歷史演講 PAST TALKS

歡迎參加，敬請公告

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

I will give a brief overview of the Arakelov geometry as developed by Gillet, Soulé and Bismut ([3]) and I will recall some elements of the classical spectral theory of Laplacians on compact Kähler manifold ([1]) and I will focus on the construction holomorphic analytic torsion ([2]). The main goal of this talk is the extension of the classical spectral theory of Laplacians to a large class of singular metrics. As an application, I will consider the class of the canonical metrics on P1. These metrics are defined by the combinatorial structure of P1 viewed as a toric manifold, but unfortunately they are singular. I will introduce the notion of "canonical" holomorphic analytic torsion in this situation and give some explicit computations. Références [1] Nicole Berline, Ezra Getzler, and Michèle Vergne. Heat kernels and Dirac operators. Grundlehren Text Editions. Springer-Verlag, Berlin, 2004. Corrected reprint of the 1992 original. [2] D. B. Ray and I. M. Singer. Analytic torsion for complex manifolds. Ann. of Math. (2), 98 :154–177, 1973. [3] C. Soulé. Lectures on Arakelov geometry, volume 33 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1992. With the collaboration of D. Abramovich, J.-F. Burnol and J. Kramer.

摘要: We consider the fractional stochastic heat equations with space-time white noise in one dimension and make sense solution in mild form. Therefore, the existence, uniqueness and moment estimates appeal to stochastic Young’s inequality. We will try to cover things in detail.

歡迎參加，敬請公告！

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

We consider the equivalence problem between the code rate capacity and information rate capacity. This equivalence relation, known as the channel coding theorem, plays a fundamental and prominent role in every communication systems. In this talk, we first introduce basic notions in Information Theory, and exhibit the basic relation between information rate capacity and code rate capacity. Then we present a specific system in Diffusion-Based Molecular Communication (DBMC), and discuss the role of channel coding theorems in DBMC.

如附件 歡迎參加，謝謝！

如附件 歡迎參加，謝謝！

歡迎參加，謝謝！

We consider a financial market model consisting of one bank account, N risky stocks and d economic factors. We assume that the mean returns of the risky stocks are affected by the economic factors which evolve as a continuous-time Markov chain with with finite states. An investor who observes only asset prices tries to maximize the discounted expected CRRA utility of the consumption of his wealth (as well as that of terminal wealth in the case of finite time horizon). Hence this is an optimal consumption/ investment problem under partial information. We treat the problem both on finite and infinite time horizons.

如附檔 歡迎參加，謝謝！

Consider a unital $C^*$-algebra or a von Neumann algebra $\mathcal{A}$ equipped with a positive linear functional $\phi:\mathcal{A}\to\mathbb{C}$ such that $\phi(1)=1$. A new type of independence called freeness can be defined by replacing tensor products with free products. This subject parallel to classical probability theory is called free probability theory. Later connections to random matrix theory and other fields in mathematics were established. The free additive convolution, analogous to classical convolution, is a binary operation on the set of probability measures on the real line. The notion of free convolution and multiplicative convolution was introduced by Voiculescu in 1985. It was shown that any probability measure on $\mathbb{R}$ generates a partially defined free convolution semigroup, as proposed by Bercovici and Voiculescu in their study of the free central limit theorem. In this talk, we will talk about the supports of measures in multiplicative semigroups. Some recent progresses in free probability and random matrix theory will be discussed as well.

In the talk, I will shortly introduce two approaches for semiconductor quantumdots. First, we present a theoretical model based on the current spin density functional theory for the three vertically aligned semiconductor quantum dots. This quantum dot molecule model is treated with realistic hard-wall confinement potential and external magnetic field. Using the effective-mass approximation with band nonparabolicity, the many-body Hamiltonian results in a cubic eigenvalue problem from a finite difference discretization. A self-consistent algorithm for solving the Schr¨odinger-Poisson system by using the Jacobi-Davidson method and GMRES is given to illustrate the Kohn-Sham orbitals and energies of six electrons in the molecule with some magnetic fields. It is shown that the six electrons residing in the central dot at zero magnetic field can be changed to such that each dot contains two electrons with some feasible magneticfield. Next,weintroduce the exact diagonalization of many-electron Hamiltonian in semiconductor quantum dot structures. In this approach, the many-electron wave function is expanded in a basis of Slater determinants constructed from numerical wave functions of the single-electron Hamiltonian with the nonparabolic effectivemass approximation which results in a cubic eigenvalue problem from a finite difference discretization. Numerical results reveal that a good convergence can be achieved by means of a few single-electron basis states.

