Scientific seminar Recent progress in PDE and Number Theory
Hà Nội 6/12/2016

Time: Tuesday, December 6, 2016
Location: Room 301, Building A5, Institute of Mathematics, 18 Hoang Quoc Viet, Cau Giay, Hanoi

9h30: Opening

9h40: Zhu Chen-bo (Dept. Math, NUS, Singapore):

Title: From conjugate classes of square matrices to smooth representations of classical groups

Abstract: Ever since the beginning of representation theory (near the end of the nineteenth century), it has been a common knowledge that conjugate classes of a group carry critical information about representations of the group. For a Lie group, this is called the orbit method, first introduced by A.A. Kirillov in the 1960’s for nilpotent Lie groups and more recently expounded by David Vogan for reductive Lie groups, which aims for a tight link between irreducible unitary representations and coadjoint orbits.
This talk is about smooth representations of classical groups with a similar message that geometry of conjugate classes and their interrelations have important consequences for their representation theory. I will examine two fun facts from linear algebra (conjugacy of A with A^t, and the relationship of AB with BA, for matrices A, B of sizes m x n and n x m) and then discuss related phenomena on branching rules and singularities of infinite-dimensional representations.

10h40: Nguyen Duy Tan (Inst. of Math, Hanoi):

Title: Massey products in Galois cohomology

Abstract: In this talk I will discuss a conjecture that Massey products in the Galois cohomology of an arbitrary field always vanish. I will mainly discuss the case of triple Massey products, for which this conjecture is known to hold. Some applications to the structure of absolute Galois groups and some results in the case of higher Massey products will be also mentioned.

11h40: Bao Weizhu: PhD program at the NUS

14h00: Bao Weizhu (Dept. Math, NUS, Singapore) :

Title: Mathematical Models and Numerical Methods for Bose-Einstein condensation

Abstract: In this talk, I will review our results on mathematical analysis and numerical simulation for Bose-Einstein condensation (BEC). As preparatory steps, we take the three-dimesnional (3D) Gross-Pitaevskii equation (GPE), scale it to obtain a three-parameter model and use an approach well known in the physical literature to reduce it to 2D and 1D GPEs in certain limiting regimes. Mathematical analysis results on ground state and dynamics of BEC are presented.
Efficient and accurate numerical methods for computing ground state and dynamics of BEC are reviewed. In addition, comparison with experimental results on collapse and explosion in 3D is reported. Finnally, extensions to BEC with an angular momentum rotation and/or long-range anisotropic dipole-dipole interaction as well as spin-orbit BEC are discussed.

15h00: Dinh Nho Hao (Inst. of Math, Hanoi):

Title: Stable reconstruction of the initial condition in parabolic equations
Abstract: Reconstruction of the initial condition in parabolic equations from various observations is an important problem in heat transfer processes and it has many applications, e.g., in diffusion processes, image processing, weather forecast ... This problem is unfortunately severely ill-posed: a small perturbation in the data may cause arbitrary large errors in the solution. Therefore, stability estimates and stable methods for it are desirable.
This talk consists of two parts. In the first part we present our stability estimates of Holder type for backward semi-linear parabolic equations with time-dependent coefficients and locally Lipschitz source. The second part is devoted to numerical methods for stably reconstructing the initial condition in parabolic equations from various observations. We present the most difficult problem when the observations are taken only in a part of the boundary. We propose a least squares method for it and discretize the optimization problem by FDM, FEM, BEM and prove the convergence of the scheme. Some numerical examples are presented for showing the efficiency of the algorithms.  
This work has been done in the collaboration with Nguyen Van Duc, Nguyen Van Thang (University of Vinh), Nguyen Thi Ngoc Oanh (Thai Nguyen University) and Phan Xuan Thanh (Hanoi University of Science and Technology).