
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 TexasPan 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 allfocusing, alldefocusing and mixed types. By using the KPhierarchy reduction method based on the Sato theory, we construct a general formula in Gramtype 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 TSplines
Speaker: Thomas W. Sederberg (Brigham Young University, USA)
Abstract: TSplines were invented in 2003 to address the major limitations of the NURBS freeform surface representation which is the industry standard in computer aided. TSplines provide a watertight model of arbitrary topology that allows for local refinement. TSplines also simplify the model creation procedure for designers. In addition, TSplines 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 BousquetMelou, 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 apossibly randomizedverification 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 MonteCarlo 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 matrixtimesvector products + $n^{1+o(1)}$ arithmetic operations in the field. For example, if the $n\times n$ input matrix is sparse with $n^{1+o(1)}$ nonzero entries, our rank certificate can be verified in $n^{1+o(1)}$ field operations. This extends also to integer matrices with only an extra $\logA^{1+o(1)}$ factor.
All our certificates are based on interactive verification protocols with the interaction removed by a FiatShamir 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 JeanGuillaume 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 computerassisted proof for nonlinear 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 PragerSynge¡¯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 Selfadjoint Differential Operators
Speaker: Xuefeng Liu (Waseda University, Japan)
Abstract:In this talk, a uniform framework to provide guaranteed eigenvalue bounds for selfadjoint differential operators is proposed. In this framework, both conforming and nonconforming finite element methods (FEMs) are adopted to construct explicit eigenvalue bounds, even in the case that the eigenfunction has a singularity around the reentrant 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 CrouzeixRaviart FEMs and the FujinoMorley FEMs, along with explicit a priori error estimation, are used to provide explicit eigenvalue bounds for the Laplacian and the Biharmonic operators, respectively. Further, LehmannGoerisch¡¯s theorem is applied to give dramatically improved highprecision 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 nonlinear partial differential equations. 
Time: Thursday, August 28, 2014 at 15:00 –16:00 p.m. Room: N210
Title: Solution Verification for Nonlinear Elliptic Partial Differential Equations
Speaker: Xuefeng Liu (Waseda University, Japan)
Abstract: The computerassisted proof based on verified computation is becoming a powerful tool to investigate the solution existence and uniqueness of nonlinear partial differential equations (PDEs). For the nonlinear 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 fixedpoint 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 firstorder 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, worstcase to averagecase 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 preypredator 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 $2m1$ 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 RocheNewton (RICAMLinz, 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 "Erdöstype 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: Gammapositivity 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 gammapositive, which implies both unimodality and symmetry. Moreover, we find a combinatorial interpretation for its gammacoefficients 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 highdimensional 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:GaussNewton 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 GaussNewton iteration step. We prove that a resampled 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 GaussNewton method has better performance over the other algorithms.