Title and Abstract

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Donghyeop Lee (Korea University)

Regularity and Zero-Dimensional Schemes

We are interested in the relationship between the Castelnuovo–Mumford regularity, which measures algebraic complexity, and the geometric properties of zero-diemsional schemes. If 𝛤  is a zero-dimensional scheme, then its Castelnuovo-Mumford regularity can be understood as the number 1+d where d is the smallest degree for which the restrictions of homogeneous polynomials of degree d are sufficient to express all regular functions on 𝛤. If the scheme is reduced, this is equivalent to knowing the degree of interpolating polynomials. In this talk, we describe the geometric configuration of 𝛤 when its regularity is close to the known upper bound  (d-n-1)/(t(𝛤)+ 2) where t(𝛤) denotes the smallest integer t such that 𝛤 admits a (t + 2)-secant t(𝛤)-dimensional space. This is a joint work with Euisung Park.

Hyobeen Kim (G-LAMP, Chonnam National University)

The complexity of adaptably k-colouring edge-coloured uniform-hypergraphs

Given an r-uniform hypergraph G and a colouring p : E(G) → [k] of the hyperedges with up to k-colours, an adapted k-coloring of (G,p) is a vertex map 𝜙: V(G) → [k] such that for every hyperedge e, there is some vertex in e such that 𝜙(v) ≠ p(e). We show that for k,r ≥ 2, the problem of deciding if an instance edge coloured graph (G,p) has an adapted k-colouring is NP-complete unless (k,r) = (2,2). In this last case, we show that the problem is polynomial time solvable.

Hyunil Choi (Pusan National University)

On a Geometric Approach to the Nevanlinna–Pick Interpolation Theorem

The classical Nevanlinna–Pick interpolation problem on the unit disc was independently solved by Nevanlinna and Pick [3, 4]. The solvability of the interpolation problem is determined by the positive semi-definiteness of the associated Pick matrix. While this criterion is elegant and powerful, its underlying geometric meaning remains somewhat obscure. This has led to various efforts to reinterpret or reprove the theorem from alternative perspectives.

In particular, Beardon and Minda provided a geometric proof for the three-point case  [1], and later, Baribeau, Rivard, and Wegert extended the argument to the general n-point case [5]. In this talk, we consider the n-point case under distance-preserving assumptions and present a potential-theoretic proof of the Nevanlinna–Pick interpolation theorem using Calabi's diastasis function [2]. This is a joint work with Jisoo Byun.

References

1. Beardon, A. F.; Minda, D., A multi-point Schwarz-Pick lemma, J. Anal. Math. 92 (2004), 81–104.

2. Calabi, E, Isometric imbedding of complex manifolds, Ann. of Math. (2) 58 (1953), 1–23.

3. Nevanlinna, R., Uber beschrankte Funktionen, die in gegebenen Punkten vorgeschrieben Werte annehmen, Ann. Acad. Sci. Fenn. 13 No. 1 (1919), 1–72.

4. Pick, G., Uber die Beschrankungen analytischer Funktionen, welche durch vorgegebene Funktionswerte bewirkt werden, Math. Ann. 77 (1916), 7–23.

5. Baribeau, L.; Rivard, P. and Wegert, E., On hyperbolic divided differences and the Nevanlinna-Pick problem, Comput. Methods Funct. Theory. 9 (2009), 391-405.

Jaewoo Jung (Global Basic Research Laboratory, DGIST)

Linear space arrangements having (almost) maximal Pythagoras numbers

The Pythagoras number of a real projective variety measures how many squares are needed to represent every sums of squares on it. For linear space arrangements defined by square-free monomial ideals, this invariant determines the minimal rank required for row-rank positive semidefinite (PSD) matrix completions. In this setting, the Pythagoras number is completely governed by the graph associated with the ideal. Laurent and Varvitsiotis classified the cases where the Pythagoras number remains small, corresponding to “easy” PSD completion problems. In contrast, we investigate worst-case behaviors where the Pythagoras number becomes extremely large.  

In this talk, I will introduce this problem through its connection to PSD completion theory, and explain our current progress.

Jin-Xin Zhou (Beijing Jiaotong University)

Title: Symmetry in Cayley graphs (August 21)

Cayley graphs form an important class of vertex-transitive graphs, which have been the object of study for many decades. These graphs admit a group of automorphisms that acts regularly (sharply-transitively) on vertices. Although every Cayley graph has such a ‘large’ group of automorphisms, it is difficult in general to decide the full automorphism group of a Cayley graph. In this talk, I will survey some old and new results in this area.

Title: On mixed dihedral groups and 2-arc-transitive normal covers of K_{2^n,2^n} (August 22)

In this talk, I will introduce the notation of a mixed dihedral group, which is a group H with two disjoint subgroups X and Y, each elementary abelian of order 2^n, such that H is generated by X ∪ Y, and H/H' ≅ X╳Y. We will give a graph theoretic characterization of this family of groups, and this is then used to investigate the 2-arc-transitive normal covers of the `basic' graph K_{2^n,2^n}.  (This is a joint work with Daniel R. Hawtin and Cheryl E. Praeger)\end{abstract}

Seungjae Lee (Kyungpook National University)

Ricci Curvatures on Graphs

In this talk, I will introduce the Ricci curvature in Ollivier’s sense on graphs. Although the concept of Ricci curvature is in the realm of differential geometry, which is based on differentiability, it is known that we can consider analogues of Ricci curvature on graphs. Ollivier’s Ricci curvature is based on the optimal transport theory, and recently it has been widely studied by several scholars. In particular, the concept shows that some global properties of graphs (such as diameter) are restricted by the curvature as an analogue of comparison theorems of differential geometry such as Bonnet–Myers theorem. I will provide some motivation for Ricci curvature and explain how Ollivier discretizes the concept, which originally depends on smooth structures.

WonTae Hwang (Jeonbuk National University)

Non-Jordaness of the automorphism group of

the zero-divisor graph of matrix ring over number rings

In this talk, we provide a construction of the induced subgraphs of the zero-divisor graph of M_2(R) for the ring R of algebraic integers of some number fields that are neither complete nor connected, and then describe

the structure of the induced subgraphs explicitly. As an application, we can talk about the non-Jordaness of the automorphism group of the zero-divisor graph of M_2(R). This talk is based on a joint work with Ei Thu Thu Kyaw.

Young Soo Kwon (Yeungnam University)

Classification of regular generalized Cayley maps on cyclic groups

In this talk, we will consider the complementary product of two groups and regular generalized Cayley maps on cyclic groups. For a regular generalized Cayley map M on a cyclic group Cn, the automorphism group of M is a complementary product of a dihedral group and the cyclic group Cn. So the classification of complementary products of dihedral groups and cyclic groups is quite helpful to classify all regular generalized Cayley maps on cyclic groups. Unfortunately, the classification of complementary products of dihedral groups and cyclic groups is not completed. Recently, we classify complementary products of dihedral groups and cyclic groups under the condition that cyclic group is core-free. Using this classification and some more properties of regular generalized Cayley maps on cyclic group, we classify all regular generalized Cayley maps on cyclic groups. This is joint work with Istvan Kovacs and Kan Hu.