Past Talks
Speaker: Sergey Goryainov (Hebei Normal University)
Room: College of Science M1001
Time: 2025/05/15, Thursday, 10:00 - 11:00
Divisible designs graphs (DDGs for short) were introduced in 2011 by Haemers, Kharaghani and Meulenberg as a bridge between graph theory and the theory of (group divisible) designs. Since then, around 20 constructions producing infinitely many DDGs haven been introduced. These constructions make use of many combinatorial and algebraic objects: finite geometries, Hadamard matrices, weighing matrices, designs, Cayley graphs, block matrices, strongly regular graphs and so on. Also, a number of characterisations of divisible design graphs is known. In this talk we will discuss three infinite families of DDGs recently discovered in connection with symplectic spaces. The talk is based on joint works with Anwita Bhowmik, Bart De Bruyn, Willem Haemers and Leonid Shalaginov.
Speaker: Jianfeng Hou (Fuzhou University)
Room: College of Science M1001
Time: 2025/05/08, Thursday, 10:00 - 11:00
For a nondegenerate r-graph F, large n, and t in the regime [0, cF n], where cF > 0
is a constant depending only on F, we present a general approach for determining the
maximum number of edges in an n-vertex r-graph that does not contain t + 1 vertex-disjoint
copies of F. In fact, our method results in a rainbow version of the above result
and includes a characterization of the extremal constructions.
Our approach applies to many well-studied hypergraphs (including graphs) such as
the edge-critical graphs, the Fano plane, the generalized triangles, hypergraph expansions,
the expanded triangles, and hypergraph books. Our results extend old results of Erd˝os,
Simonovits, and Moon on complete graphs, and can be viewed as a step towards a general
density version of the classical Corr´adi–Hajnal and Hajnal–Szemer´edi Theorems.
Our method relies on a novel understanding of the general properties of nondegenerate
Tur´an problems, which we refer to as smoothness and boundedness. These properties are
satisfied by a broad class of nondegenerate hypergraphs and appear to be worthy of future
exploration (Join work with Heng Li, Xizhi Liu, Long-Tu Yuan and Yixiao Zhang).
Speaker: Xianwei Xia (Suzhou University of Science and Technology)
Room: College of Science M1001
Time: 2025/04/17, Thursday, 10:00 - 11:00
In their study of the truncated sums of the classical theta functions, Andrews-Merca and Guo-Zeng posed a conjecture on truncated sums of a special case of the Jacobi triple product identity which was confirmed independently by Mao and Yee. In 2016, Chan, Ho and Mao examined the truncated series arising from two consequences of the quintuple product identity. In this talk, we establish an explicit series form with nonnegative coefficients on a new truncated sum of a special cases of the Jacobi triple product identity by taking different truncated series which is stronger than the conjecture due to Andrews-Merca and Guo-Zeng. As a corollary of our results, we obtain a new truncated sums of Jacibi’s identity which implies another conjecture given by Guo and Zeng. In addition, we determine the signs of coefficients of new truncated sums of two well-known identities derived from the quintuple product identity which can be considered as the companion results of a theorem proved by Chan, Ho and Mao.
Speaker: Congpei An (Guizhou University)
Room: College of Science M1001
Time: 2025/04/15, Tuesday, 14:00 - 15:00
We discuss the approximation of continuous functions on the unit sphere by spherical polynomials of degree n via hyperinterpolation. Hyperinterpolation of degree n is a discrete approximation of the L2-orthogonal projection of degree n with its Fourier coefficients evaluated by a positive-weight quadrature rule that exactly integrates all spherical polynomials of degree at most 2n. This talk aims to bypass this quadrature exactness assumption by replacing it with the Marcinkiewicz–Zygmund property. Consequently, hyperinterpolation can be constructed by a positive-weight quadrature rule–not necessarily with quadrature exactness. This scheme is called unfettered hyperinterpolation. We provide a reasonable error estimate for unfettered hyperinterpolation. The error estimate generally consists of two terms: a term representing the error estimate of the original hyperinterpolation of full quadrature exactness and another introduced as compensation for the loss of exactness degrees. A guide to controlling the newly introduced term in practice is provided. In particular, if the quadrature points form a quasi-Monte Carlo design, then there is a refined error estimate. Numerical experiments verify the error estimates and the practical guide.
