Date/Time : 10 Sep 2026 14:00
Venue : S16 #03-06
Speaker : Thomas Karam, Shanghai Jiao Tong University

Can we determine in a reasonably “simple” way all inequalities between notions of complexity on an object that are always true regardless of the specific choice of the object?

In some contexts, practically sufficient versions of that dream have already been meaningfully realised. In information theory, for instance, one can characterise the linear combinations of joint Shannon entropies in finite sets of variables that hold for all choices of the variables. Although not all entropy inequalities fall into that class, it is in practice sufficient for most applications.

We will focus on current progress for two notions of complexity: covering numbers of finite sets of points, and ranks of tensors, where we will essentially restrict our attention to rank notions arising from partitioning the coordinates. In the latter setting, we will particularly emphasise how the majority of rank inequalities studied so far can be classified into six types of inequalities.

Afterwards, we will delve more closely in each of these categories of inequalities, first summarising known results, then describing progress from the speaker, and finally outlining the remaining bottlenecks.

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Date/Time : 08 Sep 2026 15:00
Venue : S17 #04-05 (Seminar Room 2)
Speaker : Thomas Karam, Shanghai Jiao Tong University

It is a familiar fact that the size of a subset $A$ of $\mathbb{R}^n$ is bounded above by the product of the $n$ sizes of its one-dimensional axis-parallel projections: after all, it suffices to contain $A$ in the cartesian box that these projections enclose. Shearer’s lemma in its geometric form generalises this fact by providing sharp upper bounds on the size of $A$ in terms of the product of the sizes of its $k$-dimensional projections, for any given $1 \le k \le n$.

Yet not all mathematical notions of complexity are formulated in terms of sizes or measures. In particular many definitions involve decomposition length: given a class of objects viewed as basic building blocks, the complexity of an object is defined as the smallest number of these blocks in terms of which the object may be contained or decomposed. Compared to sizes or measures, understanding these complexity notions tends to be less straightforward, the fundamental difficulty being the general lack of uniqueness of the containments of decompositions.

Notions formulated in this other way include in particular (i) the smallest number of $d$-dimensional subspaces that suffice to contain a finite subset $A$ of $\mathbb{R}^n$ for a fixed $1 \le d \le n$, and (ii) the various rank notions on tensors defined in terms of partitions of the coordinates. In both settings we will establish new inequalities bounding the complexity of the multidimensional object in terms of the complexity of its lower-order projections, discuss the contrast with Shearer’s lemma, before concluding with an overview of the remaining difficulties standing before a comprehensive description of which such inequalities are true.

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Date/Time : 22 Apr 2026 17:00
Venue : S17 #04-04 (Seminar Room 3)
Speaker : Phan Quoc Vuong,

Blow-up of a graph is a useful trick in extremal graph theory, replacing every vertex with a finite collection of copies so that the copies of two vertices are adjacent if and only if the originals are. In this talk, I will explain how to extend the blow-up trick to general combinatorial objects, and discuss about its potential application, illustrated by a recent result related to hypergraph Turan problem.

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Date/Time : 22 Apr 2026 16:00
Venue : S17 #04-04 (Seminar Room 3)
Speaker : Swee Hong Chan, Rutgers University

A sequence of nonnegative real numbers a_1, a_2,… , a_n, is log-concave if a_i^2 >= a_{i-1}a_{i+1} for all i ranging from 2 to n-1. Examples of log-concave inequalities range from inequalities that are readily provable, such as the binomial coefficients a_i = \binom{n}{i}, to intricate inequalities that have taken decades to resolve, such as the number of independent sets a_i in a matroid M with i elements (otherwise known as the first Mason’s conjecture; and was resolved by June Huh in 2010 in a remarkable breakthrough). It is then natural to ask if it can be shown that the latter type of inequalities is intrinsically more challenging than the former. In this talk, we provide a rigorous framework to answer this type of questions, by employing a combination of combinatorics, complexity theory, and geometry.

This is a joint work with Igor Pak.

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Date/Time : 08 Apr 2026 15:00
Venue : S17 #04-04 (Seminar Room 3)
Speaker : Rao Rui, National University of Singapore

Consider the family of all k-element subsets of {1,…,n}. A subfamily F is called intersecting if any two of its sets have non empty intersection. Huang and Zhao proved that when n>2k, every intersecting family F contains an element i in [n] that belongs to at most {n-2 \choose k-2} members of F. This bound is tight, being attained by 1-star: all k-sets containing a fixed element.

We show that, for n sufficiently large (depending on k), all but at most k+1 elements of [n] have the same property: each of those elements appears in at most {n-2 \choose k-2} sets of F. The number k+1 is tight because of the Hilton-Milner construction of non trivial intersecting family.

This is joint work with Hao Huang.

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Date/Time : 08 Apr 2026 16:00
Venue : S17 #04-04 (Seminar Room 3)
Speaker : Ho Boon Suan, National University of Singapore

Given a bipartite graph G with parts of size m and n, Ehrenborg conjectured in 2006 that the number of spanning trees in G is at most the product of all vertex degrees, divided by mn. We will discuss a linear-algebraic proof of this bound, including a characterization of when equality holds.

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Date/Time : 18 Mar 2026 16:00
Venue : S17 #05-11 (Seminar Room 5)
Speaker : Fan Chang, Nankai University

In this talk, I will present a short and elementary proof of the Poincaré inequality for increasing subsets of the hypercube, recently proved by Fei and Ferreira Pinto Jr. This result implies an $O(n^2)$ upper bound on the mixing time of censored random walks, improving on the earlier $O(n^3)$ bound of Ding and Mossel. This is joint work with Guowei Sun and Lei Yu.

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Date/Time : 11 Feb 2026 16:00
Venue : S17 #05-12 (Seminar Room 4)
Speaker : Jing Yifan, Ohio States University

We show that for all integers $2\le s\le t$, any $K_{s,t}$-free subset of $[N]$ with size $\Omega(n^{1-1/s})$ must contain a nontrivial solution to every fixed translation-invariant linear equation in at least five variables. This extends earlier results for Sidon sets due to Conlon-Fox-Sudakov-Zhao and Prendiville to the full family of $K_{s,t}$-free sets.

We also study the corresponding problem in vector spaces over finite fields. In $\mathbb F_q^n$ we obtain stronger quantitative bounds, including polylogarithmic savings, by combining Fourier-analytic transference with polynomial-method input from the arithmetic cycle-removal lemma of Fox-Lov\’asz-Sauermann. Joint work with Cosmin Pohoata and Max Wenqiang Xu.

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Date/Time : 06 Feb 2026 16:00
Venue : S17 #04-06 (Seminar Room 1)
Speaker : Fang Yijia, National University of Singapore

Euclidean Ramsey theory asks if we can find specific geometric structures in colored space. A refined version of this, known as the canonical Ramsey property, asks: Does every finite coloring of Euclidean space contain a congruent copy of a shape that is either monochromatic or rainbow (all in different colors)?

While this was known for specific shapes, such as acute and right triangles, the question of whether all triangles satisfy this property remained open. In this talk, I will present a solution to this problem posed by Gehér, Sagdeev, and Tóth. We prove that 4-dimensional space is sufficient to guarantee the canonical property for any triangle. The proof uses rotation arguments and a particular configuration that force monochromatic structures.

(Joint work with Gennian Ge, Yang Shu, Qian Xu, Zixiang Xu, and Dilong Yang.)

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