Inhyeok Choi (Cornell)
Many infinite clusters from percolation on many groups
Given a reasonably homogeneous graph G, what does the resulting graph look like after removing each edge independently with probability 1-p? When G is a planar grid, the resulting graph either has many finite clusters (when $p \le 1/2$) or a unique infinite cluster (when $p > 1/2$) almost surely.
For $G=\mathbb{Z}^{d}$, it seems difficult to find a value of $p$ resulting in infinitely many infinite clusters. It is indeed a theorem by Burton and Keane, later generalized by Gandolfi, Keane and Newman to Cayley graphs of amenable groups, that such $p$ does not exist. Benjamini and Schramm conjectured the converse: for every Cayley graph of a non-amenable group, does there exist $p$ for which the resulting graph has infinitely many infinite clusters? Hutchcroft answered this affirmatively for word-hyperbolic groups—that is, groups that are themselves hyperbolic.
In this talk, I will present an answer to the conjecture for relatively hyperbolic groups and mapping class groups of surfaces. The argument is based on a connection between Hutchcroft’s observation and ideas from geometric group theory. This is joint work in progress with Donggyun Seo.
June E Huh (Princeton)
Lorentzian polynomials, volume polynomials, and matroids over triangular hyperfields
[Workshop Lecture 1] In the first talk, I will focus on two concrete questions about projections of geometric objects in 4-dimensional spaces. The first question concerens convex bodies in R^4: Let (p_12, p_13, p_14, p_23, p_24, p_34) be the areas of the six coordinate projections of a convex body in R^4 to R^2. Which six numbers arise in this way? The second question concerns algebraic surfaces in (P^1)^4: Let (p_12, p_13, p_14, p_23, p_24, p_34) be the degrees of the six coordinate projections from an irreducible surface in (P^1)^4 to (P^1)^2. Which six numbers arise in this way? The answers to these questions are governed by the Plücker relations for the Grassmannian Gr(2,4) over the triangular hyperfield T_2. These results suggest a general conjecture regarding homology classes of irreducible surfaces in smooth projective varieties (based on joint work with Daoji Huang, Mateusz Michalek, Botong Wang, Shouda Wang).
[Workshop Lecture 2] I will give an overview of the intricate relationships among Grassmannians over hyperfields and the theory of Lorentzian polynomials. The main result is the identification of the space of Lorentzian polynomials with a given support and the corresponding matroid strata in the Grassmannain over the triangular hyperfield, up to homeomorphism (based on joint work with Matt Baker, Mario Kummer, Oliver Lorscheid).
[Workshop Lecture 3] I will focus on volume polynomials, a distinguished class of Lorentzian polynomials with remarkable analytic and combinatorial properties that arise from projective varieties. I will discuss their applications to algebraic matroids, introduce the new class of analytic matroids, and pose a number of questions (based on joint work with Lukas Grund, Mateusz Michalek, Henrik Süss, Botong Wang).
This series is intended for general mathematical audience.
Ji Oon Lee (KAIST)
Free energy of spherical spin glass models
In this talk, I will consider various spherical spin glass models with 2-spin interactions, including the spherical Sherrington-Kirkpatrick (SK) model, where the spin variables are uniformly distributed on a hypersphere. I will introduce a general strategy for the analysis of the free energy in spherical spin glass model based on recent results of random matrix theory, and explain several distinct phase transitions for the limit and the fluctuation of the free energy. I will also discuss the relation between the free energy of the spin glass models and the signal detection problem in statistical learning theory.
Yair Minsky (Yale)
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Jinyoung Park (NYU Courant)
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Youngtak Sohn (Brown)
Phase transitions of random constraint satisfaction problems
Consider a graph with n vertices and for each pair of vertices, connect an edge independently with probability d/n where the average degree d is fixed. This is the infamous Erdős–Rényi graph G(n,d/n). A natural question is: when is this graph k-colorable? That is, when can its vertices be colored using k colors such that no two adjacent vertices share the same color?
This colorability question belongs to the broader class of random constraint satisfaction problems (CSPs). Many random CSPs, including random graph coloring, are conjectured to exhibit a sharp phase transition: there exists a threshold d* where the probability that G(n,d/n) is k-colorable abruptly drops from near 1 to near 0 as d crosses d*. Inspired by insights from statistical physics, particularly spin glass theory, physicists have proposed a detailed conjectural phase diagram describing the intricate geometry of the solution space for these random CSPs. However, rigorous verification of these conjectures remains a significant mathematical challenge.
In this talk, I will review the conjectural phase diagram of random CSPs. Then, I will describe the recent progress in understanding the global and local geometry of solutions, focusing particularly on a boolean satisfiability model known as the random regular NAE-SAT model. This is joint work with Danny Nam and Allan Sly.
Michel Talagrand (CNRS)
[Colloquium] Mysteries in high dimension
We explore some facets of the counterintuitive high dimensional spaces.
[Workshop Lecture 1] Chaining, a long story
This gives a survey of the main results concerning upper and lower bounds for stochastic processes.
[Workshop Lecture 2] My favorite problem: creating convexity in a few steps
We explore a mysterious question which brings forwards our abysmal ignorance about basic questions.