Date/Time : 04 Oct 2023 13:00
Venue : via Zoom
Speaker : Shengtong Zhang, Stanford University

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Meeting ID: 810 4646 2775
Passcode: 23571113

In this talk, we start with Nikiforov’s spectral version of Turan’s theorem. This leads us to an intriguing conjecture of Bollobas and Nikiforov on the first two eigenvalues of graphs. Despite its simple appearance, the conjecture has resisted attacks for over 15 years. We survey several partial results on this conjecture, up to some recent progress by the speaker confirming the conjecture in the case of regular graphs. Based on https://arxiv.org/abs/2309.08184.

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Date/Time : 20 Sep 2023 15:00
Venue : S16 #04-35
Speaker : Goh Jun Le, National University of Singapore

We apply and reframe statements about graph decompositions, Latin squares, affine planes, etc. to statements about reductions between problems. In particular, we exhibit a statement (depending on n) which is equivalent to the existence of a finite affine plane of order n (open for many n). I expect this talk to be accessible to undergraduates.

Joint work with David Belanger and Damir Dzhafarov.

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Date/Time : 13 Sep 2023 14:00
Venue : S17 #05-11 (SR5)
Speaker : Jan Dobrowolski, University of Manchester

We study the relation R of collinearity of triples of points on a smooth cubic surface X. Our aim is to understand sequences of triples (A,B,C) of finite sets with |A|=|B|=|C| whose Cartesian products have asymptotically maximal possible size of intersection with the relation R (i.e. growing roughly as quickly as the square of the size of the sets A,B,C). This is an instance of the so-called Elekes-Szabó problem, which is closely related to the sum-product phenomenon. It is well-known that such configurations can be built on planar cubic curves, hence we can find them in an intersection of X with a plane. We prove this is essentially the only way in which such configurations on X arise, i.e. every such configuration on X concentrates on its intersection with a plane.

This is a joint work with Martin Bays and Tingxiang Zou.

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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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