Please find the talks before December 31, 2017 at Previsous Talks

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• 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

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

Abstract: Á¿×Ó¾À´íÂëÊÇÒ»ÖÖÄÜ¹»¿Ë·þÁ¿×ÓÐÅµÀÔëÉùµÄ±àÂë·½°¸£¬Ëü¿ÉÒÔÊ¹Á¿×Ó¼ÆËã»úÔÚÓÐÔëÉùµÄ»·¾³ÖÐÓÐÐ§¼ÆËã£¬Ò²ÄÜÊ¹Á¿×ÓÏûÏ¢ÔÚ´øÔëÉùµÄÁ¿×ÓÐÅµÀÉÏÊµÏÖ¿É¿¿Í¨ÐÅ£¬Òò´ËÁ¿×Ó¾À´íÂëÊÇÁ¿×ÓÐÅµÀ±àÂëÖÐ×îÖØÒªµÄÑÐ¾¿ÄÚÈÝÖ®Ò»¡£ ±¾±¨¸æ¶ÔÄ¿Ç°Á¿×Ó¾À´íÂëµÄÑÐ¾¿×öÒ»×ÛÊö£¬ÖØµã½éÉÜÎÈ¶¨×ÓÂë¼°ÆäÖØÒªµÄ¼¸¸ö×ÓÀà£ºCSSÂë£¬Á¿×ÓBCHÂë£¬Á¿×ÓMDSÂëµÈ¡£

• Time: Thursday, April 19, 2018 at 14:00 –15:00 p.m.    Room: N205

Title:Partitions related to mock theta functions

Speaker: Liuquan Wang  (Wuhan University)

Abstract: Let $\omega(q)$ be the third order mock theta function due to Ramanujan and Watson. In his 2007 paper in Invent. Math., Andrews interpreted the coefficients of $\omega(q)$ as a partition function $p_{\omega}(n)$. He also introduced the concepts of odd rank and $k$-marked odd Durfee symbols for this kind of partitions. In this talk, we will present some new congruences satisfied by $p_{\omega}(n)$ and the smallest parts function associated to it. Meanwhile, we will discuss the arithmetic properties of odd ranks and $k$-marked odd Durfee symbols. Applying the properties we find, we give proofs to Andrews¡¯ conjectures on the parity of $k$-marked odd Durfee symbols.

• Time: Thursday, April 26, 2018 at 15:00 –16:00 p.m.    Room: N219

Title:On the chordality of polynomial sets in triangular decomposition in top-down style

Abstract: In this talk, we show the connections between chordal graphs which permit perfect elimination orderings on their vertexes from graph theory and triangular decomposition which decomposes polynomial sets into triangular set from symbolic computation and present the chordal graph structures of polynomial sets appearing in triangular decomposition in top-down style when the input polynomial set has a chordal associated graph. In particular, we show that the associated graph of one specific triangular set in any algorithm for triangular decomposition in top-down style is a subgraph of that chordal graph and that all the triangular sets computed by Wang's method for triangular decomposition have associated graphs which are subgraphs of that chordal graph. Furthermore, the associated graphs of polynomial sets can be used to describe their sparsity with respect to thee variables, and we present a refined algorithm for efficient triangular decomposition for sparse polynomial sets in this sense.
This talk is based on the joint work with Yang Bai.

• Time: Monday, May 14, 2018 at 15:00 –16:00 p.m.    Room: N219

Title:Introduction to Elliptic Curves

Speaker: Ilias Kotsireas  (Wilfrid Laurier University, Canada)

Abstract: We will present an accessible introduction to the vast theory of Elliptic Curves, from an elementary standpoint, i.e. by looking for points with integer/rational coordinates in polynomial equations. Among other things, we will explain the concept of the rank of an elliptic curve, the group law for elliptic curves, Noam Elkies' record rank elliptic curve and the Birch and Swinnerton-Dyer conjecture. No previous knowledge of Elliptic Curves is necessary, this is designed to be a self-contained talk.