Speaker: Andrey Kupavskii (Moscow Institute of Physics and Technology)
Room: College of Science M4009
Time: 2025/04/10, Thursday, 10:00 - 11:00
I will speak about the recent progress on the Erdos-Ko-Rado type results for different structures that we have managed to obtain using the method of spread approximations. I’ll mostly focus on partitions and, in particular, the resolution of the Meagher-Moura question on partially t-intersecting families of partitions.
Speaker: Zixiang Xu (Extremal Combinatorics and Probability Group, Institute for Basic Science)
Room: College of Science M1001
Time: 2025/04/03, Thursday, 10:00 - 11:00
The Vapnik-Chervonenkis (VC) dimension is a fundamental concept in learning theory that has found increasing applications in extremal combinatorics. Over the years, significant progress has been made in understanding the interplay between VC-dimension theory and extremal combinatorics. A cornerstone result from the 1970s, known as the Sauer-Shelah Lemma, precisely characterizes the maximum size of a non-uniform set system with bounded VC-dimension. However, the analogous problem for uniform set systems remains a major open question.
In this talk, I will present recent advances on this problem, including improved bounds and new structural insights. This is based on joint work with Ting-Wei Chao, Gennian Ge, Chi Hoi Yip, Shengtong Zhang, and Xiaochen Zhao.
Speaker: Zilong Wang (Xidian University)
Room: College of Science M1001
Time: 2025/03/27, Thursday, 11:00 - 12:00
With the development of MIMO communication technology, the sequences for channel estimation assigned to the same user are required to be orthogonal, while the magnitude of the inner product of sequences assigned to different users is expected to be as small as possible. Suppose that there are K users and each user has M mutually orthogonal normalized sequences of length L. In this paper, a lower bound of the maximum magnitude of the inner product between sequences assigned to the different users is given. It is also proved that the magnitude of the inner product between any pair of sequences for different users is equal, when the lower bound is met. Such sequence sets meeting the lower bound are referred to as mutually unbiased sets of orthogonal vectors (MUSOVs), and two infinite class of MUSOVs is constructed by combinatorial methods.
Speaker: Li Liu (Qufu Normal University)
Room: College of Science M1001
Time: 2025/03/27, Thursday, 10:00 - 11:00
The Eulerian polynomials and derangement polynomials arise often in combinatorics,
algebra and geometry. It is well known that the Eulerian polynomials and derangement polynomials form Sturm sequences, respectively. In this paper, we give new sufficient conditions for the interlacing property of recurrence sequences of polynomials. As applications, we show some interesting interlacing sequences among the Eulerian polynomials, the derangement polynomials and those generalized polynomials for colored permutations.
Speaker: Ferdinand Ihringer (SUSTech)
Room: College of Science M1001
Time: 2025/03/20, Thursday, 10:00 - 11:00
In 2017 Gil Kalai asked the speaker about the largest family of $t$-intersecting spanning trees of the complete graph on n vertices. We will
discuss these questions: What motivates this problem? Which common techniques for EKR-problems fail? How can we actually solve the problem?
Based on joint work with Peter Frankl, Glenn Hurlbert, Andrey Kupavskii, Nathan Lindzey, Karen Meagher, and Venkata Raghu Tej Pantangi.
Speaker: Guangzhou Chen (Henan Normal University)
Room: College of Science M1001
Time: 2025/03/13, Thursday, 10:00 - 11:00
For positive integers s, t, m and n, the Zarankiewicz number Zs,t(m, n) is usually defined to be the maximum number of edges in a bipartite graph with parts of sizes m and n that has no complete bipartite subgraph containing s vertices in the part of size m and t vertices in the part of size n. In this talk, we obtain some new exact Zarankiewicz numbers by using pairwise balanced designs.
Speaker: Jimeng Xiao (SUSTech)
Room: College of Science M4009
Time: 2025/03/06, Thursday, 10:00 - 11:00
Intersection problems are of great interest, and importance, in extremal combinatorics.
The Erdos-Ko-Rado theorem is one of the most important results in intersection problems.
In this talk, I will introduce Katona’s proof of Erdos-Ko-Rado theorem, which Erdos described as a ‘Book Proof’.
