This work is concerned with an optimal selling rule for a large
position of stock in a market. Selling a large block of stock in a short
period typically depresses the market, which would result in a poor filling
price. In addition, the large selling intensity makes the regime more
likely to be poor state in the market. In this paper, regime switching
and depressing terms associated with selling intensity are considered on
a set of geometric Brownian models to capture movements of underlying
asset. We also consider the liquidation strategy to sell much smaller
number of shares in a long period. The goal is to maximize the overall
return under state constraints. The corresponding value function with
the selling strategy is shown to be a unique viscosity solution to the
associated HJB equations. Optimal liquidation rules are characterized
by a finite difference method. A numerical example is given to illustrate
the result.

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

The polygonality in free groups concerns the existence of a van Kampen diagram on a closed surface with certain restrictions. By formulating polygonality in terms of corresponding Whitehead graphs, we prove that the double of a rank-two free group is one-ended if and only if it contains a hyperbolic surface group. For the free groups of higher ranks, we resolve the case when the amalgamating list of words have a regular Whitehead graph; a key ingredient is Edmonds and Seymour's characterizations of perfect matching polytopes. This is partially joint work with Sang-il Oum and with Henry Wilton.

Statistical tests are introduced for distinguishing between short-range dependent
time series with changes in mean, and long-range dependent time series, with the former
making the null hypothesis. The tests are based on estimation of long- (or short-) range
dependence parameter after removing changes in mean from the series. The focus is on
GPH (Geweke and Porter-Hudak (1983)) and local Whittle estimation methods in the
spectral domain. Theoretical properties of the resulting estimators are established when
testing for zero or one break, and small sample properties of the tests are examined in
simulations. The tests are applied to several real data sets.
The introduced tests also improve on the BHKS (Berkes, Horv¶ath, Kokoszka and
Shao (2006)) test which is the only other available test for the considered problem. It
is argued that the BHKS test su®ers from low power against long-range dependence
alternatives and that this happens because the BHKS test statistic involves estimation
of the sum of the time series covariances. The BHKS test could be readily improved
by considering its R/S-like regression version which estimates long-range dependence
parameter and which does not involve the sum of covariances. Yet better alternatives
are to use more powerful estimation methods (such as GPH or local Whittle) and lead
to the tests introduced here.

In 2006 at MSRI, nine tropical geometers and combinatorialists met and announced the list of ten key open problems in (algebraic and combinatorial side of) tropical geometry. Axiomatization of tropical oriented matroids was one of them. After the work of Develin and Sturmfels, tropical oriented matroids were conjectured to be in bijection with subdivisions of product of simplices as well as with tropical pseudohyperplane arrangements. Ardila and Develin defined tropical oriented matroid, and showed one direction that tropical oriented matroids encode subdivision of product of simplices. Recently, in joint work with Oh, we proved that every triangulation of product of simplices encodes a tropical oriented matroid.

In this talk, I will give a survey on this topic, and discuss this well known conjecture. I will also suggest a new class of combinatorial objects that may describe all subdivisions of a bigger class of polytopes.

The Helly number of a collection of sets is the size of its largest inclusionwise minimal subfamily with empty intersection. The precise conditions that lead to bounded Helly numbers have been studied since the 1920′s, when Helly showed that the Helly number of any collection of compact convex sets in R^d has Helly number at most d+1.

I will discuss a proof that any collection of subsets of R^d where the intersection of any subfamily consists of at most r connected components, each of which is contractible, has Helly number at most r(d+1). I will show how this implies, in a unified manner, quantitative bounds for several Helly-type theorems in geometric transversal theory.

Our main ingredients are a new variant of the nerve, a “homological nerve theorem” for this structure and an extension of a projection theorem of Kalai and Meshulam.

This is joint work with Eric Colin de Verdiere and Gregory Ginot.

Hilbert schemes of points on surfaces are the crepant resolutions of the symmetric products of surfaces. It can be studied via the orbifold theory or as moduli spaces of stable sheaves on surfaces. I will give a brief review of the study of the cohomology ring of the Hilbert schemes of points on surfaces by the method of vertex operator algebras. I will also discuss the extremal Gromov-Witten invariants of the Hilbert schemes. The former is a joint work with Z.B. Qin and W. Q. Wang. The latter is a joint work with J. Li.

Since the work of Gottsche on Betti numbers of Hilbert schemes of points on surfaces and Nakajima and Grojnowski's independent discovery of Heisenberg algebra actions on the cohomology ring of the Hilbert schemes, there have been a lot of research on the cohomology ring structure via the infinite dimensional Lie algebras. In this talk, I will explain how some infinite dimensional Lie algebras come to play a fundamental role in the seemly totally unrelated research area of topology and geometry of Hilbert schemes.

There are, so far, two classes of moduli examples which naturally carry symmetric obstruction theories: Moduli of stable objects in the abelian category of a) coherent sheaves on a Calabi-Yau threefold; and b) representations of a quiver with relations given by a superpotential. In this talk, we present one more such class motivated by the work of Diaconescu: Moduli of stable objects in the abelian category of coherent twisted quiver sheaves on a fixed projective smooth curve. It will arise as a curve counting on a holomorphic symplectic quotient described by a quiver and as an application of quasimap construction. The quasimap construction is joint with I. Ciocan-Fontanine and D. Maulik. When the symplectic quotients are the Hilbert schemes of points on the plane, the study was carried by D.E. Diaconescu.

