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What can we learn about sphere packings by studying independent sets? What can we learn about independent sets from studying sphere packings? We will give an overview of various recent results on independents and sphere packings, each of which drew some inspiration from work on the other. Our focus will be on the interplay of ideas in these two settings. Time permitting, we will discuss results concerning the existence of large independent sets on locally sparse graphs; Markov chain mixing times for the hard core model and hard sphere model; and phase transitions (and lack thereof) for the hard sphere model and related models in Euclidean space and beyond.
Counting and sampling are fundamental algorithmic primitives in high-dimensional statistics and computer science. For many models, there is an enormous literature studying the worst-case computational complexity of these tasks and how they connect to phase transitions the underlying system undergoes as its parameters (e.g., ""temperature"") are varied. Their average-case complexity is comparatively far less understood.
We study the problem of estimating the partition function of Ising models with random interactions. These are fundamental probability distributions originating in statistical physics that form a useful sandbox for new algorithms and mathematical techniques. We give the first QPTAS for the Sherrington-Kirkpatrick model and the first FPTAS for the antiferromagnetic Ising model on the uniformly random d-regular graph, for all inverse temperatures up to their respective replica symmetry breaking thresholds; beyond these thresholds, it is believed that no such algorithms exist. To achieve these results, we study the locations of the zeros of the partition function, drawing inspiration from the seminal Lee-Yang program. More specifically, we establish the absence of Fishers zeros via a combination of Jensen's Formula, the second moment method, and small subgraph conditioning.
Joint work with Ferenc Bencs, Brice Huang, Daniel Z. Lee, and Guus Regts.
We present here the first part of Jorge Garza-Vargas’s three-part mini-course on the strong convergence phenomenon, from our ICM Satellite Conference on Spectral Theory, High-Dimensional Expansion, and Pseudorandomness. The course introduces and motivates the notion of strong convergence, discusses recent proof techniques, and showcases concrete applications.
High-dimensional expansion (HDX) is a generalization of expansion in graphs to higher dimensions (i.e., hypergraphs). In this first of three lectures from our ICM satellite conference, Mitali Bafna defines the spectral notion of HDX, proves the trickle-down theorem, and defines random walks on HDX.
In this presentation from this fall’s joint boot camp for the programs on Spectral Theory Beyond Graphs and on Pseudorandomness & High-Dimensional Expansion, William Hoza presents an introduction to the “L vs. BPL” problem, which asks whether randomness is ever necessary for space-efficient computation.