• Time: Monday, June 30, 2014 at 15:00 –16:00 p.m.    Room: N514

Title: General Soliton Solution to the Vector Nonlinear Schrödinger Equation

Speaker: Baofeng Feng  (University of Texas-Pan American, USA)

Abstract: In the present talk, we consider general soliton solution to the vector nonlinear Schrödinger (NLS) equation of all possible combinations of nonlinearities including all-focusing, all-defocusing and mixed types. By using the KP-hierarchy reduction method based on the Sato theory, we construct a general formula in Gram-type determinants for the vector NLS equation. The condition for the reality of the soliton solution with all possible combinations of nonlinearities is elucidated.

• Time: Tuesday, July 22, 2014 at 9:30 –10:30 a.m.    Room: N226

Title: The Theory and Applications of T-Splines

Speaker: Thomas W. Sederberg  (Brigham Young University, USA)

Abstract: T-Splines were invented in 2003 to address the major limitations of the NURBS free-form surface representation which is the industry standard in computer aided. T-Splines provide a watertight model of arbitrary topology that allows for local refinement. T-Splines also simplify the model creation procedure for designers. In addition, T-Splines are emerging as the representation of choice for IsoGeometric Analysis.

• Time: Monday, July 28, 2014 at 10:00 –11:00 a.m.    Room: N420

Title: Classification of Restricted Lattice Walks in 3D

Speaker: Manuel Kauers  (RISC, Johannes Kepler University, Austria)

Abstract: We consider walks in the positive octant $N^3$ that start at the origin and consist of exactly n steps, where each step is taken from some prescribed step set $S$. Depending on the choice of $S$, the generating function for such walks may be $D$-finite or not. Together with Alin Bostan, Mireille Bousquet-Melou, and Steve Melczer, we have started to classify the step sets $S$ according to the nature of the generating function. In the talk, we present the results we have obtained so far, and we explain the computational and combinatorial tools we employed to obtain these results.

• Time: Thursday, August 7, 2014 at 10:00 –11:00 a.m.    Room: N420

Title: A Survey of Alternating Permutations

Speaker: Richard Stanley  (MIT, USA)

Abstract: A permutation $a_1,a_2,\dots,a_n$ of $1,2,\dots,n$ is called alternating if $a_1>a_2< a_3 > a_4 < \cdots$. The number of alternating permutations of $1,2,\dots,n$ is denoted $E_n$ and is called an Euler number. The most striking result about alternating permutations is the generating function $$\sum_{n\geq 0}E_n\frac{x^n}{n!} = \sec x+\tan x,$$ found by Désiré André in 1879. We will discuss this result and how it leads to the subject of "combinatorial trigonometry''. We will then survey some further aspects of alternating permutations, including some other objects that are counted by $E_n$, the use of the representation theory of the symmetric group to count certain classes of alternating permutations, and the statistical properties of the longest alternating subsequence of a random permutation.

• Time: Thursday, August 7, 2014 at 15:00 –16:00 p.m.    Room: N420

Title: Essentially Optimal Interactive Certificates in Linear Algebra

Speaker: Erich Kaltofen  (North Carolina State University, USA)

