Rational homology projective planes are normal projective surfaces having same Betti numbers with the complex projective plane. The first half of the talk will be devoted to the introduction to the topic: examples, relations with other classification problems. For the remaining part of the talk, I will present the recent results on the algebraic Montgomery-Yang problem. 

컴퓨터의 발명은 인간에게 많은 양의 수학적 계산을 짧은 시간에 해낼 수 있는 능력을 가져다주었다.  이러한 변화는, 주어진 문제의 답의 존재성에 더 관심을 두는 수학의 전통적인 접근법과는 다르게 답을 제한된 계산 자원 (시간, 메모리 등)을 이용하여 효율적으로 계산하는 방법에 대한 새로운 문제의 중요성을 대두시키게 된다.

이 강연에서는 효율적으로 계산 가능한 문제 (P)와 효율적으로 검증가능한 문제 (NP)에 대해 알아보고, approximation algorithm, randomized algorithm 등 효율적인 계산이 어려운 문제들에 대한 접근법과 암호론 등에서의 응용에 대해 알아본다.

We discuss how to efficiently compute shortest and approximate shortest paths in graphs, thereby focussing on shortest path query processing. The algorithm computing (approximate) shortest path queries is allowed to access a pre-computed data structure (often called distance oracle) in order to improve the query time. Several questions regarding such data structures are of particular interest: How can they be computed efficiently? What amount of storage is necessary? How much improvement of the query time is possible? How good is the approximation quality of the query result? What are the tradeoffs between pre-computation time, storage, query time, and approximation quality?

For general dense graphs, the tradeoff between the storage requirement and the approximation quality is known up to constant factors. We discuss both the lower and the upper bound (by Thorup and Zwick). For specific types of sparse graphs, however, the exact tradeoff is not known; the general tradeoff can be improved: there are special data structures of a certain size that enable query algorithms to return distances of higher quality than what the general tradeoff would predict. We outline the state of the art of distance oracles for planar graphs and other classes of sparse graphs. We then prove that this improvement for some classes of sparse graphs cannot be extended to all sparse graphs: there is a three-way relationship between space, query time, and approximation quality for general sparse graphs. If time permits, we also cover space- and time-efficient data structures for certain complex networks with power-law degree sequences.

Joint work with Wei Chen, Shinichi Honiden, Michael Houle, Ken-ichi Kawarabayashi, Shang-Hua Teng, Elad Verbin, Yajun Wang, Martin Wolff, and Wei Yu.

A J-holomorphic curve is a map from a Riemann surface to an almost complex manifold (M,J) whose differential preserves almost complex structures. The concept of J-holomorphic curves is a powerful tool to study symplectic manifolds. A symplectic manifold always admits an almost complex structure J and J can be chosen to be "tamed" by the symplectic structure. In this case, J-holomorphic curves behave well and we can study symplectic manifolds by studying J-holomorphic curves. In this talk I will explain some results obtained by using J-holomorphic curves.

 The constant demand for increasingly accurate, efficient, and robust numerical methods, which can handle acoustic, elastodynamic and electromagnetic wave propagations in unbouded domains, spurs the search for improvements in artificial boundary conditions. In the last decade, the perfectly matched layer (PML) approach has proved a flexible and accurate method for the simulation of waves in unbounded media. Standard PML formulations, however, usually require wave equations stated in their standard second-order form to be reformulated as firstorder systems, thereby introducing many additional unknowns. To circumvent this cumbersome and somewhat expensive step we propose instead a simple PML formulation directly in its second-order form in 3D. Our formulation requires fewer auxiliary unknowns than previous formulations. Starting from a high-order local nonreflecting boundary condition (NRBC) for single scattering, we derive a local NRBC for time-dependent multiple scattering problems, which is completely local both in space and time. To do so, we first develop a high order exterior evaluation formula for a purely outgoing wave field, given its values and those of certain auxiliary functions needed for the local NRBC on the artificial boundary. By combining that evaluation formula with the decomposition of the total scattered field into purely outgoing contributions, we obtain the first exact, completely local, NRBC for time-dependent multiple scattering. The accuracy, stability and efficiency of this new local NRBC is evaluated by coupling it to standard finite element or finite difference methods.

In this talk, I present an innovative nonparametric Bayes methodology for ex-ibly characterizing the relationship between a continuous response and multiple predictors.

