Department of Math., Ajou University

Room 621 (or Room 311 or Zoom), Paldal hall, Ajou University

Contact information : 

Soojin Cho (조수진 chosj@ajou.ac.kr)

Suyoung Choi (최수영 schoi@ajou.ac.kr)

Boram Park (박보람 borampark@ajou.ac.kr)

Homoon Ryu (류호문 ryuhomoon@ajou.ac.kr) 

This seminar has been supported by the research fund of Ajou university since September of 2015.

2025

1. Shinya Fujita (Yokohama City University) April 29(Tues), 2025. 11:00 -- 12:00 (팔달관 435호)

• Title:  Contractible paths containing prescribed vertices in highly connected triangle-free graphs 

• Abstract: Let m,k be integers with m≥ 1, k≥ 2. For a k-connected graph G, a subgraph H of G is k-removable if G-V(H) is still a k-connected graph. A graph is triangle-free if it contains no triangle as a subgraph.

In this talk, I introduce my recent result which states that if G is a k-connected triangle-free graph with minimum degree at least k+(m-1)/2, then for any vertex v∈ V(G), there exists a path P of order m starting from v such that G-V(P) is a  (k-1)-connected graph. This result is obtained by showing a stronger statement concerning the existence of  k-removable paths in k-connected triangle-free graphs. We also show that if G is a k-connected triangle-free graph with minimum degree at least k+1, then G contains a k-removable edge. Our results confirm a conjecture due to Luo et al. concerning the existence of a k-removable path of order m in a k-connected bipartite graph for all odd m together with the case m=2.

2024

5. Byung-Hak Hwang (황병학 고등과학원) June 24(Mon), 2024. 16:30 -- 17:30 (Zoom)

• Title: The theory of noncommutative symmetric functions

• Abstract: Given a symmetric function, its expansion into specific bases may reveal algebraic, geometric, and/or probabilistic meanings. Hence, finding a combinatorial interpretation of the coefficients is an important problem in algebraic combinatorics. Noncommutative symmetric function theory is a useful tool for this purpose.

In this talk, I will provide an introduction to the theory of noncommutative symmetric functions. If time permits, I will also discuss a generalization of the theory and present two recurrences for the e-coefficients of chromatic quasisymmetric functions.

4. Mikiya Masuda (Osaka Central Advanced Mathematical Institute) June 18(Tue), 2024. 16:30 -- 17:30 (Zoom)

• Title: REGULAR SEMISIMPLE HESSENBERG VARIETIES OF DOUBLE LOLLIPOP TYPE

• Abstract: The solution of Shareshian-Wachs conjecture by Brosnan-Chow (and Guay- Paquet unpublished) linked together graded chromatic symmetric functions on unit interval graphs (combinatorics) and the cohomology of regular semisimple Hessen- berg varieties (geometry). Regular semisimple Hessenberg varieties are subvarieties of the full flag variety Fl(n) parametrized by Hessenberg functions h: [n] → [n] (equivalently Dyck paths). The family of connected regular semisimple Hessenberg varieties X(h) contains Fl(n) itself (as the largest h) and the permutohedral variety Permn (as the smallest h). The cohomology rings of both Fl(n) and Permn are generated in degree two as is well-known but the cohomology ring of X(h) is not necessarily generated in degree two. It turns out that it is generated in degree two if and only if h is of double lollipop type. I will explain the idea of the proof. This is joint work with Takashi Sato.

3. Hemanshu Kaul (Illinois Institute of Technology) May 24(Fri), 2024. 13:30 -- 14:30 (Room 621)

• Title: A Polynomial Method for Counting Colorings of Sparse Graphs

• Abstract: Conjectures and results related to the number of colorings of families of planar graphs have a long history. In this talk we give an overview of this history and our results that generalize this from two perspectives - more general notions of colorings and more general families of sparse graphs.

The notion of S-labeling, where S is a subset of the symmetric group, is a common generalization of signed k-coloring, signed Z_k-coloring, DP-coloring, group coloring, and coloring of gained graphs that was introduced in 2019 by Jin, Wong, and Zhu. We present a unified and simple polynomial method for giving exponential lower bounds on the number of colorings of a sparse S-labeled graph. 

We apply these to give exponential lower bounds on the number of DP-colorings (and consequently, number of list colorings, and usual colorings) of families of planar graphs, and on the number of colorings of signed graphs. These lower bounds either improve previously known results, or are first known such results. 

This is joint work with Samantha Dahlberg and Jeffrey Mudrock.

2.  Jihyeug Jang (장지혁 성균관대학교) May 16(Thu), 2024. 16:30 -- 17:30

• Title: Lattice on permutation tableaux

• Abstract: In this talk we propose a poset on permutation tableaux and show that it is semidistributive and trim. This poset contains the Tamari lattice as a sublattice. We show that the tableaux of a given shape form an interval of this lattice. We also characterize its join and meet irreducibles and the elements belonging to the longest chains. We also present many further directions. This is a joint work with Sylvie Corteel and Baptiste Rognerud.

1. Woo-Seok Jung (정우석 서울시립대학교) April 8(Mon), 2024. 16:00 -- 17:00

• Title: Quasisymmetric functions and supercharacters

• Abstract: A supercharacter theory of a group G is a pair (cl(S), ch(S)) where cl(S) is a set partition of G and ch(S) is a set partition of the set of irreducible characters of G satisfying certain conditions. Each supercharacter theory gives rise to a subspace of the class function space of G, called the supercharacter function space. We identify the Hopf algebra QSym_C of quasisymmetric functions over the field of complex numbers as a direct sum of certain supercharacter function spaces.

As an application, we established product and coproduct rules for two bases of QSym_F of quasisymmetric functions over F, where F is either C(q,t) or C(q). One basis, {D_{alpha}(q,t)}, is derived from natural bases of supercharacter function spaces and yields notable bases of QSym_C upon appropriate specializations of q and t. The other basis consists of the quasisymmetric Hall-Littlewood functions introduced by Hivert.

This is joint work with Young-Tak Oh.

2023

7. Jaeseong Oh (오재성 연세대학교) September 14(Thu), 2023. 15:00 -- 16:00

• Title: $\alpha$-chromatic symmetric functions

• Abstract: We introduce a generalization of Stanley's chromatic symmetric function involving additional parameter $\alpha$. In this context, we formulate positivity conjectures and theorems for the $\alpha$-chromatic symmetric functions using two bases defined by binomial coefficients. This is based on ongoing cowork with Jim Haglund and Meesue Yoo.

6. Minho Cho (조민호 IBS ECOPRO) March 30(Thu), 2023. 11:00 -- 12:00

• Title: Strong Erd\H{o}s-Hajnal property on chordal graphs and its variants

• Abstract: A graph class $\mathcal{G}$ has the strong Erd\H{o}s--Hajnal property (SEH-property) if there is a constant $c=c(\mathcal{G}) > 0$ such that for every member $G$ of $\mathcal{G}$, either $G$ or its complement has $K_{m, m}$ as a subgraph where $m \geq \left\lfloor c|V(G)|\right\rfloor$. We prove that the class of chordal graphs satisfy SEH-property with constant $c = 2/9$.

On the other hand, a strengthening of SEH-property which we call the colorful Erd\H{o}s--Hajnal property was discussed in geometric settings by Alon et al.(2005) and by Fox et al.(2012). Inspired by their results, we show that for every pair $F_1, F_2$ of subtree families of the same size in a tree $T$ with $k$ leaves, there exists subfamilies $F'_1 \subseteq F_1$ and $F'_2 \subseteq F_2$ of size $\theta \left( \frac{\ln k}{k} \left| F_1 \right|\right)$ such that either every pair of representatives from distinct subfamilies intersect or every such pair do not intersect. Our results are asymptotically optimal.

Finally, we propose some questions on Erd\H{o}s-Hajnal type properties of various graph classes. Joint work with Andreas~Holmsen,~Jinha~Kim~and~Minki~Kim.

5. Morteza Hasanvand (Sharif University of Technology and Yokohama National University) March 23(Thu), 2023. 11:00 -- 12:00

• Title: The List Square Coloring Conjecture fails for cubic graphs and their line graphs

• Abstract: Kostochka and Woodall (2001) conjectured that the square of every graph has the same chromatic number and list chromatic number. In 2015 Kim and Park disproved this conjecture for non-bipartite and bipartite graphs. It was asked by several authors whether this conjecture holds for bipartite graphs with small degrees, claw-free graphs, or line graphs. In this talk, we introduce cubic counterexamples to this conjecture to solve three open problems posed by Kim and Park (2015), Kim, Kwon, and Park (2015), and Dai, Wang, Yang, and Yu (2018).

4. Sun Kim (김선 전북대학교) March 17(Fri), 2023. 16:00 -- 17:00

• Title: Ramanujan's modular équations and related partition identities

• Abstract: Ramanujan's modular equations of prime degrees 3, 5, 7, 11 and 23 are associated with elegant colored partition theorems. In 2005, S. O. Warnaar established a general identity which implies the modular equations of degrees 3 and 7. In this talk, we show a generalization of the remaining modular equations of degrees 5, 11 and 23. In addition, we discuss many partition identities associated with these modular equations and their generalizations. 

3. Kyeong-Dong Park (박경동 경상대학교) February 24(Fri), 2023. 14:00 -- 15:00

• Title: Colored fans of spherical varieties and classification of Gorenstein Fano group compactifications

• Abstract:  For a reductive algebraic group G, a normal G-variety is called spherical if it contains an open B-orbit, where B is a fixed Borel subgroup of G. The class of spherical varieties contains several important families which are studied independently, for example, toric varieties, flag varieties, horospherical varieties, symmetric varieties, and group embeddings. By the Luna-Vust theory, spherical varieties admit a simple combinatorial description in a given equivariant birational class. The normal equivariant embeddings of a given spherical homogeneous space are classified by combinatorial objects called colored fans, which generalize the fans appearing in the classification of toric varieties. As an application, I give a classification of Gorenstein Fano equivariant compactifications of semisimple complex Lie groups with rank two.
  
