Date/Time : 18 Dec 2024 14:00
Venue : S17 #04-04 (SR3)
Speaker : Wenjie Fang, Université Gustave Eiffel

We consider the number of occurrences of subwords (non-consecutive sub-sequences) in a given word. We first introduce the notion of subword entropy that measures the maximal number of occurrences among all possible subwords. We then give upper and lower bounds of minimal subword entropy for words of fixed length in a fixed alphabet, and show that minimal subword entropy per letter has a limit value. A better upper bound of minimal subword entropy for a binary alphabet is then given by looking at certain families of periodic words. We also give some conjectures based on experimental observations.

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Date/Time : 05 Dec 2024 14:00
Venue : S17 #04-04 (SR3)
Speaker : Hong Liu, Institute for Basic Science, South Korea

A well-known theorem of Nikiforov asserts that any graph with a positive $K_r$-density contains a logarithmic blowup of $K_r$. We explore variants of Nikiforov’s result and investigate when positive clique density condition implies the existence of a significantly larger blowup of a clique. Our results study such problems for families of ordered graphs with forbidden induced monotone path, obtaining optimal bounds. As corollaries, we strengthen a result of Pach and Tomon, and resolve a conjecture of Tomon in a strong form. To find a large blowup, we reduce the embedding problem to a certain Ramsey problem. For optimal lower bound constructions, we make use of concentration of measure and the isodiametric inequality on high dimensional spheres.

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Date/Time : 13 Nov 2024 14:00
Venue : S17 #04-05 (SR2)
Speaker : Tuan Tran, University of Science and Technology of China

Consider a multipartite graph G with maximum degree at most n-o(n), parts V1,…,Vk of size n each, and every vertex has at most o(n) neighbors in any part Vi. Loh and Sudakov proved that any such G has an independent transversal. They further conjectured that the vertex set of G can be decomposed into pairwise disjoint independent transversals. We resolve this conjecture approximately by showing that G contains n-o(n) pairwise disjoint independent transversals. As applications, we give approximate answers to questions of Yuster, and of Fischer, Kühn, and Osthus.

Joint work with Debsoumya Chakraborti.

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Date/Time : 23 Oct 2024 14:00
Venue : S17 #04-04 (SR3)
Speaker : Wei Xin, University of Science and Technology of China

Separating hash families (SHFs) are useful combinatorial structures which generalize several well-studied objects in cryptography and coding theory, including perfect hash families, frameproof codes and parent-identifying codes. The classical upper and lower bounds for the extremum size of SHFs as a function of the alphabet size do not match. In 2008, Blackburn et al. conjectured that the upper bound has the correct exponent.Before our work, the only nontrivial known cases of this conjecture was by Shangguan and Ge in 2016, when the type sum of the SHF goes no larger than 4.

In this talk, I will discuss our recent proof of all the cases of Blackburn’s conjecture when the upper bound exponent is equal to 2. We do this by transforming the problem into finding a large common solution-free set of a large collection of related linear equations. The new idea is to use an algorithm to create a link from each equation to a class of equations with some good quality in finite many steps, while the good quality can be passed backwards through the link.

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Date/Time : 16 Oct 2024 14:00
Venue : S17 #04-04 (SR3)
Speaker : Zhang Xin, The University of Hong Kong

It is a discovery of Margulis in 1970s that congruence quotients of SL_2(Z) can be used to construct expanders, which are certain sparse but highly connected graphs and are ideal models for network building. The Super Approximation Conjec- ture of Salehi-Golsefidy and Varju gives a precise prediction on which more general subgroups of SL_d(Z) have this property.

In this talk, I will survey the history of this conjecture, and describe a recent progress by Tang Jincheng and myself that all Zariski dense subgroups of SL_2(Z) × SL_2(Z) or SL_2(Z) w Z_2 have this property. This progress relies on the development of a key tool in arithmetic combinatorics conjectured by Salehi-Golsefidy.

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Date/Time : 09 Oct 2024 14:00
Venue : S17 #04-05 (SR2)
Speaker : Ho Boon Suan, National University of Singapore

When traversing a graph, each subsequent node is typically chosen from the set of unvisited nodes adjacent to the already visited ones. In this talk, I will explore how to count the number of possible traversals, particularly in graphs with a lot of symmetry. We will see how these counting problems lead to interesting hypergeometric identities and how computational methods can help in establishing such identities. The talk is intended for a general mathematical audience.

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Date/Time : 02 Oct 2024 14:00
Venue : via Zoom
Speaker : Jing Yu, Fudan University

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https://nus-sg.zoom.us/j/81046462775?pwd=aDhaYnhQOTgybzdYS21yajlQdnBmZz09

Meeting ID: 810 4646 2775
Passcode: 23571113

In 1995, Levin and Linial, London, and Rabinovich conjectured that every connected graph $G$ of polynomial growth admits an injective homomorphism to the $n$-dimensional grid graph for some $n$. Moreover, they conjected that if every ball of radius $r$ in $G$ contains at most $O(r^\rho)$ vertices, then one can take $n = O(\rho)$. Krauthgamer and Lee confirmed the first part of this conjecture and refuted the second in 2007. By constructing some finite expander graphs, they showed best possible upper bound on $n$ is $O(\rho \log \rho)$.

Prompted by these results, Papasoglu asked whether a graph $G$ of polynomial growth admits a coarse embedding into a grid graph. We give an affirmative answer to this question. Moreover, it turns out that the dimension of the grid graph only needs to be linear in the asymptotic growth rate of $G$, which confirms the original Levin–Linial–London–Rabinovich conjecture “on the large scale.”

Besides, we find an alternative proof of the result of Papasoglu that graphs of polynomial growth rate $\rho < \infty$ have asymptotic dimension at most $\rho$.

Furthermore, our proof works in the Borel setting and shows that Borel graphs of polynomial growth rate $\rho < \infty$ have Borel asymptotic dimension at most $\rho$.

(This is joint work with Anton Bernshteyn.)

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Date/Time : 18 Sep 2024 12:00
Venue : via Zoom
Speaker : Peter van Hintum, Institute for Advanced Study

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https://nus-sg.zoom.us/j/81046462775?pwd=aDhaYnhQOTgybzdYS21yajlQdnBmZz09

Meeting ID: 810 4646 2775
Passcode: 23571113

The Brunn-Minkowski inequality is a foundational result in convex geometry asserting that if sets A and B in R^n have the same volume and t is some number in (0,1), then |tA+(1-t)B|>|A|. We’ll prove the following folklore conjecture on the stability of the Brunn-Minkowski inequality that if in fact |tA+(1-t)B|<(1+d) |A|, then there is a common convex hull K containing A and B so that |K|<(1+ O(sqrt(d)))|A| and |co(A)|<(1+O(d))|A|. We’ll additionally describe how this result neatly fits in the additive combinatorics study of sets with small sumset. (based on joint work with Alessio Figalli and Marius Tiba)

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Date/Time : 11 Sep 2024 14:00
Venue : S17 #04-05 (SR2)
Speaker : Tran Chieu-Minh, National University of Singapore

A locally compact group is nonunimodular if it has no Haar measure (i.e. every left Haar measure m is not right invariant). Considering doubling ratio m(AA)/m(A) initially looks strange on such exotic setting but turns out to be crucial even for the purpose of studying doubling on unimodular locally compact groups. In this talk I will discuss this issue further including a recently published result in this direction. ( Based on joint works with Yifan Jing and Ruixiang Zhang).

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