Abstract: Certificates to a linear algebra computation are additional data structures for each output, which can be used by a--possibly randomized--verification algorithm that proves the correctness of each output. The certificates are essentially optimal if the time (and space) complexity of verification is essentially linear in the input size $N$, meaning $N$ times a factor $N^{o(1)}$, that is, a factor $N^{\eta(N)}$ with $\lim_{N\to \infty} \eta(N) = 0$.
We give algorithms that compute essentially optimal certificates for the positive semidefiniteness, Frobenius form, characteristic and minimal polynomial of an $n\times n$ dense integer matrix A. Our certificates can be verified in Monte-Carlo bit complexity $(n^2 \log ||A||)^{1+o(1)}$, where $\log ||A||$ is the bit size of the integer entries, solving an open problem in [Kaltofen, Nehring, Saunders, Proc. ISSAC 2011] subject to computational hardness assumptions.
Second, we give algorithms that compute certificates for the rank of sparse or structured $n\times n$ matrices over an abstract field, whose Monte Carlo verification complexity is 2 matrix-times-vector products + $n^{1+o(1)}$ arithmetic operations in the field. For example, if the $n\times n$ input matrix is sparse with $n^{1+o(1)}$ non-zero entries, our rank certificate can be verified in $n^{1+o(1)}$ field operations. This extends also to integer matrices with only an extra $\log||A||^{1+o(1)}$ factor.
All our certificates are based on interactive verification protocols with the interaction removed by a Fiat-Shamir identification heuristic. The validity of our verification procedure is subject to standard computational hardness assumptions from cryptography. Our certificates improve on those by Goldwasser, Kalai and Rothblum 2008 and Thaler 2012 for our problems in the prover complexity, and are independent of the circuit that computes them thus detecting programming errors in them.
This is joint work with Jean-Guillaume Dumas at the University of Grenoble.

• Time: Tuesday, August 26, 2014 at 15:00 –16:00 p.m.    Room: N210

Title: Verified Computation Based on Finite Element Method: From Qualitative Error Analysis to Quantitative One

Speaker: Xuefeng Liu  (Waseda University, Japan)

Abstract: In computer-assisted proof for non-linear differential equations, verified computation based on the finite element method plays an important role. Particularly, the quantitative error analysis is much more important than the classical qualitative one. That is, not only the convergence order but also the explicit error estimation of approximate solutions are desired. This talk contains the following topics. a) Verified computation and interval arithmetic computation; b) The quantitative error estimation for the finite element method in solving partial differential equations; c) The technique of hypercircle equation from Prager-Synge¡¯s theorem to deal with the singularity of solutions; d) The error estimation for various interpolation error constants, which are reduced to eigenvalue problems.

• Time: Wensday, August 27, 2014 at 15:00 –16:00 p.m.    Room: N210

Title: Verified Eigenvalue Bounds for Self-adjoint Differential Operators

Speaker: Xuefeng Liu  (Waseda University, Japan)

Abstract:In this talk, a uniform framework to provide guaranteed eigenvalue bounds for self-adjoint differential operators is proposed. In this framework, both conforming and non-conforming finite element methods (FEMs) are adopted to construct explicit eigenvalue bounds, even in the case that the eigenfunction has a singularity around the re-entrant corners of domains.
As concrete examples, the conforming Lagrange finite element is used to give eigenvalue bounds for the Laplacian defined over polygonal domains of general shapes. The Crouzeix-Raviart FEMs and the Fujino-Morley FEMs, along with explicit a priori error estimation, are used to provide explicit eigenvalue bounds for the Laplacian and the Bi-harmonic operators, respectively. Further, Lehmann-Goerisch¡¯s theorem is applied to give dramatically improved high-precision eigenvalue bounds. As the computation is performed under the interval arithmetic, the obtained eigenvalue bounds are mathematically correct and thus can be used in solution existence verification for certain non-linear partial differential equations.

• Time: Thursday, August 28, 2014 at 15:00 –16:00 p.m.    Room: N210

Title: Solution Verification for Non-linear Elliptic Partial Differential Equations

Speaker: Xuefeng Liu  (Waseda University, Japan)

Abstract: The computer-assisted proof based on verified computation is becoming a powerful tool to investigate the solution existence and uniqueness of non-linear partial differential equations (PDEs). For the non-linear elliptic PDEs, several methods have proposed to verify the solution existence, for example, the methods of M. Plum, M. Nakao, S. Oishi, respectively. Each of these methods is based on the fixed-point theorem and the spectrum estimation for elliptic partial differential operators. In this talk, I would like to give a brief introduction of these methods and more efforts are paid to the spectrum estimation and the quantitative error estimation.

• Time: Thursday, September 25, 2014 at 15:00 –16:00 p.m.    Room: N205

Title: Minimal Universal Denominators for Linear Difference Equations

Speaker: Qinghu Hou  (Nankai University, China)

Abstract: We provide minimal universal denominators for linear difference equations with fixed leading and trailing coefficients. In the case of first-order equations, they are factors of Abramov¡¯s universal denominators. While in the case of higher order equations, we show that Abramov¡¯s universal denominators are minimal.