  

2.  Byung Hee An  (안병희 경북대학교) February 23(Thu), 2023. 17:00 -- 18:00 

•  Title: Homotopy invariants of graph braid groups

• Abstract: In this talk, I will present recent results on configuration spaces of graph braid groups including edge stabilizations and formalities, asymptotic homologies, and so on.

I will also introduce some questions and future directions.

1. Hyunsuk Moon  (문현석 고등과학원) February 16(Thu), 2023. 14:00 -- 15:00 

•  Title:  A new bound for the Real Waring rank of Monomials

• Abstract:  The Waring decomposition is the way of representing a homogeneous polynomial as the sum of power of linear forms. It is related to the tensor rank, secant variety and optimization problems. In this talk, we will introduce the known results about Waring rank and open problems related to it. Also, we will mainly consider the Waring rank of monomials over the real and the rational numbers. We give a new upper bound for it by establishing a way in which one can take a structured apolar set for any given monomial. Our bound is also lower than any other known general bounds for the real Waring rank.

2022

8.  Jaeho Shin (신재호 서울대학교) December 19(Mon), 2022. 15:00 -- 16:00 

•  Title:  The Circuit-Cocircuit Intersection Conjecture for Intersection Size $k \leq 6$

• Abstract: Oxley conjectured (1992) that if a matroid has a circuit-cocircuit intersection of size $k \geq 4$, then it has a circuit-cocircuit intersection of size $k-2$. In the first half of the talk, I will talk about the general theory of matroid and introduce necessary terminology. With this, in the second half, I will provide the sketch of the proof.

7. Stijn Cambie (IBS 기초과학연구원) November 16(Wed), 2022. 11:00 -- 12:00 

•  Title:  Packing Variants of Colourings

• Abstract: Graph colouring is an influential and classic topic in graph theory, which is related to e.g. frequency assignment, resource allocation and scheduling problems. There do exist many variants; next to the classical vertex colouring, among others, odd colouring and conflict-free colouring. For certain types of colourings, one can consider a list version. Here every vertex gets assigned its own list of possible colours. Dvorak and Postle also considered a correspondence version of list colouring, nowadays sometimes called DP-colouring.
  Sometimes it is natural to consider multiple of these and the best one can aim for is a packing of disjoint solutions/ list colourings. We investigate this natural strengthening, the list packing problem. This study was already suggested 25 years ago by Alon, Fellows and Hare.
  In the elementary talk, accompanied by many pictures, we give an introduction to the concept of packing of list-colourings and give some intuition behind it. Next, we present some of the basic and surprising results for list- and DP-colouring. Finally, we wonder about the question for variants which have not been explored before.

6. Seok Hyun Byun  (변석현 Clemson University) November 3(Thu), 2022. 09:00 -- 10:00

•  Title:  Various proofs of MacMahon’s Theorem and related problems

• Abstract: More than a hundred years ago, MacMahon showed that the number of boxed plane partitions is given by a simple product formula. It is well-known that the boxed plane partition is in bijection with several combinatorial objects. In the first part of the talk, we review these bijections and various proofs of MacMahon’s theorem using these bijections. One of the goals (of the first part) is to provide some enumeration techniques to audiences.

  In the second part of the talk, we focus on the lozenge tiling form of MacMahon’s theorem. One way to generalize the theorem is by counting the number of tilings of hexagonal regions with some defects (or holes). We review previous results on generalizations of MacMahon’s theorem or related problems in this direction. At the end of the talk, I will introduce some recent developments and explain how they relate to previous results.

5. Semin Yoo  (유세민 KIAS 고등과학원) October 6(Thu), 2022. 14:00 -- 15:00

•  Title:  Weak Bruhat interval modules of the $0$-Hecke algebras for genomic Schur functions

• Abstract: In 2017, Yong and Pechenik \cite{PY} introduced the \textit{genomic Schur function} as a natural deformation of the ordinary Schur function in the context of the K-theory of Grassmannians. They showed that genomic Schur functions consist of a basis of the ring of symmetric functions, although they are not Schur-positive in general. Recently, Pechenik \cite{Pe} proved that genomic Schur functions are expanded positively in terms of fundamental quasisymmetric functions, implying the possibility of the existence of nice $0$-Hecke modules corresponding to genomic Schur functions under quasisymmetric characteristics. Since the mid-2010s, there have been considerable attempts to provide a representation-theoretic interpretation of noteworthy quasisymmetric functions by constructing appropriate $0$-Hecke modules. Very recently, Jung, Kim, Lee, and Oh \cite{JKLO} introduced \emph{weak Bruhat interval modules} to provide a unified method to study those $H_{n}(0)$-modules. In this talk, we construct new $0$-Hecke modules whose quasisymmetric characteristics are the homogeneous components of genomic Schur functions and study their structural properties. Furthermore, we decompose our modules into weak Bruhat interval modules and find the projective cover of each summand of the decomposition. This is joint work with Young-Hun Kim.

4. Joonkyung Lee (이준경 Hanyang  University 한양대학교) August 19(Fri), 2022. 16:00 -- 17:00

•  Title:  Ramsey multiplicity and common graphs

• Abstract:  A graph $H$ is common if the number of monochromatic copies of $H$ in a 2-edge-colouring of the complete graph is minimised by the random colouring. Burr and Rosta, extending a famous conjecture by Erd\H{o}s, conjectured that every graph is common. The conjectures by Erd\H{o}s and by Burr and Rosta were disproved by Thomason and by Sidorenko, respectively, in the late 1980s. 

Collecting new examples for common graphs had not seen much progress since then, although a few more graphs are verified to be common by the flag algebra method or the recent progress on Sidorenko's conjecture. I will give an overview on both old and new results in the area. 

Based on joint work with Andrzej Grzesik, Jangsoo Kim, Sejin Ko, Bernard Lidický, and Jan Volec.

3. Cheolwon Heo (허철원 Sungkyunkwan  University 성균관대학교) July 05(Tue), 2022. 15:00 -- 16:00

•  Title:  The Complexity of the Matroid-homomorphism problems

• Abstract: In this talk, we introduce homomorphisms between binary matroids that generalize graph homomorphisms.

For a binary matroid $N$, we prove a complexity dichotomy for the problem $\rm{Hom}_\mathbb{M}(N)$ of deciding if a binary matroid $M$ admits a homomorphism to $N$. 

The problem is polynomial time solvable if $N$ has a loop or has no circuits of odd length, and is otherwise $\rm{NP}$-complete. 

We also get dichotomies for the list, extension, and retraction versions of the problem.

This is joint work with Hyobin Kim and Mark Siggers at Kyungpook National University.

2. Jinha Kim (김진하 IBS 기초과학연구원) May 24(Tue), 2022. 15:00 -- 16:00

•  Title:  Fractional Helly theorem for Cartesian products of convex sets

• Abstract: Helly's theorem and its variants show that for a family of convex sets in Euclidean space, local intersection patterns influence global intersection patterns. A classical result of Eckhoff in 1988 provided an optimal fractional Helly theorem for axis-aligned boxes, which are Cartesian products of line segments. Answering a question raised by B\'ar\'any and Kalai, and independently Lew, we generalize Eckhoff's result to Cartesian products of convex sets in all dimensions.
  In particular, we prove that given $\alpha \in (1-\frac{1}{t^d},1]$ and a finite family $\mathcal{F}$ of Cartesian products of convex sets $\prod_{i\in[t]}A_i$ in $\mathbb{R}^{td}$ with $A_i\subset \mathbb{R}^d$, if at least $\alpha$-fraction of the $(d+1)$-tuples in $\mathcal{F}$ are intersecting, then at least $(1-(t^d(1-\alpha))^{1/(d+1)})$-fraction of sets in $\mathcal{F}$ are intersecting. This is a special case of a more general result on intersections of $d$-Leray complexes. We also provide a construction showing that our result on $d$-Leray complexes is optimal. Interestingly, the extremal example is representable as a family of Cartesian products of convex sets, implying that the bound $\alpha>1-\frac{1}{t^d}$ and the fraction $(1-(t^d(1-\alpha))^{1/(d+1)})$ above are also best possible.
  This is based on the joint work with Debsoumya Chakraborti, Jaehoon Kim, Minki Kim and Hong Liu.

1. Hayan Nam (남하얀 Duksung Women's University 덕성여대) April 26(Tue), 2022. 15:30 -- 16:30

•  Title: Counting the number of certain partitions by using numerical semigroups

• Abstract : A partition of $n$ is an expression of $n$ as the sum of positive integers. By assigning non-negative numbers to the boundary of a partition's Ferrers diagram, we get the numerical set corresponding to the partition. Considering several invariants of numerical semigroups, we classify partitions under those invariants and count the number of partitions corresponding to numerical semigroups with fixed invariants. 

 2021

3. Jongbaek Song(송종백 KIAS 고등과학원) November 19(Fri), 2021. 16:00 -- 17:00

•  Title: f-vectors, h-vectors and toric varieties

• Abstract : The fundamental theorem in toric geometry establishes a one-to-one correspondence between the category of projective toric varieties and the category of lattice polytopes. This raises the question of how to read off various topological invariants of a toric variety from the corresponding lattice polytope. In this talk, we will look at the cohomology of a toric variety and how it relates to the face numbers of the associated polytope. This is based on the joint work with Seonjeong Park. 