Then I will show some further applications of Katona cycle methods in forbidden subposet problems. This is based on joint work of Casey Tompkins and Gyula Katona.
Speaker: Ziqing Xiang (SUSTech)
Room: College of Science M1001
Time: 2025/02/27, Thursday, 10:00 - 11:00
Distance-regular graphs are essentially P-polynomial association schemes. For general graphs, although we cannot expect the existence of such nice structure, we can consinder its coherent configuration, which is a generalization of association scheme. In this talk, we show how the coherent configuration relates to the structure of the graph. In particular, we settle an open problem by van Dam and Haemers in 2003 by demonstrating the existence of a reasonable matrix, which corresponds to a particular element in the Bose-Mesner algebra, whose spectrum determines the structure of a random graph.
Speaker: Yibo Gao (Peking University)
Room: College of Science M1001
Time: 2025/02/20, Thursday, 10:00 - 11:00
Algebraic combinatorics is the research area that connects algebra to combinatorics. In this talk, we will provide a very gentle introduction to this area. In the first half, we give an example of studying algebraic structures using combinatorics. Specifically, Schubert polynomials, which are the polynomial representatives of the Schubert classes that form a basis of the cohomology ring of the flag variety, enjoys rich combinatorial properties. In the second half of the talk, we give an example of using algebra to solve combinatorial questions. Namely, we establish the Sperner property of the weak Bruhat order via sl2 representations.
Speaker: Shuxing Li (University of Delaware)
Room: College of Science M1001
Time: 2024/12/26, Thursday, 15:00 - 16:00
An (mn)-generalized bent function is a function from Zn
2 to Zm so that its
associated Fourier transformations have constant absolute value. It is known
that an (mn)-generalized bent function exists whenever one of the following
holds:
(1) both m and n are even.
(2) 4 m.
On the other hand, all known results suggest that for (mn) pair that fails
to satisfy both of the above conditions, (mn)-generalized bent function does
not exist. In this talk, we will discuss the recent nonexistence result of (m4)
generalized bent functions with m being odd. This result crucially relies on
analyzing vanishing sums of complex roots of unity.
This is joint work with Ka Hin Leung (National University of Singapore)
and Songtao Mao (Johns Hopkins University).
Speaker: Bo Zhou (South China Normal University)
Room: College of Science M1001
Time: 2024/12/26, Thursday, 14:00 - 15:00
Spectral graph theory deals with the study of the graph properties by using linear algebra tools and techniques via an associated marix for a graph. The most frequently used are the adjacency matrix, the Laplacian, the signless Laplacian, \and\ the distance matrix. We discuss some recent works on the \alpha-spectral radius, the Laplacian eigenvalue distribution and the second largest distance eigenvalue.
Speaker: Yuefeng Yang (China University of Geosciences)
Room: College of Science M4009
Time: 2024/12/19, Thursday, 14:00 - 15:00
As a directed version of distance-regular graphs, the concept of weakly distance regular digraphs was introduced in 2003. In this talk, I will introduce the development of this family of digraphs.
Speaker: Hongyi Huang (University of Bristol )
Room: College of Science M1001
Time: 2024/12/12, Thursday, 09:00 - 10:00
Let G be a permutation group on a finite set Ω. A base for G is a subset of Ω with trivial pointwise stabiliser, and the base size of G is the minimal size of a base forG. This classical invariant has been studied intensively since the early years of group theory in the 19th century, finding a wide range of applications.
In 2020, Burness and Giudici defined the Saxl graph of G with base size 2, whose vertices are the points of Ω, and where two vertices are adjacent if and only if they form a base for G. Later in the same year, I reported on my study on Saxl graphs at the Discrete Mathematics Seminar of SUSTech, which was my undergraduate thesis and eventually turned to a joint paper with Jiyong Chen.
Here at the same place, after my PhD thesis submission, I will review the main open problems and new development of this direction over the past four years, highlighting a recent generalisation with Saul Freedman, Melissa Lee and Kamilla Rekvényi.
Speaker: Yifan Jing (The Ohio State University)
Room: College of Science M1001
Time: 2024/12/10, Tuesday, 09:00 - 10:00
Confirming a conjecture by Breuillard and Green, we show that for every epsilon>0 there is a delta>0 such that if A is an open subset of SO(3,R) with measure at most delta, then the measure of A^2 is strictly greater than (4-epsilon) times the measure of A. This is joint work with Chieu-Minh Tran and Ruixiang Zhang.