• Time: Thursday, October 30, 2014 at 15:00 –16:00 p.m.    Room: N219

Title: Mixed Volume Computation and Solving Polynomial Systems

Speaker: Tien Yien Li  (Michigan State University, USA)

Abstract: In the last few decades, the homotopy continuation method has been established in the U.S. for finding the full set of isolated zeros to a polynomial system numerically. The method involves first solving a trivial system, and then deforming these solutions along smooth paths to the solutions of the target system. Recently, modeling the sparse structure of a polynomial system by its Newton polytopes leads to a major computational breakthrough. Based on an elegant method for computing the mixed volume, the new polyhedral homotopy can find all isolated zeros of a polynomial system much efficiently. The method has been successfully implemented and proved to be very powerful in many occasions, especially when the systems are sparse. We will elaborate the method in this talk.

• Time: Wensday, November 19, 2014 at 10:00 –11:00 p.m.    Room: N420

Title: The Algorithmic Revolution in Geometry of Numbers

Speaker: Phong Nguyen  (INRIA, France and Tsinghua University, China)

Abstract: In the past 30 years, there has been significant progress in the study of algorithmic questions in geometry of numbers. This research has used or revisited many classical mathematical results. In this talk, we survey the connection between algorithmic and mathematical aspects of geometry of numbers. Examples include random lattices and distributions, worst-case to average-case reductions and lattice algorithms.

• Time: Thursday, November 20, 2014 at 15:00 –16:00 p.m.    Room: N205

Title: Computer Aided Analysis for the Global Stability and the Existence of Limit Cycles of ODE's

Speaker: Zhengyi Lu  (Sichuan Normal University, China)

Abstract: Symbolic computation for dealing with the global stability and the existence of periodic orbits of ODE's are shown.
Wu's well ordering principle is applied to the prey-predator chain systems. An algorithm for real root isolation of polynomial systems is proposed and used to check the uniqueness of a positive equilibrium which implies the global stability of the corresponding monotone systems. The positive definiteness of a class of polynomials from the global stability analysis of discrete di usion systems is proved. Small amplitude limit cycles for Kolmogorov systems are constructed based on the Liapunov method and the algorithm for real root isolation. Center and focus problems and the algorithmic construction for multiple limit cycles for 3D systems are mentioned..

• Time: Monday, December 15, 2014 at 15:00 –16:00 p.m.    Room: N205

Title: Melham's Conjecture on Sums of Odd Powers of Fibonacci Numbers

Speaker: Arthur L. B. Yang  (Center for Combinatorics, Nankai University, China)

Abstract: Let $F_n$ denote the $n$-th Fibonacci number and $L_n$ denote the $n$-th Lucas number. Melham conjectured that for any $n, m\geq 1$, the sum $$L_1L_3L_5\cdots L_{2m+1}\sum_{r=1}^n F_{2r}^{2m+1}$$ can be expressed as $(F_{2n+1}-1)^2P(F_{2n+1})$, where $P(x)$ is a polynomial of degree $2m-1$ with integer coefficients. Based on a formula due to Prodinger, we give an affirmative answer to Melham's conjecture.
This is a joint work with Brian Y. Sun and Matthew H.Y. Xie.

• Time: Tuesday, Juanary 20, 2015 at 15:00 –16:00 p.m.    Room: N205

Title: Incidence Geometry and Erdös Type Problems

Speaker: Oliver Roche-Newton  (RICAM-Linz, Austrian Academy of Sciences, Austria)

Abstract: A famous problem of Erdös in the field of discrete geometry is the following: given a set of $n$ points in the plane, what lower bounds can we obtain for the number of distinct distances determined by pairs of points from the set. One can generate many other interesting questions about bounds for certain configurations determined by a finite set of points - these are often described as "Erdös-type problems" and most of the best answers that have been obtained have come from progress in the field of incidence geometry. This talk will give an introduction to these two closely related areas, and touch upon some new results of this nature in the finite field setting.