2. Jaeseong Oh(오재성 KIAS 고등과학원) September 10(Fri), 2021. 16:00 -- 17:00

•  Title: A permutation interpretation of the transition matrix between the Specht and the $\mathfrak{sl}_2$ web base

• Abstract : We introduce a new class of permutations each of whose cycle is a variant of an André permutation. We show that these permutations provide a combinatorial interpretation of the transition matrix between the Specht and the $\mathfrak{sl}_2$ web bases of the irreducible $\mathfrak{S}_{2n}$-representation of shape $(n,n)$. We will discuss the enumerative properties of these permutations.
  This is based on the joint work with Byung-Hak Hwang and Jihyeug Jang.

1. Donghyun Kim(김동현 Sungkyunkwan  University 성균관대학교) August 6(Fri), 2021. 16:00 -- 17:00

•  Title: Inhomogeneous TASEP on a ring and Schubert polynomials 

• Abstract : The inhomogeneous totally asymmetric simple exclusion process (or TASEP) is a Markov chain on the set of permutations, in which adjacent numbers i and j swap places at rate $x_i - y_j$ if the larger number is clockwise of the smaller. Conjecturally, steady state probabilities can be written as a positive sum of (double) Schubert polynomials. We will start by giving some background on this model, including Cantini's result showing that the inhomogeneous TASEP is a solvable lattice model. We will then use his result to show that a large number of states
        -- those corresponding to the "evil-avoiding" permutations (permutations avoiding patterns 2413, 4132, 4213, 3214)
      -- have steady state probabilities which are proportional to a product of Schubert polynomials.
  Based on joint work with Lauren Williams.

2020

5. Ilkyoo Choi(최일규 Hankuk University of Foreign Studies 한국외국어대학교) December 1(Tue), 2020. 09:30 -- 10:30

•  Title: Flexibility of Planar Graphs 

• Abstract : Oftentimes in chromatic graph theory, precoloring techniques are utilized in order to obtain the desired coloring result. For example, Thomassen's proof for 5-choosability of planar graphs actually shows that two adjacent vertices on the same face can be precolored. In this vein, we investigate a precoloring extension problem formalized by Dvorak, Norin, and Postle named {\it flexibility}. Given a list assignment $L$ on a graph $G$, an {\it $L$-request} is a function on a subset $S$ of the vertices that indicates a preferred color in $L(v)$ for each vertex $v\in S$. A graph $G$ is {\it $\varepsilon$-flexible for list size $k$} if given a $k$-list assignment $L$ and an $L$-request, there is an $L$-coloring of $G$ satisfying an $\varepsilon$-fraction of the requests in $S$. We survey known results regarding this new concept, and prove some new results regarding flexibility of planar graphs.
  
  

4.  Taekgyu Hwang (황택규 Ajou University 아주대학교) November  3(Tue), 2020. 16:00 -- 17:00

•  Title: The Gromov width of graph associahedra

• Abstract : The Gromov width of a symplectic manifold is an invariant that measures the maximal size of a ball which can be symplectically embedded. I will explain a formula for the Gromov width of  symplectic toric manifolds obtained from connected simple graphs. This talk is based on a joint work with Suyoung Choi.

3.   JinHoo Ahn (안진후 KIAS 키아스) October 8(Thu), 2020. 15:00 -- 16:00

•  Title : Model theoretic approaches on graphs and groups

• Abstract : Model theory is a branch of mathematical logic which studies mathematical theories and structures in the language of first order logic and classifies them into categories of model theoretic properties. It concerns almost all structures in math such as algebraic structures like groups and fields, graphs, and even set theory. Surprisingly, in some cases, it is enough to observe a model theoretic property of theory only in the area of graphs or groups when checking the existence of a counterexample. This is possible because there is a method, so called Mekler’s construction, to produce graphs and groups having the same model theoretic properties from any given structure. In this talk, we explain how to build such graphs and groups in two parts; first, from any structure to a graph, then a graph to a group.

2.   ByungHak Hwang(황병학 Sungkyunkwan  University 성균관대학교) September  22(Tue), 2020. 14:00 -- 15:00

• Title : Chromatic quasisymmetric functions and noncommutative P-symmetric functions

• Abstract : Chromatic quasisymmetric functions are purely combinatorial objects, introduced by Stanely and Shareshian--Wachs, but they arise in algebra and geometry.  Since their connection with Hessenberg varieties was established, they have received more and more attention. In combinatorial viewpoint, the most important question concerning them is about positivity. Especially the conjecture that chromatic quasisymmetric functions are e-positive is one of the most famous long-standing open problems in algebraic combinatorics. In this talk, I introduce a noncommutative counterpart of chromatic quasisymmetric functions, and show various positivity of chromatic quasisymmetric functions, including some partial results for the $e$-positivity conjecture. I also present a refined $e$-positivity conjecture.

1.   Meesue Yoo (류미수 Chungbuk   University  충북대학교) May 27 (Wed), 2020.  15:00 -- 16:00

• • Title:   Counting standard barely set-valued tableaux of shifted shapes 

  • Abstract : In this work, we prove the CDE property of the trapezoidal shifted shapes by counting standard barely set-valued tableaux via q-integral method.  A standard barely set-valued tableau of shape $\lambda$ is a filling of the Young diagram $\lambda$ with integers $1,2,\dots,|\lambda|+1$ such that the integers are increasing in each row and column, and every cell contains one integer except one cell that contains two integers. Counting standard barely set-valued tableaux is closely related to proving Young's lattice has the coincidental down-degree expectations (CDE) property. Using $q$-integral techniques we give a formula for the number of standard barely set-valued tableaux of arbitrary shifted shapes. We then prove a conjecture of Reiner, Tenner and Yong on the CDE property of trapezoidal shifted shape $(n,n-2,\dots,n-2k)$. This is joint work with Jang Soo Kim and Michael Schlosser. 

2019

12. Anton A. Ayzenberg(National Research University Higher School of Economics,Russia) November 12 (Tue), 2019. 17:00 -- 18:00

• • Title: Quoric manifolds and torus actions of complexity one.

  • If the compact torus T^k acts effectively on a smooth 2n-manifold X, with nonempty finite set of fixed points, we call the number n-k the complexity of the action. We are interested in actions of complexity one and the structure of their orbit spaces. I will give a brief overview of the results we have so far and concentrate on a particular class of examples: 8-dimensional quoric manifolds. A motivating example is the action of T^3 on HP^2.
    Quoric manifolds were introduced by Jeremy Hopkinson in his doctoral thesis, and the term "quoric" stands as the abbreviation for "quaternionic quasitoric". These are 4n-manifolds with "locally standard actions'' of Sp(1)^n, the quaternionic torus, such that the orbit space is a simple polytope. Many notions from toric topology have their analogues in quaternionic case, however, since the acting group is noncommutative, one should proceed with certain care in this generalization. I think, there are many natural questions about quoric manifolds, topological, geometrical, and combinatorial, which haven't been answered yet.

11. Ji Sun Huh(허지선 Sungkyunkwan  University 성균관대학교) October 17 (Thu), 2019. 15:00 -- 16:00

• • Title: A bijection between simultaneous core partitions and rational Motzkin paths

  • For a positive integer $t$, a partition $\lambda$ is a $t$-core partition if it has no box of hook length $t$. The study of simultaneous core partitions began with the work of Anderson.  In this talk, we first propose an $(s+d, d)$-abacus for $(s,s+d,\dots,s+pd)$-core partitions, and then establish a bijection between the $(s,s+d, \dots, s+pd)$-core partitions and rational Motzkin paths of type $(s+d, -d)$. This bijection gives a lattice path interpretation of the $(s,s+d, \dots, s+pd)$-core partitions and counts them with a closed formula. Also we enumerate $(s, s+1, \dots, s+p)$-core partitions with $k$ corners and self-conjugate $(s,s+1,\dots, s+p)$-core partitions.

10. Xin Zhang (Xidian University, China) September 30 (Mon), 2019.  15:00--16:00

• • Title: 1-Planar Graphs: from Structure to Coloring

  • Abstract:  A graph is 1-planar if it admits a drawing in the plane so that each edge is crossed by at most one another edge. The notion of 1-planarity is among the most natural and most studied generalizations of graph planarity and the study of 1-planar graphs has driven increasing attention in the areas of graph theory, graph algorithms, graph drawing, and computational geometry. In this talk, I will first show some structural and chromatic results on 1-planar graphs and its subclasses including NIC-planar graphs, IC-planar graphs and outer-1-planar graphs. Meanwhile, I will introduce how to apply the discharging method when we are working on non-planar and 1-planar graphs.

9. Ringi Kim (김린기 KAIST) July 10 (Wed), 2019. 14:00--15:00

• • Title: Decomposition of a planar graph into a forest and a 3-choosable subgraph 

  • Abstract: Recently, Grytczuk and Zhu proved that every planar graph G contains a matching M such that G-M is 4-choosable. In this talk, we show that every planar graph G contains a forest F such that G-E(F) is 3-choosable. We also show that a forest cannot be replaced by a subgraph of maximum degree at most 3 or a star forest. This is joint work with Seog-Jin Kim and Xuding Zhu.

8. Minki Kim (김민기 Technion) May 16 (Thu), 2019. 16:30--17:30

• • Title: Rainbow independent sets in graphs

  • Abstract: Let $F = (F_1, \ldots, F_m)$ be a collection of (not neccessarily distinct) sets. A (partial) rainbow set for $F$ is a set of the form $R = \{x_{i_1}, \ldots, x_{i_k}\}$ of distinct elements, where $1 \leq i_1 < \cdots < i_k \leq m$ and $x_{i_j}$ is an element of $F_{i_j}$. We are interested in the following question: given sufficiently many independent sets of size $n$ in a graph belonging to a certain class, there exists a rainbow independent set of size $n$. In this talk, I will present our recent results on this question, mainly about $H$-(induced) free graphs. This is joint work with Ron Aharoni, Joseph Briggs and Jinha Kim. 

