Date/Time : 06 Sep 2023 13:00
Venue : S17 #05-11 (SR5)
Speaker : Yunkun Zhou, Stanford University

Degeneracy plays an important role in understanding Turán- and Ramsey-type properties of graphs. Unfortunately, the usual hypergraphical generalization of degeneracy fails to capture these properties. We define the skeletal degeneracy of a k-uniform hypergraph as the degeneracy of its 1-skeleton (i.e., the graph formed by replacing every k-edge by a k-clique). We prove that skeletal degeneracy controls hypergraph Turán and Ramsey numbers in a similar manner to (graphical) degeneracy.

Specifically, we show that k-uniform hypergraphs with bounded skeletal degeneracy have linear Ramsey number. This is the hypergraph analogue of the Burr-Erdős conjecture (proved by Lee). In addition, we give upper and lower bounds of the same shape for the Turán number of a k-uniform k-partite hypergraph in terms of its skeletal degeneracy. The proofs of both results use the technique of dependent random choice. In addition, the proof of our Ramsey result uses the `random greedy process’ introduced by Lee in his resolution of the Burr-Erdős conjecture.

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Date/Time : 30 Aug 2023 13:00
Venue : S17 #05-11 (SR5)
Speaker : Max Wenqiang Xu, Stanford University

Multiplicative energy of a set A is defined to be the total number of solutions to the equation: $m_1m_2=n_1n_2$ where the m_i’s and n_i’s are in the set A.

The first application is to use this quantity to study the product set of set A. Yunkun Zhou and I managed to solve some conjectures of Elekes-Ruzsa by studying this quantity.

The second application is to use this quantity to prove central limit theorems for a random model of arithmetic functions. The random model nowadays is called random multiplicative functions. This part is mainly based on joint work with K. Soundararajan.

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Date/Time : 23 Aug 2023 13:00
Venue : S17 #04-04 (SR3)
Speaker : Lim Jeck, California Institute of Technology

An old problem in discrete geometry, originating with Kupitz, asks whether there is a fixed natural number k such that every finite set of points in the plane has a line through at least two of its points where the number of points on either side of this line differ by at most k. We give a negative answer to a natural variant of this problem, showing that for every natural number k there exists a finite set of points in the plane together with a pseudoline arrangement such that each pseudoline contains at least two points and there is a pseudoline through any pair of points where the number of points on either side of each pseudoline differ by at least k.

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Date/Time : 16 Aug 2023 14:00
Venue : S17 #04-05 (SR2)
Speaker : Yifan Jing, University of Oxford

Let G be a connected compact group equipped with the normalised Haar measure μ. Our first result shows that given α, β > 0, there is a constant c = c(α, β) > 0 such that for any compact sets A, B ⊆ G with αμ(B)≥μ(A)≥μ(B) and μ(A)+μ(B)≤1−β, there exist b_1, …, b_c∈ B such that μ(A · {b_1,…,b_c}) ≥ μ(A) + μ(B). A special case of this, that is, when G is the d-dimensional torus confirms a recent conjecture of Bollobas, Leader and Tiba.

We also prove a quantitatively stronger version of such a result in the discrete setting of d -dimensional Euclidean space. Thus, given d ∈ N, we show that there exists c = c(d) > 0 such that for any finite, non-empty set A ⊆ R^d which is not contained in a translate of a hyperplane, one can find a_1,…,a_c∈ A satisfying
|A + {a_1, . . . , a_c}| ≥ (d + 1)|A| − O_d(1). The main term here is optimal and recovers the bounds given by Freiman’s lemma up to the O_d(1) error term.

(Joint with Akshat Mudgal)

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Date/Time : 12 Apr 2023 16:00
Venue : via Zoom
Speaker : Tran Chieu-Minh, National University of Singapore

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Meeting ID: 810 4646 2775
Passcode: 23571113

The following question was described in Ben Green’s list of open problems, but also attributed to discussions with Emmanuel Breuillard: When A \subseteq SO(3, R) has sufficiently small normalized Haar measure, do we have \mu(AA) > 3.99 \mu(A)?
This problem is often considered the first serious obstacle in generalizing results of additive combinatorics to nonabelian settings when we also want sharp bounds.

In this talk, I will discuss a recent joint work with Jing Yifan and Zhang Ruixiang where we combine techniques from model theory, probability theory, and harmonic analysis to answer it positively.

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Date/Time : 05 Apr 2023 16:00
Venue : via Zoom
Speaker : Noam Lifshitz, Hebrew University

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Meeting ID: 810 4646 2775
Passcode: 23571113

A subset A of a group is said to be product free if for all elements a, b in A their product ab is not in A. In 1985 Babai and Sos posed the problem of determining the largest product free sets in the alternating group. We solve this problem completely when n is sufficiently large.

Our proof combines representation theory with a recent machinery we developed called `hypercontractvity for global functions’.

Based on joint works with Keevash, Long, and Minzer.

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Date/Time : 29 Mar 2023 16:00
Venue : via Zoom
Speaker : Xizhi Liu, University of Warick

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Meeting ID: 810 4646 2775
Passcode: 23571113

We present a general approach for determining the maximum number of edges in an n-vertex r-uniform hypergraph that does not contain t+1 vertex-disjoint copies of F, when n is large, t = o(n), and F is nondegenerate. This extends old results of Simonovits and Moon on complete graphs.

Based on joint work with Jianfeng Hou, Heng Li, Long-Tu Yuan, and Yixiao Zhang.

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Date/Time : 22 Mar 2023 15:00
Venue : via Zoom
Speaker : Huy Tuan Pham, Stanford University

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Meeting ID: 810 4646 2775
Passcode: 23571113

A graph is Ramsey if its largest clique or independent set is of size logarithmic in the number of vertices. While almost all graphs are Ramsey, there is still no known explicit construction of Ramsey graphs. We discuss recent progress on finding these “dark matter” of graphs. Along the way, we study some fundamental problems in additive combinatorics, and discover that group structure is superfluous for these problems. Joint work with David Conlon, Jacob Fox, and Liana Yepremyan.

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Date/Time : 15 Mar 2023 10:00
Venue : via Zoom
Speaker : Cosmin Pohoata, Institute for Advanced Study

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Meeting ID: 810 4646 2775
Passcode: 23571113

Suppose we are given matchings M_1,….,M_N of size t in some r-uniform hypergraph, and let us think of each matching having a different color. How large does N need to be (in terms of t and r) such that we can always find a rainbow matching of size t? This problem was first introduced by Aharoni and Berger, and has since been studied by several different authors. While interesting for its own sake in the context of finding transversals in Latin Squares (e.g. the Ryser–Brualdi–Stein Conjecture), it is perhaps not too surprising that this question is also connected with different sides of additive combinatorics. In this talk, we will survey some of these connections, discuss some recent results that more or less completely answer the original question when one of the parameters t or r is fixed, and then end with some (fairly arbitrary) speculations/open problems. Joint work with Lisa Sauermann and Dmitrii Zakharov.

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