Multilinear and multiparameter spectral multipliers on stratified groups



摘要:In this talk, we will introduce Hormander-type multiplier theorems for multilinear and multiparameter spectral multipliers on product spaces of stratified groups $G =G1 × · · · × GM$.

Spherical Function Regularization for Parallel MRI Reconstruction



摘要:In this talk, we introduce such a regularization based on spherical function bases. To perform this regularization, we represent efficient recurrence formulas for spherical Bessel functions and associated Legendre functions. Numerically, we study the solution of the model with non-linear ADMM. We perform various numerical simulations to demonstrate the efficacy of the proposed model in parallel MRI reconstruction.

Constructions of involutions over finite fields



摘要:An involution over finite fields is a permutation polynomial whose inverse is itself. Owing to this property, involutions over finite fields have been widely used in applications such as cryptography and coding theory. Following the idea in [1] to characterize the involutory behavior of the generalized cyclotomic mappings,this paper gives a more concise criterion for $x^rh(x^s)\in \mathbb{F}_q[x]$ being involutions over the finite field $\mathbb{F}_q$, where $r\geq 1$ and $s\,|\, (q-1)$. By using this criterion we propose a general method to construct involutions of the form $x^rh(x^s)$ over $\mathbb{F}_q$ from given involutions over some subgroups of $\mathbb{F}_q^*$ by solving congruent and linear equations over finite fields. Then, many classes of explicit involutions of the form $x^rh(x^s)$ over $\mathbb{F}_q$ are obtained.













Geometry and Mesh Paramerization in Natural Image Stitching



摘要:Natural image stitching (NIS) is a key point in many computer vision applications, such as panoramic images, $360^{\circ}$ videos, stereoscopic panoramas and virtual reality. Nowadays, the ubiquity of smart mobile devices, such as phones and tablets, enables users to casually capture stitching materials with a sweep of a camera. It brings not only some great opportunities but also many novel challenges for NIS. In this talk, we will introduce some representative warping/deforming methods in NIS that tackle such challenges (alignment accuracy and shape distortion). They are divided into two categories: warps using geometry paramerization (APAP, SPHP, QH) and deformations using mesh paramerization (GSP, GHP). The first ones are essentially modifications of the classic projective (homography) warp from different geometric points of view. The second ones are following the framework of energy minimization, where different energies are defined for addressing issues of alignment, distortion and smoothness. QH and GHP are joint works with Tianli Liao, Chao Wang and Yifang Xu.

The multi-degree of coverings on Lie groups



摘要:In mathematical physics covering maps on Lie groups are essential to extend the constructions of various physical models associated with simply connected Lie groups, to that associated with non-simply connected Lie groups。 In this talk we introduce for each covering map of Lie groups an invariant, called the multi-degree of the covering; present a method to evaluate the invariant; and apply the results to solve two topological problems arising from the studies of the Wess-Zumino-Witten models and the topological Gauge theories. The main tool of our approach is the Chow rings of Lie groups, introduced by Grothendieck in 1958.

A Miyaoka-Yau type inequality of surfaces in characteristic $p>0$



摘要:For minimal smooth projective surfaces $S$ of general type, we prove $K^2_S\le 32\chi(\mathcal{O}_S)$ and give examples of $S$ with $$K^2_S=32\chi(\mathcal{O}_S).$$ This proves that $\chi(\mathcal{O}_S)>0$ holds for all smooth projective minimal surfaces $S$ of general type, which answers completely a question of Shepherd-Barron. Our key observation is that such Miyaoka-Yau type inequality follows slope inequalities of a fiberation $f:S\to C$. However, we will gives examples of $f:S\to C$ with non-smooth generic fibers of arithmetic genus $g\ge 2$ such that $$K^2_{S/C}<\frac{4g-4}{g}{\rm deg}f_*\omega_{S/C},$$ which are counterexamples of Xiao's slope inequality in case of positive characteristic. This is a joint work with Gu Yi and Zhou Mingshuo.

On mathematical construction of quantum field theories: Feynman Geometries



摘要:We introduce a notion of Feynman geometry on which quantum field theories could be properly defined. A strong Feynman geometry is a geometry when the vector space of $A_\infty$ structures is finite dimensional. A weak Feynman geometry is a geometry when the vector space of $A_\infty$ structures is infinite dimensional while the relevant operators are of trace-class. We construct families of Feynman geometries with "continuum" as their limit.