7. Eunjeong Lee (이은정 IBS-CGP 기초과학연구원 기하학 수리물리 연구단), May 1 (Wed), 2019. 16:00--17:00

• • Title: Smooth toric Richardson varieties

  • Abstract: The standard action of a complex torus $T = (\mathbb{C}^{\ast})^n$ on the complex vector space~$\mathbb{C}^n$ induces an action of $(\mathbb{C}^{\ast})^n$ on the full flag variety $\mathcal{F}\ell(\mathbb{C}^n)$.  The set of fixed points can be identified with the set $\mathfrak{S}_n$  of permutations on~$\{1,2,\dots,n\}$. For given permutations $v$ and $w$ in $\mathfrak{S}_n$ with $v \leq w$, we define the Richardson variety $X_w^v$, which is a $T$-invariant subvariety of the full flag variety $\mathcal{F}\ell(\mathbb{C}^n)$ and the fixed point set is identified with permutations $\{z  \in \mathfrak{S}_n \mid v \leq w \leq w\}$. The moment map image of the Richardson variety $X^v_w$ is the convex hull of points $(z(1),\dots,z(n)) \in \mathbb{Z}^n$ for permutations $v^{-1} \leq z \leq w^{-1}$.  We study sufficient condition on $v$ and $w$ such that the Richardson variety $X_w^v$ to be a smooth toric variety. In particular, we show that smooth toric Richardson varieties are Bott manifolds.  This is joint work with Mikiya Masuda and Seonjeong Park.

6. Eun-Kyung Cho (조은경 Hankuk University of Foreign Studies 한국외국어대학교), March 15 (Fri), 2019. 13:30 -- 14:30

• • Title: A characterization of the groups whose proper Schur rings are commutative

  • Abstract:  Let $\mathds{Z}[G]$ be the group ring of a group $G$ over $\mathds{Z}$. A Schur ring $\mathscr{A}$ over $G$ is a subring of $\mathds{Z}[G]$ which is determined by a certain partition of $G$ (called a Schur partition). An $\mathscr{A}$-subgroup is a subgroup of $G$ that is a union of some elements from the Schur partition corresponding to $\mathscr{A}$. We define a Schur ring $\mathscr{A}$ to be {\it Dedekind} if the formal sum of every $\mathscr{A}$-subgroup is contained in the center of $\mathscr{A}$. Then the class of Dedekind groups is related to the class of Dedekind Schur rings in the sense that all the Schur rings over a Dedekind group are Dedekind Schur rings. A Schur ring is {\it proper} if it is not the group ring. We prove in this talk that all the proper Schur rings over a group $G$ are Dedekind Schur rings if and only if $G$ is a Dedekind group or a dihedral group of order $8$ or $2$ times a Fermat prime. As a corollary of this result, we prove that all the proper Schur rings over a group $G$ are commutative if and only if $G$ is an abelian group, the quaternion group, or a dihedral group of order $8$ or $2$ times a Fermat prime. Also, we prove that all the proper Schur rings over a group $G$ are symmetric if and only if $G$ is a Boolean group or a cyclic group of order $4$ or a Fermat prime.

5. Jongbaek Song (송종백 KAIST 카이스트), February 18 (Mon), 2019. 16:30 -- 17:30

• • Title: The q-CW structure on singular toric varieties and its application to Schubert Calculus

  • Abstract: The CW-complex structure of certain spaces can be too complicated for computational purposes. In this talk, we discuss the concept of a q-CW structure which enables us to detect torsion in its integral cohomology. Then we apply this to a certain singular toric variety, which gives  us some information about Schubert calculus of a full flag variety. 

4. Yunhyung Cho (조윤형 Sungkyunkwan University 성균관대학교), February 11 (Mon), 2019. 16:30 -- 17:30

• • Title: Flag varieties, polytopes, and integrable systems

  • Abstract: In this talk, we discuss a certain integrable system, called a Gelfand-Cetlin system defined on a flag variety of type A, B, and D. We first illustrate the similarity and the difference between a GC system and a toric moment map, and explain an algorithm which allows us to “read off” the topology of a Gelfand-Cetlin fiber from the combinatorics of its image, called a Gelfand-Cetlin polytope. We also discuss so-called a string polytope, which can be thought of as a generalization of a Gelfand-Cetlin polytope, and related open questions. This is based on joint work with Yoosik Kim, and partially with Yoosik Kim, Eunjeong Lee and Kyeong-Dong Park.

3. [내부세미나] Seonjeong Park (박선정 Ajou University 아주대학교), February 11 (Mon), 2019. 14:00 -- 15:30

• • Title: 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 Bruhat interval polytopes and some open problems. 

2. Hanchul Park (박한철 Jeju National University 제주대학교),  January 25 (Fri), 2019. 15:00 -- 16:00

• • Title: Simplicial cohomology and topology of real toric manifolds 2

  • Abstract: Toric topology is the study of topological spaces with particularly good torus symmetry. Those spaces, called (real) toric spaces, can be represented by a simplicial sphere $K$ and additional combinatorial information. Real toric manifolds and real moment-angle manifolds are examples of real toric spaces.
    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.
    In this series of talks, we study the cup product for $H^*(M^\mathbb{R};R)$ recently computed by Choi and the speaker. To do this, we briefly review simplicial (co)homology and Hochster formula for real moment-angle complexes, and explain the method to compute the cup product using the simplicial cohomology.

1. Hanchul Park (박한철 Jeju National University 제주대학교),  January 24 (Thu), 2019. 15:00 -- 16:00

• • Title: Simplicial cohomology and topology of real toric manifolds 1

  • Abstract: Toric topology is the study of topological spaces with particularly good torus symmetry. Those spaces, called (real) toric spaces, can be represented by a simplicial sphere $K$ and additional combinatorial information. Real toric manifolds and real moment-angle manifolds are examples of real toric spaces.
    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.
    In this series of talks, we study the cup product for $H^*(M^\mathbb{R};R)$ recently computed by Choi and the speaker. To do this, we briefly review simplicial (co)homology and Hochster formula for real moment-angle complexes, and explain the method to compute the cup product using the simplicial cohomology.

2018

9. Yandong Bai (Northwestern Polytechnical University, China), December14 (Fri), 2018. 15:30 -- 16:20

• • Title: Bipartitions of digraphs

  • Abstract:  Stiebitz asked in 1995 the following question. For any two positive integers $s$ and $t$, does there exist a finite integer $f(s,t)$ such that every digraph with minimum outdegree at least $f(s,t)$ admits a bipartition $(A,B)$ such that $A$ induces a subdigraph with minimum outdegree at least $s$ and $B$ induces a subdigraph with minimum outdegree at least $t$? In this talk, we give an affirmative answer for tournaments, multipartite tournaments and digraphs with bounded maximum indegree. In particular, we show that for every $\varepsilon$ with $0<\varepsilon<1/2$, there exists an integer $\delta_{0}$ such that every tournament $T$ with minimum outdegree at least $\delta_{0}$ admits a bipartition $(A,B)$ satisfying that $-1\leqslant |A|-|B| \leqslant 1$ and each vertex of $T$ has at least $(1/2-\varepsilon)$ of its outneighbors in both $A$ and $B$.

8. Tatsuya Horiguchi (Osaka University, Japan), October 18 (Thu), 2018. 15:00 -- 16:00

• • Title: Hessenberg varieties and hyperplane arrangements

  • Abstract:  Hessenberg varieties are subvarieties of flag varieties. Its topology makes connection with many research areas such as geometric representation of the Weyl group, the quantum cohomology of flag varieties, the chromatic quasi-symmetric function in graph theory, and hyperplane arrangements. In this talk, I will talk about the connection between Hessenberg varieties and hypeplane arrangements. This is joint work with Takuro Abe, Mikiya Masuda, Satoshi Murai, and Takashi Sato. 

7. Boram Park (박보람 Ajou University 아주대학교), August 21 (Tue), 2018. 16:30 -- 17:00

• • 내부세미나 (Some results on acyclic coloring in graph theory)

6. Suyoung Choi (최수영 Ajou University 아주대학교), August 21 (Tue), 2018. 16:00 -- 16:30

• • 내부세미나 (Acyclic coloring in toric topology)

5. Henry Liu (Sun Yat-sen University, China), August 21 (Tue), 2018. 15:00 -- 16:00

• • Title:  Degree powers in graphs with a forbidden forest

  • Abstract: (PDF) For a graph $G$ with degree sequence $d_1,\dots , d_n$, and for a positive integer $p$, let $e_p(G)=\sum_{i=1}^n d_i^p$. In 2000, Caro and Yuster [A Tur\'an type problem concerning the powers of the degrees of a graph, Electron.~J.~Combin.~7 (2000), R47] introduced the following Tur\'an type problem: \emph{Given a positive integer $p$ and a graph $H$, determine the function $\textup{ex}_p(n,H)$, which is the maximum value of $e_p(G)$ taken over all graphs $G$ on $n$ vertices that do not contain $H$ as a subgraph.} Obviously, we have $\textup{ex}_1(n,H)=2\textup{ex}(n,H)$, where $\textup{ex}(n,H)$ denotes the classical Tur\'an function. Previous results on this problem, obtained by various authors, include the determination of the function $\textup{ex}_p(n,H)$ when $H$ is a complete graph, a cycle, a path, and a star. In this talk, we shall present some new results for the function $\textup{ex}_p(n,H)$ when $H$ is a certain type of forest, namely, a linear forest, a star forest, and a broom (i.e., a path with a star at one end). This talk is based on joint work with Yongxin Lan (Nankai University), Zhongmei Qin (Chang'an University) and Yongtang Shi (Nankai University).

