# Schedule

- Program Booklet (다운로드)

### 8월 28일 (수요일)

Time | Speaker | Content | Notes |
---|---|---|---|

14:00 - 14:30 | Opening/Registration | ||

좌장 | 엄상일 (IBS) | ||

14:30 - 15:30 | 김정한 | Asymptotic bounds of Ramsey Numbers | Abstract |

15:30 - 16:00 | coffee break | ||

좌장 | 박종육 (경북대학교) | ||

16:00 - 16:30 | 신희성 | 102-avoiding inversion sequences | Abstract |

16:30 - 17:00 | 이현우 | Random matchings in linear hypergraphs | Abstract |

17:00 - 17:30 | 이재호 | Towards a classification of 1-homogeneous graphs with positive intersection number $a_1$ | Abstract |

### 8월 29일 (목요일)

Time | Speaker | Content | Notes |
---|---|---|---|

좌장 | 김석진 (건국대학교) | ||

9:30 - 10:30 | 김민기 | Extensions of the colorful Helly theorem for $d$-collapsible and $d$-Leray complexes | Abstract |

10:30 - 11:00 | coffee break | ||

좌장 | 이준경 (연세대학교) | ||

11:00 - 11:30 | 서재현 | Transversal Hamilton paths and cycles of arbitrary orientations in tournaments | Abstract |

11:30 - 13:30 | lunch | ||

좌장 | 이은정 (충북대학교) | ||

13:30 - 14:30 | 최수영 | Toric Colorability of Graphs of Simplicial $d$-Polytopes with $𝑑+4$ vertices | Abstract |

14:30 - 15:00 | coffee break | ||

좌장 | 김진하 (전남대학교) | ||

15:00 - 15:30 | 윤영한 | Alternating $\mathcal{B}$-permutations arising from toric topology | Abstract |

15:30 - 16:00 | 정준호 | Partitions of ordered partitions and Bott manifolds | Abstract |

16:00 - 16:30 | coffee break | ||

좌장 | 김상욱 (전남대학교) | ||

16:30 - 17:00 | 임선혁 | Homotopy types of Vietoris-Rips complexes and their connection to hyperconvexity | Abstract |

17:00 - 17:30 | 백지선 | On the extremal number of face-incidence graphs | Abstract |

### 8월 30일 (금요일)

Time | Speaker | Content | Notes |
---|---|---|---|

좌장 | 서승현 (강원대학교) | ||

9:30 - 10:30 | 김동현 | Lusztig $q$ weight multiplicities via affine crystals | Abstract |

10:30 - 11:00 | coffee break | ||

11:00 - 11:30 | 송민호 | Combinatorics of orthogonal polynomials on the unit circle | Abstract |

11:30 - 13:30 | lunch | ||

좌장 | 류미수 (충북대학교) | ||

13:30 - 14:30 | 김장수 | Enumeration of multiplex juggling card sequences using generalized $q$-derivatives | Abstract |

14:30 - 15:00 | 김동규 | Two ways to generalize matroids with coefficients | Abstract |

## 초청연사 (invited speakers)

- 김정한 (고등과학원)

**Title**: Asymptotic bounds of Ramsey Numbers

**Abstract**: Ramsey numbers, denoted as $R(s,t)$, are fundamental in graph theory, representing the smallest number of vertices $n$ such that every graph on $n$ vertices either contains a clique of size $s$ or an independent set of size $t$. Recent developments in Ramsey theory have focused on finding asymptotic bounds for Ramsey numbers.

In this talk, we survey asymptotic bounds of Ramsey Numbers $R(3,t)$ and $R(4,t)$, including significant contributions of Sam Mattheus and Jacques Verstraete on $R(4,t)$.

- 김민기 (광주과학기술원)

**Title**: Extensions of the colorful Helly theorem for $d$-collapsible and $d$-Leray complexes

**Abstract**: We present extensions of the colorful Helly theorem for $d$-collapsible and $d$-Leray complexes, providing a common generalization to the topological colorful Helly theorem by Kalai and Meshulam, the very colorful Helly theorem by Arocha et al., and the semi-intersecting colorful Helly theorem by Karasev and Montejano. As an application, we obtain a strengthened version of Tverberg’s theorem. This is joint work with Alan Lew.