• Time: Thursday, August 27, 2015 at 15:00 –16:00 p.m.    Room: N205

Title: Gamma-positivity for the Eulerian distribution on separable permutations

Speaker: Shishuo Fu  (College of Mathematics and Statistics, Chongqing Univeristy, China)

Abstract: In this talk, we introduce the descent polynomial on separable permutations (3142 and 2413 avoiding). We prove that it is gamma-positive, which implies both unimodality and symmetry. Moreover, we find a combinatorial interpretation for its gamma-coefficients which is comparable with that of Eulerian polynomials. This is joint work with Zhicong Lin and Jiang Zeng.

• Time: Tuesday, December 6, 2016 at 9:30 –10:30 a.m.    Room: N420

Title: Multidimensional Wavelets and Framelets

Speaker: Bin Han  (University of Alberta, Canada)

Abstract: One of the current active research directions in wavelet analysis is on multidimensional wavelets and framelets, which are of interest for high-dimensional problems such as image processing and computer graphics. The study of multidimensional wavelets and framelets is closely related to multivariate Laurent polynomials, algebraic geometric, symbolic computing, and symmetry groups. In this talk, we first introduce some basic theory on multidimensional wavelets and framelets. Then we shall present some recent progresses on their constructions. Next we shall discuss some major tools and techniques used from other areas in mathematics for the study of multivariate wavelets and framelets. Finally we shall address some difficulties and challenges on the construction of multidimensional wavelets and framelets as well as some open problems on this topic.

• Time: Tuesday, December 6, 2016 at 10:45 –11:45 a.m.    Room: N420

Title:Gauss-Newton Method for Phase Retrieval

Speaker: Zhiqiang Xu  (Academy of Mathematics and Systems Science, CAS, China)

Abstract: In this talk, we introduce a concrete algorithm for phase retrieval, which we refer to as GaussNewton algorithm. In short, this algorithm starts with a good initial estimation, which is obtained by a modified spectral method, and then update the iteration point by a Gauss-Newton iteration step. We prove that a re-sampled version of this algorithm quadratically converges to the solution for the real case with the number of random measurements being nearly minimal. Numerical experiments also show that Gauss-Newton method has better performance over the other algorithms.

• Time: Monday, Feburary 20, 2017 at 14:00 –15:00 p.m.    Room: N210

Title:Polynomials with only real zeros in combinatorics

Speaker: Yi Wang  (Dalian University of Technology, China)

Abstract: Polynomials with only real zeros arise often in combinatorics. Our interest in such polynomials was originally due to its implication about unimodality and log-concavity. In this talk we establish some sufficient conditions to the reality of zeros of polynomial sequences satisfying certain recurrence relations and then apply them to solve several open problems.

• Time: Monday, March 27, 2017 at 14:00 –15:00 p.m.    Room: N205

Title:Smith normal form and combinatorics

Speaker: Lili Mu  (Liaoning Normal University, China)

Abstract: This talk surveys some combinatorial aspects of Smith normal form. The discussion includes Smith normal form of Laplacian matrices, random integer matrices and matrices associated with Youngs Lattice and partitions. we then give some examples of Smith normal form arising from three interesting cases of Varchenko matrices of hyperplane arrangements.

• Time: Tuesday, April 18, 2017 at 10:00 –11:00 a.m.    Room: N202

Title:The Riemann Hypothesis in terms of eigenvalues of certain almost triangular Hankel matrices

Speaker: Yuri Matiyasevich  (St.Petersburg Department of Steklov Institute of Mathematics of Russian Academy of Sciencies, Russia)

Abstract: The famous Riemann Hypothesis (RH) is one of the most important open problem in Number Theory. As many other outstanding problems, RH has many equivalent statements. Ten years ago the speaker reformulated the Riemann Hypothesis as statements about the eigenvalues of certain Hankel matrices, entries of which are defined via the Taylor series coefficients of Riemann's zeta function. Numerical calculations revealed some very interesting visual patterns in the behaviour of the eigenvalues and allowed the speaker to state a number of new conjectures related to the RH.
Recently computations have been performed on supercomputers. This led to new conjectures about the finer structure of the eigenvalues and eigenvectors and to conjectures that are (formally) stronger than RH. Further refinement of these conjectures would require extensive computations on more powerful computers than those that were available to the speaker.