4. Soumen Sarkar (Indian Institute of Technology Madras, India), June 12 (Tue), 2018. 16:30 -- 17:30

• • Title: Higher equivariant and invariant topological complexity

  • Abstract: In this talk I will introduce the concepts of higher equivariant and invariant topological complexity and discuss their properties. Then I will compare them with equivariant LS-category. We give lower and upper bounds for these new invariant. This is a joint work with Marzieh Bayeh.

3. Hong Liu (University of Warwick, UK), March 29 (Thu), 2018. 16:00 -- 17:00

• • Title : Edges not in any monochromatic copy of a fixed graph

  • Abstract : (Joint work with Oleg Pikhurko and Maryam Sharifzadeh) For a sequence $(H_i)_{i=1}^k$ of graphs, let $\rm{nim}(n;H_1,\ldots, H_k)$ denote the maximum number of edges not contained in any monochromatic copy of $H_i$ in colour $i$, for any colour $i$, over all $k$-edge-colourings of~$K_n$. When each $H_i$ is connected and non-bipartite, we introduce a variant of Ramsey number that determines the limit of $\rm{nim}(n;H_1,\ldots, H_k)/{n\choose 2}$ as $n\to\infty$ and prove the corresponding stability result. Furthermore, if each $H_i$ is what we call \emph{homomorphism-critical} (in particular if each $H_i$ is a clique), then we determine $\rm{nim}(n;H_1,\ldots, H_k)$ exactly for all sufficiently large~$n$. The special case $\rm{nim}(n;K_3,K_3,K_3)$ of our result answers a question of Ma. For bipartite graphs, we mainly concentrate on the two-colour symmetric case (i.e., when $k=2$ and $H_1=H_2$). It is trivial to see that $\rm{nim}(n;H,H)$ is at least $\rm{ex}(n,H)$, the maximum size of an $H$-free graph on $n$ vertices. Keevash and Sudakov showed that equality holds if $H$ is the $4$-cycle and $n$ is large; recently Ma extended their result to an infinite family of bipartite graphs. We provide a larger family of bipartite graphs for which $\rm{nim}(n;H,H)=\ex(n,H)$. For a general bipartite graph $H$, we show that $\rm{nim}(n;H,H)$ is always within a constant additive error from ex(n,H), i.e.,~min(n;,H,H)=ex(n,H)+OH(1).

2. Sun-Young Nam (남선영 KIAS 고등과학원), February 8 (Thu), 2018. 16:00 -- 17:00

• • Title : Combinatorial interpretations for S3-symmetry of shifted Littlewood-Richardson coefficients

  • Abstract : The shifted Littlewood-Richardson coefficient is the structure constant appearing in the Schur P-expansion of the product of two Schur P-polynomials. The purpose of this talk is to give a combinatorial interpretation of an S3-symmetric rule that shifted Littlewood-Richardson coefficients satisfy. To begin with, we briefly introduce classical theory of symmetric polynomials: Young tableaux, Schur polynomials and Littlwood-Richardson coefficients. We then introduce a shifted analogue of them. Combining it with theory of shifted jeu de taquin developed by Sagan and Worley, we conclude that shifted Littlewood-Richardson coefficients has S3-action.

1. Ringi Kim (김린기 KAIST), January 17 (Wed), 2018. 14:00 -- 15:00

• • Title : The domination number of a tournament

  • Abstract : For a tournament T, a vertex set S in V(T) is a dominating set if every vertex outside of S has an in-neighbor in S. The domination number of T is the size of a minimum dominating set of T. In this talk, I will introduce conjectures and recent results about domination number.

2017

4. SangWook Kim (김상욱 Chonnam National University 전남대학교), December 8 (Fri), 2017. 14:00 -- 15:00

• • Title : Posets and complete intersections

  • Abstract : The columns of the incidence matrix of a connected acyclic directed graph D generate an affine semigroup under addition. If D has v vertices and e arcs, then the nullspace of the incidence matrix has dimension e-v+1.  The Laurent monomials ti tj-1 corresponding to arcs (i,j) of D generate a multiplicative semigroup SD. The kernel of the homomorphism ϕ: [u1, ... , ue] →  [t1, ... , tv, t1-1, ... , tv-1], where ϕ(ur)= ti tj-1 if ur= (i,j), is called the {toric ideal} ID. It is known that ID is generated by binomials and that a minimal generating set contains at least e-v+1 binomials. If ID is generated by exactly e-v+1 binomials, then D is called a {complete intersection}.  In this talk, we show that the semigroup ring corresponding to the Hasse diagram of a poset P is a complete intersection if and only if the Hasse diagram of P can be obtained from unicyclic posets by repeated gluings along a chains. If time permits, we also discuss type B and C analogues.  This is a joint work with Walter Morris.     

3. Henry Liu (Central Sough University, China), June 7 (Wed), 2017. 15:00 -- 16:00   

• • Title : Connected subgraphs in edge-coloured graphs

  • Abstract : Whenever the edges of a graph G on n vertices are coloured with r colours, on how many vertices can we find a monochromatic, connected subgraph? In this talk, we consider this Ramsey theory type question and its variants, such as replacing "monochromatic" by "s-coloured" (for some s ≤  r); asking for a specific type of connected subgraph; and putting a condition on the r-colouring of G. Many open questions will be presented. This talk is based on a joint survey with Shinya Fujita (Yokohama City University, Japan) and Colton Magnant (Georgia Southern University, USA).

2. Tadashi Sakuma (Yamagata University, Japan), January 16 (Mon), 2017. 16:30 -- 17:15

• • Title : Similarities and dissimilarities between the blocking and anti-blocking polyhedra

  • Abstract : The study of similarities and dissimilarities between the blocking and anti-blocking polyhedra began with a series of celebrated papers by Fulkerson (1970, 1971, 1972), and it has grown up a mature theory by significant contributions of Lehman, Lovász, Padberg, and others in 1970s and 1980s. Even today, this theory still shows a big progression such as the perfect graph theorem of Seymour et al. (2006). In this paper, we survey the current status of this research field with a focus on the conjecture of Conforti & Cornuéjols and the conjecuture of Grinstead.

1. Shinya Fujita (Yokohama City University, Japan), January 16 (Mon), 2017. 15:30 -- 16:15

• • Title : Color degree and monochromatic degree conditions for short properly colored cycles

  • Abstract : For an edge-colored graph, its minimum color degree is defined as the minimum number of colors appearing on the edges incident to a vertex and its maximum monochromatic degree is defined as the maximum number of edges incident to a vertex with a same color. A cycle is called properly colored if every two of its adjacent edges have distinct colors. In this work, we give a complete solution to a problem on the minimum color degree condition for the existence of properly colored cycles. We also introduce some new results on the minimum color degree and maximum monochromatic degree conditions for an edge-colored complete graph to contain properly colored triangles, for an edgecolored complete bipartite graph to contain properly colored cycles of length 4, and those passing through a given vertex or edge, respectively. This is joint work with Ruonan Li (University of Twente) and Shenggui Zhang (Northwestern Polytechnical University).

2016

13. Ji Sun Huh (허지선 Yonsei University 연세대학교), September 19 (Mon), 2016. 15:00 -- 16:00

• • Title : Strong semi-complete digraphs

  • Abstract : A digraph is strongly connected if for all pair x, y, distinct vertices, has a path from x to y. A simple digraph is semi-complete if for every pair x and y of distinct vertices at least one of the arcs (x,y), (y,x) is present. In this talk, we give a combinatorial interpretation of a relation between the number of labeled strong semi-complete digraphs and the number of labeled colored graphs.

12. Rinki Kim (김린기 University of Waterloo, Canada), September 7 (Wed), 2016. 15:00 -- 16:00

• • Title : Unavoidable subtournaments in tournaments with large chromatic number

  • Abstract : For a tournament T, the chromatic number of T is the minimum number of transitive sets with union V(T). We say a set ℋ of tournaments is heroic if there exists c such that every tournament excluding all members of ℋ has chromatic number at most c. Berger et al. explicitly characterized all heroic sets of size one. In this talk, we study heroic sets of size two.

11. Hanchul Park (박한철 KIAS 고등과학원), August 18 (Thu), 2016. 17:00 -- 17:50

• • Title : Simplicial wedges and classification of toric varieties

  • Abstract : In this talk, we review the concept of the simplicial wedge construction and characteristic maps, and explain the algorithm to find all toric varieties of given Picard number. This is an ongoing work jointly with Suyoung Choi.

10.Matthias Franz (University of Western Ontario, Canada), August 18 (Thu), 2016. 16:00 -- 16:50

• • Title : Big polygon spaces

  • Abstract : Polygon spaces are configuration spaces of polygons with prescribed edge lengths. We present a related family of compact orientable manifolds, called big polygon spaces. They come with a canonical torus action, whose fixed point set is a polygon space. Big polygon spaces are particularly interesting because they provide the only known examples of maximal syzygies in equivariant cohomology. We will therefore briefly review the theory of syzygies in equivariant cohomology and its relation to the equivariant Poincaré pairing and the "GKM method".

9. Shizuo Kaji (Yamaguchi University, Japan), March 31 (Thu), 2016. 18:00 -- 19:00

• • Title : Computational approach of real toric variety

  • Abstract : In the talk, we present how to use the maple package for computing the rational Betti number of real toric varieties associated to Weyl chambers.

