The radial mean body of parameter $p>-1$ of a convex body $K \subseteq \mathbb R^n$ is a radial set $R_p K$ that was introduced by Gardner and Zhang in 1998.

They proved that if $p \geq 0$, then $R_p K$ is convex, and conjectured that this holds also for $p \in (-1, 0)$.

We prove that if $K \subseteq \mathbb R^2$ is a convex body in the plane, then $R_p K$ is convex for every $p > (-1,0)$.

Although Vietoris–Rips (VR) complexes are frequently used in topological data analysis to approximate the “shape” of a dataset, their theoretical properties are not fully understood. In the case of the circle, these complexes show an interesting progression of homotopy types as the scale increases, moving from the circle S^1 to S^3 to S^5, and so on. However, much less is known about VR complexes of higher-dimensional spheres.

I will present work exploring the VR complexes of the n-sphere S^n and show how the appearance of nontrivial homotopy groups of these complexes can be controlled by the covering properties of S^n and real projective space RP^n. Specifically, if the first nontrivial homotopy group of a VR complex of S^n at scale π - t occurs in dimension k, then S^n can be covered by 2k + 2 balls of radius t, but there is no covering of RP^n by k balls of radius t/2. This is joint work with Henry Adams and Žiga Virk.

Modern complex systems often involve multiple interacting agents in a shared environment, e.g., transportation systems, power systems, swarm robotics, and human-robot interactions. Controlling these multi-agent systems (MASs) requires the characterization of agents’ interactions to account for their interdependent self-interests and coupled agents’ constraints such as collision avoidance and/or limited shared resources. To enable interaction awareness and human-like reasoning processes, game-theoretic control has been explored in the recent development of autonomous systems operating in multi-agent environments. However, fundamental challenges, including solution existence, algorithm convergence, scalability, and incomplete information, still remain to be addressed before the game-theoretic approaches could be sufficiently practical to be employed in a broad range of autonomous system applications. Possible solutions to addressing these challenges will be discussed in this talk, using autonomous driving as an application example.

We establish a multiversion of real and complex Gaussian hypercontractivity. More precisely, our result generalizes Nelson’s hypercontractivity in the real setting and the works of Beckner, Weissler, Janson, and Epperson in the complex setting to several functions. The proof relies on heat semigroup methods, where we construct an interpolation map that connects the inequality at the endpoints. As a consequence, we derive sharp multiversion of the Hausdorff-Young inequality and the log-Sobolev inequality. This is joint work with Paata Ivanisvili.

Given a valuation over a singularity, it is a fundamental problem whether it has finitely generated associated graded rings. This problem has deep connection with the theory of K-stability and moduli, where the finite generation of certain minimizing valuations were shown. For klt singularities, we propose the study of Kollár valuations, which are valuations with finitely generated associated graded rings that induces klt degenerations. We show that the locus of Kollár valuations is path connected. We discuss some open questions and examples. Based on joint work with Chenyang Xu.

One-dimensional persistent homology is arguably the most important and heavily used tool in topological data analysis. Additional information can be extracted from datasets by studying multi-dimensional persistence modules and by utilizing cohomological ideas. In this talk, we introduce a certain 2-dimensional persistence module structure associated with the persistent cohomology ring, where one parameter is the cup-length and the other is the filtration parameter. We show that this new persistence structure, called the persistent cup module, is stable.
In addition, we consider a generalized notion of persistent invariants, which extends the standard rank invariant, Puuska's rank invariant induced by epi-mono-preserving invariants of abelian categories, and the recently-defined persistent cup-length invariant, and we establish their stability. This generalized notion of persistent invariants also enables us to lift the LS-category of topological spaces to a novel stable persistent invariant, called the persistent LS-category invariant.

This is Part II of Benjamin’s talk.

We discuss two problems in stochastic geometry:

1. Sylvester's problem is a classical question that asks for the probability that d+2 random points in R^d are in convex position, that is, that none of them is in the convex hull of the others. We compute this probability when the points are the steps of a random walk. Remarkably, the probability depends only on dimension and is independent of the increment distribution of the walk, provided that it satisfies a mild nondegeneracy condition.

2. An exact formula for the mean volume of the convex hull of d-dimensional Brownian motion at a given time has been known since 2014. What can be said about the inverse volume process? In other words, how much time, on average, is required for the convex hull to attain a given volume? We establish two-sided bounds that capture the correct order of asymptotic growth in the dimension.

We discuss work on the epsilon multiplicity which shows that the epsilon multiplicity exists and is a limit of Amao multiplicities in the most general possible case. These are results from joint work with Cutkosky and more recent work from last month.