• Time: Tuesday, April 18, 2017 at 11:00 –12:00 a.m.    Room: N202

Title:Further results on Hilbert's Tenth Problem

Speaker: Zhi-Wei Sun  (Nanjing University, China)

Abstract: Hilbert's Tenth Problem (HTP) asks for an effective algorithm to test whether an arbitrary polynomial equation $P(x_1, ... ,x_n)=0$ (with integer coefficients) has solutions over the ring $\mathbb{Z}$ of the integers. This was finally solved by Matiyasevich in 1970 negatively. In this talk we introduce the speaker's further results on HTP. In particular, we present a sketch of the proof of the speaker's main result that there is no effective algorithm to determine whether an arbitrary polynomial equation $P(x_1, ... ,x_{11})=0$ (with integer coefficients) in 11 unknowns has integral solutions or not.

• Time: Thursday, May 4, 2017 at 14:00 –15:00 p.m.    Room: N205

Title:ÃÜÂëÑ§ÖÐ¼¸Àà·ÇÏßÐÔÐòÁÐ×ÛÊö

Speaker: ÆÝÎÄ·å  (½â·Å¾üÐÅÏ¢¹¤³Ì´óÑ§)

• Time: Monday, June 5, 2017 at 14:00 –15:00 p.m.    Room: N202

Title:On the summability of formal solutions of singular PDEs

Speaker: Changgui Zhang  (Université de Sciences et Technologies de Lille, France)

Abstract: Via the Stokes phenomenon, the Borel-Laplace summation method plays a central role for the analytic description of the Galois differential group associated with a singular linear ODE. In the talk, we deal with a family of totally characteristic type PDEs, establishing the summability of their formal solutions in suitable Gevrey spaces. This is a joint work with Chen H. and Luo Z..

• Time: Monday, June 5, 2017 at 15:30 –16:30 p.m.    Room: N202

Title:Algebraic and computational aspects of tensors

Speaker: Ke Ye  (Univsersity of Chichago, USA)

Abstract: Tensors are direct generalizations of matrices. They appear in almost every branch of mathematics and engineering. Three of the most important problems about tensors are: 1) compute the rank of a tensor 2) decompose a tensor into a sum of rank one tensors 3) Comon¡¯s conjecture for symmetric tensors. In this talk, I will try to convince the audience that algebra can be used to study tensors. Examples for this purpose include structured matrix decomposition problem, bilinear complexity problem, tensor networks states, Hankel tensors and tensor eigenvalue problems. In these examples, I will explain how algebraic tools are used to answer the three problems mentioned above.

• Time: Monday, June 19, 2017 at 14:00 –15:00 p.m.    Room: N202

Title:Automation in interactive theorem proving

Speaker: Bohua Zhan  (MIT, USA)

Abstract: Interactive theorem proving involves using proof assistants to verify, with human guidance, proofs of either mathematical theorems or correctness of computer programs. In this talk, I will give a brief overview of the history of this field, with an emphasis on automation techniques. I will then discuss my own work on a new heuristic theorem prover called auto2 for the proof assistant Isabelle.

• Time: Monday, August 21, 2017 at 9:00 –10:00 a.m.    Room: N205

Title: µÈ¼¸ºÎÔìÐÍÓë·ÖÎöÐÂ½øÕ¹

Speaker: Ðì¸Ú  (º¼ÖÝµç×Ó¿Æ¼¼´óÑ§¼ÆËã»úÑ§Ôº)

• Time: Wensday, September 6, 2017 at 14:00 –15:00 p.m.    Room: N210

Title: Several Combinatorial Problems Concerning Tableaux

Speaker: Peter L. Guo  (ÄÏ¿ª´óÑ§)

Abstract: Tableaux have numerous applications in combinatorics, representation theory and algebraic geometry. In this talk, we shall discuss several combinatorial problems related to tableaux. For example, are there tableau formulas for Schubert/Grothendieck polynomials? We also talk about some our recent work on tableaux.

