“數學及其交叉應用”青年學者研讨會

報告時間：  2016年4月30日 下午  14:00-17:30

報告地點： 科技樓南樓702

報告人：葛化彬 

（華科數學2006年本科畢業， 2012年北京大學博士畢業， 現工作單位：北京交通大學 ）

題目:  On the structure of harmonic-Einstein equations

摘要:  In this talk, we study the structure of the Gromov-Hausdorff limit (X,d) of a sequence of Harmonic-Einstein manifolds. We show that the singular set of X is closed with codimension 2 and the convergence is smooth on the regular part. This is a joint work with Dr. Wenshuai Jiang.

報告人：牛原玲

（華科數學2006年本科畢業，2011年華科數學博士畢業，現工作單位： 中南大學）

題目: Modelling biochemical reaction systems by stochastic differential equations with reflection

摘要: In this work, we gave a new framework for modelling and simulating biochemical reaction systems by stochastic differential equations with reflection not in a heuristic way but in a mathematical way. The model is computationally efficient compared with the discrete-state Markov chain approach, and it ensures that both analytic and numerical solutions remain in a biologically plausible region. Specifically, our model mathematically ensures that species numbers lie in the domain $D$, which is a physical constraint for biochemical reactions, in contrast to the previous models. The domain $D$ is actually obtained according to the structure of the corresponding chemical Langevin equations, i.e., the boundary is inherent in the biochemical reaction system. A variant of projection method was employed to solve the reflected stochastic differential equation model, and it includes three simple steps, i.e., Euler-Maruyama method was applied to the equations first, and then check whether or not the point lies within the domain $D$, and if not perform an orthogonal projection. It is found that the projection onto the closure $\bar{D}$ is the solution to a convex quadratic programming problem. Thus, existing methods for the convex quadratic programming problem can be employed for the orthogonal projection map. Numerical tests on several important problems in biological systems confirmed the efficiency and accuracy of this approach.

報告人：徐旭

（華科數學2006年本科畢業，2011年中科院博士畢業， 現工作單位： 武漢大學）

題目：A combinatorial Yamabe problem on two and three dimensional manifolds 

摘要：In this talk, we will introduce a new definition of discrete curvature on two and three dimensional triangulated manifolds, which is a modification of the well-known discrete curvature on these manifolds. The new definition is more natural and respects the scaling exactly the same way as Gauss curvature does. Moreover, the new discrete curvature can be used to approximate the Gauss curvature on surfaces. Then we study the corresponding constant curvature problem, which is called the combinatorial Yamabe problem, by the corresponding combinatorial versions of Ricci flow and Calabi flow for surfaces and Yamabe flow for 3-dimensional manifolds.  As a byproduct, we get a generalization of Thurston's criterion for the existence of circle packing metric with constant combinatorial curvature. The basic tools are the discrete maximal principle and variational principle.

報告人：周達

（華科數學2006年本科畢業，北京大學2011年博士畢業， 現工作單位： 廈門大學）

題目：Some quantitative methods for characterizing reversible cell lineage 

摘要：The paradigm of cell plasticity suggests reversible relations of different cellular phenotypes, which extends the conventional theory of cell hierarchy. This talk will present some quantitative characteristics of reversible cell lineage in comparison to hierarchical cell lineage. Two types of stochastic models will be discussed: forward-time branching process and backward-time coalescent process. For the former, we will show the advantage of the reversible model in capturing both the steady-state and transient dynamics. For the latter, we will present a coalescent-based statistical method for inferring the de-differentiation rate from differentiated cell to stem cell.

報告人：周科（華科數學2006年本科畢業，2009年山東大學碩士畢業，2014年香港中文大學博士畢業，  現工作單位： 湖南大學）

題目：DYNAMIC MEAN-LPM AND MEAN-CVAR PORTFOLIO OPTIMIZATION IN CONTINUOUS-TIME

摘要：We investigate in this paper dynamic mean-downside risk portfolio optimization problems in continuous-time, where the downside risk measures can be either the lower-partial moments (LPM) or the conditional value-at-risk (CVaR). Our contributions are two-fold, in both building up tractable formulations and deriving corresponding analytical solutions. By imposing a limit funding level on the terminal wealth, we conquer the ill-posedness exhibited in a class of mean-downside risk portfolio models. For a general market setting, we prove the existence and uniqueness of the Lagrangian multiplies, which is a key step in applying the martingale approach, and establish a theoretical foundation for developing efficient numerical solution approaches. Moreover, for situations where the opportunity set of the market setting is deterministic, we derive analytical portfolio policies for both dynamic mean-LPM and mean-CVaR formulations

報告人：李東方（華科數學2006年本科畢業，2011年華科數學博士畢業， 現工作單位： 88858cc永利）

題目: Unconditionally optimal error estimates of linearized L1-Galerkin FEMs for time fractional nonlinear Schr\"{o}dinger equations

摘要：In this paper, a linearized L1-Galerkin finite element method is proposed to solve the multi-dimensional time fractional nonlinear Schr\"{o}dinger equation. We proved that the finite element approximations in $L^2$-norm and $L^\infty$-norm are bounded, in terms of temporal-spatial error splitting argument. Then, optimal error estimates of the numerical scheme are obtained unconditionally, while previous works on nonlinear time fractional differential equations always required certain temporal stepsize conditions. Numerical examples in both two and three dimensional spaces are given to confirm our theoretical results.