Abe, Takuro (Rikkyo University)
Title: Face algebra and logarithmic derivation modules of graphic arrangements
Abstract: In the recent work by Satoshi Murai, we introduced a relation between the module of logarithmic differential 1-forms of graphic arrangements and the cover ideal of that graph. The latter is famous Stanley-Reisner object, so it is natural to expect the same for the logarithmic derivation modules of graphic arrangements. In this talk, we give it, i.e., we show that they are isomorphic to the face algebra of some simplicial poset coming from separators and connected components of the graph. We also give some applications from Stanley-Reisner theory. This is a joint work with Satoshi Murai.
Clarke, Oliver (Durham University)
Title: A Tropical Approach to Rigidity
Abstract: In this talk I will introduce Rigidity Theory: the study of frameworks and their motions, which sits at difficult intersection of algebraic geometry and combinatorics. I will explain how the problem of counting realisations of frameworks can be translated into a question about Tropical intersection products, which allows us to apply new tools and techniques to this problem. This talk is based on joint works with S. Dewar, M. Gallet, G. Grasegger, D. Green Tripp, J. Maxwell, A. Nixon, Y. Ren, B. Smith.
Negishi, Ryo (Rikkyo University)
Title: GKZ systems and Gauss-Manin connections of projective hypersurfaces
Abstract: It is known that the periods of a family of projective hypersurfaces satisfy certain GKZ systems. In general, however, these GKZ systems are reducible, and their solution spaces contain solutions that do not arise as periods. This leads to the problem of determining which part of the GKZ system corresponds to the Gauss–Manin connection. In this talk, we study families of projective hypersurfaces containing a Fermat variety as a fiber, and describe the part of the associated GKZ system corresponding to the Gauss–Manin connection. We give a description of this part in terms of Fourier transform and minimal extension. We also show that it can be realized as the image of suitable contiguity operators.
Nakamura, Yusuke (Nagoya University)
Title: Ehrhart theory on periodic graphs: reciprocity laws
Abstract: A periodic graph is defined as a graph on which the lattice Z^N acts freely such that its quotient graph is a finite graph. Periodic graphs are objects of study in mathematical crystallography, and they also appear naturally in geometric group theory as Cayley graphs of virtually abelian groups. The growth sequence b(n) of a graph is defined as the number of vertices within a graph distance of n or less from a starting vertex. In this talk, we discuss phenomena analogous to Ehrhart theory.
Sugawara, Sakumi (Hokkaido University)
Title: Topological invariants of combinatorial line arrangements
Abstract: Combinatorial line arrangements are abstractions of the notion of line arrangements in the projective plane. In this talk, we discuss their topological properties. In particular, we focus on the complement of topological realizations. We show that the cohomology ring of the complement is isomorphic to the Orlik-Solomon algebra, as in classical line arrangements. In contrast, we show that the complement may have a complicated fundamental group and homotopy type that do not occur in the classical case.
Takayama, Nobuki (Kobe University)
Title: An algorithm to classify Appell and Lauricella D-modules
Abstract: We present an algorithm to classify the parameter spaces of Appell and Lauricella hypergeometric D-modules into isomorphism classes as D-modules and to construct the corresponding isomorphisms.
This algorithm is based on the following four results:
(1) In the paper "Isomorphism classes of A-hypergeometric systems" (2001), Mutsumi Saito demonstrated that GKZ hypergeometric systems can be classified into isomorphism classes of D-modules based on certain geometric conditions on their parameter spaces.
This result can be implemented as an algorithm using algorithms for hyperplane arrangements, Hilbert bases, and Presburger arithmetic.
(2) An algorithm for restricting D-module morphisms.
(3) An algorithm for determining isomorphisms between holonomic D-modules, developed by H. Tai and U. Walther (2001).
(4) Algorithms for constructing contiguity relations.
In this talk, we will explain these algorithms and present several examples.
This is joint work with Hiromasa Nakayama.
Additionally, AI was utilized in the design and implementation of the algorithms.
Telen, Simon (Max Planck Institute for Mathematics in the Sciences)
Title: Principal Matroid Determinants
Abstract: We develop a theory of principal determinants and hypergeometric systems for realizable matroids. Our framework parallels the toric theory of Gel'fand, Kapranov, and Zelevinsky (GKZ), but with the combinatorics of matroids and their flats replacing the usual role of polytopes and their faces. In this analogy, the toric variety is replaced by a reciprocal linear space. The principal A-determinant is replaced by the principal matroid determinant, defined as a specialization of a resultant. The GKZ hypergeometric system is replaced by the matroid hypergeometric system, a holonomic D-module of combinatorial nature whose singular locus is conjectured to be the principal matroid determinant. Joint work with Saiei-Jaeyeong Matsubara-Heo.
Tanaka, Kentaro (Nagoya University)
Title: Space of prime congruences in tropical geometry
Abstract: Tropical geometry is often regarded as algebraic geometry over the tropical semifield. To make this slogan rigorous, several attempts have been made to develop tropical geometry from an algebraic perspective, including tropical scheme theory. Joó and Mincheva defined the dimension of a tropical algebra in terms of chains of prime congruences, rather than prime ideals. They computed the dimensions of certain tropical algebras by representing prime congruences using matrices with real coefficients. In this talk, we propose a different way to represent prime congruences and study the geometry of the space of prime congruences. As an application of our framework, we give a necessary and sufficient condition for a prime congruence to be finitely generated.
Tsuchiya, Akiyoshi (Toho University)
Title: On the Structure and Classification of Gorenstein Simplices
Abstract: Gorenstein simplices form an important class of lattice polytopes arising naturally in combinatorics and toric geometry. In this talk, I will discuss structural and classification results for Gorenstein simplices. In particular, I will explain a connection between extremal Gorenstein simplices and binary linear codes, and present some recent developments toward a more general classification of Gorenstein simplices.
Yoshinaga, Masahiko (Osaka University)
Title: Chambers of rational plane curve arrangements with real intersections
Abstract: A celebrated theorem of Zaslavsky asserts that the number of chambers of a real hyperplane arrangement is equal to the sum of the Betti numbers of its complexified complement. This beautiful formula fails in general for hypersurface arrangements, even for plane curve arrangements. Nevertheless, we will prove that an analogous formula holds for arrangements of rational plane curves with real intersections. We will also discuss further cohomological aspects of their chambers.