Numerous empirical studies have shown that Geometric Levy Models (GLM) are able to fit extremely well daily returns of financial assets. On the other hand, in spite of their importance and popularity nowadays, few works have considered intraday data. In this paper, we fill this gap by analyzing the ability of two popular GLM (the Variance Gamma and Normal Inverse Gaussian) to fit the statistical features of intraday data at different sampling frequencies. Using high-frequency transaction data of the equity market, we find out that the estimator of the volatility parameter is quite stable at a wide range of intraday frequencies, in a sharp contrast to the estimator of the kurtosis parameter, which is more sensitive to market microstructure. By taking into account a microstructure noise component and then characterizing its effect in the estimation results, we put forward a heuristic method to determine suitable sampling frequencies for which the Levy model could serve as a good probabilistic approximation of the underlying price process. We also assess the performance of the two most favored estimation methods, the Method of Moments Estimators (MME) and the Maximum Likelihood Estimators (MLE), when dealing with high- frequency data sampled from a GLM. We show that neither high-frequency sampling nor MLE will result in a significant reduction of the estimation error of the volatility parameter. On the contrary, the estimation error of the parameter controlling the kurtosis of log returns can be significantly reduced by using MLE or intraday data. The theoretical infill asymptotic behavior of the bias and standard error of the MME are also provided with and without a microstructure noise component.

A moment angle complex is realized as a subspace of a polydisk (D^2)^m and the natural action of a torus (S^1)^m on (D^2)^m leaves the moment angle complex invariant. The Buchstaber invariant $s(K)$ is defined to be the maximum integer for which there is a subtorus of dimension $s(K)$ acting freely on the moment-angle complex associated with a finite simplicial complex $K$. Analogously, its real version $s_{\mathbb{R}}(K)$ can also be defined by using the real moment-angle complex instead of the moment-angle complex. The goal of this talk is to provide some properties of the (real) Buchstaber invariants $s_{\mathbb{R}}(K)$ of a seleton of a simplex in two ways.(Y.Fukukawa, M.Masuda)

We construct a basis the space of  weakly holomorphic Drinfeld modular forms.

and observe some arithmetic properties of coefficients of the basis elements.

Precisely we find that the basis elements satisty some generating function and find some duality between coefficients of the basis elements.

In this talk I will give an overview of contact topology and an important invariant of contact manifolds, namely contact homology. Contact homology is an invariant in the spirit of Floer homology: its chain complex is freely generated by so-called periodic Reeb orbits, which are closed orbits of a distinguished vector field. The differential is defined by counting certain holomorphic curves. Contact homology and its generalization, symplectic field theory, are able to detect the non-existence of fillings. I will describe how the algebraic structure of these theories can be used to show such a property.

To be announced

A Bayesian network is a graphical model that encodes probabilistic relationships among variables of interest. In this seminar I will talk about the definitions and properties of Bayesian networks. I will also describe Bayesian statistical methods for using data to estimate Bayesian network parameters. I will describe methods for learning both the parameters and structure of a Bayesian network. In addition, I will talk about the applications of the Bayesian network methods to classification tasks. If time permits, I will describe recent studies based on log-linear models for learning Bayesian network structures.

We will focus on characterizing operators up to unitary equivalence or similarity. We will use concepts and ideas fom complex geometry and harmonic analysis.

In this talk, we consider mathematically and computationally optimal control problems for stochastic partial differential equations under the Neumann boundary condition. The control objective is to minimize the expectation of a cost functional, and the control is of the deterministic, boundary value type. Mathematically, we prove the existence of an optimal solution and a Lagrange multiplier; we represent the input data in their Karhunen-Loµeve (K-L) expansions and deduce the deterministic optimality system of equations. Computationally, we approximate the finite element solution of the optimality system and estimate its error through the discretizations of both the probability space and the spatial space.

Sophus Lie developed the theory of Lie groups and their Lie algebras. The theory of semisimple Lie algebra is useful in many parts of mathematics and physics. Many results of Lie algebra are corresponding the thing of Lie group such as representation theory and the classification problem. The semisimple Lie algebras over C are in one to one correspondence with compact and simply connected Lie groups. In the period 1888 - 1894 much of the structure of Lie algebras over C was developed, including W. Killing's discovery of "Killing form" in E. Cartan's thesis. And, Weyl's Theorem was orginally proven using integration on compact Lie groups. An algebraic proof of Weyl's theorem was found in 1935 by Casimir and van der Waerden.
The semisimple Lie algebras over C were first classified by Wilhelm Killing though his proof lacked rigor. His proof was made rigorous by Elie Cartan (1894) in his Ph.D. thesis, who also classified semisimple real Lie algebras. This was subsequently refined, and the present classification by Dynkin diagrams was given by then 22-year old Eugene Dynkin in 1947. Some minor modifications have been made (notably by J. P. Serre), but the proof is unchanged in its essentials.
In this talk, we discuss the classification problem of finite dimensional semisimple Lie algebras over C. And we give some relevant concepts of classification problem such as Dynkin diagram and root systems and some algebraic property of Lie algebras.