Goals are (1) to estimate the conditional response distribution addressing the distributional changes across the predictor space, and (2) to identify important predictors for the response distribution change both within local regions and globally. We rst introduce the probit stick-breaking process (PSBP) as a prior for an uncountable collection of predictor-dependent random distributions and propose a PSBP mixture (PSBPM) of normal regressions for mod-eling the conditional distributions. A global variable selection structure is incorporated to discard unimportant predictors, while allowing estimation of posterior inclusion probabili-

ties. An ecient stochastic search sampling algorithm is proposed for posterior computation.

The methods are illustrated through simulation and applied to an epidemiologic study.

1. 금번 금융위기에 대한 금융공학적 접근:

주택관련 파생상품을 중심으로 발생된 2008년 금융위기의 발생 원인과 해결 과정에서 금융공학의 역할을 분석하고, 금융공학의 미래에 대해 전망한다.

2. 금융시장에서 수학의 중요성 확대:

금융시장의  다양한 분석 방법과 최근 조류에 대한 조망을 통해 금융시장에서 수학이 필요한 이유와 비중 확대 가능성을 제시한다.

There are some important PDEs in fluid dynamics, such as heat equation, wave equation, and Stokes equation, etc. Numerical methods to solve those PDEs has been developed and now there are two kind of method widely used. Finite Difference and Finite element method.  Finite element method can be described as follows. first, formulate given equations as weak equations, and determine the space in which we seek for original solution. second, find admissible finite dimensional function space in which we seek for numerical solution. third, discretize weak formulations and reduce original problem as a set of linear equations

I will give a brief introduction to invariant metrics (the Kobayashi, Carath\'eodory, Bergman, and Sibony metric) and explain how the asymptotic behavior of the metrics near the boundary of a domain is related to the geometry of the boundary. The talk will be accessible to graduate students.

I will give an introduction to Stein complex manifolds. I will describe basic examples and non examples for domains in complex Euclidean space. Trickier examples arise from the interplay with isolated singularities of surfaces, an interesting topic in its own. I will assume familiarity with smooth manifolds, and interest in the field of several complex variables. 

In this talk, I will introduce high-dimensional data analysis and related problems. Traditionally it is considered as a statistical problem, but due to its innate difficulty, often described as the curse of dimensionality, it produces many challenging and interesting mathematical problems and more and more mathematicians are interested in its geometry and analysis, considering data sets as discrete or sampled continuous geometric structures embedded in high-dimensional spaces. With such view point, I'll explain Laplacian, eigenfunctions and heat equation on data sets and graphs and talk about their applications.

Coisotropic submanifolds generalize real hypersurfaces,
which occur in mechanical systems as energy level sets. Leafwise xed
points correspond to trajectories for which a given perturbation of the
system results in a phase shift. Recently, the problem of nding lower
bounds on the number of such points has caught a lot of attention. I
will discuss a bound that in many cases is optimal. As an application,
I obtain a presymplectic non-embedding result.

The fundamental sufficient condition for the existence of a proper 3-coloring of the vertices of a planar graph G was proved by Grötzsch more than 50 years ago, and it requires that G has no triangles (cycles of length 3). This talk discusses conjectures for other possible sufficient conditions, some of which have stubbornly resisted proofs for decades, and also various recent partial results. A conjecture in a different direction deals with a stronger 3-colorability property, which for a planar graph turns out to be equivalent to triangle-freeness, but here it is unknown whether the assumption of planarity may be weakened.

http://mathsci.kaist.ac.kr/~manifold/Arithmetics.html

We consider a correlation between the default intensities to incorporate dependency
between multivariate Cox process. Assuming that each obligor has its own default
intensity process, we use multivariate shot noise intensity process where jumps (i.e.
magnitude of contribution of primary events to default intensities) occur collaterally
and their sizes are correlated. A homogeneous Poisson process is used to describe
collateral event jumps in default intensities and the Farlie-Gumbel-Morgenstern (FGM)
copulas are used to produce correlations between jump sizes. Using a bivariate Cox
process with exponential margins for FGM copulas, we derive joint survival/default
probabilities and conditional default probabilities. As an example of pricing credit
derivatives, we calculate credit default swaps (CDS) rates, assuming that a zero-coupon
default-free bond price follows a generalised Cox-Ingersoll-Ross (CIR) model. Standard
martingale theory is used to derive the joint Laplace transforms.

