Department Seminars & Colloquia
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A classical problem regarding pointwise behavior of the free Schr\"odinger equation is to determine the optimal exponent $s$ for which
$\lim_{t\to 0} e^{it\Delta}f(x)=f(x) \text{ a.e. }x\in \mathbb R^d,$ whenever $f\in H^s(\mathbb R^d)$. This problem was settled for $d=1$ but still remains open for the higher dimensions even though some progresses have been made for $d=2$. In this talk we consider the convergence problem for Schr\"odinger equations with quadratic potentials, which include the hermite Schr\"odinger equation. This problem is also closely related to pointwise convergence along variable curves. We will show that these problems are more or less equivalent to that of free Schr\odinger equation and discuss on equivalence of related time-space estimates. Part of this talk is based on a joint work with Keith Rogers.
We introduce knots and links in dimension three and their deformation in dimension four which was first introduced by Fox and Milnor. We discuss its key role in understanding the topology of 4-dimensional manifolds and the interplay with 3-dimensional topology. We discuss briefly recent developments, including joint work with Kent Orr which presents new L2-theoretic methods for amenable groups.
Keeping the previous notation, these satisfy the stronger condition n>=2c+1.
Here we will talk about special cases of the (in)famous Hartshorne Conjecture.
Lecture III: Defective manifolds. There are two kinds of such: secant
defective and dual defective. We shall investigate both and see that they
are actually related.
The weakly over-penalized symmetric interior penalty (WOPSIP) method, which belongs to the family of discontinuous Galerkin methods, was introduced for second order elliptic problems by Brenner et al. in 2008. We will discuss a preconditioner for the WOPSIP method that is based on balancing domain decomposition by constraints (BDDC). Theoretical results on the condition number estimate of the preconditioned system will be presented along with numerical results. This is joint work with Susanne C. Brenner and Li-yeng Sung.
Matrix analogues of embedded surfaces, as well as discrete versions of curvature and the Euler characteristic will be presented, and the new notions, related to a novel algebraic formulation of classical differential geometry in terms of Poisson/Nambu brackets, will be illustrated by several examples.
This paper presents a mathematical analysis of the time harmonic
electrical complex potential in an electrical conducting object. In fact, there is little previous work on the complex elliptic partial differential equation (PDE) and it is difficult for us to sketch distributions of complex potential and current density for a given admittivity distribution inside the conducting object. In order to develop admittivity imaging reconstruction methods, we need to study the complex elliptic PDE and its solution. In this paper, we will introduce auxiliary real potentials which enable us to predict distributions of the complex potential and its current density. Above all, in this paper, we are interested in the interrelation between angular frequency $\om$, conductivity and permittivity distributions $\sigma$, $\ep$ inside the conducting object, and the complex potential. Although auxiliary potentials are quite useful for understanding the behavior of the complex potential, it is still difficult to see the influence of $\om$, $\sigma$, and $\ep$ to the complex potential. So we will try to investigate relations between the complex potential and $u_{\sigma}$, $u_{\ep}$, where $u_{\sigma}$ and $u_{\ep}$ are well-known solutions to the standard elliptic PDE with coefficient $\sigma$ and $\epsilon$, respectively. In particular, we will observe that the imaginary $h_{\gamma^\om}$ is closely related with the difference $u_\sigma - u_\ep$. Finally, as an application, this observation will be applied to multi-frequency Trans-Admittance Scanner in order to understand the interpretation of the frequency difference methods for breast cancer diagnosis. It is shown that the weighted difference method proposed in \cite{klswz} is actually based on the difference Neumann data $\f{\p u_\sigma}{\p \n} - \f{\p u_\ep}{\p \n}$.
Interactions between mathematics and physics are incrasing in the application of mathematics to problems in physics as well as in solving mathematical problems inspierd by physics. In this talk, mathematical frameworks to describe quantum mechanics will be introduced. Some known results and heuristic arguments to understand many-particle systems will also be explained.
E6-1 #3433
Discrete Math
Yoomi Rho (University of Incheon)
L(j,k) Labelings of Direct Product of Complete Graphs
E6-1 #3433
Discrete Math
An $L(j,k)$ labeling of a graph is a vertex labeling such that the difference of the labels of any two adjacent vertices is at least $j$ and that of any two vertices of distance 2 is at least $k$. The minimum of the spans of all $L(j,k)$-labelings of $G$ is denoted by $\lambda_k^j(G)$. Recently Haque and Jha \cite{HJ} proved if $G$ is a direct product of complete graphs, then $\lambda_k^j(G)$ coincide with the trivial lower bound $(N-1)k$ where $N$ is the order of $G$ when $\frac{j}{k}$ is within a certain bound.
