Place: Room 1114 (Building of Math School, 理科一号楼)
Time: Usually at 15:00 –16:00 on Friday.
Tea time: Friday 16:00–16:30, Room 1384 (Building of Math School, 理科一号楼).
Contact: Huijun Fan (范辉军) (Email: fanhj@math.pku.edu.cn) or Xiaobo Liu (刘小博) (Email: xbliu@math.pku.edu.cn)
Spring Semester 2018:
2018/03/09
Title：Solutions of SPDE as zeros of maps on scaled path spaces
Speaker: Professor Rockner, Bielefeld University, Germany
Abstract: It has been recently shown that the solutions of a large class of stochastic partial differential equations (SPDE) can be obtained as zeros of a properly defined map on a path space equipped with a norm which is “scaled“ by the exponential of a function-valued Brownian motion. In the talk this result will be reviewed and connected to current developments about the case where the underlying SPDE is a gradient flow, perturbed by linear multiplicative noise. In this case it follows from the above result and by applying methods from the calculus of variations that the solution minimizes a certain explicit convex functional on the path space. Applications include stochastic porous media equations, stochastic nonlinear parabolic equations (as e.g. the stochastic Cauchy problem for the p-Laplacian) and in the non-gradient case also stochastic nonlinear transport equations. (Invited by Liu Yong）
2018/03/16
Title: Spherical Tarski’s plank problem
Speaker: Prof. Jiang Zilin,
Abstract: A zone of width ω on the unit sphere is defined as the set of points within spherical distance ω/2 of a given great circle. Zones can be thought of as the spherical analogue of planks. In this talk we show that the total width of any (finite) collection of zones covering the unit sphere is at least π, answering a question of Fejes Tóth from 1973. This is joint work with Alexandr Polyanskii. （Invited by Dong Zijing）
2018/03/20
Title: The space of positive scalar curvature metrics
Speaker: Pro. Bernhard Hanke
Abstract: The construction and classification of Riemannian metrics satisfying certain curvature bounds are of fundamental interest in differential geometry. In this talk we specialize to metrics of positive scalar curvature, which are characterized by a simple volume growth condition of balls with small radii. Research during the last couple of years, based on a variety of methods, revealed the topological complexity of the space of all such metrics on a fixed manifold. We will give an overview of this development.
2018/03/23
Title: The Riemann Hypothesis and Mathematical Physics;
Speaker: Prof. Charles Newman, Courant Institute of Mathematical Sciences at New York University.
Abstract: In both analytic number theory (the Riemann Hypothesis) and mathematical physics (Ising models and Euclidean field theories) the following complex analysis issue arises. For ρa finite positive measure on the real line R, let H(z; ρ, λ) denote the Fourier transform of \exp{λ u2} dρ(u), i.e., the integral over R of \exp{izu + λu2} dρ(u) extended from real to complex z, for those λ (including all λ < 0) where this is possible. The issue is to determine for variousρ's thoseλ's for which all zeros of H in the complex plane are real. We will discuss some old and new theorems about this issue. ( Brief CV: Charles Newman is Silver Professor of Mathematics at the Courant Institute of Mathematical Sciences at New York University. He holds a PhD and an MA from Princeton University and two BS degrees from MIT. Newman is a Fellow of the American Mathematical Society, a Fellow of the Institute of Mathematical Statistics, a member of the International Association of Mathematical Physicists, a member of the US National Academy of Sciences, a member of the American Academy of Arts and Sciences, and a member of the Brazilian Academy of Sciences.)
Title: Maps between 3-manifold groups.
Speaker: Daniel Groves (University of Illinois at Chicago)
Abstract: In recent years there has been a lot of work on the structure of 3-manifolds and on questions about maps between them. I will describe some of this work, and state some work in progress towards answering some open questions, and towards developing a structure theory for the set of homomorphisms from an arbitrary finitely generated group to the set of all 3-manifold groups.
