Consider a rooted d-regular tree with \ell layers, where each vertex is colored either blue or green. Starting from the root, the color propagates down the tree so that each child inherits its parent’s color but flips with probability 30%. Now, suppose you only observe the colors of the leaves—can you infer the color of the root?

This setting describes a broadcasting process on trees, where in general we have q possible 'colors' and a transition matrix specifying the probability that a child receives color a given that its parent has color b. The associated inference problem is known as the Tree Reconstruction Problem.

A classical result, the Kesten–Stigum bound, characterizes a sharp threshold: above the bound, simply counting the colors at the leaves provides enough information to make a reliable guess of the root color, whereas below it, counting reconstruction is impossible.

In our recent work, we identify the Kesten–Stigum bound as a threshold of computational complexity. Specifically, we show that while it may still be statistically possible to infer the root color below the bound, any algorithm achieving this must overcome a complexity barrier.

I will aim to make this talk accessible to a broad audience, beyond probability.

This is a joint work with Elchanan Mossel.

A weakening of Frobenius splitting, Quasi-F-Splittings have proven to be a vital invariant in the study of varieties in positive characteristic, with numerous applications to arithmetic and birational geometry. This weaker condition extends the application of Frobenius to study singularities of arithmetically supersingular varieties, encompassing a much broader class of examples.  In this talk I'll provide an overview of Quasi-F-Splittings and introduce a local analogue, Quasi-F-Purity. I will also discuss how quasi-F-pure hypersurfaces are "as close to being F-pure as possible" by computing the F-pure Threshold of an arbitrary quasi-F-pure hypersurface. This talk includes joint work with Jack J Garzella. 

This talk will focus on low-energy resolvent expansions for a wide class of scattering settings.  We concentrate on operators acting on \(\mathbb{R}^2\), with particular attention to the Dirichlet or Neumann Laplacian on the exterior of an obstacle and to Aharonov-Bohm operators.  We give applications to long-time behavior of the solutions of the wave equation and low-energy behavior of the scattering phase.

This is based on joint work with Kiril Datchev.  A portion is also with Mengxuan Yang and Pedro Morales.

Frames for a Hilbert space allow for a linear and stable reconstruction of a vector from linear measurements. In many real-world applications, sensors are set up such that any measurement above and below a certain threshold would be clipped as the signal gets saturated. We study the recovery of a vector from such measurements which is called declipping or saturation recovery. Phase retrieval is the problem of recovering a vector where only the intensity of each linear measurement of a signal is available and the phase information is lost. Using a frame theoretic approach to saturation recovery, we characterize when saturation recovery of all vectors in the unit ball is possible and then compare some of the known results in phase retrieval with saturation recovery. This is joint work with Wedad Alharbi, Daniel Freeman, Brody Johnson, and Nirina Randrianarivony.

An algebraic variety over a field k is said to be rational if its function field is a purely transcendental extension of k. In this talk, we work over the real numbers and study the rationality question for a class of conic bundle threefolds; the varieties we consider all become rational over the complex numbers, but this rationality construction does not in general descend to R. This talk is based on joint work with Sarah Frei, Soumya Sankar, Bianca Viray, and Isabel Vogt, and on joint work with Mattie Ji.

Finding general solutions to a mechanical system is often far too much to ask. Instead, we look for more tractable, special solutions, such as the equilibria. If the system is symmetric with respect to a group action then we can also look for the relative equilibria: these are solutions contained to a group orbit. Famous examples include the circular solutions of Euler and Lagrange in the 3-body problem. In this talk I will present a new formalism for finding relative equilibria by defining a 'web structure' on shape space, and demonstrate this by classifying the relative equilibria for the spherical 3-body problem.

We introduce a mixed characteristic analog of log canonical centers in characteristic \(0\) and centers of \(F\)-purity in positive characteristic, which we call centers of perfectoid purity. We show that their existence detects (the failure of) normality of the ring. We also show the existence of a special center of perfectoid purity that detects the perfectoid purity of \(R\), analogously to the splitting prime of Aberbach and Enescu, and investigate its behavior under étale morphisms.

The Euclidean distance geometry (EDG) problem concerns the reconstruction of point configurations in R^n from prior partial knowledge of pairwise inter-point distances, often accompanied by assumptions on the prior being noisy, incomplete or unlabeled. Instances of this problem appear across diverse domains, including dimensionality reduction techniques in machine learning, predicting molecular conformations in computational chemistry, and sensor network localization for acoustic vision. This talk provides a concise (inexhaustive) overview of the typical priors that arise in EDG problems. We will then turn to a specific instance motivated by Cryo-Electron Microscopy (cryo-EM), where recovering the 3-dimensional structure of proteins can be reformulated as an EDG problem with partially labeled distances. For this problem, we will outline a generic recovery result and present a recovery algorithm that is polynomial-time in fixed dimension. The algorithm achieves exact recovery when distances are noiseless and is robust to small levels of noise.