Speaker: Chao Yang (Guangdong University of Foreign Studies)
Room: College of Science M1001
Time: 2024/12/05, Thursday, 09:00 - 10:00
Recently, Greenfeld and Tao disproved the conjecture that translational tilings of a single tile can always be periodic (Ann. Math. 200(2024), 301-363). In another paper (to appear in J. Eur. Math. Soc.), they also show that if the dimension $n$ is part of the input, the translational tiling for subsets of $Z^n$ with one tile is undecidable. These two results are solid evidence for the conjecture that translational tiling of $Z^n$ with a monotile is undecidable, for some fixed $n$. In general, we study the decidability or undecidability of the following problem: let $k$ and $n$ be fixed positive integers, and a tile is a finite subset of $Z^n$, given a set $S$ of $k$ tiles in $Z^n$, is there an algorithm to decide whether $Z^n$ can be tiled by translated copies of tiles in the set $S$? In this talk, we report some recent progress on the undecidability of this problem based on the joint work with Zhujun Zhang.
Speaker: Ryoh Fuji-Hara (University of Tsukuba)
Room: College of Science M4009
Time: 2024/11/29, Friday, 10:00 - 11:00
We are inspired by an application in deep learning called the DropConnect method, which sparses edges in a complete bipartite graph (connections) to avoid overfitting, and its application to experimental design to improve estimation accuracy. We propose a combinatorial problem called Spanning Bipartite Block Design (SBBD) and show how to construct designs that satisfy the combinatorial requirements of SBBD.
Speaker: Yandong Bai (Northwestern Polytechnical University )
Room: College of Science M1001
Time: 2024/11/28, Thursday, 09:00 - 10:00
Burr and Erdős conjectured in 1976 that for two integers $k>\ell\geqslant 0$ satisfying that $k\mathbb{Z}+\ell$ contains some even integer, an $n$-vertex graph containing no cycles of length $\ell$ mod $k$ can contain at most a linear number of edges on $n$. Bollobás confirmed this conjecture in 1977 and then Erdős proposed the problem of determining the exact value of the maximum number of edges in such a graph. For the above $k$ and $\ell$, define $c_{\ell,k}$ to be the least constant such that every $n$-vertex graph with at least $c_{\ell,k}\cdot n$ edges contains a cycle of length $\ell$ mod $k$. The precise (or asymptotic) values of $c_{\ell,k}$ are known for very few pairs $\ell$ and $k$. We precisely determine the maximum number of edges in a graph containing no cycles of length 1 mod 3. In particular, we show that every $n$-vertex graph with at least $\frac{5}{3}(n-1)$ edges contains a cycle of length 1 mod 3, unless $9\,\mid\,(n-1)$ and each block of the graph is a Petersen graph. As a corollary, we get that $c_{1,3}=\frac{5}{3}$. This is the last remaining class modulo $k$ for $1\leqslant k\leq 4$.
Speaker: Wei Wang (Xi'an Jiaotong University)
Room: College of Science M1001
Time: 2024/11/21, Thursday, 09:00 - 10:00
An invariant for cospectral graphs is a property shared by all graphs. In this
talk, we give three new invariants for cospectral graphs, characterized by
their arithmetic nature. Based on these, we are able to show that under certain
conditions, every graph cospectral with a graph $G$ is determined by its
generalized spectrum. This is a joint work with Yizhe Ji, Quanyu Tang and Hao
Zhang.
Speaker: Xin Zhang (Xidian University)
Room: College of Science M1001
Time: 2024/11/07, Thursday, 09:00 - 10:00
A cooperative coloring in a family G_1,G_2,…,G_m (not necessarily distinct) of graphs that all share the same vertex set V is defined as a process of selecting one independent set I_i from each graph G_i for every i∈ [m]: = {1,2,…,m}, in such a way that the union of all these independent sets covers the entire vertex set V. The notion of cooperative coloring was initially introduced by Aharoni et al in 2015 and has since garnered significant attention and extensive research. In this talk I will introduce the relationships among the cooperative coloring, the adapted coloring, the list coloring, and the independent transversal, and survey known results on this topic. In addition, I will present our new results on the cooperative colorings of hypergraphs (see arXiv:2408.03727). Joint work with Xuqing Bai, Bi Li, and Weichan Liu.