8. Donghun Lee (이동헌 삼성전자, Princeton University, USA), March 29 (Thu), 2016. 16:30 -- 17:30

• • supported by Ajou center for mathematics in industry

  • Title : 수학적으로 알파고를 바라보기

  • Abstract : 알파고 사건으로 한국에서 다양한 관점의 논란과 토의가 이루어졌지만, 유달리 그 수학적인 함의를 다루지는 못한 것 같다. 수학의 관점에서 알파고라는 바둑 프로그램 안에 어떠한 사실과 가정이 들어가 있는지, 인공지능 연구자로서 간추려낸 알파고의 수학을 살펴보도록 하자. 인공지능에도 수학은 들어가 있다.

7. Suyoung Choi (최수영 Ajou University 아주대학교), March 8 (Thu), 2016. 17:00 -- 18:00

• • 내부 세미나 (Type B representation on the first cohomology)

  • Abstract : We study the first cohomology group of the real toric variety corresponding to Weyl chmaber of Type B.

6. Michitaka Furuya (Tokyo University of Science, Japan), February 29 (Mon), 2016. 16:00 -- 16:50

• • Title : Some approaches for comparing rainbow domination numbers

  • Abstract : Let k ≥ 1 be an integer, and set [k]:={1, … ,k}. Let G be a graph. A function f : V(G) → 2[k] is a k-rainbow dominating function (or k-RDF) of G if ∪y ∈ NG(x)f(y)=[k] for all x ∈ V(G) with f(x)= ∅ . The minimum weight w(f):= ∑x ∈ V(G)|f(x)| of a k-RDF f of G is called the k-rainbow domination number and denoted by Γrk(G). It has been known that every graph G satisfies Γrk(G) ≤ |V(G)|, and for k ≥ 4, the bound is sharp because Γrk(Pn)=n. In particular, when we focus on upper bounds for the rainbow domination, there is no difference between Γrk and Γrk' with k ≠ k'. In this talk, we give some approaches for finding potentially difference.

5. Kenta Ozeki (National Institute of Informatics, Japan), February 29 (Mon), 2016. 15:00 -- 15:50

• • Title : Polychromatic colorings of plane graphs and graphs on surfaces

  • Abstract : For a graph G on a surface (or on the plane), a polychromatic k-coloring of G is a vertex k-coloring such that there appear all k colors in every face of G. In this talk, I will show several results from the case k = 3 or 4, including a necessary and sufficient conditions for the case of quadrangulations. This is joint work with Atsuhiro Nakamoto (Yokohama National University) and Kenta Noguchi (Tokyo Denki University).

4. Soojin Cho (조수진 Ajou University 아주대학교), February 12 (Fri), 2016. 15:30 -- 17:30

• • 내부 세미나 (Character of representation)

  • Abstract : We study the character of representations.

3. Eunjeong Lee (이은정 KAIST), February 3 (Wed), 2016. 16:00 -- 17:00

• • Title : A combinatorial description of untwistedness of a Grossberg--Karshon twisted cube

Abstract : Let G be a complex semisimple simply connected Lie group. Let λ be a dominant weight for G and Ι=(i1, i2, …, in) a word decomposition for an element w =si1 si2 … sin of the Weyl group of G, where the si are the simple reflections. In the 1990s, Grossberg and Karshon introduced a virtual lattice polytope associated to λ and Ι, which they called a twisted cube, whose lattice points encode the multiplicities of a certain G-representation associated to λ and w. We introduce the notion of hesitant λ-walks and then prove that the associated Grossberg--Karshon twisted cube is untwisted precisely when Ι is hesitant-λ-walk-avoiding. This is joint work with M. Harada.

2. Li Cai (Chinese Academy of Science, China), January 25 (Mon), 2016. 17:00 -- 18:00

• • Title : On the integral cohomology of a small cover

  • Abstract : A small cover is a closed manifold admitting a special 2-torus action, such that the quotient space is a simple convex polytope. The mod 2 cohomology of a small cover has a presentation as the quotient of the associated Stanley-Reisner ring, which is Cohen-Macaulay (over the field of mod 2 integers), by a linear system of parameters. In this talk, we shall analyse the 2-torsion elements in the integral cohomology group of a small cover, using Bockstein homomorphisms.

1. Suyoung Choi (최수영 Ajou University 아주대학교), January 21 (Wed), 2016. 15:30 -- 17:30

• • 내부 세미나 (Sn+1-action on the toric variety associated to Weyl group of type An)

  • Abstract : We discuss the Sn+1-action on the complex and real toric varieties associated to Weyl group of type An

2015

14. Shizuo Kaji (Yamaguchi University, Japan), December 31 (Thu), 2015. 15:00 -- 16:00

• • supported by Ajou center for mathematics in industry

  • Title : A topological algorithm for shape deformation in computer graphics

  • Abstract : In computer graphics, mathematics has been a fundamental toolbox. The Navier-Stokes equations is indispensable for generating clouds and fires, reproducing kernel Hilbert space is used for mixing different facial expressions, and simplicial complex and piecewise linear map (PL-map, in short) provide a natural framework for shape manipulation. In this talk, I will discuss an algorithm to blend/deform shapes based on PL-map and Lie theory. A shape is represented by a simplicial complex and its deformation by a PL-map. Then, creating an animation boils down to finding a nice path in the space of 3-dimensional PL-maps.

13. Seonhwa Kim (김선화 IBS-CGP, 기하학 수리물리 연구단, Institute for Basic Science), December 31 (Thu), 2015. 11:00 -- 12:30

• • Title : Volume of hypercubes clipped by hyperplanes and combinatorial identities

  • Abstract : There was an elegant expression for the volume of hypercube [0,1]n clipped by a hyper-plane. We generalize the formula to the case of more than one hyperplane. Furthermore we derive several combinatorial identities from the volume expressions of clipped hyper-cubes.

12. Seonjeong Park (박선정 NIMS 국가수리과학연구소), November 3 (Tue), 2015. 17:00 -- 18:00

• • Title : Real toric manifolds over pseudograph associahedra

  • Abstract : A graph associahedron is a simple convex polytope whose facets correspond to proper connected induced subgraphs. Recently, S. Choi and H. Park computed the rational Betti numbers of real toric manifolds over graph associahedron. In this talk, we focus on a pseudograph associahedron which was introduced by Carr, Devadoss, and Forcey as a generalization of a graph associahedron. And then we discuss how to compute the rational Betti numbers of real toric manifolds over pseudograph associahedra.

11. Dong Ho Moon (문동호 Sejong University 세종대학교), October 19 (Mon), 2015. 17:00 -- 18:00

• • Title : A Schur-Weyl Duality Approach to Walking on Representation Graphs

  • Abstract : A representation graph is a (directed) Cayley graph which encodes the representation theory of a group. I will present how we study combinatorics of representation graphs of finite groups using Schur-Weyl dualities.

10. Kang-Ju Lee (이강주 Seoul National University 서울대학교), October 6 (Tue), 2015. 17:45 -- 18:45

• • Title : Path-intersection matrix and applications to network

  • Abstract : (LINK)

9. Atsuhiro Nakamoto (Yokohama National University, Japan), September 15 (Tue), 2015. 17:00 -- 18:00

• • Title : Domination number of graphs on surfaces

  • Abstract : Let G be a graph and let S be a subset of the vertex set of G. We say that S is a dominating set of G if for each vertex v of G, v or a neighbor of v is contained in S. The domination number of G, denoted by γ(G), is the minimum cardinality of all dominating sets of G.

  • Finding the domination number of a given graph is a classical and important problem in graph theory, and many people are interested in it. 

  • In our talk, we introduce several results on dominating sets of planar graphs and graphs embeddable on fixed surfaces. In particular, we describe some elegant proofs for them using a graph coloring, and discuss whether they can be used for solving other related problems.

8. Stephanie van Willigenburg (Univerisity of British Columbia, Canada), June 30 (Tue), 2015. 11:00 -- 12:00

• • Title : Maximal supports and Schur-positivity among connected skew shapes

  • Abstract : The Schur-positivity order on skew shapes B and A is denoted by B<A if the difference of their respective Schur functions is a positive linear combination of Schur functions. It is an open problem to determine those connected skew shapes that are maximal with respect to this ordering.  In this talk we see that to determine the maximal connected skew shapes in the Schur-positivity order it is enough to consider a special class of ribbon shapes. We also explicitly determine the support for these ribbon shapes. This is joint work with Peter McNamara and assumes no prior knowledge.

7. Sun-Young Nam (남선영 Sogang Univerisity 서강대학교), June 17 (Wed), 2015. 11:00 -- 12:00

• • Title : Schur P-functions and shifted Littlewood-Richardson coefficients

  • Abstract : In this talk, we provide bijections among three combinatorial models for shifted Littlewood- Richardson coefficients: 

(a) Littlewood-Richardson-Stembridge tableaux due to Stembridge, 

(b) λ-good semistandard decomposition tableaux due to Cho, and 

(c) shifted Littlewood-Richardson decomposition tableaux due to Grantcharov, Jung, Kang, Kashiwara, and Kim.

6. Uijin Jung(정의진 Ajou University 아주대학교), May 13 (Wed), 2015. 17:00 -- 18:00

• • 내부 세미나 (Combinatorics arising from symbolic dynamics: Digraphs, shifts of finite type, Perron eigenvectors and PASEP)

  • Abstract : In this introductory seminar, we will review basic concepts of symbolic dynamics, which form a subclass of discrete dynamical systems, and combinatorial objects related to symbolic dynamics. We will focus on how notions from directed graphs corresponds to those from shifts of finite type.

5. Kyoungsuk Park (박경숙 Ajou University 아주대학교), April 8 (Wed), 2015. 16:30 -- 18:00

• • 내부 세미나 (Eulerian numbers, tableaux, and the Betti numbers of a toric variety)

  • Abstract : 이번 세미나는 지난 4월 1일 세미나의 연속으로, Eulerian numbers, tableaux, and the Betti numbers of a toric variety 논문의 Combinatorial part를 다룰 예정입니다.