In recent years, the Bakry-Emery Ricci curvature has been received a lot of attentions in stochastic analysis and geometric analysis. In this series of lectures, I will present some recent results in the study of various problems in geometric analysis and stochastic analysis on Riemannian manifolds with weighted measure. After an introduction of the Bakry-Emery Ricci curvature and related materials, I will present the following topics in my lectures: Lecture 3. W-entropy formula and the optimal transport problems on manifolds with weighted measure

We are concerned with the FitzHugh-Nagumo equations and some reactoin-diffusion systems with conservation property in a bounded domain with the Neumann boundary condition. Those systems share the common features such that the Turing instability takes places and the stationary equations can be reduced to a scalar elliptic equation with a nonlocal term, whose solution is provided by a critical point of an energy functional. Our aim is to show the gradient-like property of each system and the coincidence of the dimension of the unstable manifold of any equilibrium solution and the Morse index of the corresponding critical point of the energy.

Abstract: The Kardar-Parisi-Zhang (KPZ) equation is a nonlinear stochastic partial differential equation, which is actually ill-posed because of inconsistency between the nonlinearity and the roughness of the noise. We introduce a renormalization procedure for the KPZ equation, which is appropriate from the view point to identify the invariant measures. The Cole-Hopf transform applied to this approximating equation leads to a linear stochastic heat equation (SHE) with a smeared noise having an extra complex nonlinear term. Under the time average, this complex term can be replaced by a simple linear term, at least under the situation that the corresponding tilt process is stationary. As a result, it is shown that the distribution of a two-sided geometric Brownian motion with a height shift given by Lebesgue measure is invariant under the evolution determined by the SHE.

Drift-diffusion equations are important models with many applications in biology, engineering and physics. Cross-diffusion terms may describe the exclusion of size/volume effects. In this lecture, I shall introduce the derivation of such models and traveling wave solutions of a two-species chemotaxis model.

Abstract: In recent years, the Bakry-Emery Ricci curvature has been received a lot of attentions in stochastic analysis and geometric analysis. In this series of lectures, I will present some recent results in the study of various problems in geometric analysis and stochastic analysis on Riemannian manifolds with weighted measure. After an introduction of the Bakry-Emery Ricci curvature and related materials, I will present the following topics in my lectures: Lecture 1. Liouville theorems and Cheeger-Gromoll splitting theorem on manifolds with weighted measure Lecture 2. Harnack inequalities, eigenvalue and heat kernel estimates of Witten Laplacian on manifolds

Description of this intensive short course Modern engineering disciplines require computations and insight which cannot be obtained through analytical solutions alone. To cope with the demand for solutions to such real life problems, engineers and scientists have dedicated large amounts of time and effort to employ modern computational technology to aid them in their engineering analysis. Such efforts have led to the creation of simulation tools such as Fluent, ANSYS to name a few. While these tools have been beneficial, the limit to which we can use such tools to solve real life problems is limited by the speed at which these computations can be performed. With increasing demands placed upon engineers for increased fidelity in their computations, there is a demand for larger scale simulation results. Hence, modern engineers require parallel computing to obtain suitable “modern” results. The goal of this intense workshop is to apply a problem-centred approach to parallel computing using GPU devices. Students will be presented with everyday (common) problems in mathematics and science and will be required to use the GPU to solve these problems. Hence, the approach used during this workshop is a form of Problem Based Learning. As such, during activities, students will be grouped into pairs and will work together to solve these potentially challenging problems.

A 7-dimensional Riemannian manifold (M,\varphi) is called a G_2 manifold if it admits globally defined tensor(\varphi) which is parallel with respect to the Levi-Civita connection. In this talk I will first give introduction to G_2 structure, and then report recent research on the relations between G_2 and (almost) contact structure on (M,\varphi). This is a joint work with Firat Arikan and Sema Salur.

歡迎參加，謝謝！

歡迎參加,謝謝!