Speaker: Xujun Liu (Xi'an Jiaotong Liverpool University)
Room: College of Science M1001
Time: 2024/10/24, Thursday, 09:00 - 10:00
For a sequence S = (s1,…,sk) of non-decreasing positive integers, a
packing S-coloring of a graph G is a partition of its vertex set V (G) into
V1,…,Vk such that for every pair of distinct vertices u,v ∈ Vi the distance
between u and v is at least si +1, where 1 ≤ i ≤ k. The packing chromatic
number, χp(G), of a graph G is defined to be the smallest integer k such that
G has a packing (1,2,…,k)-coloring. Gastineau and Togni asked an open
question “Is it true that the 1-subdivision (D(G)) of any subcubic graph G
has packing chromatic number at most 5?” and later Breˇ sar, Klavˇ zar, Rall,
and Wash conjectured that it is true. Furthermore, Gastineau and Togni
also asked “Is it true that every subcubic graph except the Petersen graph
is packing (1,2,2,2,2,2)-colorable?”.
In this talk, we will discuss some recent developments on above-mentioned
problems
Speaker: Bocong Chen (South China University of Technology)
Room: College of Science M1001
Time: 2024/10/17, Thursday, 09:00 - 10:00
Let $\mathcal{C}$ be an arbitrary simple-root cyclic code and let $\mathcal{G}$ be the subgroup of ${\rm Aut}(\mathcal{C})$ (the automorphism group of $\mathcal{C}$) generated by the multiplier, the cyclic shift and the scalar multiplications.In this talk, an explicit formula, in some cases an upper bound, for the number of orbits of $\mathcal{G}$ on $\mathcal{C}\backslash {\bf 0}$ is established. An explicit upper bound on the number of non-zero weights of $\mathcal{C}$ is consequently derived and a necessary and sufficient condition for the code $\mathcal{C}$ meeting the bound is exhibited. Many examples are presented to show that our new upper bounds are tight and are strictly less than the upper bounds in [Chen and Zhang, IEEE-TIT, 2023]. In addition, for two special classes of cyclic codes, smaller upper bounds on the number of non-zero weights of such codes are obtained by replacing $\mathcal{G}$ with larger subgroups of the automorphism groups of these codes.
As a byproduct, our main results suggest a new way to find few-weight cyclic codes. This talk is based on joint works with Yuqing Fu, Hongwei Liu and Guanghui Zhang.
Speaker: Zhicong Lin (Shandong Unversity)
Room: College of Science M1001
Time: 2024/10/10, Thursday, 10:00 - 11:00
Gamma-positivity implies unimodality for palindromic polynomials. The idea behind it stems from work of Foata, Schutzenberger and Strehl on the Eulerian polynomials. I will present my work on this theme from the view point of permutation statistics, from q-Eulerian polynomials to Gessel’s gamma-positivity conjecture, to our recent exciting connections found between multiset Eulerian polynomials and weakly increasing trees, as well as some of my favorite conjectures.
Speaker: Akihiro Munemasa (Tohoku University)
Room: College of Science M1001
Time: 2024/09/19, Thursday, 10:00 - 11:00
Distance biregular graphs are a generalization of biparte distance regular graphs in the sense that the parameters with respect a vertex may depend on the bipartite half to which it belongs. In 1994, C. Delorme gave constructions of distance biregular graphs whose sets of vertices of a bipartite half is the set of points of the 3-dimensional affine space over a finite field. The other half can be described as the set of hyperplanes of the affine space, with some parallel classes deleted. Then the incidence graph of this incidence structure is a distance biregular graph. In this talk, we give a characterization of the set of parallel classes of hyperplanes to be deleted, in order for the resulting incidence graph is distance biregular (and further, distance regular). We also discuss possible generalization of this construction.