Isoparametric Polynomials and sums of Squares



摘要:Hilbert’s 17th problem asks that whether every nonnegative polynomial can be a sum of squares of rational functions. It has been answered affirmatively by Artin. However, as to the question whether a given nonnegative polynomial is a sum of squares of polynomials is still a central question in real algebraic geometry. In this paper, we solve this question completely for the nonnegative polynomials (associated with isoparametric polynomials, initiated by E. Cartan) which define the focal submanifolds of the corresponding isoparametric hypersurfaces.




摘要:分离性理论是调和分析中的一个崭新分支,在偏微分方程、解析数论与几何测度论发挥重要作用. 从数学上来讲,decoupling 理论反映了频率支在不同区域上函数相加所发生的“干涉模式”及傅立叶空间中谱几何如何影响物理空间中发生结构性干涉. 该方向着力利用decoupling方法研究调和分析中限制性猜想及局部光滑性猜想、数论中的指数求和猜想及丢番图方程整数解、紧致Riemann流形上非线性色散方程、几何测度论与代数几何中的代数多项式零点分解定理与Falconer距离集猜想等相关问题。





Safety Verification of Nonlinear Hybrid Systems Based on Bilinear Programming



摘要:In safety verification of hybrid systems, barrier certificates are generated by solving the verification conditions derived from non-negative representations of different types. In this talk, we present a new computational method, sequential linear programming projection, for directly solving the set of verification conditions represented by the Krivine-Vasilescu-Handelman's positivstellensatz. The key idea is to decompose it into two successive optimization problems that refine the desired barrier certificate and those undetermined multipliers, respectively, and solve it in an iterative scheme. The most important benefit of the proposed approach lies in that it is much more effective than the LP relaxation method in producing real barrier certificates, and possesses a much lower computational complexity than the popular sum of square relaxation methods, which is demonstrated by the theoretical analysis on complexity and the experiment on a set of examples gathered from the literature.

Telescopers for Differential Forms with one parameter



摘要:We introduce the notion of telescopers for differential forms. Precisely, let $$\omega=\sum f_{i_1,\cdots,i_p} dx_{i_1}\wedge dx_{i_2}\wedge \cdots \wedge dx_{i_p}$$ be a differential $p$-form, where $f_{i_1,\cdots,i_p}$ is $D$-finite over $k(x_1,\cdots,x_n,t)$. A nonzero operator $L \in k(t)[\partial_t]$ is called a telescoper for $\omega$ if $L(\omega)=d\eta$ for some differential $p-1$-form $\eta$. In this talk, we present a sufficient and necessary condition for a given differential $p$-form having a telescoper and develop an algorithm to compute a telescoper if it exists. We also give an algorithm to decide whether a given differential $p$-form has a telescoper or not.This is joint work with Shaoshi Chen, Ziming Li, Michael F. Singer and Stephen Watt.

Real Root Isolation of Bivariate Real Analytic Functions



摘要:In this talk, we present a new method for isolating real roots of a bivariate analytic function system. Our method is a subdivision method which is based on analyzing the local geometrical properties of the given system. We propose the concept of the orthogonal monotone system in a box and use it to determine the uniqueness and the existence of a simple real zero of the system in the box. We implement our method to isolate the real zeros of a given bivariate system. The experiments show the effectivity and efficiency of our method, especially for polynomial systems with high degrees and sparse terms. This is a joint work with Junyi Wen.




摘要:碰撞检测(Collision Detection)是计算机辅助设计和制造、计算机图形学、虚拟现实、数控技术、机器人学、分子动力学模拟等诸多领域的重要问题。我们将以两几何体的三个逐渐深入的几何关系——相对位置、交线形态、交体构型为目标,介绍其逐层深入的代数判定条件,并给出连续碰撞检测的符号算法,以此探索符号计算在碰撞检测及构型分析相关的工业环境中的应用。

Gosper's Algorithm and Its Multivariate Extensions



摘要:Gosper's algorithm for indefinite hypergeometric summation is a fundamental algorithm in Symbolic Summation and also forms the core in the Wilf-Zeilberger theory of mechanical proving of combinatorial identities. How to extend Gosper's algorithm to the multivariate case is a challenging problem. In this talk, we will overview some previous developments towards this extension and present our recent results which solve the problem in the rational case.