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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Date/Time : 04 Sep 2024 14:00
Venue : S17 #04-05 (SR2)
Speaker : Peng Fei, National University of Singapore

A quasi-kernel of a digraph is an independent set Q such that every vertex can reach Q in at most two steps. The 48-year-old small QK conjecture says that every sink-free digraph contains a quasi-kernel of size at most n/2. Spiro posed the large QK conjecture, which states that every digraph contains a quasi-kernel Q such that |N^-[Q]| ≥ n/2, and showed that it follows from the small QK conjecture. We show that the converse is true with a weaker constant. We also show that every sink-free digraph contains a quasi-kernel of size at most (1−1/k)n, where k is the digraph’s kernel-perfect number, generalizing a result by Kostochka, Luo and Shan. Joint work with Ai and Liu.

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Date/Time : 28 Aug 2024 14:00
Venue : S17 #05-12 (SR4)
Speaker : Lim Jeck, California Institute of Technology

We show that if l_1, … , l_k are algebraic numbers, then, for any finite subset A of ℂ,

|A + l_1·A + … + l_k·A| >= H(l_1,…,l_k)|A|-o(|A|),

where H(l_1,…,l_k) is an explicit constant that is best possible. The case k=1 already confirms a conjecture of Krachun and Petrov. Joint work with David Conlon.

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Date/Time : 21 Aug 2024 14:00
Venue : S17 #05-12 (SR4)
Speaker : Yang Tianchi, National University of Singapore

The Turan-type problem is fundamental in extremal graph theory, guiding much of the field’s progress. In this talk, we extend these classic results to hypergraphs, focusing on bipartite structures like 4-cycles and complete bipartite graphs $K_{s,t}$. Our result highlights a rare instance in hypergraph Turan problems where the solution depends on the parity of the uniformity.

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Date/Time : 14 Aug 2024 14:00
Venue : S17 #04-04 (SR3)
Speaker : Jie Han, Beijing Institute of Technology

Random graphs are important objects in discrete probability theory. 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.

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Date/Time : 08 May 2024 14:00
Venue : S17 #04-05 (SR2)
Speaker : Xiaofan Yuan, Arizona State University

Resolving a recent problem of Bell, Frieze, and Marbach, we establish the threshold result of Frankston–Kahn–Narayanan–Park in the rainbow setting. This is joint work with Jie Han.

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Date/Time : 12 Apr 2024 16:00
Venue : S17 #04-04 (SR3)
Speaker : Yan Wang, Shanghai Jiao Tong University

A maximum matching in a graph can be found in polynomial time. However, Karp shows that it is NP-hard to find even for 3-uniform hypergraphs. Much effort has been devoted to finding good sufficient conditions for the existence of a large matching in uniform hypergraphs, including Dirac type conditions. In this talk, I will talk about some recent progress on hypergraph matching problems.

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Date/Time : 15 Nov 2023 10:00
Venue : via Zoom
Speaker : Alex Cohen, Massachusetts Institute of Technology

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

Meeting ID: 810 4646 2775
Passcode: 23571113

Heilbronn’s triangle problem asks, how small is the smallest triangle formed by a set of points? Suppose n points are placed in the unit square and Delta is the smallest area triangle formed by three of these points. It is not too hard to see that Delta < C n^{-1}. Komlos, Pintz, and Szemeredi proved in 1981 that Delta << C n^{-8/7} by building on an ingenious method of Roth. We improve the bound to Delta < C n^{-8/7-1/2000}. The improvement comes from establishing new connections between Heilbronn’s problem and projection theory.

All joint with Dmitrii Zakharov and Cosmin Pohoata.

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