This is a learning notes of Cedric Villani’s paper in LNM 1813. It will review the links between optimal transportation problems, evolution PDE’s and functional inequalities, through examples. The second part will be continued at Academia Sinica on 15:30 ~ 17:00, Monday, November 25th, 2013

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

摘要： We study the steady fractional Navier-Stokes equations in R^3 with 1/2 < s < 1. We show that if 1/2 < s < 5/6, the suitable weak solution is regular away from a compact singular set with zero (5-6s)-Hausdorff measure and if s >5/6 or s=5/6, it is regular.

In this talk, we will study the following Schro ̈dinger-Poisson systems: {█(-∆u + u + K (x)∅u = a (x)|u| p^(-2) u in R^3;@-∆∅ = K (x) u^2 in R^3 ;)┤ where 2 < p < 4, K and a are bounded nonnegative functions. By using the fibering maps and studying some properties of the Nehari manifold, we will obtain the existence of positive solutions for the above Schro ̈dinger-Poisson systems.

摘要: This is a continuation of our previous discussion on Solomon’s paper. We will continue to discuss the average speed of a 1-d random walk in random environment. We explore the stationarity of the sequence of ladder random variables. Based on this observation, then the ergodic theorem can be applied to derive the average speed of the random walk. 參考： Fred Solomon (1975) Random Walks in a Random Environment, Ann. Probab, Vol 3 P1-31 H. Kesten, M.V. Kozlov and F. Spitzer (1975), A Limit Law for Random Walk in a Random Environment. 30, 145-168 Pal Revesz (1990) Random Walk in Random and Non-random Environments, World Scientific

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

Discussion in the Math Department

Without the definition of It^o integral, we are able to derive "It^o's formula", in the proof we show that the Hitsuda-Skorokhod integral appear naturally. In this talk we shall show that the Hitsuda-Skorokhod integral may be defined for any Gaussian and Non- Gaussian L evy functionals.

The theory of distributions (or generalized functions) introduced by L. Schwartz may be reformulated by replacing the Lebesgue measure by a canonical Gauss measure. Then we extend the one-dimensional Theory of distributions to infinite dimensions and reformulate the Gaussian White Noise Calculus initiated by Hida.

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

The theory of canonical heights for a morphism is quite well known. For a dominant rational map, it is more difficult to define and study canonical height functions. Hence, we focus on monomial maps. We develop a new canonical height function for monomial maps. In certain cases, this new canonical height function has nice properties. For example, Northcott style finiteness property.

This is a continuation of our previous discussion on Solomon’s paper. We will continue to discuss the average speed of a 1-d random walk in random environment. We explore the stationarity of the sequence of ladder random variables. Based on this observation, then the ergodic theorem can be applied to derive the average speed of the random walk. The paper by Kesten-Kozlov-Spitzer is on the CLT of the random walk. We will briefly discuss the paper if the time allows.

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

In several talks, I plan to give an introduction to the random walks in a random environment. To mention a simple model, we consider the simple random walk on 1-d lattice moves one step to the right with probability p and to the left with probability (1-p). In a random environment (denoted by (pi(ω)) , the random walk with current location at i moves one step to the right on the lattice with probability pi (ω) and to the left with probability (1-pi(ω)). This is a simple example of a random walk in a random environment. The behaviors of a simple random walk are well known and rich . Its average speed (in large time) is 2p-1. The fluctuation (along the average speed) is normal. The large deviation principle holds. There are many more rich results (see Revesz). The behaviors of the random walk in a random environment will depend on the random property of the environment. Similar to the simple random walk, the goal is to understand its average speed, its fluctuation, and its large deviation properties. This seminar provides a chance to learn interesting results of the subject. We will start with the discussion on the Solomon’s paper, which proves the first result and stimulates further studies.

Abstract A basic result in the theory of the Sobolev spaces is the L^p-differentiability of Sobolev functions - essentially a consequence of the Lebesgue differentiation theorem. Less known is the use of the Sobolev embedding theorem to improve this result to so-called L^{p-star}-differentiability. In this talk I will discuss a recent result on characterizing Sobolev spaces that turns out to be related to this result, which was quite surprising for us. This talk is partially based on joint work with my graduate advisor, Giovanni Leoni.

摘要： We present a non- vanishing result of central L-values for canonical Hecke charaters of imginary quadratic fields with quadratic twists. This result extends previous works of Rohrlich, Villegas and Tonghai Yang.