- 최수영 (아주대학교)

**Title**: Toric Colorability of Graphs of Simplicial $d$-Polytopes with $𝑑+4$ vertices

**Abstract**: The 1-skeleton of a convex polytope $P$ is called the graph of $P$. A graph of a simplicial $d$-polytope is said to be $\textit{toric colorable}$ if there is a vertex coloring $\lambda \colon V(G) \to \mathbb{Z}^d$ such that ${v_1, \ldots, v_d}$ forms a face of $P$ implies that ${\lambda(v_1), \ldots, \lambda(v_d)}$ is unimodular.

In this talk, we discuss the toric colorability of graphs of simplicial $d$-polytopes with $d+4$ vertices.

- 김동현 (서울대학교)

**Title**: Lusztig $q$ weight multiplicities via affine crystals

**Abstract**: Lusztig $q$ weight multiplicity is a polynomial in $q$ whose positivity has been verified by linking it to a specific affine Kazhdan-Lusztig polynomial. However, a combinatorial formula beyond type A has not been known until recently.

In $2019$, Lee proposed a combinatorial formula for type C using a novel combinatorial concept known as semistandard oscillating tableaux. We will outline the proof of Lee’s conjecture and discuss how it can be extended to type B spin weights case.

Based on joint work with Hyeonjae Choi and Seung Jin Lee.

- 김장수 (성균관대학교)

**Title**: Enumeration of multiplex juggling card sequences using generalized $q$-derivatives

**Abstract**: In $2019$, Butler, Choi, Kim, and Seo introduced a new type of juggling card that represents multiplex juggling patterns in a natural bijective way. They conjectured a formula for the generating function for the number of multiplex juggling cards with capacity. In this paper we prove their conjecture. More generally, we find an explicit formula for the generating function with any capacity. We also find an expression for the generating function for multiplex juggling card sequences by introducing a generalization of the $q$-derivative operator. As a consequence, we show that this generating function is a rational function.

## 일반강연 (Constributed Talks)

- 신희성 (인하대학교)

**Title**: 102-avoiding Inversion Sequences

**Abstract**: A sequence $(e_1,e_2,\cdots ,e_n)$ is an inversion sequences if $0 \le e_i < i$ for all $i = 1,…,n$. We say that an inversion sequences $e = (e_1, e_2, \cdots , e_n)$ $\textit{contains}$ the pattern 102 if there exist some indices $i < j < k$ such that $e_j < e_i < e_k$. Otherwise, $e$ is said to $\textit{avoid}$ the pattern 102. In this talk, we will construct a correspondence between the set of 2-Schröder paths without peaks and valleys ending with a diagonal step and the set of 102-avoiding inversion sequences. This is the joint work with JiSun Huh, Sangwook Kim, and Seunghyun Seo.

- 이현우 (KAIST)

**Title**:Random matchings in linear hypergraphs

**Abstract**: For a given hypergraph $H$ and a vertex $v \in V (H)$, consider a random matching $M$ chosen uniformly from the set of all matchings in $H$. In 1995, Kahn conjectured that if $H$ is a $d$-regular linear $k$-uniform hypergraph, the probability that $M$ does not cover $v$ is $(1+o_d(1))d^{−1/k}$ for all vertices $v \in V (H)$. This conjecture was proved for $k = 2$ by Kahn and Kim in 1998.

We disprove this conjecture for all $k \ge 3$. For infinitely many values of $d$, we construct $d$-regular linear $k$-uniform hypergraph $H$ containing two vertices $v_1$ and $v_2$ such that $P(v_1 \not\in M) = 1 − \frac{(1+o_d(1))}{d^{k-2}}$ and $P(v\not\in M) = \frac{(1+o_d(1))}{d+1}$. The gap between $\mathcal{P}(v\not\in M)$ and $\mathcal{P}(v \not\in M)$ in this $H$ is best possible. In the course of proving this, we also prove a hypergraph analog of Godsil’s result on matching polynomials and paths in graphs, which is of independent interest.