Speaker: Sabrina Lato (Umea University)
Room: College of Science M1001
Time: 2024/09/12, Thursday, 10:00 - 11:00
Recently, Evra, Feigon, Maurischat, and Parzanchevksi defined a notion of Cayley incidence graphs to explicitly construct bipartite biregular expanders. Other interesting bipartite biregular graphs can be constructed with considerably less work, as for instance the difference set construction of the Fano plane can be viewed as a Cayley incidence graph. Cayley graphs are a key concept in algebraic graph theory, and this new notion gives a similar structure to biregular bipartite graphs, which we use to explore some of the theory. Since a biregular bipartite graph can be viewed as a uniform regular hypergraph, these Cayley incidence graphs relate to the notions of Cayley hypergraphs that have been defined by several authors previously. This is joint work with Arnbjörg Soffía Árnadóttir, Alexey Gordeev, Tovohery Randrianarisoa, and Joannes Vermant.
Speaker: Jie Han (Beijing Institute of Technology)
Room: College of Science M1001
Time: 2024/05/30, Thursday, 10:00 - 11:00
There has been a growing interest in extending graph theory theorems into the random graph setting. We will survey this line of problems and mention some recent developments.
Speaker: Felix Lazebnik (University of Delaware)
Room: College of Science M1001
Time: 2024/05/23, Thursday, 10:00 - 11:00
In this talk I will describe the ideas behind 1) a new lower bound on Turan number of 14-cycle, due to A. Terlep and J. Williford;
2) a new proof for the lower bound on the girth of graphs CD(k, q), due to V. Taranchuk; 3) a construction of polynomial graphs with small automorphism group and its corollaries, due to V. Taranchuk and myself. At the end, several related open questions will be mentioned.
Speaker: Jiaxi Nie (Fudan University)
Room: College of Science M1001
Time: 2024/05/16, Thursday, 10:00 - 11:00
Let $\mathbb{F}$ be an $r$-uniform hypergraph. The random Tur'an number $\mathrm{ex}(G^r_{n,p},\mathbb{F})$ is the maximum number of edges in an $\mathbb{F}$-free subgraph of $G^r_{n,p}$, where $G^r_{n,p}$ is the Erd\H{o}s-R'enyi random $r$-graph with parameter $p$. Let $C^r_{\ell}$ denote the $r$-uniform linear cycle of length $\ell$. For $p\ge n^{-r+2+o(1)}$, Mubayi and Yepremyan showed that $\mathrm{ex}(G^r_{n,p},C^r_{2\ell})\le\max{p^{\frac{1}{2\ell-1}}n^{1+\frac{r-1}{2\ell-1}+o(1)},pn^{r-1+o(1)}}$. This upper bound is not tight when $p\le n^{-r+2+\frac{1}{2\ell-2}+o(1)}$. Recently, we close the gap for $r\ge 4$. More precisely, we show that $\mathrm{ex}(G^r_{n,p},C^r_{2\ell})=\Theta(pn^{r-1})$ when $p\ge n^{-r+2+\frac{1}{2\ell-1}+o(1)}$. Similar results have recently been obtained independently in a different way by Mubayi and Yepremyan. For $r=3$, we significantly improve Mubayi and Yepremyan’s upper bound.
Speaker: Meng Zhao (SUSTech)
Room: College of Science M1001
Time: 2024/05/09, Thursday, 10:00 - 11:00
Given a group $G$ and a positive integer $h$, we denote by $BH(G,h)$ the set of all $G$-invariant Butson-Hadamard matrices with entries being $h$-th roots. The group invariant Butson-Hadamard matrices are closely related to the generalized Hadamard matrices, the generalized bent functions, and also the cyclic $h$-th roots. In this talk, we focus on $BH(G,h)$ with $G=\mathbb{Z}_8$ and $\mathbb{Z}_9$, and give a complete classification of these matrices using vanishing sums of roots of unity and the field descent method.
Speaker: Chong Shangguan (Shandong University)
Room: College of Science M1001
Time: 2024/04/25, Thursday, 10:00 - 11:00
In this talk I will give a brief introduction to chromatic thresholds and homomorphism thresholds of graphs, and survey some recent progress. In particular, I will discuss how they are related to the theory of VC dimensions.
Speaker: Yongnan Ye (Wenzhou University)
Room: College of Science M1001
Time: 2024/04/18, Thursday, 10:00 - 11:00
In this talk, we present a survey on the study of David-Barton identities and Stirling permutations. By using an alternate formulation of the David-Barton identity which relates the alternating run polynomials to Eulerian polynomials, we find that for any gamma-positive polynomial, there exists a David-Barton type identity. We then discuss some enumerative polynomials on Stirling permutations, including ascent-plateau polynomials, left ascent-plateau polynomials and flag ascent-plateau polynomials.