4. Soojin Cho (조수진 Ajou University 아주대학교), April 1 (Wed), 2015. 16:30 -- 18:00

• • 내부 세미나 (Eulerian numbers, tableaux, and the Betti numbers of a toric variety)

  • Abstract : Eulerian numbers, tableaux와 Betti numbers of a toric variety에 대하여 알아본다.

3. Kang-Ju Lee (이강주 Seoul National University 서울대학교), March 27 (Fri), 2015. 16:30 -- 18:30

• • Title : High-Dimensional Trees and Matroids

  • Abstract : Part 1. High-dimensional trees We introduce high-dimensional trees as a high-dimensional analogue of spanning trees. G. Kalai (1983) initiated a study of these objects, giving a formula for the high-dimensional tree numbers of standard simplexes as a high-dimensional analogue of Cayley's fomrula (for the number of spanning trees in complete graphs). We also present our formula for high-dimensional weighted tree numbers.
    Part 2. Introduction to matroid theory We see the definition of a matroid and some terminologies, examples, and invariants from matroid theory.
    Part 3. Weighted tree numbers of matroid complexes We give a new formula for the weighted high-dimensional tree numbers of matroid complexes. This formula is derived from our result that the spectra of the weighted combinatorial Laplacians of matroid complexes are polynomials in the weights. In the formula, Crapo's β-invariant appears as the key factor relating weighted combinatorial Laplacians and weighted tree numbers for matroid complexes. As an application of our formula, we answer the question posed by R. Adin (1992) for weighted tree numbers of complete colorful complexes.
    This is a joint work with Woong Kook (SNU).

2. Suyoung Choi (최수영 Ajou University 아주대학교), March 16 (Mon), 2015. 18:30 -- 19:30

• • 내부 세미나 (On permutohedron)

  • Abstract : Permutohedron에 대해 소개하고 Permutohedron이 가지는 위상적, 조합적 성질에 대하여 알아본다.

1. Shinya Fujita (Yokohama City University, Japan), January 26 (Mon), 2015. 15:30 -- 16:30

• • Title : Safe Set Problem on Graphs

  • Abstract : For a connected graph G=(V(G),E(G)), a vertex subset S of V(G) is a safe set if for every component C of G[S], |C| ≥ |D| holds for every component D of G-S such that there exists an edge between C and D; and moreover, if G[S] is connected, then S is called a connected safe set. In this talk, we discuss the minimum sizes of safe sets and connected safe sets in connected graphs. This is joint work with Gary MacGillivray (University of Victoria) and Tadashi Sakuma (Yamagata University).

2014

4. Hiroaki Ishida (RIMS, Japan), October 7 (Tue), 2014. 16:30 -- 18:30

• • Title : Simplicial spheres and moment-angle manifolds

  • Abstract : A moment-angle complex $\mathcal{Z}_K$ is a topological space equipped with an $(S^1)^m$-action, defined for an abstract simplicial complex $K$ on the vertex set $\{1,\dots, m\}$. It has been shown that a moment-angle complex $\mathcal{Z}_K$ is a topological manifold if the geometric realization of $K$ is homeomorphic to sphere, and it admits a smooth structure invariant under the torus action if $K$ is a star-shaped sphere. However, any necessary and sufficient condition for a simplicial complex $K$ so that $\mathcal{Z}_K$ admits a structure of smooth $(S^1)^m$-manifold.

3. Sinuk Kang (강신욱 NIMS), May 30 (Fri), 2014. 16:30 -- 18:30

• • Lecture note : (Slide 3) (Slide 4) (Matlab Code 2)

2. Sinuk Kang (강신욱 NIMS), May 26 (Mon), 2014. 16:30 -- 18:30

• • Intensive series lectures for undergraduate course students (4 Lectures)

  • Title : Introduction to Mathematical signal processing : Fourier anlaysis

  • Abstract : The aim of this series of lectures is to show how mathematics is involved in our daily life by introducing mathematical theories of signal processing. Fourier analysis has played a fundamental and crucial role in development of the signal processing. We glance through examples of practical applications in the area, especially in audio/image processing, and see how Fourier analysis comes into play. We cover various topics which benefit from Fourier analysis, such as basis/frame theory, Shannon’s sampling theorem, wavelets etc. Both theoretical and practical aspect of the topics are to be discussed.  Basic linear algebra and analysis is prerequisite for the lectures. The lectures are intended for auditors attending all, however auditors interested in a particular topic according to the schedule below are also welcome to attend just that lecture. Link

  • Lecture note : (Slide 1) (Slide 2) (Matlab Code)

1. Li Cai (Kyushu University, Japan), March 7 (Fri), 2014. 16:30 -- 17:30

• • Title : On the hierarchy in certain aspherical real moment-angle manifolds

  • Abstract : The Borel Conjecture states that any map inducing a homotopy equivalence between two closed aspherical manifolds is homotopic to a homeomorphism. It is known that aspherical real moment-angle manifolds and their quotients provide a big class of examples supporting the Borel Conjecture, when the dimension is not 4. In this talk we try to extend Haken-Waldhausen's work in dim 3 to all dimensions for these manifolds, and it turns out that there is a natural hierarchy for them.

2013

5. Seonjeong Park (박선정 NIMS), December 20 (Fri), 2013. 11:00 -- 12:00

• • Title : Introduction to topological data analysis

  • Abstract : Data is a finite set of discrete noisy points, sampled from an unknown space, and embedded in a high dimensional space. Topological data analysis recovers the topology of the sampled space. In this talk, we discuss two computational methods in topological data analysis: persistent homology and mapper.

4. Yunhyoung Cho (조윤형 KIAS), November 25 (Mon), 2013. 17:00 -- 18:00

• • Title : Unimodality of the Betti numbers for Hamiltonian circle action with isolated fixed points

  • Abstract : Link

3. Hye Jin Jang (장혜진 POSTECH), November 11 (Mon), 2013. 17:00 -- 18:00

• • Title : 매듭 동형군 소개

  • Abstract : 매듭 동형군(Knot concordance group)의 정의를 소개하고, 매듭 동형군 연구를 통해 해결할 수 있는 4차원 위상수학 문제에 관해 논한다. 매듭 동형군 분류를 위해 사용할 수 있는 기본적인 불변량을 소개한다.

2. Kyoungsuk Park (박경숙 Ajou University 아주대학교), November 4 (Mon), 2013. 17:00 -- 18:30

• • Title : A combinatorial proof of a symmetry of (t,q)-Eulerian numbers of type B and D

  • Abstract : A symmetry of (t,q)-Eulerian numbers of type B is combinatorially proved using permutation tableaux of type B; we define a bijection preserving many important statistics on the set of permutation tableaux of type B. This bijection proves a symmetry of the generating polynomial $\hat{D}_{n, k}(p,q,r)$ of number of crossings, (two types of) alignments, and hence q-Eulerian numbers of type A (defined by L. Williams). By considering the restriction of our bijection on permutations of type D, we were lead to define a new statistic on the set of permutations of type D and (t,q)-Eulerian numbers of type D, which is proved to have a nice symmetry also. We conjecture that our new statistic is in the family of Eulerian statistics for permutations of type D.

1. Hyun Woong Cho (조현웅 KAIST), October 7 (Mon), 2013. 17:00 -- 18:30

• • Title : Periodicity of real Buchstaber invariant

  • Abstract : In this talk, I'll explain the conjecture in the Y.Fukukawa and M.Masuda's paper "Buchstaber Invariants of Skeleta of a Simplex". The conjecture is about the real Buchstaber invariant. The Buchstaber invariant s(K) is defined to be the maximum integer for which there is a subtorus of dimension s(K) acting freely on the moment-angle complex associated with a finite simplicial complex K. Analogously, its real version s_R(K) can be also be defined by using the real moment-angle complex. In the above paper, the authors make some computation of s_R(K) when K is a skeleton of a simplex and give a conjecture about some number which is closely related with s_R(K). I'll summarize the paper briefly. After that, we shall consider some naive reasons to think that the conjecture is true. Also I'll explain some trial to prove it.

2012

16. Ihyeok Seo (서이혁 KIAS), October 23 (Tue), 2012. 16:30 -- 17:30

• • Title : Topics in unique continuation and Schrodinger equations

  • Abstract : Link

15. Suyoung Choi (최수영 Ajou University 아주대학교), October 9 (Tue), 2012. 16:30 -- 17:30

• • 내부세미나 (Title : Topology of real Bott manifolds)

  • Abstract : A real Bott manifold is a closed smooth manifold obtained as the total space of an iterated RP^1-bundles starting with a point, where each RP1-bundle is the projectivization of the Whitney sum of two real line bundles. A 2-dimensional torus and a Klein bottle provide examples of real Bott manifolds.  It is shown that the diffeomorphism types of real Bott manifolds can be completely characterized in terms of three simple matrix operations on square binary matrices symmetrically permutable to strict upper triangular form. This characterization can be visualized combinatorially in terms of graph operations on directed acyclic graphs.  Using this combinatorial interpretation, in this talk, we discuss about several properties of real Bott manifolds, and introduce recent works on this topic.

14. 성찬영 (건국대학교), September 18 (Tue), 2012. 16:30 -- 17:30

• • Title : Geometry and physics of Einstein metrics

13. Sung Rak Choi (최성락 POSTECH), September 12 (Wed), 2012. 15:00 -- 16:50,

• • Lecture 1

    • Title : The minimal model program; birational geometry of algebraic varieties

    • Abstract : I will give an introduction and overview of the history of the minimal model program.

  • Lecture 2

    • Title : The minimal model program and beyond

    • Abstract :I will discuss the recent developments of the minimal model program (MMP). As their applications, I will explain how the MMP can be used to study the structure of the various cones of divisors of Fano varieties.

