In this talk we will introduce the notion of a Kaplansky devissage and show its key properties. We will use these properties to prove that any projective module is some direct sum of countably generated projective submodules. We then define the direct sum property for modules and show that any countably generated module with the direct sum property is free. We finally show that any projective module over a local ring has the direct sum property letting us conclude that all projective modules over local rings are free.

Diffusion model is an emerging generative modeling technique, achieving the state-of-the-art performances in image and video synthesis, scientific simulation, inverse problems, and offline reinforcement learning. Yet, existing statistical analysis of diffusion models often requires restrictive theoretical assumptions or is suboptimal. In this talk, we present two recent works from our group toward closing these gaps between diffusion models and the theoretical limits in standard nonparametric and high-dimensional statistical settings, and discuss some future directions.

1) For subGaussian distributions on $\mathbb{R}^d$ with $\beta$-Hölder smooth densities ($\beta\le 2$), we show that the sampling distribution of diffusion models can be minimax optimal under the total variation distance, even if the score is learned from noise perturbed training samples with noise multiplicity equal to one and without density lower bound assumptions.

2) For conditional sampling under i.i.d. priors and noisy linear observations, we show that diffusion models (also known as stochastic localization) can successfully sample from the posterior distribution, provided the signal-to-noise ratio exceeds a computational threshold predicted by prior work on approximate message passing (Barbier et al., 2020). This improves previous thresholds established in the stochastic localization literature, and enhances the sampling accuracy of dominant noisy inverse sampling techniques used in machine learning - albeit in a stylized theoretical model.

(Based on works with Kaihong Zhang, Heqi Yin, Feng Liang arXiv: 2402.15602, and with Han Cui, and Zhiyuan Yu arXiv: 2407.10763)

Group synchronization is a mathematical framework used in a variety of applications, such as computer vision, to situate a set of objects given their pairwise relative positions and orientations subjected to noise. More formally, synchronization estimates a set of group elements given some of their noisy pairwise ratios. In this talk I will present an entirely new view on the task of group synchronization by considering the natural higher-order structures that relate the relative orientations of triples or n-wise sets of objects. Examples of these structures include triples of so-called ‘common lines’ in cryo-EM and trifocal tensors in multi-view geometry. Thus far very little mathematical or computational work has explored synchronizing these higher-order measurements. I will introduce the problem of higher-order group synchronization and discuss the formal foundations of synchronizability in this setting. Then I will present a message passing algorithm to solve the problem and compare its performance to classical pairwise synchronization algorithms. 

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.