Organizers : Jeff Kahn, Bhargav Narayanan, Swee Hong Chan, and Caleb Fong
Next Talk:
Date: October 19, 2026 at 2:00pm
Speaker: Huy Tuan Pham (UChicago)
Title: Random graphs: The extreme tail
Abstract: Since their inception, random graphs have been a central topic in probability and combinatorics, offering a remarkably rich landscape for studying how complex structures emerge from randomness. We will discuss the following fundamental questions: What is the probability that the random graph fails to contain certain structures? For instance, what is the probability that the random graph is triangle-free?
Studying different regimes of such containment probability leads to the heart of different subjects with rich understanding and fundamental phenomena. While the study of thresholds occupies the typical regime of the containment probability, understanding precise asymptotics in the extreme tail is a central topic in the study of large deviations. I will discuss recent progress in this regime in joint work with Matthew Kwan, and describe connections with other developments in modern combinatorics and probability.
Date: October 5, 2026 at 2:00pm
Speaker: Ahmed Bou-Rabee (UPenn)
Title: Eulerian walkers on the square lattice
Abstract : In the Eulerian walker model, also called rotor walk, every site of the square lattice starts with an arrow pointing at one of its four neighbors. A walker repeatedly turns the arrow at its current site clockwise and steps in the new direction. Priezzhev, Dhar, Dhar, and Krishnamurthy introduced this walk as a model of self-organized criticality and conjectured that, for independent uniform initial arrows, the region explored in the first steps has radius of order t^(2/3). I will explain the recent proof of this conjecture.
Date: September 28, 2026 at 2:00pm
Speaker: Tom Waknine (Technion)
Title: Scale-sensitive combinatorial dimension and uniform laws of large numbers.
Abstract : We study scale-sensitive combinatorial dimensions for classes of bounded real-valued functions, focusing on their relationship with uniform laws of large numbers and empirical covering numbers.
Our main result gives a sharp scale-sensitive characterization of uniform laws of large numbers for real-valued function classes, improving the classical result of Bartlett and Long (1998) by removing a factor-of-two loss and achieving an equivalence at the optimal scale. The key technical ingredient is a direct bound on empirical covering numbers in terms of scale-sensitive combinatorial dimension. This yields sharp asymptotic metric-entropy bounds and resolves questions of Alon–Ben-David–Cesa-Bianchi–Haussler (1997) and Rudelson–Vershynin (2006).
The talk will present the combinatorial structure behind these results, including the role of partial concept classes, disambiguation, and scale-sensitive analogues of Sauer–Shelah type arguments.
Date: September 21, 2026 at 2:00pm
Speaker: Mayank Pandey (Courant)
Title: Large value estimates for exponential sums Some applications of nilsequences to point counting.
Abstract: We discuss some recent work with Alex de Faveri on large value estimates for exponential sums. We breach the mean value theorem barrier for certain special phases.
Date: September 14, 2026 at 2:00pm
Speaker: Imre Leader (Cambridge)
Title: The Maze Problem
Abstract: Does there exist a universal sequence for all mazes on the two-dimensional integer lattice? This fundamental problem remains unsolved, despite several interesting results relating the question to random walks and other fundamental phenomena. We will discuss the background on this problem, as well as presenting some recent results, which are joint work with Mariaclara Ragosta.