歷史演講 PAST TALKS

摘要： We consider a Parabolic Anderson Model (PAM) with color noise. Using a transformation, the equation becomes an ordinary parabolic differential equation. We derive an expression for the solution. This can be used to derive Feymann-Kac formula for the moments of the solution.

參考: Brownian Motion and Stochastic Calculus 2/E, Ioannis Karatzas and Steven E. Shreve.

In this talk, three-dimensional viscous primitive equations are considered. Primitive equations are also called hydrostatic Navier-Stokes equations which are the governing equations of the motion of flows in geophysical fluid dynamics. The global in time strong solution for these models has been established by Cao and Titi and Kobelkov independently. In this talk, several time-stepping numerical schemes have been studied. We are interested in the study and discussion of the long-time stability of these numerical schemes, which is important for the study of the climate. In this talk, we will start with some examples which give that the long time behavior of numerical schemes could be problematic. Then, we will introduce the primitive equations, present the semi-discretized schemes and analyze and discuss these numerical schemes. This talk is based on the joint work with Chun-Hsuing Hsia.

In this talk, I briefly introduce some collective phenomena arising from complex systems and discuss some mathematical issues related to collective mode of complex systems.

參考: Brownian Motion and Stochastic Calculus 2/E, Ioannis Karatzas and Steven E. Shreve.

Organizer : 陳建隆 (Jann-Long Chern) 教授 (中央大學數學系)

A method based on a differential algebraic equation (DAE) approach is employed to find stationary solutions of the Poisson-Nernst-Planck equations with steric effects (PNP-steric equations). Under appropriates boundary conditions, the equivalence of the PNP-steric equations and a corresponding system of DAEs is shown. Solving this system of DAEs leads to the existence of stationary solutions of PNP-steric equations. We show that for suitable range of the parameters, the steric effect can produces infinitely many discontinuous stationary solutions. Moreover under a stronger intra-species steric effect, we prove that a smooth solution converges to a constant stationary solution. This is a joint work with Mach Nguyet Minh.

數學建模專題演講 Abstract: Data mining is a powerful tool that can be used to analyze large quantities of data and discover potentially helpful patterns or hidden rules. Data mining is widely used in many fields such as marketing, yield enhancement, fault location, health care management, customer behavior analysis, and knowledge management. In this talk, basic concept and several selected techniques of data mining will be covered. In addition, how to formulate real world problems into mathematical models will be discussed. Finally, some applications in business using data mining techniques will be presented with cases study.

參考: Brownian Motion and Stochastic Calculus 2/E, Ioannis Karatzas and Steven E. Shreve.

In this talk, we give a product formula of Hermite polynomials and show the relation between the real Wiener-Itô chaos and the complex Wiener-Itô chaos (or: multiple integrals), and then we extent the fourth moment theorem (or say: Nualart-Peccati criterion) to the complex multiple Wiener-Itô integrals. We also prove that the complex Hermite polynomials (or say: Hermite-Laguerre-Itô polynomials) are the eigenfunctions of complex Ornstein-Uhlenbeck operators. This talk is based on the following joint articles with Yong CHEN (陳勇). [1] Chen, Y., Liu Y., On the eigenfunctions of the complex Ornstein-Uhlenbeck operators, Kyoto J. Math., Vol. 54(3), 577-596, (2014) [2] Chen Y., Liu Y., On the fourth Moment Theorem for the Complex Multiple Wiener-Itô Integrals, Preprint (2015)

參考: Brownian Motion and Stochastic Calculus 2/E, Ioannis Karatzas and Steven E. Shreve.

When a population grows old, it is always interesting to know what happened to it in the past. The coalescence problem provides a way to understand the ancestry of the individuals in the population. Here, we pick some individuals at random from the current generation by simple random sampling without replacement and trace their lines of descent backward in time till they meet for the first time. We call the common ancestor of these chosen individuals at the coalescent time their most recent common ancestor. The coalescence problem is to investigate the limit behaviors of some characteristics of this most recent common ancestor such as the generation number and the type in a population with various types of individuals.

Free probability, arising from the theory of operator algebras, is a mathematical theory that studies random variables in non-commutative probability spaces. This theory was initiated by D. Voiculescu around 1985 in order to study an important unsolved problem in operator algebra, and is often regarded as a non-commutative parallelism of classical probability theory. Independence in non-commutative probability spaces (or in quantum physics) is called freeness, the analogue of the classical notion of independence. Surprisingly, the random matrix models play a major role in the study of free probability theory. Connections to other theories, such as classical probability theory, combinatorics, wireless communication, large deviations, quantum information theory, etc. were established later on. In this talk, we will briefly introduce fundamental results in free probability and random matrix theories, and their connections to classical probability theory. More precisely, we will talk about the free counterpart of limit theorems, infinitely divisible distributions along with their Lévy-Hinčin representations, Lévy processes, stale laws, etc. If time permits, we will also introduce bi-free probability, the newly developed theory in free probability, and its recent developments.

參考: Brownian Motion and Stochastic Calculus 2/E, Ioannis Karatzas and Steven E. Shreve.

Abstract: In this talk I will discuss the properties solutions of some degenerate parabolic equations which arise in various physical and geometric problems. In particular I will discuss the asymptotic behaviour of solutions of such equations.

Algorithms for overcoming the curse of dimensionality for certain Hamilton-Jacobi equations arising in control theory and elsewhere ( a recent work by Darbon and Osher).

參考: Brownian Motion and Stochastic Calculus 2/E, Ioannis Karatzas and Steven E. Shreve.