• Time: Thursday, May 31, 2018 at 16:00 –17:00 p.m.    Room: N219

Title:Continued fractions in positive characteristic and automatic sequences

Speaker: Ò¦¼ÒÑà  (Çå»ª´óÑ§ÊýÑ§Ïµ)

Abstract: For an algebraic continued fraction in positive characteristic, what can we read from its partial quotients? Such a question is natural, since by the famous CKMFR's theorem, the coefficients of an algebraic power series over a finite field form an automatic sequence. Certain algebraic continued fractions are such that the sequence of the leading coefficients of the partial quotients is automatic. Here we give some rather general families of such sequences. Moreover, inspired by these examples, we give several criteria on automatic sequences, which allow us to obtain new families of automatic sequences in an arbitrary finite field.

• Time: Thursday, June 7, 2018 at 10:00 –1:00 a.m.    Room: N205

Title:Interlacing eigenvalues of Hermitian matrices

Speaker: ÁõÀö  (Çú¸·Ê¦·¶´óÑ§ÊýÑ§¿ÆÑ§Ñ§Ôº)

Abstract:In this talk, we first give a short survey of classical interlacing properties of Hermitian matrices and then present some applications to (normalized) Laplacian matrices of graphs.

• Time: Wednesday, June 27, 2018 at 15:00 –16:00 p.m.    Room: N202

Title:Root geometry of polynomials generated by a recursion of order two

Abstract:We consider the root distribution of a sequence of univariate polynomials satisfying a recurrence of order two with linear polynomial coefficients. We show that the set of non-isolated limits of zeros of the polynomials is either an arc, or a circle, or a `lollipop'', or an interval. As an application, we discover a sufficient and necessary condition for the universal real-rootedness of the polynomials, subject to certain sign condition on the coefficients of the recurrence.

• Time: Wednesday, June 27, 2018 at 16:00 –17:00 p.m.    Room: N202

Title:Some number theoretical results concerning automatic sequences

Speaker: ºúâùÄþ  (»ªÖÐ¿Æ¼¼´óÑ§ÊýÑ§ÓëÍ³¼ÆÑ§Ôº)

Abstract:Automatic sequences arise naturally in various contexts. The study of automatic sequences lies at the interface of number theory, combinatorics on words, dynamic system, logic and theoretical computer science. In this talk I will first give several equivalent definitions and classical examples of automatic sequences. Then I will present important theorems in this domain. Finally I will talk about our works related to infinite products, completely multiplicative functions, transcendence in positive characteristics, and coefficient extraction.

• Time: Thursday, June 28, 2018 at 10:00 –11:00 a.m.    Room: N202

Title:A lower bound on the positive semidefinite rank of convex bodies

Speaker: Safey El Din, Mohab  (UPMC, Université Paris 06, INRIA, France)

Abstract:The positive semidefinite rank of a convex body $C$ is the size of its smallest positive semidefinite formulation. We show that the positive semidefinite rank of any convex body $C$ is at least $\sqrt{\log d}$ where $d$ is the smallest degree of a polynomial that vanishes on the boundary of the polar of $C$. This improves on the existing bound which relies on results from quantifier elimination. Our proof relies on the B\'ezout bound applied to the Karush-Kuhn-Tucker conditions of optimality. We discuss the connection with the algebraic degree of semidefinite programming and show that the bound is tight (up to constant factor) for random spectrahedra of suitable dimension.
Joint work with H. Fawzi (University of Cambridge, UK).

• Time: Friday, June 29, 2018 at 11:00 –12:00 a.m.    Room: N202

Title: Quantum codes from linear codes over finite chain rings

Speaker: Xiusheng Liu  (Hubei Polytechnic University, China)

Abstract:In this paper, we provide two methods of constructing quantum codes from linear codes over finite chain rings. The first one is derived from the Calderbank-Shor-Steane (CSS) construction applied to self-dual codes over finite chain rings. The second construction is derived from the CSS construction applied to Gray images of the linear codes over finite chain ring $\mathbb{F}_{p^{2m}}+u \mathbb{F}_{p^{2m}}$. The good parameters of quantum codes from cyclic codes over finite chain rings are obtained.
This is joint work with Hualu Liu.

