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 ≤ 1/2) or a unique infinite cluster (when p > 1/2) almost surely.
For G = Zd, 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 R4: Let (p12, p13, p14, p23, p24, p34) be the areas of the six coordinate projections of a convex body in R4 to R2. Which six numbers arise in this way? The second question concerns algebraic surfaces in (P1)4: Let (p12, p13, p14, p23, p24, p34) be the degrees of the six coordinate projections from an irreducible surface in (P1)4 to (P1)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 T2. 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)
[Workshop Lecture 1] Current questions in 3-manifolds
The study of 3-manifolds and their geometric structures has been a tremendous success over several decades. Some interesting structural questions remain: how, quantitatively, do we compare the hyperbolic structure of a 3-manifold to its topological description? What uniform control do we have over deformation spaces of hyperbolic structures? What is the interaction between topological structures such as foliations and flows with geometric structures? What lessons can we learn from 3-manifolds that apply to other settings, in other dimensions and ranks? I will give a biased overview of these topics.
[Workshop Lecture 2] Uniformity in deformation theory of Kleinian groups
In this second talk I will focus on the question of uniform models for hyperbolic 3-manifolds, and more specifically on the structure of Thurston's skinning map, which is a basic tool for describing the gluing process that constructs and controls hyperbolic structures. Some of this corresponds to joint work with Bromberg, Canary, and Kent.
Jinyoung Park (NYU Courant)
[Workshop Lecture 1] Thresholds
We will walk through some basics of the random graph theory, aiming to understand a high-level motivation for the "second" Kahn-Kalai Conjecture. The second Kahn-Kalai Conjecture specifically concerns "thresholds" for subgraph containment problems for Erdos-Renyi random graphs, which has been a central interest in the area of probabilistic combinatorics.
[Workshop Lecture 2] p-smallness
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 notion of increasing families generalizes lots of interesting properties. Talagrand introduced the notion of "p-smallness" as an explicit certificate to show the p-biased product measure of a given increasing family F is small. We will introduce this notion and various problems related to it.
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.