2018/03/30
Title: Nonlinear wave equations from a geometric point of view
Speaker: Prof. Yu Pin, Yau Math. Sci. Center, Tsinghua University
Abstract:The discussion of the Lorentzian geometry behind various wave equations, such as the Maxwell equations, the Yang-Mills equations and the Einstein field equations. In particular, we will apply this idea to study the formation of shocks for quasilinear wave equations in higher dimensions
2018/04/06 (Holiday)
2018/04/13
Title: Spectrum and Behaviors of Vlasov-Poisson-Boltzmann equations
Speaker: Prof. Hai-Liang Li,School of Mathematics, Capital Normal University
Abstract:The Vlasov-Poisson-Boltzmann equations can be used to model the transport of charged particles (one carrier or two carriers) under the influence of electrostatic potential force. It is not well understood yet how the electrostatic potential force and/or the mutual interaction between charged particles of different type shall affect the spectrum structures and the asymptotical behaviors of global solutions. In this talk, we present the recent progress on the analysis on the spectrum structure and optimal pointwise space-time behaviors of three dimentional Vlasov-Poisson-Boltzmann equations, and the nonlinear stability of planar wave pattern for the bipolar Vlasov-Poisson-Boltzmann equations, such as including shock profile, rarefaction wave and contact discontinuity. These works are joint with Tong Yang, Ming-Ying Zhong, Yi Wang, and Teng Wang.
2018/04/20
Title: Alternating Direction Methods of Multipliers for Optimization Problems Involving Nonconvex Functions
Speaker: Han Deren
Abstract : The efficiency of the classic alternating direction method of multipliers has been exhibited by various applications for large scale separable optimization problems, both for convex objective functions and for nonconvex objective functions. While there are a lot of convergence analysis for the convex case, the nonconvex case is still an open problem and the research for this case is in its infancy. In this talk, we consider two classes of optimization problems involving nonconvex functions. The first case is the “strongly+weakly” convex model and the second on is the general nonconvex model. For both cases, by using different analysis techniques, we prove the global convergence of the algorithms, and under some further conditions on the problem’s data, we also analyze the convergence rate. （Invited by Wen Zaiwen）
(简介：韩德仁，教授，博士生导师。2002年获南京大学计算数学博士学位。从事大规模优化问题、变分不等式问题的数值方法的研究工作。在Mathematical Programming, Numerische Mathematik, SIAM Journal on Numerical Analysis, Mathematics of Computation等计算数学、运筹学重要杂志以发表多篇学术论文，入选江苏省333高层次人才培养工程、江苏省“青蓝工程”中青年学术带头人。担任中国运筹学会理事、数学规划分会常务理事；《计算数学》、《Journal of the Operations Research Society of China》编委。）
2018/04/27
Title:From harmonic maps to the nonlinear supersymmetric sigma model of quantum field theory -At the interface of geometry, analysis and theoretical physics
Speaker: Prof. Zhu Miaomiao (Shanghai Jiao Tong University)
Abstract: The theory of harmonic maps from Riemann surfaces is one of the most classical and important variational problems arising from geometry (minimal surfaces in Riemannian manifolds, pseudo-holomorphic curves in symplectic manifolds etc.) and theoretical physics (nonlinear sigma model from quantum field theory etc.). In this talk, we shall explore the extent to which the classical methods of nonlinear geometric analysis developed for the study of harmonic maps can be generalized to PDEs of more general type occurring in geometry and theoretical physics, with particular focus on issues of compactness, regularity and existence.
2018/05/04
2018/05/11 |occupied by Chenyang Xu
Title: Rationality of algebraic varieties
Speaker: Prof. Oliver Debarre, école normale supérieure, France
Abstract: In algebraic geometry, one says that an algebraic variety $X$ defined over a field $K$ is rational if it is very close to being the affine space $K^n$. This means that one can parametrize the points of $X$ by $K^n$, in an almost one-to-one fashion. Deciding whether an algebraic variety, given by polynomial equations, is rational is in general very difficult. I will start with very classical and elementary examples (like the rational parametrization of the circle) and move on to spectacular results obtained recently on the behavior of rationality in a family: given a family $(X_t)$ of algebraic varieties, is the set of $t$ for which $X_t$ is rational open, closed?
2018/05/17
Title:K-stability and geometric nonlinear problems
Speaker:Prof. Akito Futaki, Tsinghua University
Abstact:The Yau-Tian-Donaldson conjecture claims the equivalence between the existence of constant scalar curvature metric and K-stability on polarized K?hler manifolds. The special case for Fano manifolds is the claim for the existence of K?hler-Einstein metrics, and has been confirmed affirmatively by Chen-Donaldson-Sun and Tian. The conceptual idea is explained using moment map picture. In this talk we see there are other geometric nonlinear problems which fit in similar pictures, and the well-known obstructions can be obtained. As a typical example we take up Cahen-Gutt moment map which has been studied in the study of deformation quantizations.