Algorithms for overcoming the curse of dimensionality for certain Hamilton-Jacobi equations arising in control theory and elsewhere ( a recent work by Darbon and Osher).

大綱： 用邏輯分析找尋自己的未來!簡述金融產業與數學相關的領域，如:CFP、CFA與精算師等。 介紹: 本場演講適合尋找未來、深造與就業方向的大學生。演講者為本系系友現任職國泰人壽。

參考: Brownian Motion and Stochastic Calculus 2/E, Ioannis Karatzas and Steven E. Shreve.

In this talk we consider a Kolmogorov operator K_0 in a Hilbert space H, related to a SPDE with jumps. We show that K_0 is essentially m-dissipative in the space L^p ([0,T]×H;ν), p≧1, where ν(dt,dx)=ν_t (dx)dt and the family {ν_t }_([0,T]) is a solution of the Fokker-Planck equation given by K_0. We also prove that uniqueness of solutions to the Fokker-Planck equation. This result is a generalization of the SPDE considered in [V. Bogachev, G. Da Prato, M. Röcker, Fokker-Planck equations and maximal dissipativity for Kolmogorov operators with time dependent singular drifts in Hilbert space, Journal of Functional Analysis, 256(2009) 1269-1298] to SPDE with jumps.

參考: Brownian Motion and Stochastic Calculus 2/E, Ioannis Karatzas and Steven E. Shreve.

We consider stochastic heat equations [SHEs] on the interval with Dirichlet boundary condition or Neumann boundary condition. Kim and Khoshnevisan prove that the growth of the energy is approximately exponential to the noise level to some power and then Foodun and Joseph improve few of their results and more delicate results are discussed in the paper of Foodun and Nualart.

A set of lines in R^n is called equiangular if the angle between each pair of lines is the same. We address the question of determining the maximum size of equiangular line sets in R^n, using semidefinite programming to improve the upper bounds on this quantity. Improvements are obtain in dimensions 23<=n <=136. In particular, we show that the maximum number of equiangular lines in R^n is 276 for all 24 <=n <=41 and 344 for n=43. This provides a partial resolution of the conjecture set forth by Lemmens and Seidel (1973). We also study the existence problem for tight spherical designs of harmonic index T. We prove the nonexistence of tight {8,4} designs by using the theory of elliptic diophantine equations, and the semidefinite programming method of eliminating some 2-angular systems for small dimensions.

Recently, a wide variety of Poisson-Boltzmann (PB) type equations have been established to approach the phenomena of ion transport in various electrolyte solutions. Due to finite sized effect on the electric double layer at the surface, Andelman et. al. [PRL, 1997] have investigated a modified Poisson-Boltzmann (MPB) equation for electrolyte solutions with two types of ions having the same size. The MPB equation is a good model for describing the effect of ions' radii on the behavior of the electric double layer. However, in the case when different size ion species occupy electrolyte solutions, the MPB equation seems unavailable. The main purpose of this talk is to introduce a new PB equation with finite size effects, called the PB_ns equation. To briefly study this model, we are going to focus on the case of one cation and one anion species and formally derive a limiting form of the PB_ns equation which is more general than the MPB equation. In particular, when the size of all ions are the same, we show that solutions of the PB_ns equation and the MPB equation have the same asymptotic behavior in the limit of zero Debye length.

A common variation of the sum-product problem is the question of determining whether certain sets defined by a combination of additive and multiplicative operations are guaranteed to be large. Recent years have seen progress in this area, with a number of essentially optimal results being established. A natural follow-up problem concerns the structural question of when these lower bounds can be attained. This talk will give an introduction to problems of this type, discussing some of the many open problems in the area and giving some details of the few results that are known.

演講因故取消

參考: Brownian Motion and Stochastic Calculus 2/E, Ioannis Karatzas and Steven E. Shreve.

In this talk, we establish the global transonic solutions for hydrodynamic escape problem (HEP), which is characterized by a free boundary value problem of Euler equation with gravity and heat. The global existence of transonic solutions to HEP is established by the generalized Glimm's method. The range of the hydrodynamical escape region in planetary atmospheres will also be discussed.

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Abstract: Under the existence of the traveling waves in some reaction-diffusion (RD) systems, the stability of traveling waves means the asymptotic stability, or the so-called orbital stability due to the translation invarience of traveling waves. The stability analysis are based on the spectrum distribution of the operator which is the RD system linearized around the traveling waves. In the first part of this talk, some basic theories of stability will be reviewed. In the second part, we will consider stability of traveling waves in the Lotka-Volterra competition-diffusion systems as the example been studied currently.

參考: Brownian Motion and Stochastic Calculus 2/E, Ioannis Karatzas and Steven E. Shreve.

In Euclidean space, the fundamental theorem of curves states that any curve is characterized by its curvature and torsion up to a rigid motion. This results has been widely developed to many fields of geometry. However, in Heisenberg group, as a model case of CR geometry, there are few such results presented. By defining the appropriate invariants, we show the analogous result in the Heisenberg groups; moreover, we also classify horizontally regular curves by the order of curves.

參考: Brownian Motion and Stochastic Calculus 2/E, Ioannis Karatzas and Steven E. Shreve.

In this talk we present a few results in two directions. On the one hand, we obtain a maximum principle for second order elliptic and parabolic equations with unbounded operator via the support theorem for multivalued SDEs; on the other hand in the opposite direction we characterize the infinitesimal generator of the solutions of multivalued SDEs by using elliptic equations with unbounded operator.