• Time: Wensday, September 6, 2017 at 15:30 –16:30 p.m.    Room: N210

Title: Over and Under Approximations of Reachable Sets Within Hamilton-Jacobi Framework

Abstract: For dynamical systems, reachable sets can be described by solutions of Hamilton-Jacobi equations. In this paper, we discuss a methodology to compute approximations, defined by zero sub-level sets of polynomials, of time-bounded reachable sets (i.e., flowpipes) with arbitrary bounded errors for polynomial dynamical systems via solving derived Hamilton-Jacobi equations with inequality constraints. We start with evolution functions for describing the flowpipes of systems, and find their explicit Taylor expansions with respect to time. Then, we prove the existence of polynomial approximations to evolution functions with arbitrary bounded errors by investigating solutions of corresponding partial differential equations with derived inequality constraints, which shows the applicability of this methodology to obtain both over and under approximations of reachable sets with arbitrary precisions in Hausdorff metric. Afterwards, we propose two methods to compute polynomial template based evolution functions with constraints via using sum-of-squares decomposition and quantifier elimination, respectively. We test these two methods on some examples with comparisons to the advection operator based method. The computation and comparison results show that the QE based method to certain extent has better performance than the SOS based method and the advection operator based method.

• Time: Wensday, September 13, 2017 at 9:30 –10:30 a.m.    Room: N205

Title: Tropical geometry and its applications

Speaker: Yue Ren  (Max Planck Institute for Mathematics in the Sciences,Germany)

Abstract: Tropical varieties are balanced polyhedral complexes that arise in several contexts. In Geometry, they are commonly regarded as combinatorial shadows of their algebraic counterparts; in combinatorics, they appear in the study of realizable matroids; and in optimization, they appear as parameter domains in which the optimal solution is not unique.
In this talk, we will discuss the different equivalent definitions for tropical varieties and the different applications of tropical geometry that they entail. Moreover, we will discuss how tropical varieties can be computed, and, in doing so, highlight some bread-and-butter techniques of computer algebra.

• Time: Tuesday, October 10, 2017 at 15:00 –16:00 p.m.    Room: N205

Title: On $q$-series and the $q$-partial differential equations

Speaker: ÁõÖÎ¹ú  (»ª¶«Ê¦·¶´óÑ§)

Abstract: A $q$-partial derivative of a function of several variables is its $q$-derivative with respect to one of these variables, regarding other variables as constants. A $q$-partial differential equation is an equation containing unknown multivariable functions and their $q$-partial derivatives, which is a $q$-extension of the ordinary partial differential equation. The $q$-partial differential equation is a completely new research topic, which reveal some surpring connections between $q$-series and the analytic functions of several complex variables. In this talk, I will intoduce some research results in the $q$-partial differential equations.

• Time: Monday, October 30, 2017 at 15:00 –16:00 p.m.    Room: N205

Title:Some generalizations of a supercongruence of van Hamme

Speaker: ¹ù¾üÎ°  (»´ÒõÊ¦·¶Ñ§Ôº)

Abstract: In 1997, van Hamme conjectured that Ramanujan's formula for $1/\pi$ has a nice $p$-analogue. This result was proved by Mortenson using a $6F5$ transformation, and was reproved by Zudilin via the Wilf¨CZeilberger method. In this talk, we propose a conjectural generalization of van Hamme's supercongruence and prove some special cases.

• Time: Tuesday, November 21, 2017 at 14:00 –15:00 p.m.    Room: N205

Title:Éî¶ÈÑ§Ï°ÔÚÍ¼ÏñÊÓÆµ´¦ÀíÖÐµÄÓ¦ÓÃ

Speaker: Áõ‚Æ  (ÖÐ¹ú¿ÆÑ§ÔºÐÅÏ¢¹¤³ÌÑÐ¾¿Ëù)