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.

- Age-at-death, time-until-death random variables
- Some parametric survival models
- Tabulation of basic mortality functions

Hamiltonian cycle in a graph G is a cycle, which contains every vertex of the graph G. The problem of existence of a hamiltonian cycle in a graph is a well known NP-complete problem. While some theoretical necessary and sufficient conditions are known, to date, but no practical characterization of hamiltonian graphs has been found. There are several ways how to generalize the notion of a hamiltonian cycle.

For any integer r>1, an r-trestle is a 2-connected graph F with maximum degree ∆(F)≤ r. We say that a graph G has an r-trestle if G contains a spanning subgraph which is an r-trestle. This concept can be viewed as an interesting variation on the notion of Hamilton cycle. Another such variation is a concept of k-walks, where a k-walk in a graph G is a closed spanning walk visiting each vertex at most k times, where k is a positive integer.

We present several results and problems concerning mainly with k-walks and r-trestles and relations between them.

- Cash Flow Modeling: Concepts and Applications
- Interest Rates: Definitions and relationships
- Valuation Principles

In an attempt to transfer the loss rate risks in motor insurance to the capital market, we use the tranche technique to hedge the motor insurance risks. Though this application is new, this transaction is based on a concept similar to CDOs. The pricing methods of the tranches are illustrated, and possible suggestions to improve the pricing method and the design of these new securities follow.

It is a task of central importance in understanding an ambient closed
3-manifold to find an essential surface - that is, a
$\pi_1$-injectively immersed closed surface of non-positive Euler
characteristic. Extending this question, one naturally asks what
conditions guarantee or prohibit the existence of essential surfaces
in non-positively curved cubical complexes. In this talk, I will
survey results on this latter question specialized in the case of
right-angled Artin groups. This talk will be self-contained and all
the necessary definitions will be given.

- Actuary의 정의와 그 역할
- Actuary 시험제도 및 처우
- 우수 Actuary들을 양성할 수 있는 국내 대학에서의 교육 프로그램 개발 방안

A total coloring is a combination of a vertex coloring and an edge coloring of a graph: every vertex and every edge is assigned a color and any two adjacent/incident objects must receive distinct colors. One of the main open problems in the area of graph colorings is the Total Coloring Conjecture of Behzad and Vizing from the 1960’s asserting that every graph has a total coloring with at most D+2 colors where D is its maximum degree.

Fractional colorings are linear relaxation of ordinary colorings. In the setting of fractional total colorings, the Total Coloring Conjecture was proven by Kilakos and Reed. In the talk, we will present a proof of the following recent conjecture of Reed:

For every real ε>0 and integer D, there exists g such that every graph with maximum degree D and girth at least g has total fractional chromatic number at most D+1+ε.

For D=3 and D=4,6,8,10,…, we prove the conjecture in a stronger
form: there exists g such that every graph with maximum degree D and girth at least g has total fractional chromatic number equal to D+1.

Joint work with Tomás Kaiser, František Kardoš, Andrew King and Jean-Sebastien Sereni.

We present recent developments of syzygy theory and applications of Koszul cohomology to 
the geometry of complex projective varieties. The lectures will begin with a review of definitions
 and basic properties, examples, and will end with a discussion on the curve case. Voisin's 
description of Koszul cohomology in terms of Hilbert schemes will also be discussed during 
the lectures.
The main source is the joint book with J. Nagel with the same title, published by the AMS 
(Univ. Lect. Series 52)

We present recent developments of syzygy theory and applications of Koszul cohomology to
the geometry of complex projective varieties. The lectures will begin with a review of definitions and basic properties, examples, and will end with a discussion on the curve case. Voisin's
description of Koszul cohomology in terms of Hilbert schemes will also be discussed during the lectures.
The main source is the joint book with J. Nagel with the same title, published by the AMS
(Univ. Lect. Series 52)

The Catalan number $\frac{1}{n+1}{2n \choose n}$ is perhaps the most frequently occurred number in combinatorics. Richard Stanley has collected more than 170 combinatorial objects counted by the Catalan number. Noncrossing partition, which has received great attention recently, is one of these, so called, Catalan objects. Noncrossing partitions are generalized to each
finite Coxeter group. In this talk, we will interpret noncrossing partitions of type B in terms of noncrossing partitions of type A. As applications, we can prove interesting properties of noncrossing partitions of type B.

