To be announced

I plan to explain the interactions between diffusion and the spatial inhomogeneity in mathematical ecology. Main examples used for illustration include the classic logistic equation and the Lotka-Volterra competition systems. Starting from eigenvalue problems for indefinite weights, we will study the various interesting phenomena associated with the (single) logistic equation as well as the competition systems in a systematic way. More realistic models, such as directed movements and taxis, will be discussed.

To be announced

우리나라에서는 경영학이 '문과'학문으로 분류되어 있다. 그러다 보니 으레 경영학을 인문학이나 사회과학처럼 사람과 사람의 관계만 다루거나, 수치화 하기 어려운 부분을 다루는 학문으로 여기는 경향이 있다. 이로 인해 경영에는 수학적 논리나 산술적 분석 능력보다는 사람을 다루는 능력이 더 필요하다는 일반적인 인식을 가지고 있다. 아쉽게도 현대 경영은 이런 일반인들의 정서와 상당한 간극이 있다. 왜 현대 경영 현장에서 수학자들을 원하는지 그리고 현대 경영에서 어떻게 수학적 이론이 활용되어 경영현장에 응용되는지를 알아본다. 외국 글러벌 기업에서 활약하는 수학자들의 역할과 마케팅, 세일즈, 유통, 기획실 등 다양한 분야에 응용되는 수학적 이론을 소개한다.

Since the EA crisis, the Korean government has pursued capital account
liberalization aggressively. Although the swap basis has increased
significantly since onset of the global credit crisis, both rates are
highly correlated.
In fact, the markets for TBs, swaps and foreign exchange rates are closely
interconnected, and deep financial linkages have been established.
The direct implication of the deep financial linkages is a sharp rise in
non-core foreign currency liabilities.
Non-core liabilities, i.e., interbank liabilities not reflected in the
monetary aggregates, are vulnerable to credit shocks.
Deleveraging started immediately after the Lehman Brothers collapse, and
double drain was unprecedented.

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

A Monte Carlo method is one of the most frequently used methods to price
financial exotic derivatives. It can be used for almost all financial
derivatives easily except American style ones. Especially for path-
dependent exotics, it can be a most useful method since we can easily give
conditions to generated sample paths. However, generating sample paths with
daily grids for giving conditions could waste the performance. Alternative
solution using a probability density will be introduced and applied to
pricing an ELS, one of the most exotic derivatives in Korea.

Consisting of time-frequency shifts of functions in L^2(R), the Gabor system (also known as the Weyl-Heisenberg system) provides the ground for time-frequency analysis of data. We consider multiwindow Gabor systems (G_N; a, b) with N compactly supported windows and rational sampling density N/ab. We give another set of necessary and suffcient conditions for two multiwindow Gabor systems to form a pair of dual frames in addition to the Zibulski-Zeevi and Janssen conditions. Our conditions come from the back transform of Zibulski-Zeevi condition to the time domain but are more informative to construct window functions. For example, the masks satisfying unitary extension principle(UEP) condition generate a tight Gabor system when restricted on [0,2] with a=1 and b=1. As another application, we show that a multiwindow Gabor system (G_N; 1, 1) forms an orthonormal basis if and only if it has only one window(N=1) which is a sum of characteristic functions whose supports `essentially' form a Lebesgue measurable partition of the unit interval. Our criteria also provide a rich family of multiwindow dual Gabor frames and multiwindow tight Gabor frames for the particular choices of lattice parameters, number and support of the windows.

Consisting of time-frequency shifts of functions in L^2(R), the Gabor system (also known as the Weyl-Heisenberg system) provides the ground for time-frequency analysis of data. We consider multiwindow Gabor systems (G_N; a, b) with N compactly supported windows and rational sampling density N/ab. We give another set of necessary and suffcient conditions for two multiwindow Gabor systems to form a pair of dual frames in addition to the Zibulski-Zeevi and Janssen conditions. Our conditions come from the back transform of Zibulski-Zeevi condition to the time domain but are more informative to construct window functions. For example, the masks satisfying unitary extension principle(UEP) condition generate a tight Gabor system when restricted on [0,2] with a=1 and b=1. As another application, we show that a multiwindow Gabor system (G_N; 1, 1) forms an orthonormal basis if and only if it has only one window(N=1) which is a sum of characteristic functions whose supports `essentially' form a Lebesgue measurable partition of the unit interval. Our criteria also provide a rich family of multiwindow dual Gabor frames and multiwindow tight Gabor frames for the particular choices of lattice parameters, number and support of the windows.

Volatility is one of the most important factors in stock and derivative markets.
In this lecture, we introduce several derivatives on volatility and explain how
to hedge volatility risk. We also explain how to price volatility derivatives.

