A dynamic coloring of a graph G is a proper coloring of the vertex set V(G) such that for each vertex of degree at least 2, its neighbors receive at least two distinct colors. A  dynamic k-coloring of a graph is a dynamic coloring with k colors. Note that the gap χ_d(G) – χ(G) could be arbitrarily large for some graphs. An interesting problem is to study which graphs have small values of χ_d(G) – χ(G).
One of the most interesting problems about dynamic chromatic numbers is to find upper bounds of χ_d(G)$  for planar graphs G. Lin and Zhao (2010) and Fan, Lai, and Chen (recently) showed that for every planar graph G, we have χ_d(G)≤5, and it was conjectured that χ_d(G)≤4 if G is a planar graph other than C₅. (Note that χ_d(C₅)=5.)
As a partial answer, Meng, Miao, Su, and Li (2006)  showed that the dynamic chromatic number of Pseudo-Halin graphs, which are planar graphs, are at most 4, and Kim and Park (2011) showed that χ_d(G)≤4 if G is a planar graph with girth at least 7.
In this talk we settle the above conjecture that χ_d≤4 if G is a planar graph other than C₅. We also study the corresponding list coloring called a list dynamic coloring.
This is joint work with Seog-Jin Kim and Won-Jin Park.

글로벌 금융위기 및 최근 미국과 유럽의 재정위기 이후 파생상품에 대한 부정적 인식이 증가하고 있다. 장외 파생상품 및 파생결합증권의 규제강화로 상대적으로 장내파생상품 시장이 비중이 증가하고 있으나, 파생상품 전체적으로는 성장세가 둔화되고 있다. 본 보고서에서는 장내 및 장외파생상품, 그리고 최근 급격히 증가하고 있는 파생결합증권의 최근 동향 및 위험요인을 살펴본다. 신용위험(credit risk), 거래상대방 위험(counter-party risk), 극단적 상황에서의 손실 위험(extreme-event risk) 등 파생상품의 위험요인과 관련해서 금융투자회사의 위험관리의 개선과제를 제시한다. 또한 금융혁신(financial innovation), 위험분산(risk sharing), 가격발견(price discovery) 기능 등 파생상품의 순기능을 재점검해본다.

A description of the product structure of the cohomology of polyhedral products is given in terms of the stable splitting. Several applications will be discussed. This report is based on joint work with Martin Bendersky, Fred Cohen and Sam Gitler.

This survey talk will be a review results obtained in collaboration with Mattias Franz and Nigel Ray. As singular toric varieties, weighted projective spaces have an action of a real torus. The equivariant cohomology with respect to this action is computed to be isomorphic to the ring of piecewise polynomials on the defining fan. The theory is seen to parallel that for smooth toric varieties with the role of the Stanley-Reisner ring replaced by the ring of piecewise polynomials. If time permits, an alternative presentation of weighted projective spaces as iterated Thom complexes will be discussed briefly. Further collaboration with Mattias Franz, Nigel Ray and Dietrich Notbohm yields in a complete topological classification of weighted projective spaces.

Certain natural subspaces of a product of CW complexes, called polyhedral products, play an important role in a variety of different fields including: toric varieties, toric manifolds/orbifolds, intersections of quadrics, homotopy theory, algebraic combinatorics complements of subspace arrangements, robotics and group theory. The talk will survey certain combinatorics based constructions on polyhedral products which allow for a description of the cohomology rings of certain families of toric manifolds in a particularly compact form. The new constructions will be related to a generalization of the basic Davis-Januszkiewicz construction of toric manifolds. This report is based on joint work with Martin Bendersky, Fred Cohen and Sam Gitler.

