・Enrique Artal Bartolo (Universidad de Zaragoza)

Title: Complex Orbifolds
Abstract: In this first talk we introduce the concept of orbifold, mainly in the complex case, and orbifold maps. We continue with definition and examples of orbifold fundamental group and orbifold covering, orbifold Seifert-van Kampen theorem and orbifold curves.

Title: Quotient singularities and weighted projective spaces
Abstract: We define (abelian and non-abelian) quotient singularities and we stress that they can be viewed as germs of varieties and germs of orbifolds. With these ideas in mind we will define weighted projective spaces and their quotients, relating their class group with the orbifold fundamental group.

Title: Symmetric curves and line arrangements
Abstract: One of the goals of complex orbifolds is to understand symmetric curves studying their quotients in an orbifold defined by the automorphism group. We will study the quotient of some symmetric curves, with special attention to some line arrangements, e.g., regular-polygon arrangements, icosidodecahedron arrangement, etc

・Toshizumi Fukui (Saitama University)

Title: Singularities of mixed polynomials with Newton polyhedrons
Abstract: Oka introduced the concept of mixed polynomials and started to investigate how a study for singularities of mixed polynomials similar to the study of singularities of polynomials is possible. We introduce a mixed toric modification as a mixed analogy of toric modifications and discuss when this provides an analogy of resolutions of singularities defined by mixed polynomials. A mixed toric modification is associated with a mixed fan, which is a notion we introduce in the paper. They provide several combinatorial data for singularities of mixed polynomials. We define the notion of mixed Newton non-degeneracy for mixed polynomials and show that a mixed toric modification provides a semi-algebraic or real algebraic analogue of resolutions of singularities under mixed Newton non-degeneracy condition. Our approach allows us a combinatorial description of the topology of singularities of mixed polynomials, which are mixed Newton non-degenerate, and we show a formulas for the Euler characteristics and the monodromy zeta function of nearby fibers. We also show how the dual graphs of analogy of resolution of singularities of such mixed polynomial are obtained in low dimensions.

・Tatsuya Horiguchi (National Institute of Technology, Ube College)

Title: Regular nilpotent partial Hessenberg varieties
Abstract: Hessenberg varieties are defined as subvarieties of full flag varieties, which are introduced by DeMari-Procesi-Shayman. These varieties include Springer fibers, Peterson varieties, and permutohedral varieties. Interestingly, the cohomology rings of regular nilpotent Hessenberg varieties can be described in terms of logarithmic derivation modules for ideal arrangements. Hessenberg varieties are naturally generalized to subvarieties of partial flag varieties. We call them partial Hessenberg varieties. In this talk, I will talk about a cohomology relation between regular nilpotent Hessenberg varieties and regular nilpotent partial Hessenberg varieties. If time permits, I will explain that the cohomology rings of regular nilpotent partial Hessenberg varieties are described in terms of invariants of logarithmic derivation modules for ideal arrangements under certain Weyl group actions.

・Yuya Koda (Keio University)

Title: On the Powell Conjecture for the Heegaard Splittings of the 3-Sphere I, II, III
Abstract: The Powell Conjecture suggests a specific finite generating set for the mapping class group of Heegaard splittings of the 3-sphere. This conjecture remains an open problem, closely tied to studies on knots, mapping class groups, and singularities. In this series of talks, we will give a brief overview of the conjecture’s background and present a proof for the case where the genus of the Heegaard splitting is 3. This result provides an alternative proof of the work by Freedman and Scharlemann. Additionally, we will discuss a potential framework for extending this approach to higher genera. In particular, for the genus-4 Heegaard splitting, we will introduce a straightforward sufficient condition to support the conjecture. This talk is based on joint work with Sangbum Cho (Hanyang University) and Jung Hoon Lee (Jeonbuk National University).

・Takayuki Koike (Osaka Metropolitan University)

Title: Formal principle for line bundles on neighborhoods of an analytic subset of a compact Kähler manifold
Abstract: We investigate the formal principle for holomorphic line bundles on neighborhoods of an analytic subset of a complex manifold mainly when it can be realized as an open subset of a compact Kähler manifold.

・Kazuaki Miyatani (Tokushima University)

Title: Dwork family, hypergeometric functions and arithmetic
Abstract: In the first part of this talk, we survey the Dwork family, emphasizing its relationship with hypergeometric functions. In the second part, we discuss our previous work and related topics on the connection between (a generalization of) the Dwork family and hypergeometric functions in the context of arithmetic geometry.

・Naoyuki Monden (Okayama University)

Title: On generating mapping class groups by pseudo-Anosov elements
Abstract: The mapping class group of a closed oriented surface is generated by two elements. We prove that the mapping class group is generated by two conjugate pseudo-Anosov elements with arbitrarily large dilatations if the genus is greater than or equal to nine. From earlier work of Lanier and Margalit, the mapping class group is generated by infinitely many conjugate pseudo-Anosov elements with arbitrarily large dilatations. Our result is a refinement of this result in the sense that two such conjugate pseudo-Anosov elements suffice to generate the mapping class group if the genus is greater than or equal to nine.

・Yusuke Nakamura (Nagoya University)

Title: Rationality of the growth series of virtually abelian groups
Abstract: As joint work with Takuya Inoue, we have investigated "Ehrhart theory on periodic graphs". In this talk, I will discuss some applications of this theory to combinatorial group theory of virtually abelian groups.

・Taiki Takatsu (Tokyo University of Sciences)

Title: On virtual cohomological dimensions of automorphism groups of K3 surfaces
Abstract: We will discuss Mukai’s conjecture that the virtual cohomological dimension of the automorphism group of a K3 surface is equal to the maximum rank of its Mordell-Weil groups. The action of the automorphism group on the second cohomology induces the natural action on a hyperbolic space. In this talk, we will explain that Mukai’s conjecture can be regarded as a problem of hyperbolic geometry and geometric group theory via this action. In particular, we give the formula that determines the virtual cohomological dimension of the automorphism group of a K3 surface by the covering dimension of the blown-up boundary associated with the ample cone of the K3 surface.