• 
  

12. Jang Soo Kim (김장수 University of Minnesota), August 7 (Tue), 2012. 16:30 -- 17:30

• • Title : Proofs of Two Conjectures of Kenyon and Wilson on Dyck Tilings

  • Abstract : Recently, Kenyon and Wilson introduced a certain matrix M in order to compute pairing probabilities of what they call the double-dimer model. They showed that the absolute value of each entry of the inverse matrix M-1 is equal to the number of certain Dyck tilings of a skew shape. They conjectured two formulas on the sum of the absolute values of the entries in a row or a column of M-1. In this talk we prove the two conjectures. As a consequence we obtain that the sum of the absolute values of all entries of M-1 is equal to the number of complete matchings. We also find a bijection between Dyck tilings and complete matchings.

11. Soojin Cho (조수진 Ajou University 아주대학교), June 29 (Fri), 2012. 16:30 -- 18:00

• • 내부 세미나 (Combinatorics of Coxeter Groups)

10. Soojin Cho (조수진 Ajou University 아주대학교), May 29 (Tue), 2012. 16:30 -- 18:00

• • 내부 세미나 (Combinatorics of Coxeter Groups)

9. 최재혁 (Goldman Sachs), May 22 (Tue), 2012. 16:30 -- 17:30

• • Title : 금융위기 이후 퀀트의 역할

  • Abstract : 서브 프라임 모기지로 시작된 금융위기는 금융의 각 분야에 큰 변화를 가져오고 있습니다. 특히 파생상품 시장에 도입되는 새로운 규제와 거래 방법들은 월가에서 일하는 퀀트들의 업무에도 직접적인 영향을 미치고 있습니다. 이러한 변화들이 퀀트들에게는 기회일지 위기일지, 그리고 한국에서 퀀트를 꿈꾸는 학생들에게는 어떤 의미가 있을지에 대해서 얘기를 나눠 보겠습니다.

8. Soojin Cho (조수진 Ajou University 아주대학교), May 8 (Tue), 2012. 16:30 -- 18:00

• • 내부 세미나 (Combinatorics of Coxeter Groups)

7. Suyoung Choi (최수영 Ajou University 아주대학교), April 3 (Tue), 2012. 16:30 -- 18:30

• • 내부 세미나 (Different moment angle manifolds with the same bigraded betti numbers)

6. Akihiko Miyachi (Tokyo Woman's Christian University), March 27 (Tue), 2012. 16:30 -- 17:30

• • Title : Some results on multilinear Fourier multipliers

5. Suyoung Choi (최수영 Ajou University 아주대학교), March 13 (Tue), 2012. 16:30 -- 18:30

• • 내부 세미나 (Tor algebra of polytopes)

4. Kyoungsuk Park (박경숙 Ajou University 아주대학교), February 23 (Thu), 2012. 16:00 -- 17:30

• • 내부 세미나 (Poset topology; a basic)

3. Young Woo Choi (최영우 Ajou University 아주대학교), February 9 (Thu), 2012. 16:00 -- 17:30,

• • 내부 세미나 (Combinatorics of Coxeter Groups Chap.4 III)

2. Young Woo Choi (최영우 Ajou University 아주대학교), February 2 (Thu), 2012. 16:00 -- 17:30,

• • 내부 세미나 (Combinatorics of Coxeter Groups Chap.4 II)

1. Young Woo Choi (최영우 Ajou University 아주대학교), January 26 (Thu), 2012. 16:00 -- 17:30,

• • 내부 세미나 (Combinatorics of Coxeter Groups Chap.4 I)

2011

16. Shizuo Kaji (Yamaguchi University), November 23 (Wed), 2011. 16:30 -- 17:30

• • Title: The Weyl group action on the GKM graph of a G-manifold

  • Abstract: GKM theory provides a way to study the topology of manifolds with "good" torus action by means of the combinatorics of a graph. For example, the torus equivariant cohomology can be described purely combinatorially.  In this talk, I will talk about a certain kind of symmetry on the graph which arises from a Lie group action on the manifold. The main result is a combinatorial description of G-equivariant cohomology of the G-manifold.

15. Anna Abczynski (Bonn University), November 18 (Fri), 2011. 16:30 -- 17:30

• • Title : On the Polarisation of Manifolds

  • Abstract : In this talk I will explain how modified surgery theory can be used to study the polarised structure set. The polarised structure set of some manifold M is the set of diffeomorphism classes of manifolds whose cohomology ring is isomorphic to the one of M, preserving Pontrjagin and Stiefel-Whitney classes. I'll give an upper bound for the order of the polarised structure set of a complex 4-dimensional Spin Bott manifold.

14. 유환철 (KIAS), November 2 (Wed), 2011. 15:00 -- 17:00

13. 유환철 (KIAS), October 26 (Wed), 2011. 15:00 -- 17:00

• • Series Lectures on hyperplane arrangements (4 lectures)

  • Title : Hyperplane arrangements, Weyl groups and Schubert varieties

  • Abstract : I will talk about some basic materials on hyperplane arrangements, Weyl groups, and Scubert varieties. I will define important invariants, including characteristic polynomial, and look at some interesting examples of hyperplane arrangements. In particular I will talk about root hyperplane arrangements of Weyl groups. Using properties of (signed) graphical arrangements, I will show how the characteristic polynomials factor. I will also briefly talk about Schubert varieties and their relation to Weyl groups if time allows. The materials are going to be used when I present the result of the joint work with Suho Oh and Alexander Postnikov that links Schubert varieties in the full flag manifolds G/B with hyperplane arrangements. The Schubert varieties are parameterized by elements of the Weyl group, and we associate a hyperplane arrangement to each of the elements. Our result is that the generating function of regions of the arrangement coincides with the Poincare polynomial of the corresponding Schubert variety if and only if the Schubert variety is rationally smooth. We also describe how they factor in terms of certain invariants of the hyperplane arrangements.

12. Suyoung Choi (최수영 Ajou University 아주대학교), October 12 (Wed), 2011. 15:00 -- 16:30

• • 내부 세미나 (Small remarks on Bruhat order complex of symmetric groups)

11. 정은경 (아주대학교), October 5 (Wed), 2011. 15:00 -- 16:30

• • 내부 세미나 (Combinatorics of Coxeter groups. Chapter 3. II)

10. 정은경 (아주대학교), September 28 (Wed), 2011. 16:00 -- 17:30

• • 내부 세미나 (Combinatorics of Coxeter groups. Chapter 3. I)

9. Suyoung Choi (최수영 Ajou University 아주대학교), September 14 (Wed), 2011. 16:00 -- 17:30

• • 내부 세미나 ('Bruhat order complex)

8. 유미수 (서울대학교), August 25 (Thu), 2011. 14:00 -- 15:30

• • 내부 세미나 (Combinatorics of Coxeter groups. Chapter 2)

7. Shintaro Kuroki (KAIST), August 24 (Wed), 2011. 16:00 -- 17:00

• • Title : On cohomological rigidity problems of CP-towers

  • Abstract : The cohomological rigidity problem of the set of manifolds \mathcal{M} asks whether cohomology rings determine the diffeomorphism (or homeomorphism) types of \mathcal{M}. Of course for general manifolds the answer of this problem is NO. However, the answer is sometimes YES if we restricted the class of manifolds. In particular, for toric manifolds or more restricted Bott manifolds this problem is still open. In this talk, we study this problem for a little bit wider class, called CP-tower, than the Bott manifolds.

6. Seonjeong Park (박선정 KAIST), August 11 (Thu), 2011. 15:00 -- 16:30

• • 내부 세미나 (cohomological rigidity of small covers)

5. 유미수 (서울대학교), July 28 (Thu), 2011. 17:00 -- 18:00

• • Title : Combinatorics of Macdonald polynomials and the space of diagonal harmonics

  • Abstract : In 1988, Macdonald introduced a remarkable new basis for the space of symmetric functions, denoted by P_{\lambda}(X;q,t), which are called "Macdonald polynomials". They specialize to many of the well-known bases for the symmetric functions such as Schur functions, Hall-Littlewood functions, Jack symmetric functions and zonal functions. Since their introduction, Macdonald polynomials have been intensely studied and have found applications in special function theory, representation theory, algebraic geometry, group theory and quantum mechanics.a
    In this talk, we introduce Macdonald polynomials and combinatorial formulas concerning them including Haglund-Haiman-Loehr's. Also, we concern the space of diagonal harmonics and combinatorics of Dyck path and parking functions.

4. Soojin Cho (조수진 Ajou University 아주대학교), July 10 (Thu), 2011. 15:00 -- 16:30

• • 내부 세미나 (Combinatorics of Coxeter groups. Chapter 1. III))

3. Suyoung Choi (최수영 Ajou University 아주대학교), Soojin Cho (조수진 Ajou University 아주대학교), June 28 (Tue), 2011. 15:00 -- 18:00

• • 내부 세미나 (Finite reflection group과 Coxeter graph 와의 관계에 대해서. III)

  • 내부 세미나 (Combinatorics of Coxeter groups. Chapter 1. II)

2. Suyoung Choi (최수영 Ajou University 아주대학교), Soojin Cho (조수진 Ajou University 아주대학교), June 22 (Wed), 2011. 15:00 -- 18:00

• • 내부 세미나 (Finite reflection group과 Coxeter graph 와의 관계에 대해서. II)

  • 내부 세미나 (Combinatorics of Coxeter groups. Chapter 1. I)

1. Suyoung Choi (최수영 Ajou University 아주대학교), June 14 (Tue), 2011. 15:00 -- 16:30

• • 내부 세미나 (Finite reflection group과 Coxeter graph 와의 관계에 대해서. I)