본 강연은 KMRS(KIAST Math Research Station)에서 제공하는 집중 강연으로 기하학적 입장에서 유도되어지는 편미분 방정식을 소개하는 것을 목적으로 하고 있다. 이제까지 주로 연구 되어진 다양한 편미분 방정식은 주로 물리학적인 문제들에서 유도 되어진 것이다. 본 강연을 통해서 기존의 편미분 방정식을 보다 기하학적인 관점에서 이해할 뿐 아니라 기하학적인 문제를 편미분 방정식의 형태로 소개 하고자 한다.   This lecture series is provided by KMRS and aiming to introduce PDEs which are derived from geometry view point. So far PDEs are mostly derived from physical view point. In these lectures PDEs will be understood in geometry view point and PDEs related to geometry will be introduced.

본 강연은 KMRS(KIAST Math Research Station)에서 제공하는 집중 강연으로 기하학적 입장에서 유도되어지는 편미분 방정식을 소개하는 것을 목적으로 하고 있다. 이제까지 주로 연구 되어진 다양한 편미분 방정식은 주로 물리학적인 문제들에서 유도 되어진 것이다. 본 강연을 통해서 기존의 편미분 방정식을 보다 기하학적인 관점에서 이해할 뿐 아니라 기하학적인 문제를 편미분 방정식의 형태로 소개 하고자 한다.  

This lecture series is provided by KMRS and aiming to introduce PDEs which are derived from geometry view point. So far PDEs are mostly derived from physical view point. In these lectures PDEs will be understood in geometry view point and PDEs related to geometry will be introduced.

본 강연은 KMRS(KIAST Math Research Station)에서 제공하는 집중 강연으로 기하학적 입장에서 유도되어지는 편미분 방정식을 소개하는 것을 목적으로 하고 있다. 이제까지 주로 연구 되어진 다양한 편미분 방정식은 주로 물리학적인 문제들에서 유도 되어진 것이다. 본 강연을 통해서 기존의 편미분 방정식을 보다 기하학적인 관점에서 이해할 뿐 아니라 기하학적인 문제를 편미분 방정식의 형태로 소개 하고자 한다.  

This lecture series is provided by KMRS and aiming to introduce PDEs which are derived from geometry view point. So far PDEs are mostly derived from physical view point. In these lectures PDEs will be understood in geometry view point and PDEs related to geometry will be introduced.

In his book, Shimura constructed the ray class field of some particular conductor over a given real quadratic field by adjoining ideal section points of Jacobian variety of a modular curve.To understand his methods, I will survey some properties of common eigen forms of Hecke operators and some theories about abelian varieties. explain some problems related to it. And then I will consider some problems related to Shimura's construction.

We define a heat kernel for Markov process and consider the problem of estimating heat kernels in both continuous and non-continuous Markov processes.

본 강연은 KMRS(KIAST Math Research Station)에서 제공하는 집중 강연으로 기하학적 입장에서 유도되어지는 편미분 방정식을 소개하는 것을 목적으로 하고 있다. 이제까지 주로 연구 되어진 다양한 편미분 방정식은 주로 물리학적인 문제들에서 유도 되어진 것이다. 본 강연을 통해서 기존의 편미분 방정식을 보다 기하학적인 관점에서 이해할 뿐 아니라 기하학적인 문제를 편미분 방정식의 형태로 소개 하고자 한다.  

This lecture series is provided by KMRS and aiming to introduce PDEs which are derived from geometry view point. So far PDEs are mostly derived from physical view point. In these lectures PDEs will be understood in geometry view point and PDEs related to geometry will be introduced.

In 1975, Y. Morita conjectured that if an abelian variety defi ned over a number fi eld has the Mumford-Tate group which does not have any non-trivial Q-rational unipotent element, then it has potential good reduction everywhere. In this talk, we explain a proof of this conjecture. The main ingredients of proof include some newly established cases of the conjecture due to Vasiu, a generalization of a criterion of Paugam on good reduction of abelian varieties, and the local-global principle of isotropy for Mumford-Tate groups of abelian varieties.

A local factor of a Dedekind zeta function can be btained as a partition function of an appropriate C*-dynamical system. For a Dedekind zeta function itself such a dynamical system is related to the Galois group of the given number field and is called a BC system. We will ee how to extend this idea in the case of Hecke L-functions.