In this paper, we suggest a new labeling method of such a graph $G$. With this method, we extend the range of $\frac{j}{k}$ such that $\lambda_k^j(G)=(N-1)k$ holds. Moreover, we obtain an upper bound of $\lambda_k^j(G)$ for the remaining cases.
In this talk we discuss the key ideas on -theory on stochastic parabolic equations and systems in an informal manner. We use only simple equations and systems to avoid technical complexity. We talk over the followings:
● A heat equation with stochastic force and its solution.
● Estimation of the solution using BDG inequality and a generalized Littlewood- Paley inequality.
● Half space domain and weights.
● A warning on systems.
● Stochastic parabolic systems.
E6-1 #1409
Discrete Math
Seog-jin Kim (Konkuk University)
Decomposition of Sparse Graphs into Forests and a Graph with Bounded Degree
E6-1 #1409
Discrete Math
Say that a graph with maximum degree at most d is d-bounded. For d>k, we prove a sharp sparseness condition for decomposability into k forests and a d-bounded graph. Consequences ar e that every graph with fractional arboricity at most k+ d/(k+d+1) has such a decomposition, and (for k=1) every graph with maximum average degree less than 2+2d/(d+2) decomposes into a forest and a d-bounded graph. When d=k+1, and when k=1 and d≤6, the d-bounded graph in the decomposition can also be required to be a forest. When k=1 and d≤2, the d-bounded forest can also be required to have at most d edges in each component.
This is joint work with A.V. Kostochka, D.B. West, H. Wu, and X. Zhu.
A time harmonic scattering problem of electromagnetic waves from a two-dimensional open cavity is considered. A variational formulation reduces the scattering problem into a bounded domain(the cavity) problem. In this talk, a stability of the solution isestablished for the bounded domain problem in the energy space.
Moreover, the stability estimate provides the explicit dependence on the high wave number.
Introduction of the "Basic Notions in Physics" seminar (hosted by Prof. Wei-Dong Ruan):
Mathematics and physics are very much intertwined, and development in mathematics has been greatly impacted by ideas from physics in the recent decades. This seminar aims at giving math students (graduate and advanced undergraduate students) and faculty an introduction to basic concepts and ideas in physics, in languages they can understand. Hopefully it will help mathematicians better understand some of the development in mathematics brought by physics ideas. We are very glad to have Prof. Stewart, an expert in theoretical physics, to give talks in the seminar.
A page in Ramanujan's lost notebook contains two identities for trigonometric sums in terms of doubly infinite series of Bessel functions. One is related to the famous "circle problem'' and the other to the equally famous "divisor problem." These relations are discussed as well as various attempts to prove the identities. Our methods also yield new identities for certain trigonometric sums, for which analogues of the circle and divisor problems are proposed. The research to be described is joint work with Sun Kim and Alexandru Zaharescu.
We establish sharp two-sided estimates for the Dirichlet heat kernel of time-dependent parabolic operators with singular drifts in a $C^{1,\alpha}$-domain in $\R^d$, where $d\ge 1$ and $\alpha \in (0, 1]$.
Our operator is $\partial_t - L - \mu \cdot \nabla_x$, where $L$ is a time-dependent uniformly elliptic divergent
operator with Dini continuous coefficients
and $\mu$ is a signed measure on $(0,\infty)\times \R^d$ belonging to the parabolic Kato class.
Along the way, a gradient estimate is
also established. In a probabilistic counterpart, we construct a Markov process corresponding to the operator $\partial_t - L - \mu \cdot \nabla_x$ and obtain two-sided estimates for the transition density of the Markov process.
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.
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.
5시부터 자연과학동 3433호에서 다과회(피자제공)이 있습니다.
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.
후원:수리과학연구정보센터, 금융수학연구센터
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.
후원:수리과학연구정보센터, 금융수학연구센터
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.
후원:수리과학연구정보센터, 금융수학연구센터
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.
후원:수리과학연구정보센터, 금융수학연구센터
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.
We will discuss the specialties due to work with internal data and will give rather clear theory based on classical pde theory. This talk includes the case when conductivity is tensor field, so called anisotropic conductivity.
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.
후원:수리과학연구정보센터, 금융수학연구센터