We study algebraic analogues of the graph Removal Lemma. Vaguely speaking, the graph Removal Lemma says that if a given graph does not contain too many subgraphs of a given kind, then all the subgraphs of this kind can be destroyed by removing few edges. In 2005, Green conjectured the following analogue of it for systems of equations over integers:

For every k x m integral matrix A with rank k and every ε>0, there exists δ>0 such that the following holds for every N and every subset S of {1,…N}: if the number of solutions of A x = 0 with x ∈ S^m is at most δ N^{m-k}, then it is possible to destroy all solutions x ∈ S^m of A x = 0 by removing at most ε N elements from the set S.

We prove this conjecture by establishing its variant for not necessarily homogenous systems of equations over finite fields. The core of our proof is a reduction of the statement to the colored version of hypergraph Removal Lemma for (k+1)-uniform hypergraphs. Independently of us, Shapira obtained the same result using a reduction to the colored version of hypergraph Removal Lemma for O(m²)-uniform hypergraphs. The talk will be self-contained and no previous knowledge of the area related to the graph Removal Lemma will be assumed.

Joint work with Oriol Serra and Lluis Vena.

subprime crisis, LTCM, KIKO, 다이아몬드 펀드 등 사례를 중심으로
금융공학과 파생상품에 대한 이해를 높이는 내용입니다.

 In this talk, we propose an orientation-matching functional minimization for image denoising and image inpainting. Following the contemporary two-step TV-Stokes algorithm, a regularized tangential vector field with zero divergence condition is first obtained. Then a novel approach to reconstruct the image is proposed. Instead of finding an image that fits the regularized normal direction from the first step, we propose to minimize an orientation matching cost measuring the alignment between the image gradient and the regularized normal direction. This functional yields a new nonlinear partial differential equation for reconstructing denoised and inpainted images. The equation has an adaptive diffusivity depending on the orientation of the regularized normal vector field, providing reconstructed images which have sharp edges and smooth regions. The additive operator splitting scheme is used for discretizing Euler-Lagrange equations. We present the results of various numerical experiments that illustrate the improvements obtained with the new functional.

The implied volatility from Black and Scholes (1973) model has been empirically tested for the forecasting performance of future volatility and commonly shown to be biased. Based on the belief that the implied volatility from option prices is the best estimate of future volatility, this study tries to find out a better model, which can derive the implied volatility from option prices, to overcome the forecasting bias from Black and Scholes (1973) model. Heston (1993)’s model which improves on the problems of Black and Scholes (1973) model the most for pricing and hedging options is one candidate, and VIX which is the expected risk neutral value of realized volatility under the discrete version is the other. This study conducts a comparative analysis on the implied volatility from Black and Scholes (1973) model, that from Heston (1993)’s model, and VIX for the forecasting performance of future volatility. From the empirical analysis on KOSPI200 option market, it is found that Heston (1993)’s implied volatility eliminates the bias mostly which Black and Scholes (1973) implied volatility has. VIX, on the other hand, does not show any improvement for the forecasting performance.

Price sensitivities(Greeks) are mathematically defined as the derivatives of a derivative security's price with respect to various model parameters. The traditional way to compute a Greek is through its finite difference approximation. However, the finite difference method has a slow convergence rate when dealing with discontinuous payoffs. To overcome this poor convergence rate, a new theoretical tool recently has been introduced in the literature. This new tool uses the so-called Malliavin calculus in order to devise efficient Monte-Carlo methods for finance. In this lecture we discuss the basic ideas of Malliavin calculus and discuss how it may be used in Mathematical Finance.

Price sensitivities(Greeks) are mathematically defined as the derivatives of a derivative security's price with respect to various model parameters. The traditional way to compute a Greek is through its finite difference approximation. However, the finite difference method has a slow convergence rate when dealing with discontinuous payoffs. To overcome this poor convergence rate, a new theoretical tool recently has been introduced in the literature. This new tool uses the so-called Malliavin calculus in order to devise efficient Monte-Carlo methods for finance. In this lecture we discuss the basic ideas of Malliavin calculus and discuss how it may be used in Mathematical Finance.