Using Todd operator one can express Euler-McLaurin formula in simple form. For recent decades, along with development of toric geometry and theory of polytopes, Euler-McLaurin formula has been generalized to a category of lattice cones and polytopes by Brion-Vergne, Karshon-Sternberg-Weitsman, Garoufalidis-Pommersheim and several others. In particular, Garoufalidis-Pommersheim expressed special values of zeta function associated to Todd operator associated to a certain cone decomposition. We extend the category of cones by Grothendieck group construction of ordinary cones. This new category contains 'virtually decomposed cones' considering cones with negative weight and the appropriate form of Todd operator construction generalizes the Euler-Mclaurin formula on it. We apply this generalization to obtain an alternating sum expression of special values of (partial) zeta functions at nonpositive integers associated (virtual) cone decomposition. This expression enables us to read polynomial behavior of special values of zeta function at nonpositive integers in some family of real quadratic fields. It is joint work with Byungheup Jun.

In this talk, I will first survey some important examples of simply connected closed positively curved manifolds as well as known nonnegatively curved manifolds and their constructions. After that, I plan to focus on exotic positively curved metrics on P_2 homeomorphic to a unit 3-sphere bundle over a 4-sphere, which have been recently constructed by Grove, Verdiani, and Ziller. Finally, I will consider some interesting problems related to positively curved manifolds and manifolds with almost positive sectional curvatures.

본 강연은 KMRS(KIAST Math Research Station)에서 제공하는 집중 강연으로 기하학적 입장에서 유도되어지는 편미분 방정식을 소개하는 것을 목적으로 하고 있다. 이제까지 주로 연구 되어진 다양한 편미분 방정식은 주로 물리학적인 문제들에서 유도 되어진 것이다. 본 강연을 통해서 기존의 편미분 방정식을 보다 기하학적인 관점에서 이해할 뿐 아니라 기하학적인 문제를 편미분 방정식의 형태로 소개 하고자 한다.  

This lecture series is provided by KMRS and aiming to introduce PDEs which are derived from geometry view point. So far PDEs are mostly derived from physical view point. In these lectures PDEs will be understood in geometry view point and PDEs related to geometry will be introduced.

In view of scheme language, we start from the beginning of the theory of elliptic curves and geometric modular forms, and further cover the topics about Jacobians and Galois representation as well as modularity problems.

Wavelets are useful for many applications including signal/image processing. Tensor product has been a predominant method in constructing multivariate wavelets. In this talk, I will first provide a brief overview of the wavelet analysis and the use of tensor product in constructing multivariate wavelets. Then I will introduce a new alternative to tensor product, to which we refer as Coset Sum. We will discuss the similarity and difference between the two methods. We will also see that some of known limitations of tensor product can be overcome by Coset Sum, albeit in a limited sense.

Since the inception of Donaldson theory and Seiberg-Witten theory in late 20 century, the invariants induced from these gauge theories, in particular, Seiberg-Witten invariants have become so powerful tools in study of smooth and symplectic 4-manifolds. Nevertheless, it has not been much known that one can distinguish smooth and symplectic 4-manifolds to some extent using Seiberg-Witten invariants. In this talk I'd like to show that there exist an infinite family of non-simply connected, non-diffeomorphic, but homeomorphic, 4-manifolds with the same Seiberg-Witten invariants. The main techniques used in the construction are a knot surgery technique and a covering method.

A path cover of a graph is a set of disjoint paths such that every vertex in the graph appears in one of the paths.
We prove an upper bound for the minimum size of a path cover in a connected
4-regular graph with n vertices, confirming a conjecture by Graffiti.pc.
We also determine the minimum number of vertices in a connected k-regular graph that is not Hamiltonian, and we solve the analogous problem for Hamiltonian paths.