We will consider a program launched by C. S. Morawetz about the existence of weak solutions to two-dimensional, steady, irrotational transonic flow equations. Among other things we will consider an analysis of the boundary conditions for the viscous problem and an analysis of the entropy functions which are generated by a Tricomi-like mixed-type equation.

수리과학과에는 30여명의 교수님이 계시고 교수님들의 연구 분야는 모두 다른데 각 교수님의 연구 분야를 자세히 들을 기회가 부족했기 때문에 이번 학기부터 대학원 진학을 희망하는 학부생과 지도교수를 정하지 않은 대학원 신입생에게 연구 분야 소개 행사를 시작했습니다.

11월 23일에는 Weidong Ruan 교수님이 소개하십니다.

Our department has about 30 professors with various research areas but it was rather difficult to listen to research areas in detail. Starting from this semester, we have events to introduce research areas to undergraduate students who are interested in graduate schools or the first year graduate students who did not yet decide advisors.  

On Nov. 23, Prof. Weidong Ruan will introduce his research area.

Laguerre histories are certain colored Motzkin paths with some weight for each elementary steps. In this talk, we study two famous bijections between permutations and Laguerre histories, made by Francon-Viennot and Foata-Zeilberger. This two bijections are enable us to give permutations as combinatorial interpretations of continued fractions. The former is associated in linear statistics and the latter be in cyclic statistics. Using two mappings, we are able to make various results about several statistics of permutations.

The $G_{2}$-manifolds are Riemannian manifolds with the speical holonomy group $G_{2}$. Here  $G_{2}$ is one of the exceptional Lie groups, which is the automorphism group of octonions. In this talk, we introduce the geometry of $G_{2}$-manifolds and the duality theory motivated from M-theory in Physics. We also discuss recent progress on the geometry from the analogous point of view to the symplectic geometry. 

To be announced

학사과정 무학과 학생의 학과 예비신청에 도움을 주기 위한 수리과학과 설명회가 열립니다. 많은 참석 바랍니다.

(피자제공)

I will discuss some computational problems that arise naturally and frequently in algebraic geometry. Esepcially I will explain how to formulate correct notion of limits of algebraic varieties, algorithms for computing the limits, and the cost of such computations. I will give various applications to several different areas, including a recent result in birational geometry of moduli space of stable curves.

Since 1970s, the study of topological spaces with torus actions has played an important role in various areas of mathematics, such as algebraic topology, algebraic geometry, convex geometry, symplectic geometry, combinatorics, and commutative algebra. As it becomes widespread, recently a field of activity called ``Toric topology'' has emerged. Davis and Januszkiewicz first introduced a certain universal $T^m$-manifold whose orbit space is a combinatorial simple polytope P. This manifold has the following universal property : for every quasitoric manifold $M^{2n} \to P$ there is a principal $T^{m-n}$-bundle $Z_P \to M^{2n}$ whose composite map is the orbit map for $Z_P$. This space $Z_P$ is later called "moment angle manifold". Buchstaber and Taras Panov extended this space to the case of arbitrary simplicial complex, and they named those spaces the "moment-angle complexes". Recently, Zhi Lu and Taras Panov extended the construction of moment-angle complexes to simplicial posets by using some categorical language.
In this talk, we discuss some facts about simplicial posets and some backgrounds about the homological algebra related to the face ring of simplicial posets. And we investigate the study of the topology of the moment angle complexes. Finally we introduce some recent works and remaining questions.

The zero level set of a solution to the heat equation is considered. Under certain conditions on moments of the initial data, we will prove the zero level set in a ball converges to a specific graph as time goes to infinity. Solving a linear combination of the Hermite polynomials gives the graph and coefficients of the linear combination depend on moments of the initial data. Also the graphs to which the zero level set converges can be classified in some cases

Minimization of functionals related to Euler's elastica energy has a wide range of applications in computer vision and image processing. An issue is that a high order nonlinear partial differential equation needs to be solved and the conventional algorithm usually takes high computational cost. In this talk, we propose a fast and efficient numerical algorithm to solve minimization problems related to the Euler's elastica energy and show applications to variational image denoising, image inpainting, and image zooming. We reformulate the minimization problem as a constrained minimization problem, followed by an operator splitting method and relaxation. The proposed constrained minimization problem is solved by using an augmented Lagrangian approach. Numerical tests on real and synthetic cases are supplied to demonstrate the efficiency of our method.

The study of electric potentials on graphs has a close connection with probability theory. On distance-regular graphs the theory is particularly elegant, as known results concerning such graphs allow explicit calculations to be made. In this talk, I will introduce these topics and discuss some recent results that Jacobus Koolen and I have obtained, as well as some conjectures and open problems.

In this talk, I will discuss the existence of positive solutions of Hessian equations of Lane-Emden type.
In particular, I will explain the structure of solutions which are compared with the structure on Lane-Emden equations.