M,n)$. We try to establish a theory of orbit braid group. This is a joint work with Hao Li and Fengling Li.

Hanchul Park (Jeju National University)

Title: Cohomology of the graphical real toric manifold

Abstract: The topology of the real toric manifold $M^\mathbb{R}$ has been less known than that of its complex counterpart. In 1985, Jurkiewicz gave the formula for the $\mathbb{Z}_2$-cohomology ring of $M^\mathbb{R}$. Its $\mathbb{Q}$-Betti numbers were calculated by Suciu and Trevisan in their unpublished paper, and the result was strengthened for coefficient ring $R$ in which 2 is a unit by Suyoung Choi and the speaker. Recently, the cup product for $H^*(M^\mathbb{R};R)$ is computed by Choi and the speaker.

For any simple graph $G$, the graph associahedron $\triangle_G$ and the graph cubeahedron $\square_G$ are simple polytopes which support natural projective toric manifolds $X_G$ and $Y_G$ respectively. For connected $G$, they are obtained by cutting faces of the simplex and the cube respectively. In this talk, we describe the structure of $H^*(X^\mathbb{R}_G;F)$ and $H^*(Y^\mathbb{R}_G;F)$ in terms of the graph $G$, where $F$ is a field with characteristic other than 2. Note that their $\mathbb{Q}$-Betti numbers can be calculated using the graph invariants known as the \emph{$a$-number} and \emph{$b$-number} of $G$ respectively. A weird thing about $X^\mathbb{R}_G$ and $Y^\mathbb{R}_G$ is that it seems that a slight generalization of them would make the computation of the cohomology too difficult. For example, nestohedra or simple generalized permutohedra define real toric manifolds, but in general their cohomology rings are very difficult to compute.

It is suspected that $H^*(X^\mathbb{R}_G;F)$ determines $G$, and in particular, the family of $X^\mathbb{R}_G$ is $\mathbb{Q}$-cohomologically rigid.

Seonjeong Park (Ajou University)

Title: Toric Richardson varieties and Bruhat interval polytopes

Abstract: Given $v,w\in \mathfrak{S}_n$ with $v\leq w$, the Richardson variety~$X^v_w$ is the intersection of the Schubert variety$X_w$ and the opposite Schubert variety $X^v$. A Bruhat interval polytope~$Q_{v,w}$ is the convex hull of all permutation vectors $x = (x(1), x(2), . . . , x(n))$ with $v\leq x\leq w$. It is known that $Q_{v^{-1},w^{-1}}$ is the moment map image of $X^v_w\subset\mathrm{Fl}(\mathbb{C}^n)$. In this talk, we discuss the properties of a Bruhat interval polytope when the corresponding Richardson variety is a toric variety. This is joint work with Eunjeong Lee and Mikiya Masuda in progress.

Soumen Sarkar (IIT-Madras)

Title: Cohomology rings of a class of torus manifolds

Abstract: Torus manifolds are topological generalization of smooth projective toric manifolds. In this talk, I will compute the rational cohomology ring of a class of smooth locally standard torus manifolds whose orbit space is a connected sum of simple polytopes. This is a joint work with Donald Stanley.

Takashi Sato (Osaka City University)

Title: GKM-theoretical description of double coinvariant rings of pseudo-reflection groups

Abstract: The double coinvariant ring of a pseudo-reflection group is a quotient ring of the tensor product of two copies of the ring of polynomial functions on the vector space on which the pseudo-reflection group acts. It is an analogue of the equivariant cohomology rings of flag varieties. There is a natural homomorphism from the double coinvariant ring to the direct product of copies of the ring of polynomial functions, and McDaniel shows that its image is described in terms of pseudo-reflections analogously to the GKM theory. I will introduce his work and give other description which is more suitable for combinatorics.

Lisu Wu (Fudan University)

Title: Fundamental groups of small covers

Abstract: Davis and Januszkiewicz [{\em Duke Math.J.} 62(1911), pp. 417-451] proved that the fundamental group of a small cover is isomorphic to the kernel of a map from the associated Coxeter group to $\mathbb{Z}_2^n$, such description may not be sufficient in some specific problems. In this talk, I will describe this relation more explicitly based on the presentation of fundamental group we have calculated. Furthermore, our main result shows that the $\pi_1$-injectivity of a facial submanifold of a small cover only depends on the local combinatorial structure of the corresponding face. This is a joint work with Li Yu.

Li Yu (Nanjing University)

Title: On lower bounds of the sum of multigraded Betti numbers of simplicial complexes

Abstract: We find some general lower bounds of the sum of certain families of multigraded Betti numbers of any simplicial complex $K$ with a vertex coloring. These multigraded Betti numbers of $K$ are intimately related to some quotient spaces of the moment-angle complex of $K$ under some free torus actions.