The piecewise deterministic Markov processes (PDMP) theory developed by Davis(1984) is a powerful mathematical tool for examining non-di¤usion models. In this lecture, with the aid of this theory, it is shown how to derive the generators of jump diffusion processes using the Dynkin's formula (Dassios and Embrechts 1989; Rolski et al. 1998 and Dassios and Jang 2003). These generators are to be used to de-rive the general form of the Laplace transform of the distribution of jump diffusion processes and to derive relevant expressions needed to price/measure insurance and financial risks/products. For that purpose, suitable martingales are required.

Calculus was the most important field in the mathematics for several centuries. Especially, many kinds of integrals give rise to new problems in a natural way. In 18th century, people realized that there are families of totally new functions come from the antiderivative of rational functions, for instance, elliptic functions. This kind of functions has a nice property like trigonometric functions, which was known as Euler addition formula and Abel's theorem. In Riemann's great paper in 1857, he gave unbelievable ideas for classifying these integrals, without any monstrous calculations. In this talk, I will introduce some classical results and discuss about his wonderful notion of Riemann surfaces.

A catastrophic event such as flood, storm, hail, bushfire and earthquakebrings about damages in properties, motors and interruption of businesses collater-ally. Also a couple of losses incurred collaterally from the World Trade Centre (WTC)catastrophe, Hurricane Katrina and Victorian Bushfire. However it has not been developed a suitable model for insurance companies either to measure tail dependence between these collateral losses or relevant risk measures that can be used as insurance risk premiums. The first aim of this paper is to measure tail dependence between collateral losses as insurance industry is more concerned with dependence between extreme losses. The second is to calculate conditional probabilities and conditional expectations as relevant risk measures. To achieve these aims, we use bivariate compound Poisson process to count collateral losses from catastrophic events. Using a member of Farlie-Gumbel-Morgenstern copula with exponential margins, we derive explicit expressions of joint Laplace transforms of aggregate collateral losses. Fast Fourier transform is usedto obtain the joint distributions of aggregate collateral losses, with which we calculate relevant risk measures. The figures of the joint distributions of collateral losses, their contours and numerical calculations of risk measures are provided.

매듭과 소수에 대한 세미나 세번째 시간입니다.

The Fibonacci dimension fdim(G) of a graph G is introduced as the smallest integer f such that G admits an isometric embedding into
the f-dimensional Fibonacci cube. We will see combinatorial relations between the Fibonacci dimension and the isometric or lattice dimension, and establish the Fibonacci dimension for certain families of graphs.
Finally, we will discuss the problem of computing the Fibonacci dimension, namely, its NP-hardness and approximation algorithms.

Joint work with D. Eppstein and S. Klavžar.

This paper proposes a new framework which captures the systemic nature of funding liquidity risk. Using this framework we develop a set of indicators which measure different aspects of the systemic funding liquidity risk in the interbank foreign currency lending market: (i) systemic funding liquidity needs, (ii) systemic vulnerability, (iii) systemic importance and (iv) systemic liquidity shortages.
We also analyze the systemic funding liquidity risk of the Korean banking system under the new framework. The Korean banking system has become more vulnerable to the systemic funding liquidity risk of foreign currency debt since 2006. The systemic importance of foreign bank branches and the systemic vulnerability of domestic banks have simultaneously increased as the domestic banks have relied heavily on FX swap transactions with foreign bank branches to raise foreign currency funds.

A Coxeter polytope in the space X of constant curvature is a polytope whose dihedral angles are all submultiples of $\pi$. Coxeter polytopes arise as fundamental domains of discrete reflection groups acting on X. I'll talk about basic notions and properties of Coxeter polytopes.

2월 8일 콜로퀴엄에서 소개한 내용을 3번에 걸쳐 자세한 설명을 해 주시는 세미나 첫번째 시간 입니다.

매듭과 소수 두번째 세미나 입니다.

The properties of the Bose gas has been studied by many authors, and since the first experimental observation of Bose-Einstein Condensation, interests in low temperature Bose gas are renewed. In this talk, a mathematical framework to understand Bose gas will be introduced. Heuristics and rigorous proofs for the ground state energy
of Bose gas will also be explained.