- 이재호 (University of North Florida & POSTECH)

**Title**: Towards a classification of 1-homogeneous graphs with positive intersection number $a_1$

**Abstract**: Let $\Gamma$ be a graph with diameter at least two. Then Γ is said to be 1-homogeneous (in the sense of Nomura) whenever for every pair of adjacent vertices $x$ and $y$ in $\Gamma$, the distance partition of the vertex set of $\Gamma$ with respect to both $x$ and $y$ is equitable, and the parameters corresponding to equitable partitions are independent of the choice of $x$ and $y$. Assume $\Gamma$ is 1-homogeneous distance-regular with intersection number $a_1 > 0$ and diameter $D \geqslant 5$. Define $b = b_1/(\theta_1 + 1)$, where $b_1$ is the intersection number and $\theta_1$ is the second largest eigenvalue of $\Gamma$. In this talk, we show that if intersection number $c_2 \geqslant 2$, then $b \geqslant 1$ and one of the following (i)–(vi) holds:

(i) $\Gamma$ is a regular near $2D$-gon,

(ii) $\Gamma$ is a Johnson graph $J(2D, D)$,

(iii) $\Gamma$ is a halved $l$-cube where $l \in {2D, 2D+1}$,

(iv) $\Gamma$ is a folded Johnson graph $\overline{J}(4D, 2D)$,

(v) $\Gamma$ is a folded halved $(4D)$-cube,

(vi) the valency of $\Gamma$ is bounded by a function of $b$.

Moreover, we characterize 1-homogeneous graphs with classical parameters and $a_1 > 0$, as well as tight distance-regular graphs.

This is a joint work with J. Koolen, M. Abdullah, B. Gebremichel.

- 서재현 (연세대학교)

**Title**: Transversal Hamilton paths and cycles of arbitrary orientations in tournament

**Abstract**: It is well-known that a tournament always contains a directed Hamilton path. Rosenfeld conjectured that if a tournament is sufficiently large, it contains a Hamilton path of any given orientation. This conjecture was approved by Thomason, and Havet and Thomassé completely resolved it by showing there are exactly three exceptions.

We generalized this result into a transversal setting. Let $T = {T_1, . . . , T_{n−1}}$ be a collection of tournaments on a common vertex set $V$ of size $n$. We showed that if $n$ is sufficiently large, there is a Hamilton path on $V$ of any given orientation which is obtained by collecting exactly one arc from each $T_i$. Such a path is said to be transversal.

It is also a folklore that a strongly connected tournament always contains a directed Hamilton cycle. Rosenfeld made a conjecture for arbitrarily oriented Hamilton cycles in tournaments as well, which was approved by Thomason (for sufficiently large tournaments) and Zein (by specifying all the exceptions). We also showed a transversal version of this result. Together with the aforementioned result, it extends our previous research, which is on transversal generalizations of existence of directed paths and cycles in tournaments.

This is a joint work with Debsoumya Chakraborti, Jaehoon Kim, and Hyunwoo Lee.

- 윤영한 (아주대학교)

**Title**: Alternating $\mathcal{B}$-permutations arising from toric topology

**Abstract**: In this talk, we focus on the rational Betti numbers of real toric manifolds associated with chordal nestohedra. We introduce an explicit description for the Betti numbers using alternating $\mathcal{B}$-permutations for a chordal building set $\mathcal{B}$. We provide detailed computations for interesting cases of chordal nestohedra, including permutohedra, associahedra, stellohedra, Stanley-Pitman polytopes, and Hochschild polytopes. This is joint work with Suyoung Choi.

- 정준호 (충북대학교)

**Title**: Partitions of ordered partitions and Bott manifolds

**Abstract**: Bott manifolds are smooth projective toric varieties providing interesting avenues among topology, geometry, representation theory, and combinatorics. They are used to understand the geometric structure of Bott-Samelson-Demazure-Hansen (BSDH) varieties, which provide desingularizations of Schubert varieties. However, not all Bott manifolds originate from BSDH varieties. Those that do are specifically referred to as Bott manifolds of Bott-Samelson-Demazure-Hansen type. In this talk, we explore a relationship between Bott manifolds of BSDH type and partitions of ordered partitions. This talk is based on joint work with Jang Soo Kim and Eunjeong Lee.

