Mathematics

It is well known that among all convex bodies in R^n with a given surface area, the Euclidean ball has the largest inradius. We will show that this result can be reversed in the class of convex bodies with curvature at each point of their boundary bounded below by some positive constant λ (λ-convex bodies). In particular, we show that among λ-convex bodies of a given surface area,  the λ-convex lens (the intersection of two balls of radius 1/ λ) minimizes the inradius.

Synchronization is crucial for the success of many data-intensive applications. This problem involves estimating global states from relative measurements between states. While many studies have explored synchronization in different contexts using pairwise measurements, relying solely on pairwise measurements often fails to capture the full complexity of the system. In this work, we focus on a specific instance of the synchronization problem within the context of structure from motion (SfM) in computer vision, where each state represents the orientation and location of a camera. We exploit the higher-order interactions encoded in trifocal tensors and introduce the block trifocal tensor. We carefully study the mathematical properties of the block trifocal tensors and use these theoretical insights to develop an effective synchronization framework based on tensor decomposition. Experimental comparisons with state-of-the-art global synchronization methods on real datasets demonstrate the potential of this algorithm for significantly improving location estimation accuracy. To our knowledge, this is the first global SfM synchronization algorithm that directly operates on higher-order measurements. This is joint work with Joe Kileel (UT Austin) and Gilad Lerman (UMN).

I shall present an extension of Webb's simplex slicing (1996) to the analytic framework of sharp L_p bounds for centred log-concave measures on the real line, with a curious phase transition of the extremising distribution. Based on joint work with Melbourne, Roysdon and Tang.
 

We study extremal properties of spherical random polytopes, the convex
hull of random points chosen from the unit Euclidean sphere in Rn. The extremal
properties of interest are the expected values of the maximum and minimum surface area among facets.
We determine the asymptotic growth in every fixed dimension, up to absolute constants.

Inequalities for Minkowski sums of convex bodies play a central role in convex geometry and additive combinatorics. A particular  example is the Plünnecke–Ruzsa-type inequality, which also has a geometric analogue for convex bodies. In this talk, we begin by introducing such geometric inequalities and highlighting some of the open problems related to them. We then present functional analogues of Plünnecke–Ruzsa-type inequalities using the alpha-sum, a generalized sup-convolution. Our focus is on integral inequalities for alpha-concave functions, a class that extends log-concave functions. We conclude by discussing sharp constants in the case of decreasing log-concave functions supported on the positive orthant.

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.

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.

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.

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.

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.