Please find the talks before December 31, 2016 at Previsous Talks

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• 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)

• Time: Thursday, January 4, 2018 at 14:00 –15:00 p.m.    Room: N205

Title:New bounds for spherical two-distance sets and equiangular lines

Speaker: Wei-Hsuan Yu  ( Brown University, USA)

Abstract: The set of points in a metric space is called an s-distance set if pairwise distances between these points admit only $s$ distinct values. Two-distance spherical sets with the set of scalar products $a$ and $-a$, are called equiangular. The problem of determining the maximal size of $s$-distance sets in various spaces has a long history in mathematics. We determine a new method of bounding the size of an $s$-distance set in two-point homogeneous spaces via zonal spherical functions. This method allows us to prove that the maximum size of a spherical two-distance set in $n$ dimension Euclidean space is $n(n+1)/2$ with possible exceptions for some $n=(2k+1)^2-3$, where $k$ is a positive integer. We also prove the universal upper bound $2/3 n a^2$ for equiangular sets with angle $1/a$ and, employing this bound, prove a new upper bound on the size of equiangular sets in an arbitrary dimension. Finally, we classify all equiangular sets reaching this new bound.

• Time: Thursday, January 4, 2018 at 15:00 –16:00 p.m.    Room: N205

Title:Á¿×Ó¾À´íÂë

Speaker: ´÷ÕÕÅô  (Çàµº´óÑ§ÊýÑ§ÓëÍ³¼ÆÑ§Ôº)

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