• Time: Monday, July 2, 2018 at 10:00 –11:00 a.m.    Room: N202

Title: Semisoft Generalized $I_1$ Greedy Algorithm for CT Image Reconstruction

Speaker: Jiehua Zhu  (Georgia Southern University, USA)

Abstract:In this talk, a generalized $I_1$ greedy algorithm for signal and image recovery is first introduced. Its convergence and the superiority over the reweighted -minimization and greedy algorithms are demonstrated. Then this algorithm is extended as a semisoft generalized greedy algorithm by adapting the wavelet technique of semisoft thresholding. The extended algorithm is applied to image reconstruction by incorporating it into the BCPCS framework, resulting in a semisoft generalized total variation minimization algorithm for computed tomography (CT). Numerical tests indicate the proposed algorithm improves the image reconstruction for CT.

• Time: Thursday, July 5, 2018 at 15:00 –16:00 p.m.    Room: N205

Title: The Second Discriminant of a Univariate Polynomial

Speaker: Ñî¾²  (¹ãÎ÷Ãñ×å´óÑ§)

Abstract:In this talk, we define the second discriminant $D_2$ of a univariate polynomial $f$ of degree greater than $2$ as the product of the linear forms $2$ $r_k-r_i-r_j$ for all triples of roots $r_i, r_k, r_j$ of $f$ with $j>i$ and $j\neq k, k\neq i$. $D_2$ vanishes if and only if $f$ has at least one root which is equal to the average of two other roots. We show that $D_2$ can be expressed as the resultant of $f$ and a determinant formed with the derivatives of $f$, establishing a new relation between the roots and the coefficients of $f$. We prove several notable properties and present an application of $D_2$.
This work is jointly done with Professor Dongming Wang.

• Time: Thursday, July 5, 2018 at 16:00 –17:00 p.m.    Room: N205

Title: Restricted inversion sequences and enhanced $3$-noncrossing partitions

Speaker: ÁÖÖ¾´Ï  (¼¯ÃÀ´óÑ§)

Abstract:Yan and Martinez--Savage conjectured independently that inversion sequences with no weakly decreasing subsequence of length $3$ and enhanced $3$-noncrossing partitions have the same cardinality. In this talk, I will present an algebraic proof, which applies both the generating tree technique and the so-called obstinate kernel method developed by Bousquet-M\'elou. As one application of this equinumerosity, an intriguing identity involving numbers of classical and enhanced $3$-noncrossing partitions was discovered. I will also pose a related functional equation, enumerating another interesting class of restricted inversion sequences, that I could not solve.

• Time: Wednesday, July 11, 2018 at 15:00 –16:00 p.m.    Room: N205

Title: Counting polynomial sums in finite commutative rings

Speaker: Àî¼ªÓÐ  (ÉÏº£½»Í¨´óÑ§)

Abstract: Let $R$ be a finite commutative ring and $D$ be a subset of $R$. For a polynomial $f(x)$ in $R[x]$ and a positive integer $k$, we enumerate $k$-subsets $S$ in $D$ such that the polynomial sum over $S$ is a given element. In this talk, I will introduce some recent progress on this problem and its applications in combinatorics, number theory and coding theory. This talk is based on joint work with Daqing Wan.

• Time: Wednesday, July 11, 2018 at 16:00 –17:00 p.m.    Room: N205

Title: Desingularization in the $q$-Weyl algebra

Speaker: Yi Zhang  (Institute for Algebra, Johannes Kepler University Linz, Austria)

Abstract:We study the desingularization problem in the first $q$-Weyl algebra. We give an order bound for desingularized operators, and thus derive an algorithm for computing desingularized operators in the first $q$-Weyl algebra. Moreover, an algorithm is presented for computing agenerating set of the first $q$-Weyl closure of a given $q$-difference operator. As an application, we certify that several instances of the colored Jones polynomial are Laurent polynomial sequences by computing the corresponding desingularized operator. This is a joint work with Christoph Koutschan.

• Time: Wednesday, August 8, 2018 at 16:00 –17:00 p.m.    Room: N205

Title:Semi-automated proof of supercongruences on partial sums of hypergeometric series

Speaker: Áõ¼Í²Ê  (ÎÂÖÝ´óÑ§)

Abstract:Using the software package Sigma developed by Schneider, we automatically discover and prove some combinatorial identities involving harmonic numbers, from which we deduce some supercongruences on partial sums of hypergeometric series. These results confirm some conjectural generalizations of van Hamme's (B.2) and (C.2) supercongruences in some special cases, and extend van Hamme's (A.2) and (H.2) supercongruences to the cases modulo $p^4$.