Speaker: Tao Zhang (Xidian University )
Room: College of Science M1001
Time: 2024/04/11, Thursday, 10:00 - 11:00
The Fuglede conjecture establishes the relationship between spectral and tilings. Spectral is one of the core concepts in harmonic analysis. Since tilings are relatively easy to verify while spectral are more difficult, researchers are particularly interested in what kind of tiling sets are spectral sets (T-S for short). We mainly focus on the structure of tiling sets in finite abelian groups. Many researchers have considered tiling sets with special structures in finite abelian groups, such as good groups, groups with Redei property, and quasi-periodic groups. However, there are only a few good groups and groups with Redei property. Additionally, there are groups with quasi-periodic properties that do not have T-S, and there are groups with T-S that do not have quasi-periodic properties. Therefore, it is necessary to propose a more suitable property that is easy to verify, has a good structure, and satisfies T-S.
Speaker: Shishuo Fu (Chongqing University)
Room: College of Science M1001
Time: 2024/03/28, Thursday, 10:00 - 11:00
The study of sequences in partitions dates back at least to Sylvester. In this talk, we enumerate strict partitions with respect to the size, the number of parts, and the number of sequences of odd length. We write this generating function as a double sum $q$-series, which on one hand gives partition theoretical interpretation to the sum side of several identities obtained by Cao-Wang and Wang-Wang, on the other hand motivates us to look for further refinements of Euler’s partition theorem. A close relation with the $2$-measure of partitions will also be mentioned. This is a joint work with Haijun Li.
Speaker: Huajun Zhang (Shaoxing University)
Room: College of Science M4009
Time: 2024/03/21, Thursday, 10:00 - 11:00
Let $n$ and $k$ be two positive integers satisfying $n\geq 2k$. Let ${\mathcal A}$ be an intersecting family of $\binom{[n]}{k}$. The Erdős-Ko-Rado Theorem states that $|{\mathcal A}| \leq {n-1 \choose k - 1}.$ There are many generalizations of this theorem and many ways to study it have been generated as well. The generating set method, which has so far not received much attention, is a very efficient way of them. In this talk, we will introduces this method and give some of its applications.
Speaker: Yu Jin (Xiamen University)
Room: College of Science M1001
Time: 2024/03/14, Thursday, 11:00 - 12:00
Two skew diagrams are defined to be equivalent if their corresponding skew Schur functions are equal. The equivalence classes for ribbons have been classified by Billera, Thomas and van Willigenburg in 2006. In this paper, we provide a complete characterization of equivalence classes for connected skew diagrams with exactly one $2\times m$ or $m\times 2$ block of boxes for all $m\ge 2$. In particular, possible sizes of equivalence classes are one, two or four, confirming special cases of the elusive conjecture on equivalent skew connected diagrams proposed by McNamara and van Willigenburg in 2009. This is joint work with Shu Xiao Li (Shandong University).
Speaker: Haihua Deng (SUSTech)
Room: College of Science M1001
Time: 2024/03/14, Thursday, 10:00 - 11:00
Let $\Gamma$ be a simple connected graph on $n$ vertices and $C$ a code of
length $n$ whose coordinates are indexed by the vertices of $\Gamma$. We call
$C$ a $\text{\textit{storage code}}$ on $\Gamma$ if, for any codeword $c\in C$,
one can recover the information at each coordinate of $c$ by accessing its
neighbors in $\Gamma$. In 2022, A. Barg and G. Z'emor asked whether the rates
of storage codes on triangle-free graphs can be arbitrarily close to 1 and list
some candidates. Among them, we will discuss the BCH family and show that it is
of unit rate by using the polynomial method. Furthermore, we can generalize
this construction and obtain more storage codes of unit rate on triangle-free
graphs. At last, we will talk about a connection between the storage codes on
triangle-free graphs and the Ramsey number $R(3,t)$, which leads to an upper
bound for the rate of convergence of $1/(1-R(C_n))$. This is a joint work with
Hexiang Huang, Guobiao Weng and Qing Xiang.