This is a partly joint work with Gexin Yu and Rui Xu.

http://dl.dropbox.com/u/12483374/Lectures/poster.pdf

http://dl.dropbox.com/u/12483374/Lectures/poster%20seminar.pdf

The heat equation describes time variation of temperature. Poisson's equation ignores the time derivative in the heat equation so that it might describe the steady state temperature. In fact, if we consider the whole Euclidean space with dimension greater than or equal to three, solutions of the heat equation converge to a solution of Poisson's equation as time passes by. However, if the dimension is one or two, solutions of the heat equation diverge in general. In this talk, a sufficient condition to guarantee the convergence will be given. Also, if time permits, we will consider related questions for nonlinear problems. This talk is based on ongoing project with Jieun Choi and Yong Jung Kim.

A mass linear function is an affine function on a simple convex polytope whose value on the center of mass depends linearly on the positions of the supporting hyperplanes. A Delzant polytope is a simple, rational, and smooth convex polytope. In this talk, I will introduce the relationship between Delzant polytopes and symplectic toric manifolds, and then geometric implications of mass linear functions. This talk is based on the work by D. McDuff and S. Tolman, Polytopes with mass linear functions.

우리나라의 역사적 기록을 토대로 우리의 선조들이 정치, 경제, 복지, 과학분야에서 어떤 업적을 이루었는지를 재조명하는 시간을 갖는다. 아래 내용은 수학과 관련된 내용의 일부이다.

洪大容(1731-1783)

수학서 <주해수용(籌解需用)>의 내용

(1) 구체의 체적이 62,208척이다. 이 구체의 지름을 구하라.

正弦=sinA 餘弦=cosA 正切=tanA 餘切=cotA

正割=secA 餘割=cosecA 正矢=1-cosA 餘矢=1-sinA

 

正弦 30도=sin30도=0.5

正弦 25도 42분 51초=sin25.4251。=0.4338883739118

正弦 45도=sin45=0.7070167811865

(2) 甲地와 乙地는 동일한 子午眞線에 있다. 甲地는 北極出地 37도에 있고, 乙地는 36도 30분에 있다. 甲地에서 乙地로 직선으로 가는데 고뢰(鼓擂)가 12번 울리고, 종뇨(鍾鬧)가 125번 울렸다. 이 때 지구 1度의 里數와 지구의 지름, 지구의 둘레를 구하라.

The existence of topologically slice knots that are of infinite

Nonsmooth optimization problems are generally considered to be more difficult than smooth problems. Among those, optimization problem with sparsity, which has wide applicability in machine learning, satistics, and image processing, are usually structured. Hence many efficient optimization methods have been developed to solve such problems. In this talk, we introduce several optimization problems with sparsity arising in applications and optimization methods for solve them.

Basic concepts, examples and applications of quantum
(=non-commutative) probability will be presented.
Preliminaries would be a mild familiarity of random variables and
bounded linear operators on Hilbert spaces (or just matrices).

We consider a well-known combinatorial search problem. Suppose that there are n identical looking coins and some of them are counterfeit. The weights of all authentic coins are the same and known a priori. The weights of counterfeit coins vary but different from the weight of an authentic coin. Without loss of generality, we may assume the weight of authentic coins is 0. The problem is to find all counterfeit coins by weighing sets of coins (queries) on a spring scale. Finding the optimal number of queries is difficult even when there are only 2 counterfeit coins.
We introduce a polynomial time randomized algorithm to find all counterfeit coins when the number of them is known to be at most m≥2 and the weight w(c) of each counterfeit coin c satisfies α≤|w(c)|≤β for fixed constants α, β>0. The query complexity of the algorithm is O((m log n)/log m), which is optimal up to a constant factor. The algorithm uses, in part, random walks.
We will also discuss the problem of finding edges of a hidden weighted graph using a certain type of queries.

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I will start with a pivot-minor containment problem in graphs. A graph H is a pivot-minor of a graph G if H is obtained from G by a sequence of pivoting edges and vertex deletions. In recent, we have a question that any incidence graph of a tree does not have binary tree of depth at least 5 as a pivot-minor. This comes true and I gives two proofs it. First, I prove it by using the fact that an adjacency matrix of a tree is nonsingular if and only if it has a perfect matching. Second, I will discuss how this problem is related to a fundamental graph of a binary matroid. Then we can convert original problem into a graph minor containment problem and we can solve it.