2018/05/18
2018/05/25
2018/05/28
Title:Uncertainty quantification in kinetic equations
Speaker:Pro.Shi Jin,Shanghai Jiao Tong University/University of Wisconsin-Madison
Abstact: Kinetic equations describe dynamics of probability density distributions of large number of particles, and have classical applications in rarefied gas, plasma, nuclear engineering and emerging ones in biological and social sciences. Since they are not first principle equations, rather are usually derived from mean field approximations of Newton's second law, thus contain inevitably uncertainties in collision kernel, scattering coefficients, initial and boundary data, forcing and source terms. In this talk we will review a few recent results for kinetic equations with random uncertainties. We will extend hypocoercivity theory, developed for deterministic kinetic equations, to study local sensitivity, regularity, local time behavior of the solutions in the random space, and also establish corresponding theory for their numerical approximations.
2018/06/01
Title:A new and robust approach to construct energy stable schemes for gradient flows
Speaker:Jie Shen, Purdue University and Xiamen University
Abstact:We present in this talk the scalar auxiliary variable (SAV) approach and the multiple scalar auxiliary variables (MSAV) approach, to deal with nonlinear terms in a large class of gradient flows. The technique is not restricted to specific forms of the nonlinear part of the free energy, it leads to linear and unconditionally energy stable second-order (or higher-order with weak stability conditions) schemes which only require solving decoupled linear equations with constant coefficients. Hence, these schemes are extremely efficient as well as accurate. We apply the SAV approach to deal with several challenging applications which can not be easily handled by existing approaches, and present convincing numerical results to show that the new schemes are not only much more efficient and easy to implement, but also can better capture the physical properties in these models. We shall also present a convergence and error analysis under mild assumptions on the nonlinear free energy.
2018/06/08
Title:Retrospects and Prospects on Equivariant Differential Geometry
Speaker:Pro.Wu-Yi Hsiang
Abstact: In this short talk, we shall begin with a concise review on some highlights of equivariant differential geometry such as symmetry theorems on isoperimetric regions, soap bubbles, closed minimal submanifolds of spheres etc. on the one hand, and on the other hand, the construction of some outstanding new examples such as immersed soap bubbles, spherical Bernstein problems, etc. and then following with a brief overview on what kinds of natural r?les it is playing in the study of various interplays between symmetries and optimalities in the intrinsic structures of the Nature.We shall conclude with a brief discussion on its prospects of further developments, mainly by formulating some pertinent problems as well as a few new results, such as the construction of examples of Ricci flows of O(3)-hypersurfaces in E4 exhibiting the puzzling behavior of non-preservation of positive Ricci curvature everywhere.
2018/06/15
Title:Douglas-Rachford Splitting for Pathological Problems
Speaker: Pro.Wotao Yin,Department of Mathematics University of California
Abstract: First-order methods such as ADMM and Douglas-Rachford splitting (DRS) are known for their easy implementations and low per-iteration costs. What is less known is their usefulness for "solving" pathological problems, which are infeasible or feasible yet unbounded. In this talk, we establish that DRS only requires strong duality to work, even if the problem is pathological, in the sense that asymptotically iterates are approximately feasible and/or approximately optimal. Furthermore, we present a method for classifying infeasible, unbounded, and other pathological conic programs. This is joint work with Yanli Liu and Ernest Ryu.
2018/06/22
Title:From information geometry to emergent geometry
Speaker:Prof. Zhou Jian, Tsinghua University
Abstract:In this talk we will explain some surprising applications of ideas from statistics and statistical physics to some geometric problems arising from string theory. We will show that Fisher information, entropy, fractional quantum Hall effects, phase transition, Wilson renormalization theory, emergent phenomena etc, are the right language to describe the rich quantum field theory related to mirror symmetry.
2018/08/28
Title: Higher genus FJRW invariants of the Fermat cubic polynomial
Speaker: Prof. Yefeng Shen, University of Oregon.
Time: 10:00-11:30, Aug. 28th
Place: Room 1303 Science building
Abstract: Using tautological relations and axioms of Cohomological field theory, we construct an explicit Landau-Ginzburg/Calabi-Yau correspondence for Fermat cubic polynomial at all genus. The Cayley transformations of quasi-modular forms allow us to compute the higher genus FJRW invariants of the Fermat cubic polynomial from the higher genus Gromov-Witten invariants of elliptic curves. This work is joint with Jun Li and Jie Zhou.
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