- 임선혁 (성균관대학교)

**Title**: Homotopy types of Vietoris-Rips complexes and their connection to hyperconvexity

**Abstract**: The Vietoris-Rips complex, originally introduced by Leopold Vietoris in the early 1900s to develop a homology theory for metric spaces, has since found applications in various areas of mathematics. Eliyahu Rips and Mikhail Gromov further utilized it in their studies of hyperbolic groups. More recently, classifying the homotopy types of Vietoris-Rips complexes has become a significant problem in Topological Data Analysis and Global Metric Geometry. Understanding these complexes can enhance our grasp of the persistence barcode’s strength and provide lower bounds for the Gromov-Hausdorff distance between manifolds. In this talk, we will delve into these mo- tivations and introduce the precise connections between Vietoris-Rips complexes, hyperconvex metric spaces, and their homotopy types.

- 백지선 (연세대학교)

**Title**: On the extremal number of face-incidence graphs

**Abstract**: The $(k, r)$-incidence graph of a regular polytope P is the bipartite incidence graph between $k$-faces and $r$-faces of $P$. We obtain a general upper bound and a corresponding supersaturation result for the extremal number of the $(k, r)$-incidence graph of any regular polytope. This generalises recent results of Janzer and Sudakov, who obtained the same bound for hypercubes and bipartite Kneser graphs, and confirms the conjecture of Conlon and Lee on the extremal number of $K_{d,d}$-free bipartite graphs for certain $(k, r)$-incidence graphs.

Our proof, based on the reflection group method developed by Conlon and Lee, presents the method in a purely algebraic manner. As a consequence, this puts a number of results, including the Janzer-Sudakov theorem, the Conlon-Lee theorem on weakly norming graphs, and Coregliano’s theorem on Sidorenko’s conjecture, in the unified framework and simplifies all the proofs.

Joint work with David Conlon and Joonkyung Lee.

- 송민호 (성균관대학교)

**Title**: Combinatorics of orthogonal polynomials on the unit circle

**Abstract**: Orthogonal polynomials on the unit circle (OPUC) are a family of polynomials orthogonal with respect to integration on the unit circle in the complex plane. The values of these integrals can be obtained by calculating moments. Numerous combinatorial studies have explored the moments of various types of orthogonal polynomials, including classical orthogonal polynomials, Laurent biorthogonal polynomials, and orthogonal polynomials of type $R_I$.

In this talk, we first explain how OPUC relate to these other variations. Next, we study the moments of OPUC from a combinatorial perspective, providing three path interpretations: Łukasiewicz paths, gentle Motzkin paths, and Schröder paths. Using these combinatorial interpretations, we derive explicit formulas for the generalized moments of some examples of OPUC, including the circular Jacobi polynomials and the Rogers–Szegő polynomials. Furthermore, we introduce several types of generalized linearization coefficients and provide combinatorial interpretations for each of them.

- 김동규 (KAIST)

**Title**: Two ways to generalize matroids with coefficients

**Abstract**: Dress (1986) introduced matroids with coefficients offering a unified approach to ordinary matroids, representations of matroids over fields, and oriented matroids. Baker and Bowler (2019) extended this theory, whose result includes a partial field representation by Semple and Whittle (1996).

I will present two generalizations of matroids with coefficients. One is about skew-symmetric matrices and even delta-matroids, based on joint work with Tong Jin. We deduce several results on the representability of even delta-matroids as applications. The other concerns symmetric matrices and new matroid-like objects called antisymmetric matroids. It extends old results on the representability of matroids by Tutte (1958) and basis graphs of matroids by Maurer (1973). These two generalizations involve an interesting interplay between Lagrangian orthogonal/symplectic Grassmannians and combinatorics.