해석학에서의 부등식의 증명 혹은 반증명과 관련된 매우 간단하지만 유용한 방법을 소개하고, 몇 가지 예제를 통하여 dimensional analysis의 기본 개념과 활용을 살펴보고 더 관심 있는 학생들을 위하여 레퍼런스를 소개합니다.

미적분학, 선형대수학을 듣고, Cauchy-Schwartz 부등식을 이해한 학생들이면 따라올 수 있는 강연입니다.

American options can be exercised at any time before the expiration date, which makes it difficult to analyze the price and the optimal exercise boundary of an American option. In this talk we discuss the analysis and numerical computations of the optimal exercise boundary for American options.

In this presentation we introduce the so called Cell Boundary Element (CBE) methods for partial differential equations. It can be interpreted as a hy-bridized DG method. The CBE method was introduced by the speaker and
his colleagues. The method is base on 1)a local solution decomposition 2)flux continuity on intercell boundary. Therefore, the method is defined on the skeleton of a mesh generation, which will reduce degrees of freedom a lot. Moreover, the method naturally satisfies local flux conservation property.
We apply our method for the following PDEs:

• 2nd order elliptic equations
• Stokes equations
• multiscale elliptic equations

In this talk, we will introduce the conductivity recovery problem particularly when internal current density is given.

Variational methods were developed as an alternative formalism in classical mechanics and applied to various optimization problems including financial mathematics. In this introductory talk, basics of variational methods in physics will be explained and the relation between Hamilton-Jacobi methods and Hamilton-Jacobi-Bellman equation will be explored.

 Since a risk premium is determined by the riskiness of an asset and the attitude of investors to risk, one of them or both of them could predict the future return variations. As a proxy for the risk attitude of investors, I adopt the risk aversion and investigate the relationship with return predictability. More specifically, I propose the new method to estimate the timevarying risk aversion and examine its predictability for the future excess returns of the S&P
500 index. The results show that the risk aversion has a predictive power for future index returns even in a short horizon and does not lose the significance in the presence of conventional forecasting variables including dividend yield, short rate, and variance premium.
 Additionally, I test its Sharpe ratio predictability to guarantee the accuracy of the risk
aversion measure. The results regarding the Sharpe ratio prediction support the relevance of
the new measure for capturing the time-varying risk aversion.

A transversal matroid is a collection of objects that encodes maximal matchings in a bipartite graph. Generalized permutohedra is a class of polytopes obtained by deforming the permutohedron. We introduce a nice bijection that allows one to view transversal matroids as set of lattice points inside a generalized permutohedron. As a corollary, we solve a special case of the 30-year old conjecture by Stanley on matroids and pure O-sequences. The talk will be elementary and purely combinatorial.

The aim of these lectures is to provide an accessible understanding of time series models, heteroskedastic models and pure jump Levy models which are very useful in the study of financial mathematics. In Lecture 1, we first discuss the concept of stationary time series and then focus to discuss the basics of time series regression models and the ARCH and GARCH models. In Lecture 2, we first start with analyzing a very simple example of a pure jump process, viz. a compensated Poisson process which offers significant insight into the structure of pure jump Levy process. We next discuss several important results about Levy processes, such as Levy measures, the Levy-Ito decomposition, the Levy-Khintchine formula and infinite divisible distributions. We finally discuss how the asset price process can be modeled using Levy processes.

The aim of these lectures is to provide an accessible understanding of time series models, heteroskedastic models and pure jump Levy models which are very useful in the study of financial mathematics. In Lecture 1, we first discuss the concept of stationary time series and then focus to discuss the basics of time series regression models and the ARCH and GARCH models. In Lecture 2, we first start with analyzing a very simple example of a pure jump process, viz. a compensated Poisson process which offers significant insight into the structure of pure jump Levy process. We next discuss several important results about Levy processes, such as Levy measures, the Levy-Ito decomposition, the Levy-Khintchine formula and infinite divisible distributions. We finally discuss how the asset price process can be modeled using Levy processes.

In the course of proving the strong perfect graph theorem, Chudnovsky, Robertson, Seymour, and Thomas showed that every perfect graph either belongs to one of five basic classes or admits one of several decompositions. Four of the basic classes are closed under taking induced subgraphs (and have known forbidden subgraph characterizations), while the fifth one, consisting of double-split graphs, is not. A graph is doubled if it is an induced subgraph of a double-split graph. We find the forbidden induced subgraph characterization of doubled graphs; it contains 44 graphs.
This is joint work with Boris Alexeev, and Alexandra Fradkin.

To be announced