It is known that there are factors of IID for the free Ising model on the d-regular tree when it has a unique Gibbs measure and not when reconstruction holds (when it is not extremal). We construct a factor of IID for the free Ising model on the d-regular tree in (part of) its intermediate regime, where there is non-uniqueness but still extremality. The construction is via the limit of a system of stochastic differential equations.
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This is a joint work with Danny Nam and Allan Sly.

Multi-agent systems are ubiquitous in science, from the modeling of particles in Physics to prey-predator in Biology, to opinion dynamics in economics and social sciences, where the interaction law between agents yields a rich variety of collective dynamics. We consider the following inference problem for a system of interacting particles or agents: given only observed trajectories of the agents in the system, can we learn what the laws of interactions are? We would like to do this without assuming any particular form for the interaction laws, i.e. they might be ``any" function of pairwise distances.
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In this talk, we consider this problem in the case of a finite number of agents, with an increasing number of observations. We cast this as an inverse problem, and study it in the case where the interaction is governed by an (unknown) function of pairwise distances. We discuss when this problem is well-posed, and we construct estimators for the interaction kernels with provably good statistical and computational properties. We measure their performance on various examples, that include extensions to agent systems with different types of agents, second-order systems, and stochastic systems. We also conduct numerical experiments to test the large time behavior of these systems, especially in the cases where they exhibit emergent behavior.
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This talk is based on the joint work with Fei Lu, Mauro Maggioni, Jason Miller, and Ming Zhong.

Let $X$ be a compact metric space, and let $h$ be a homeomorphism
of $X$. The mean dimension of $h$ is an invariant invented by people
in topological dynamics, with no consideration of C*-algebras. The
shift on the product of copies of $[0, 1]^d$ indexed by $\mathbb{Z}$ has mean
dimension $d$.
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The radius of comparison of a C*-algebra $A$ is an invariant
introduced with no consideration of dynamics, and originally applied
to C*-algebras which are not given as crossed products. It is a
numerical measure of bad behavior in the Cuntz semigroup of $A$, and
its original use was to distinguish counterexamples to the original
formulation of the Elliott conjecture.
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It is conjectured that if $h$ is a minimal homeomorphism of a compact
metric space, then the radius of comparison of $C* (Z, X, h)$ is equal
to half the mean dimension of $h$. There is a generalization to
countable amenable groups. Considerable progress has been made on
proving that the radius of comparison of $C* (Z, X, h)$ is at most
half the mean dimension; in particular, this is known in full
generality for minimal homeomorphisms. We give the first systematic
results for the opposite inequality. We do not get the exact expected
lower bound, but, for many known examples of actions of amenable
groups with large mean dimension, we come close.
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The methods depend on ``mean cohomological independence dimension,"
Cech cohomology, and the Chern character.
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This is joint work with Ilan Hirshberg.

The classical Gauss's Theorema Egregium and the Uniformization theorem for the Riemann surfaces are illustrations of a prominent theme in geometry -- control of the global topology/geometry of a manifold through the bounds on its curvature. In the last decades, with the development of new analytic tools (Yamabe equation, mean-curvature flow etc), this idea has found numerous applications in classification problems. Application of geometric flows (specifically the Ricci flow) turned out to be particularly fruitful in the context of Kaehler (and projective algebraic) geometry. At the same time there are very few efficient analytic methods available in non-Kahler complex geometry. In this talk we will introduce the Hermitian Curvature Flow on an arbitrary compact complex manifold. We will prove a delicate version of the maximum principle for tensors along this flow and present applications to the classification problems for the complex/algebraic manifolds admitting a compatible metric with ``semipositive curvature."

I will explain the a priori estimates for the cscK equation on a compact manifold, and how to use these estimates to obtain existence when the associated energy functional is ``coercive." If time permits, I will also explain how we can hope to get existence from a more ``algebraic" condition, which might be easier to check in practice.

In a matrix completion problem, one has access to a subset of entries of a matrix and wishes to determine the missing entries subject to some constraint (e.g. a rank bound). Such problems appear in computer vision, collaborative filtering, and many other applications. In this talk, I will discuss how certain notions from nonlinear algebra can be used to understand the structure underlying various types of matrix completion problems.

Let X be a compact, smooth manifold and Diff(X) the diffeomorphism group. The topology of Diff(X) and of the classifying space BDiff(X) are of great interest. For instance, the k-th homotopy group of BDiff(X) corresponds to smooth families over the k-sphere with fibres diffeomorphic to X. By a recent result of Bustamante, Krannich and Kupers, if X has even dimension not equal to 4 and finite fundamental group, then the homotopy groups of BDiff(X) are all finitely generated. In contrast we will show that when X is a K3 surface, the second homotopy group of BDiff(X) contains a free abelian group of countably infinite rank as a direct summand. Our families are constructed using the moduli space of Einstein metrics on K3. Their non-triviality is detected using families Seiberg--Witten invariants.

Unitary Shimura varieties are moduli spaces of abelian
varieties with an action of a quadratic imaginary field, and extra
structure. In this talk, we'll discuss specific examples of unitary
Shimura varieties whose supersingular loci can be concretely described
in terms of Deligne-Lusztig varieties. By Rapoport-Zink uniformization,
much of the structure of these supersingular loci can be understood by
studying an associated moduli space of p-divisible groups (a
Rapoport-Zink space). We'll discuss the geometric structure of these
associated Rapoport-Zink spaces as well as some techniques for studying
them.

Service platforms must determine rules for matching heterogeneous demand (customers) and supply (workers) that arrive randomly over time and may be lost if forced to wait too long for a match. We show how to balance the trade-off between making a less good match quickly and waiting for a better match, at the risk of losing impatient customers and/or workers. When the objective is to maximize the cumulative value of matches over a finite-time horizon, we propose discrete-review matching policies, both for the case in which the platform has access to arrival rate parameter information and the case in which the platform does not. We show that both the blind and nonblind policies are asymptotically optimal in a high-volume setting. However, the blind policy requires frequent re-solving of a linear program. For that reason, we also investigate a blind policy that makes decisions in a greedy manner, and we are able to establish an asymptotic lower bound for the greedy, blind policy that depends on the matching values and is always higher than half of the value of an optimal policy. Next, we develop a fluid model that approximates the evolution of the stochastic model and captures explicitly the nonlinear dependence between the amount of demand and supply waiting and the distribution of their patience times. We establish a fluid limit theorem and show that the fluid limit converges to its equilibrium. Based on the fluid analysis, we propose a policy for a more general objective that additionally penalizes queue build-up.

In this talk, I will explain a mixed characteristic
analogue of Frobenius regularity and how it can be used to establish the
Minimal Model Program for threefolds in mixed characteristic.
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This is based on a joint work with Bhargav Bhatt, Linquan Ma, Zsolt Patakfalvi,
Karl Schwede, Kevin Tucker, and Joe Waldron.