Jinyoung Park (Stanford University)
September 15, 2022 at MIT
Abstract: For a finite set X, a family F of subsets of X is said to be increasing if any set A that contains B in F is also in F. The p-biased product measure of F increases as p increases from 0 to 1, and often exhibits a drastic change around a specific value, which is called a “threshold.” Thresholds of increasing families have been of great historical interest and a central focus of the study of random discrete structures (e.g. random graphs and hypergraphs), with estimation of thresholds for specific properties the subject of some of the most challenging work in the area. In 2006, Jeff Kahn and Gil Kalai conjectured that a natural (and often easy to calculate) lower bound q(F) (which we refer to as the “expectation-threshold”) for the threshold is in fact never far from its actual value. A positive answer to this conjecture enables one to narrow down the location of thresholds for any increasing properties in a tiny window. In particular, this easily implies several previously very difficult results in probabilistic combinatorics such as thresholds for perfect hypergraph matchings (Johansson–Kahn–Vu) and bounded-degree spanning trees (Montgomery). In this talk, I will present recent progress on this topic. Based on joint work with Keith Frankston, Jeff Kahn, Bhargav Narayanan, and Huy Tuan Pham.
Eva Miranda (Polytechnic University of Catalonia)
September 22, 2022 at Northeastern
Title: From dynamical chaos to logical chaos and vice-versa: Explored and unexplored paths
Abstract: Chaos was coined by Edward Lorenz in 1961 with the simple statement “Chaos: When the present determines the future, but the approximate present does not approximately determine the future”. A different sort of chaos was discovered by Cris Moore in 1990 with a 2D Turing-type dynamical system via generalized shifts. The existence of a Turing machine associated with the dynamical system added a new intrigue to the plot: the undecidability of the halting problem (Alan Turing, 1936) yielded the impossibility of logical predictions in the new models. Those 2D systems given by mappings on the square Cantor set, however, are not realized by a physical system. In this talk, I will give 3D physical (and/or “almost” physical) constructions of logical chaos using fluids. Against all expectations, the main ingredient of this construction is geometrical. It relies on a mirror unveiled in 2000 by Etnyre and Ghrist between Beltrami fields and Reeb vector fields native to contact geometry. Many questions around such construction are pending including the connection among different levels of complexity (dynamical and logical) and the (in)existence of a hierarchy among them. I will end up my talk with some new challenges and open questions.
Junehyuk Jung (Brown University)
October 27, 2022 at Brandeis
Title: Ergodicity and the number of nodal domains of eigenfunctions of the Laplacian
Abstract: Quantum Chaos concerns the relationship between a classical Hamiltonian system, whose Hamiltonian flow is chaotic, and the corresponding quantized system. When the Hamiltonian flow is the geodesic flow on a compact manifold, the corresponding quantized system is understood in terms of the eigenfunctions of the Laplace-Beltrami operators. In this talk, I will discuss how the ergodicity of the geodesic flow can affect the geometry of the nodal set (the zero set) of the eigenfunctions. The main focus of the talk is the growth of the number of the nodal domains of the eigenfunctions, as the corresponding eigenvalues tend to infinity. This presentation is based on joint work with Seung Uk Jang, and Steve Zelditch.
The colloquium meets (by default) on Thursdays at 4:30 PM Eastern (contact institutional organizers for details). The organizers include Bong Lian at Brandeis; Fabian Gundlach, Myrto Mavraki, and Assaf Shani at Harvard; Yufei Zhao at MIT; and Matan Harel, Matthew Hogancamp, and Jonathan Weitsman at Northeastern. This website is maintained by Matan Harel. The image of Boston is the property of Wikimedia user King of Hearts and is reproduced here under Creative Commons license CC BY-SA 4.0. Images of speakers are their own property and are reproduced here with permission.